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From: <noreply@so...>  20021217 16:41:11

Bugs item #655270, was opened at 20021217 11:41 You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=655270&group_id=4933 Category: None Group: None Status: Open Resolution: None Priority: 5 Submitted By: Stavros Macrakis (macrakis) Assigned to: Nobody/Anonymous (nobody) Summary: Incomplete integration Initial Comment: integrate(sin(3*asin(x))) gives ((48*x^2*%E^(LOG(x+1)+LOG(1x))+48*x^448*x^2+12) *'INTEGRATE(xx^3,x)4*x^6*%E^(LOG(x+1)+LOG(1x)) 4*x^8+4*x^65*x^4+6*x^22)/(32*x^2*%E^(LOG(x+1) +LOG(1x))+32*x^432*x^2+8) which contains 'INTEGRATE(xx^3,x), which is obviously integrable. ev(...,integrate) successfully completes the integration, and ratsimp gets it down to something nice. Similar things happen with sin(a*asin(x)) for other a's. Note that trigexpand(sin(3*asin(x))) = 3*x*(1x^2)x^3. Low priority, because the answer is correct (if not ideal) and can easily be fixed.  You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=655270&group_id=4933 
From: SourceForge.net <noreply@so...>  20060410 04:09:10

Bugs item #655270, was opened at 20021217 09:41 Message generated for change (Settings changed) made by robert_dodier You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=655270&group_id=4933 Please note that this message will contain a full copy of the comment thread, including the initial issue submission, for this request, not just the latest update. >Category: Lisp Core  Integration Group: None Status: Open Resolution: None Priority: 5 Submitted By: Stavros Macrakis (macrakis) Assigned to: Nobody/Anonymous (nobody) Summary: Incomplete integration Initial Comment: integrate(sin(3*asin(x))) gives ((48*x^2*%E^(LOG(x+1)+LOG(1x))+48*x^448*x^2+12) *'INTEGRATE(xx^3,x)4*x^6*%E^(LOG(x+1)+LOG(1x)) 4*x^8+4*x^65*x^4+6*x^22)/(32*x^2*%E^(LOG(x+1) +LOG(1x))+32*x^432*x^2+8) which contains 'INTEGRATE(xx^3,x), which is obviously integrable. ev(...,integrate) successfully completes the integration, and ratsimp gets it down to something nice. Similar things happen with sin(a*asin(x)) for other a's. Note that trigexpand(sin(3*asin(x))) = 3*x*(1x^2)x^3. Low priority, because the answer is correct (if not ideal) and can easily be fixed.  You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=655270&group_id=4933 
From: SourceForge.net <noreply@so...>  20060410 17:39:42

Bugs item #655270, was opened at 20021217 11:41 Message generated for change (Comment added) made by rtoy You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=655270&group_id=4933 Please note that this message will contain a full copy of the comment thread, including the initial issue submission, for this request, not just the latest update. Category: Lisp Core  Integration Group: None Status: Open Resolution: None Priority: 5 Submitted By: Stavros Macrakis (macrakis) Assigned to: Nobody/Anonymous (nobody) Summary: Incomplete integration Initial Comment: integrate(sin(3*asin(x))) gives ((48*x^2*%E^(LOG(x+1)+LOG(1x))+48*x^448*x^2+12) *'INTEGRATE(xx^3,x)4*x^6*%E^(LOG(x+1)+LOG(1x)) 4*x^8+4*x^65*x^4+6*x^22)/(32*x^2*%E^(LOG(x+1) +LOG(1x))+32*x^432*x^2+8) which contains 'INTEGRATE(xx^3,x), which is obviously integrable. ev(...,integrate) successfully completes the integration, and ratsimp gets it down to something nice. Similar things happen with sin(a*asin(x)) for other a's. Note that trigexpand(sin(3*asin(x))) = 3*x*(1x^2)x^3. Low priority, because the answer is correct (if not ideal) and can easily be fixed.  >Comment By: Raymond Toy (rtoy) Date: 20060410 13:39 Message: Logged In: YES user_id=28849 Tracing rischint indicates that this is a problem in the Risch integrator.  You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=655270&group_id=4933 
From: SourceForge.net <noreply@so...>  20091108 15:20:30

