Integrand size = 19, antiderivative size = 800 \[ \int \frac {\sin (c+d x)}{x^3 \left (a+b x^3\right )^2} \, dx=-\frac {d \cos (c+d x)}{2 a^2 x}-\frac {(-1)^{2/3} \sqrt [3]{b} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3}}-\frac {\sqrt [3]{b} d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{b} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}-\frac {d^2 \operatorname {CosIntegral}(d x) \sin (c)}{2 a^2}-\frac {5 b^{2/3} \operatorname {CosIntegral}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{8/3}}+\frac {5 \sqrt [3]{-1} b^{2/3} \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{8/3}}-\frac {5 (-1)^{2/3} b^{2/3} \operatorname {CosIntegral}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{8/3}}+\frac {\sin (c+d x)}{3 a b x^5}-\frac {5 \sin (c+d x)}{6 a^2 x^2}-\frac {\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a^2}-\frac {5 \sqrt [3]{-1} b^{2/3} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{8/3}}-\frac {(-1)^{2/3} \sqrt [3]{b} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3}}-\frac {5 b^{2/3} \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{8/3}}+\frac {\sqrt [3]{b} d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}-\frac {5 (-1)^{2/3} b^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{8/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}} \]
-1/9*b^(1/3)*d*Ci(a^(1/3)*d/b^(1/3)+d*x)*cos(c-a^(1/3)*d/b^(1/3))/a^(7/3)- 1/9*(-1)^(2/3)*b^(1/3)*d*Ci((-1)^(1/3)*a^(1/3)*d/b^(1/3)-d*x)*cos(c+(-1)^( 1/3)*a^(1/3)*d/b^(1/3))/a^(7/3)+1/9*(-1)^(1/3)*b^(1/3)*d*Ci((-1)^(2/3)*a^( 1/3)*d/b^(1/3)+d*x)*cos(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^(7/3)-1/2*d*cos( d*x+c)/a^2/x-1/2*d^2*cos(c)*Si(d*x)/a^2+5/9*(-1)^(1/3)*b^(2/3)*cos(c+(-1)^ (1/3)*a^(1/3)*d/b^(1/3))*Si(-(-1)^(1/3)*a^(1/3)*d/b^(1/3)+d*x)/a^(8/3)-5/9 *b^(2/3)*cos(c-a^(1/3)*d/b^(1/3))*Si(a^(1/3)*d/b^(1/3)+d*x)/a^(8/3)-5/9*(- 1)^(2/3)*b^(2/3)*cos(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))*Si((-1)^(2/3)*a^(1/3) *d/b^(1/3)+d*x)/a^(8/3)-1/2*d^2*Ci(d*x)*sin(c)/a^2-5/9*b^(2/3)*Ci(a^(1/3)* d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^(8/3)+1/9*b^(1/3)*d*Si(a^(1/3)*d /b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^(7/3)+5/9*(-1)^(1/3)*b^(2/3)*Ci(( -1)^(1/3)*a^(1/3)*d/b^(1/3)-d*x)*sin(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a^(8/ 3)+1/9*(-1)^(2/3)*b^(1/3)*d*Si(-(-1)^(1/3)*a^(1/3)*d/b^(1/3)+d*x)*sin(c+(- 1)^(1/3)*a^(1/3)*d/b^(1/3))/a^(7/3)-5/9*(-1)^(2/3)*b^(2/3)*Ci((-1)^(2/3)*a ^(1/3)*d/b^(1/3)+d*x)*sin(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^(8/3)-1/9*(-1) ^(1/3)*b^(1/3)*d*Si((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)*sin(c-(-1)^(2/3)*a^( 1/3)*d/b^(1/3))/a^(7/3)+1/3*sin(d*x+c)/a/b/x^5-5/6*sin(d*x+c)/a^2/x^2-1/3* sin(d*x+c)/b/x^5/(b*x^3+a)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 0.62 (sec) , antiderivative size = 470, normalized size of antiderivative = 0.59 \[ \int \frac {\sin (c+d x)}{x^3 \left (a+b x^3\right )^2} \, dx=\frac {\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {-5 i \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1}))-5 \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-5 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+5 i \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+d \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1})) \text {$\#$1}-i d \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}-i d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}-d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]+\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {5 i \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1}))-5 \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-5 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-5 i \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+d \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1})) \text {$\#$1}+i d \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}+i d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}-d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]-\frac {3 \left (3 a d x \cos (c+d x)+3 b d x^4 \cos (c+d x)+3 d^2 x^2 \left (a+b x^3\right ) \operatorname {CosIntegral}(d x) \sin (c)+3 a \sin (c+d x)+5 b x^3 \sin (c+d x)+3 d^2 x^2 \left (a+b x^3\right ) \cos (c) \text {Si}(d x)\right )}{x^2 \left (a+b x^3\right )}}{18 a^2} \]
(RootSum[a + b*#1^3 & , ((-5*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 5* CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 5*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + (5*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosI ntegral[d*(x - #1)]*#1 - I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - I* d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d *(x - #1)]*#1)/#1^2 & ] + RootSum[a + b*#1^3 & , ((5*I)*Cos[c + d*#1]*CosI ntegral[d*(x - #1)] - 5*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 5*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - (5*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1) ] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 + I*d*CosIntegral[d*(x - #1 )]*Sin[c + d*#1]*#1 + I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin [c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] - (3*(3*a*d*x*Cos[c + d*x] + 3*b*d*x^4*Cos[c + d*x] + 3*d^2*x^2*(a + b*x^3)*CosIntegral[d*x]*Sin[c] + 3*a*Sin[c + d*x] + 5*b*x^3*Sin[c + d*x] + 3*d^2*x^2*(a + b*x^3)*Cos[c]*S inIntegral[d*x]))/(x^2*(a + b*x^3)))/(18*a^2)
Time = 2.26 (sec) , antiderivative size = 1059, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3824, 3826, 2009, 3827, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sin (c+d x)}{x^3 \left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 3824 |
| \(\displaystyle -\frac {5 \int \frac {\sin (c+d x)}{x^6 \left (b x^3+a\right )}dx}{3 b}+\frac {d \int \frac {\cos (c+d x)}{x^5 \left (b x^3+a\right )}dx}{3 b}-\frac {\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 3826 |
| \(\displaystyle -\frac {5 \int \left (\frac {\sin (c+d x) b^2}{a^2 \left (b x^3+a\right )}-\frac {\sin (c+d x) b}{a^2 x^3}+\frac {\sin (c+d x)}{a x^6}\right )dx}{3 b}+\frac {d \int \frac {\cos (c+d x)}{x^5 \left (b x^3+a\right )}dx}{3 b}-\frac {\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \int \frac {\cos (c+d x)}{x^5 \left (b x^3+a\right )}dx}{3 b}-\frac {5 \left (\frac {b^{5/3} \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{8/3}}-\frac {\sqrt [3]{-1} b^{5/3} \sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{8/3}}+\frac {(-1)^{2/3} b^{5/3} \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{8/3}}+\frac {\sqrt [3]{-1} b^{5/3} \cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{8/3}}+\frac {b^{5/3} \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{8/3}}+\frac {(-1)^{2/3} b^{5/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{8/3}}+\frac {b d^2 \sin (c) \operatorname {CosIntegral}(d x)}{2 a^2}+\frac {b d^2 \cos (c) \text {Si}(d x)}{2 a^2}+\frac {b \sin (c+d x)}{2 a^2 x^2}+\frac {b d \cos (c+d x)}{2 a^2 x}+\frac {d^5 \cos (c) \operatorname {CosIntegral}(d x)}{120 a}-\frac {d^5 \sin (c) \text {Si}(d x)}{120 a}-\frac {d^4 \sin (c+d x)}{120 a x}+\frac {d^3 \cos (c+d x)}{120 a x^2}+\frac {d^2 \sin (c+d x)}{60 a x^3}-\frac {\sin (c+d x)}{5 a x^5}-\frac {d \cos (c+d x)}{20 a x^4}\right )}{3 b}-\frac {\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 3827 |
| \(\displaystyle \frac {d \int \left (\frac {x \cos (c+d x) b^2}{a^2 \left (b x^3+a\right )}-\frac {\cos (c+d x) b}{a^2 x^2}+\frac {\cos (c+d x)}{a x^5}\right )dx}{3 b}-\frac {5 \left (\frac {b^{5/3} \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{8/3}}-\frac {\sqrt [3]{-1} b^{5/3} \sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{8/3}}+\frac {(-1)^{2/3} b^{5/3} \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{8/3}}+\frac {\sqrt [3]{-1} b^{5/3} \cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{8/3}}+\frac {b^{5/3} \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{8/3}}+\frac {(-1)^{2/3} b^{5/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{8/3}}+\frac {b d^2 \sin (c) \operatorname {CosIntegral}(d x)}{2 a^2}+\frac {b d^2 \cos (c) \text {Si}(d x)}{2 a^2}+\frac {b \sin (c+d x)}{2 a^2 x^2}+\frac {b d \cos (c+d x)}{2 a^2 x}+\frac {d^5 \cos (c) \operatorname {CosIntegral}(d x)}{120 a}-\frac {d^5 \sin (c) \text {Si}(d x)}{120 a}-\frac {d^4 \sin (c+d x)}{120 a x}+\frac {d^3 \cos (c+d x)}{120 a x^2}+\frac {d^2 \sin (c+d x)}{60 a x^3}-\frac {\sin (c+d x)}{5 a x^5}-\frac {d \cos (c+d x)}{20 a x^4}\right )}{3 b}-\frac {\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sin (c+d x)}{3 b x^5 \left (b x^3+a\right )}-\frac {5 \left (\frac {\cos (c) \operatorname {CosIntegral}(d x) d^5}{120 a}-\frac {\sin (c) \text {Si}(d x) d^5}{120 a}-\frac {\sin (c+d x) d^4}{120 a x}+\frac {\cos (c+d x) d^3}{120 a x^2}+\frac {b \operatorname {CosIntegral}(d x) \sin (c) d^2}{2 a^2}+\frac {\sin (c+d x) d^2}{60 a x^3}+\frac {b \cos (c) \text {Si}(d x) d^2}{2 a^2}+\frac {b \cos (c+d x) d}{2 a^2 x}-\frac {\cos (c+d x) d}{20 a x^4}+\frac {b^{5/3} \operatorname {CosIntegral}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{8/3}}-\frac {\sqrt [3]{-1} b^{5/3} \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{8/3}}+\frac {(-1)^{2/3} b^{5/3} \operatorname {CosIntegral}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{8/3}}+\frac {b \sin (c+d x)}{2 a^2 x^2}-\frac {\sin (c+d x)}{5 a x^5}+\frac {\sqrt [3]{-1} b^{5/3} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{8/3}}+\frac {b^{5/3} \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{8/3}}+\frac {(-1)^{2/3} b^{5/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{8/3}}\right )}{3 b}+\frac {d \left (\frac {\cos (c) \operatorname {CosIntegral}(d x) d^4}{24 a}-\frac {\sin (c) \text {Si}(d x) d^4}{24 a}-\frac {\sin (c+d x) d^3}{24 a x}+\frac {\cos (c+d x) d^2}{24 a x^2}+\frac {b \operatorname {CosIntegral}(d x) \sin (c) d}{a^2}+\frac {\sin (c+d x) d}{12 a x^3}+\frac {b \cos (c) \text {Si}(d x) d}{a^2}+\frac {b \cos (c+d x)}{a^2 x}-\frac {\cos (c+d x)}{4 a x^4}-\frac {(-1)^{2/3} b^{4/3} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{7/3}}-\frac {b^{4/3} \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{7/3}}+\frac {\sqrt [3]{-1} b^{4/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{7/3}}-\frac {(-1)^{2/3} b^{4/3} \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{7/3}}+\frac {b^{4/3} \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{7/3}}-\frac {\sqrt [3]{-1} b^{4/3} \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{7/3}}\right )}{3 b}\) |
-1/3*Sin[c + d*x]/(b*x^5*(a + b*x^3)) - (5*(-1/20*(d*Cos[c + d*x])/(a*x^4) + (d^3*Cos[c + d*x])/(120*a*x^2) + (b*d*Cos[c + d*x])/(2*a^2*x) + (d^5*Co s[c]*CosIntegral[d*x])/(120*a) + (b*d^2*CosIntegral[d*x]*Sin[c])/(2*a^2) + (b^(5/3)*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/ 3)])/(3*a^(8/3)) - ((-1)^(1/3)*b^(5/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/ b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(3*a^(8/3)) + ((-1 )^(2/3)*b^(5/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(3*a^(8/3)) - Sin[c + d*x]/(5*a*x^5) + (d ^2*Sin[c + d*x])/(60*a*x^3) + (b*Sin[c + d*x])/(2*a^2*x^2) - (d^4*Sin[c + d*x])/(120*a*x) + (b*d^2*Cos[c]*SinIntegral[d*x])/(2*a^2) - (d^5*Sin[c]*Si nIntegral[d*x])/(120*a) + ((-1)^(1/3)*b^(5/3)*Cos[c + ((-1)^(1/3)*a^(1/3)* d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^(8/3)) + (b^(5/3)*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(8/3)) + ((-1)^(2/3)*b^(5/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^ (1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(8/3))))/(3 *b) + (d*(-1/4*Cos[c + d*x]/(a*x^4) + (d^2*Cos[c + d*x])/(24*a*x^2) + (b*C os[c + d*x])/(a^2*x) + (d^4*Cos[c]*CosIntegral[d*x])/(24*a) - ((-1)^(2/3)* b^(4/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^ (1/3)*d)/b^(1/3) - d*x])/(3*a^(7/3)) - (b^(4/3)*Cos[c - (a^(1/3)*d)/b^(1/3 )]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(7/3)) + ((-1)^(1/3)*b^...
3.2.8.3.1 Defintions of rubi rules used
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Sym bol] :> Simp[x^(m - n + 1)*(a + b*x^n)^(p + 1)*(Sin[c + d*x]/(b*n*(p + 1))) , x] + (-Simp[(m - n + 1)/(b*n*(p + 1)) Int[x^(m - n)*(a + b*x^n)^(p + 1) *Sin[c + d*x], x], x] - Simp[d/(b*n*(p + 1)) Int[x^(m - n + 1)*(a + b*x^n )^(p + 1)*Cos[c + d*x], x], x]) /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, -1] && IGtQ[n, 0] && (GtQ[m - n + 1, 0] || GtQ[n, 2]) && RationalQ[m]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Sym bol] :> Int[ExpandIntegrand[Sin[c + d*x], x^m*(a + b*x^n)^p, x], x] /; Free Q[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, - 1]) && IntegerQ[m]
Int[Cos[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Sym bol] :> Int[ExpandIntegrand[Cos[c + d*x], x^m*(a + b*x^n)^p, x], x] /; Free Q[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, - 1]) && IntegerQ[m]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.93 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.39
| method | result | size |
| risch | \(-\frac {i d^{2} \operatorname {Ei}_{1}\left (-i d x \right ) {\mathrm e}^{i c}}{4 a^{2}}-\frac {i d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-3 i \textit {\_Z}^{2} b c -i d^{3} a +i b \,c^{3}+b \,\textit {\_Z}^{3}-3 c^{2} b \textit {\_Z} \right )}{\sum }\frac {\left (-i c +\textit {\_R1} -5\right ) {\mathrm e}^{\textit {\_R1}} \operatorname {Ei}_{1}\left (-i d x -i c +\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{18 a^{2}}+\frac {i d^{2} \operatorname {Ei}_{1}\left (i d x \right ) {\mathrm e}^{-i c}}{4 a^{2}}-\frac {i d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-3 i \textit {\_Z}^{2} b c -i d^{3} a +i b \,c^{3}+b \,\textit {\_Z}^{3}-3 c^{2} b \textit {\_Z} \right )}{\sum }\frac {\left (-i c +\textit {\_R1} +5\right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {Ei}_{1}\left (i d x +i c -\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{18 a^{2}}+\frac {\left (-b \,x^{4} d^{4}-a \,d^{4} x \right ) \cos \left (d x +c \right )}{2 a^{2} x^{2} \left (d^{3} x^{3} b +a \,d^{3}\right )}-\frac {\left (5 d^{3} x^{3} b +3 a \,d^{3}\right ) \sin \left (d x +c \right )}{6 a^{2} x^{2} \left (d^{3} x^{3} b +a \,d^{3}\right )}\) | \(315\) |
| derivativedivides | \(d^{2} \left (\frac {-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{2}-\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{2}}{a^{2}}-\frac {b \,d^{3} \left (\frac {\sin \left (d x +c \right ) \left (\frac {d x +c}{3 a \,d^{3}}-\frac {c}{3 a \,d^{3}}\right )}{a \,d^{3}-c^{3} b +3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}}+\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} b c +3 c^{2} b \textit {\_Z} +a \,d^{3}-c^{3} b \right )}{\sum }\frac {-\operatorname {Si}\left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\operatorname {Ci}\left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{9 a \,d^{3} b}+\frac {\munderset {\textit {\_RR1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} b c +3 c^{2} b \textit {\_Z} +a \,d^{3}-c^{3} b \right )}{\sum }\frac {\operatorname {Si}\left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\operatorname {Ci}\left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{-\textit {\_RR1} +c}}{9 a \,d^{3} b}\right )}{a}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} b c +3 c^{2} b \textit {\_Z} +a \,d^{3}-c^{3} b \right )}{\sum }\frac {-\operatorname {Si}\left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\operatorname {Ci}\left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{3 a^{2}}\right )\) | \(388\) |
| default | \(d^{2} \left (\frac {-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )}{2}-\frac {\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{2}}{a^{2}}-\frac {b \,d^{3} \left (\frac {\sin \left (d x +c \right ) \left (\frac {d x +c}{3 a \,d^{3}}-\frac {c}{3 a \,d^{3}}\right )}{a \,d^{3}-c^{3} b +3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}}+\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} b c +3 c^{2} b \textit {\_Z} +a \,d^{3}-c^{3} b \right )}{\sum }\frac {-\operatorname {Si}\left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\operatorname {Ci}\left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{9 a \,d^{3} b}+\frac {\munderset {\textit {\_RR1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} b