Bugs item #655270, was opened at 20021217 17:41 Message generated for change (Comment added) made by crategus You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=655270&group_id=4933 Please note that this message will contain a full copy of the comment thread, including the initial issue submission, for this request, not just the latest update. Category: Lisp Core  Integration Group: None Status: Open Resolution: None Priority: 5 Private: No Submitted By: Stavros Macrakis (macrakis) Assigned to: Nobody/Anonymous (nobody) Summary: Incomplete integration Initial Comment: integrate(sin(3*asin(x))) gives ((48*x^2*%E^(LOG(x+1)+LOG(1x))+48*x^448*x^2+12) *'INTEGRATE(xx^3,x)4*x^6*%E^(LOG(x+1)+LOG(1x)) 4*x^8+4*x^65*x^4+6*x^22)/(32*x^2*%E^(LOG(x+1) +LOG(1x))+32*x^432*x^2+8) which contains 'INTEGRATE(xx^3,x), which is obviously integrable. ev(...,integrate) successfully completes the integration, and ratsimp gets it down to something nice. Similar things happen with sin(a*asin(x)) for other a's. Note that trigexpand(sin(3*asin(x))) = 3*x*(1x^2)x^3. Low priority, because the answer is correct (if not ideal) and can easily be fixed.  >Comment By: Dieter Kaiser (crategus) Date: 20091108 16:20 Message: This is again the reported bug: (%i2) integrate(sin(3*asin(x)),x); (%o2) (12*'integrate(xx^3,x)5*x^4+6*x^22)/8 We get the complete result after an extra evaluation: (%i3) %,nouns; (%o3) (5*x^4+12*(x^2/2x^4/4)+6*x^22)/8 The risch algorithm adds up integrals which are not handled at some point of the algorithm. To do this the risch algorithm calls the routine rischnoun. The problem is that rischnoun generates solvable integrals. These integrals are part of the result of risch. In general it is not possible to call the integrator or risch again to avoid the construction of solvable integrals, because we can run into infinite loops. One small extension is possible immediately. We can look for a rational expression in rischnoun and call ratint to get the integral. The following code shows this: (defun rischnoun (exp1 &optional (exp2 exp1 exp2p)) (let (($logsimp t) ($%e_to_numlog t)) (unless exp2p (setq exp1 (rzero))) (if (ratp (setq exp2 (resimplify (disrep exp2))) intvar) ;; A rational expression which can be integrated by ratint. (setq exp2 (ratint exp2 intvar)) ;; A more general integrand. Return a noun form. (setq exp2 (list '(%integrate) exp2 intvar))) `(,exp1 ,exp2))) That is the example of the bug report: (%i5) integrate(sin(3*asin(x)),x); (%o5) (4*x^46*x^2+1)/4 More results for an odd factor in the integrand: (%i10) integrate(sin(5*asin(x)),x); (%o10) (32*x^660*x^4+30*x^21)/12 (%i11) integrate(sin(7*asin(x)),x); (%o11) (96*x^8224*x^6+168*x^442*x^2+1)/12 But it don't work completely for an even factor in the integrand. The reason is, that one of the integrands is not a rational expression: (%i12) integrate(sin(2*asin(x)),x); (%o12) (sqrt(1x)*sqrt(x+1)*(5*x^25)3*'integrate(sqrt(1x)*x*sqrt(x+1),x))/6 (%i13) integrate(sin(4*asin(x)),x); (%o13) (15*'integrate(sqrt(1x)*sqrt(x+1)*(x^3x),x) +sqrt(1x)*sqrt(x+1)*(93*x^4106*x^2+13)) /60 On the other hand we get more results, e.g. the following integrals now work: (%i7) integrate(exp(asinh(x)),x); (%o7) log(sqrt(x^2+1)+x)/2+x*sqrt(x^2+1)/2+x^2/2 (%i8) integrate(exp(acosh(x)),x); (%o8) log(2*sqrt(x^21)+2*x)/2+x*sqrt(x^21)/2+(x^2/2+x)/2+(x^2/2x)/2 (%i9) integrate(exp(atanh(x)),x); (%o9) %i*log(sqrt(1x^2)+%i*x)sqrt(1x^2) To get a more general solution we had to call the integrator in rischnoun again. To do this it is necessary to extend the integrator and risch with a mechanism to avoid infinite loops. Dieter Kaiser  Comment By: Raymond Toy (rtoy) Date: 20060410 19:39 Message: Logged In: YES user_id=28849 Tracing rischint indicates that this is a problem in the Risch integrator.  You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=655270&group_id=4933 
From: SourceForge.net <noreply@so...>  20091221 00:19:33

Bugs item #655270, was opened at 20021217 17:41 Message generated for change (Comment added) made by crategus You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=655270&group_id=4933 Please note that this message will contain a full copy of the comment thread, including the initial issue submission, for this request, not just the latest update. Category: Lisp Core  Integration Group: None >Status: Closed >Resolution: Fixed Priority: 5 Private: No Submitted By: Stavros Macrakis (macrakis) Assigned to: Nobody/Anonymous (nobody) Summary: Incomplete integration Initial Comment: integrate(sin(3*asin(x))) gives ((48*x^2*%E^(LOG(x+1)+LOG(1x))+48*x^448*x^2+12) *'INTEGRATE(xx^3,x)4*x^6*%E^(LOG(x+1)+LOG(1x)) 4*x^8+4*x^65*x^4+6*x^22)/(32*x^2*%E^(LOG(x+1) +LOG(1x))+32*x^432*x^2+8) which contains 'INTEGRATE(xx^3,x), which is obviously integrable. ev(...,integrate) successfully completes the integration, and ratsimp gets it down to something nice. Similar things happen with sin(a*asin(x)) for other a's. Note that trigexpand(sin(3*asin(x))) = 3*x*(1x^2)x^3. Low priority, because the answer is correct (if not ideal) and can easily be fixed.  >Comment By: Dieter Kaiser (crategus) Date: 20091221 01:19 Message: Because of revision 1.18 of risch.lisp and revision 1.53 of sin.lisp we get the following results: This is the example of this bug report: (%i2) integrate(sin(3*asin(x)),x); (%o2) (4*x^46*x^2+1)/4 One more example which is related and works too: (%i3) integrate(sin(4*asin(x)),x); (%o3) (15*((1x^2)^(3/2)/5x^2*(1x^2)^(3/2)/5) +sqrt(1x)*sqrt(x+1)*(93*x^4106*x^2+13)) /60 This integrals work in addition: (%i4) integrate(exp(asinh(x)),x); (%o4) log(sqrt(x^2+1)+x)/2+x*sqrt(x^2+1)/2+x^2/2 (%i5) integrate(exp(acosh(x)),x); (%o5) log(2*sqrt(x^21)+2*x)/2+x*sqrt(x^21)/2+(x^2/2+x)/2+(x^2/2x)/2 (%i6) integrate(exp(asech(x)),x); (%o6) log(2*sqrt(1x^2)/abs(x)+2/abs(x))+(log(x)+x)/2+(log(x)x)/2 +sqrt(1x^2) Closing this bug report as fixed. Dieter Kaiser  Comment By: Dieter Kaiser (crategus) Date: 20091108 16:20 Message: This is again the reported bug: (%i2) integrate(sin(3*asin(x)),x); (%o2) (12*'integrate(xx^3,x)5*x^4+6*x^22)/8 We get the complete result after an extra evaluation: (%i3) %,nouns; (%o3) (5*x^4+12*(x^2/2x^4/4)+6*x^22)/8 The risch algorithm adds up integrals which are not handled at some point of the algorithm. To do this the risch algorithm calls the routine rischnoun. The problem is that rischnoun generates solvable integrals. These integrals are part of the result of risch. In general it is not possible to call the integrator or risch again to avoid the construction of solvable integrals, because we can run into infinite loops. One small extension is possible immediately. We can look for a rational expression in rischnoun and call ratint to get the integral. The following code shows this: (defun rischnoun (exp1 &optional (exp2 exp1 exp2p)) (let (($logsimp t) ($%e_to_numlog t)) (unless exp2p (setq exp1 (rzero))) (if (ratp (setq exp2 (resimplify (disrep exp2))) intvar) ;; A rational expression which can be integrated by ratint. (setq exp2 (ratint exp2 intvar)) ;; A more general integrand. Return a noun form. (setq exp2 (list '(%integrate) exp2 intvar))) `(,exp1 ,exp2))) That is the example of the bug report: (%i5) integrate(sin(3*asin(x)),x); (%o5) (4*x^46*x^2+1)/4 More results for an odd factor in the integrand: (%i10) integrate(sin(5*asin(x)),x); (%o10) (32*x^660*x^4+30*x^21)/12 (%i11) integrate(sin(7*asin(x)),x); (%o11) (96*x^8224*x^6+168*x^442*x^2+1)/12 But it don't work completely for an even factor in the integrand. The reason is, that one of the integrands is not a rational expression: (%i12) integrate(sin(2*asin(x)),x); (%o12) (sqrt(1x)*sqrt(x+1)*(5*x^25)3*'integrate(sqrt(1x)*x*sqrt(x+1),x))/6 (%i13) integrate(sin(4*asin(x)),x); (%o13) (15*'integrate(sqrt(1x)*sqrt(x+1)*(x^3x),x) +sqrt(1x)*sqrt(x+1)*(93*x^4106*x^2+13)) /60 On the other hand we get more results, e.g. the following integrals now work: (%i7) integrate(exp(asinh(x)),x); (%o7) log(sqrt(x^2+1)+x)/2+x*sqrt(x^2+1)/2+x^2/2 (%i8) integrate(exp(acosh(x)),x); (%o8) log(2*sqrt(x^21)+2*x)/2+x*sqrt(x^21)/2+(x^2/2+x)/2+(x^2/2x)/2 (%i9) integrate(exp(atanh(x)),x); (%o9) %i*log(sqrt(1x^2)+%i*x)sqrt(1x^2) To get a more general solution we had to call the integrator in rischnoun again. To do this it is necessary to extend the integrator and risch with a mechanism to avoid infinite loops. Dieter Kaiser  Comment By: Raymond Toy (rtoy) Date: 20060410 19:39 Message: Logged In: YES user_id=28849 Tracing rischint indicates that this is a problem in the Risch integrator.  You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=655270&group_id=4933 