c +3 c^{2} b \textit {\_Z} +a \,d^{3}-c^{3} b \right )}{\sum }\frac {\operatorname {Si}\left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\operatorname {Ci}\left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{-\textit {\_RR1} +c}}{9 a \,d^{3} b}\right )}{a}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} b c +3 c^{2} b \textit {\_Z} +a \,d^{3}-c^{3} b \right )}{\sum }\frac {-\operatorname {Si}\left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\operatorname {Ci}\left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{3 a^{2}}\right )\) | \(388\) |
-1/4*I*d^2/a^2*Ei(1,-I*d*x)*exp(I*c)-1/18*I*d^2/a^2*sum((-I*c+_R1-5)/(-2*I *c*_R1+_R1^2-c^2)*exp(_R1)*Ei(1,_R1-I*d*x-I*c),_R1=RootOf(-3*I*_Z^2*b*c-I* d^3*a+I*b*c^3+b*_Z^3-3*c^2*b*_Z))+1/4*I*d^2/a^2*Ei(1,I*d*x)*exp(-I*c)-1/18 *I*d^2/a^2*sum((-I*c+_R1+5)/(-2*I*c*_R1+_R1^2-c^2)*exp(-_R1)*Ei(1,I*d*x+I* c-_R1),_R1=RootOf(-3*I*_Z^2*b*c-I*d^3*a+I*b*c^3+b*_Z^3-3*c^2*b*_Z))+1/2*(- b*d^4*x^4-a*d^4*x)/a^2/x^2/(b*d^3*x^3+a*d^3)*cos(d*x+c)-1/6*(5*b*d^3*x^3+3 *a*d^3)/a^2/x^2/(b*d^3*x^3+a*d^3)*sin(d*x+c)
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 898, normalized size of antiderivative = 1.12 \[ \int \frac {\sin (c+d x)}{x^3 \left (a+b x^3\right )^2} \, dx=\text {Too large to display} \]
-1/36*(((b^2*x^5 + a*b*x^2 - sqrt(3)*(I*b^2*x^5 + I*a*b*x^2))*(I*a*d^3/b)^ (2/3) + 5*(b^2*x^5 + a*b*x^2 + sqrt(3)*(I*b^2*x^5 + I*a*b*x^2))*(I*a*d^3/b )^(1/3))*Ei(-I*d*x + 1/2*(I*a*d^3/b)^(1/3)*(-I*sqrt(3) - 1))*e^(1/2*(I*a*d ^3/b)^(1/3)*(I*sqrt(3) + 1) - I*c) + ((b^2*x^5 + a*b*x^2 - sqrt(3)*(I*b^2* x^5 + I*a*b*x^2))*(-I*a*d^3/b)^(2/3) + 5*(b^2*x^5 + a*b*x^2 + sqrt(3)*(I*b ^2*x^5 + I*a*b*x^2))*(-I*a*d^3/b)^(1/3))*Ei(I*d*x + 1/2*(-I*a*d^3/b)^(1/3) *(-I*sqrt(3) - 1))*e^(1/2*(-I*a*d^3/b)^(1/3)*(I*sqrt(3) + 1) + I*c) + ((b^ 2*x^5 + a*b*x^2 - sqrt(3)*(-I*b^2*x^5 - I*a*b*x^2))*(I*a*d^3/b)^(2/3) + 5* (b^2*x^5 + a*b*x^2 + sqrt(3)*(-I*b^2*x^5 - I*a*b*x^2))*(I*a*d^3/b)^(1/3))* Ei(-I*d*x + 1/2*(I*a*d^3/b)^(1/3)*(I*sqrt(3) - 1))*e^(1/2*(I*a*d^3/b)^(1/3 )*(-I*sqrt(3) + 1) - I*c) + ((b^2*x^5 + a*b*x^2 - sqrt(3)*(-I*b^2*x^5 - I* a*b*x^2))*(-I*a*d^3/b)^(2/3) + 5*(b^2*x^5 + a*b*x^2 + sqrt(3)*(-I*b^2*x^5 - I*a*b*x^2))*(-I*a*d^3/b)^(1/3))*Ei(I*d*x + 1/2*(-I*a*d^3/b)^(1/3)*(I*sqr t(3) - 1))*e^(1/2*(-I*a*d^3/b)^(1/3)*(-I*sqrt(3) + 1) + I*c) - 2*((b^2*x^5 + a*b*x^2)*(-I*a*d^3/b)^(2/3) + 5*(b^2*x^5 + a*b*x^2)*(-I*a*d^3/b)^(1/3)) *Ei(I*d*x + (-I*a*d^3/b)^(1/3))*e^(I*c - (-I*a*d^3/b)^(1/3)) - 2*((b^2*x^5 + a*b*x^2)*(I*a*d^3/b)^(2/3) + 5*(b^2*x^5 + a*b*x^2)*(I*a*d^3/b)^(1/3))*E i(-I*d*x + (I*a*d^3/b)^(1/3))*e^(-I*c - (I*a*d^3/b)^(1/3)) + 18*(a*b*d^3*x ^5 + a^2*d^3*x^2)*cos_integral(d*x)*sin(c) + 18*(a*b*d^3*x^5 + a^2*d^3*x^2 )*cos(c)*sin_integral(d*x) + 18*(a*b*d^2*x^4 + a^2*d^2*x)*cos(d*x + c) ...
Timed out. \[ \int \frac {\sin (c+d x)}{x^3 \left (a+b x^3\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\sin (c+d x)}{x^3 \left (a+b x^3\right )^2} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2} x^{3}} \,d x } \]
\[ \int \frac {\sin (c+d x)}{x^3 \left (a+b x^3\right )^2} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\sin (c+d x)}{x^3 \left (a+b x^3\right )^2} \, dx=\int \frac {\sin \left (c+d\,x\right )}{x^3\,{\left (b\,x^3+a\right )}^2} \,d x \]