Integrand size = 19, antiderivative size = 1163 \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^3} \, dx =\text {Too large to display} \]
-1/3*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^3-1/3*cos(c-a^(1/3)*d/b^(1/3))*Si(a^(1/3)*d/b^(1/3)+d*x)/a^3-1/3*co s(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^3 -1/3*Ci(a^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^3-1/3*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^3-1/3*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^3+ cos(c)*Si(d*x)/a^3+Ci(d*x)*sin(c)/a^3+1/54*(-1)^(2/3)*d^2*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^(7/3)/b^(2/3)- 4/27*(-1)^(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^(8/3)/b^(1/3)+4/27*(-1)^(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^(8/3)/b^(1/3)-4/27*(- 1)^(2/3)*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^(8/3)/b^(1/3)-1/54*(-1)^(1/3)*d^2*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^(7/3)/b^(2/3)+1/54*(-1)^(2/ 3)*d^2*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^(7/3)/b^(2/3)-1/54*(-1)^(1/3)*d^2*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^(7/3)/b^(2/3)+4/27*(-1)^(2/3)*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^(8/3)/b^(1/3)-4/27*d*Ci(a^(1/3)*d/b^(1/3)+d*x)*cos(c-a^(1/3)*d/b^(1/3))/ a^(8/3)/b^(1/3)+1/18*d*cos(d*x+c)/a/b^2/x^5-1/18*d*cos(d*x+c)/a^2/b/x^2...
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 0.55 (sec) , antiderivative size = 2109, normalized size of antiderivative = 1.81 \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^3} \, dx=\text {Result too large to show} \]
(-6*a^2*b*d*x*Cos[c + d*x] - 6*a*b^2*d*x^4*Cos[c + d*x] - (18*I)*b*(a + b* x^3)^2*RootSum[a + b*#1^3 & , Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*Co sIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - # 1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)] & ] + (18*I)*b*(a + b*x^3)^2*R ootSum[a + b*#1^3 & , Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegra l[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Si n[c + d*#1]*SinIntegral[d*(x - #1)] & ] - 6*a^3*d*RootSum[a + b*#1^3 & , ( Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegra l[d*(x - #1)])/#1^2 & ] - 12*a^2*b*d*x^3*RootSum[a + b*#1^3 & , (Cos[c + d *#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I *Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] - 6*a*b^2*d*x^6*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIn tegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I*Cos[c + d *#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] - 6*a^3*d*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1 )] + I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral [d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] - 12*a^2*b* d*x^3*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*Co sIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIntegral[d*(x ...
Time = 4.59 (sec) , antiderivative size = 2229, normalized size of antiderivative = 1.92, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3824, 3824, 3825, 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.
| \(\displaystyle \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 3824 |
| \(\displaystyle \frac {d \int \frac {\cos (c+d x)}{x^3 \left (b x^3+a\right )^2}dx}{6 b}-\frac {\int \frac {\sin (c+d x)}{x^4 \left (b x^3+a\right )^2}dx}{2 b}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 3824 |
| \(\displaystyle \frac {d \int \frac {\cos (c+d x)}{x^3 \left (b x^3+a\right )^2}dx}{6 b}-\frac {-\frac {2 \int \frac {\sin (c+d x)}{x^7 \left (b x^3+a\right )}dx}{b}+\frac {d \int \frac {\cos (c+d x)}{x^6 \left (b x^3+a\right )}dx}{3 b}-\frac {\sin (c+d x)}{3 b x^6 \left (a+b x^3\right )}}{2 b}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 3825 |
| \(\displaystyle -\frac {-\frac {2 \int \frac {\sin (c+d x)}{x^7 \left (b x^3+a\right )}dx}{b}+\frac {d \int \frac {\cos (c+d x)}{x^6 \left (b x^3+a\right )}dx}{3 b}-\frac {\sin (c+d x)}{3 b x^6 \left (a+b x^3\right )}}{2 b}+\frac {d \left (-\frac {5 \int \frac {\cos (c+d x)}{x^6 \left (b x^3+a\right )}dx}{3 b}-\frac {d \int \frac {\sin (c+d x)}{x^5 \left (b x^3+a\right )}dx}{3 b}-\frac {\cos (c+d x)}{3 b x^5 \left (a+b x^3\right )}\right )}{6 b}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 3826 |
| \(\displaystyle \frac {d \left (-\frac {d \int \left (\frac {x \sin (c+d x) b^2}{a^2 \left (b x^3+a\right )}-\frac {\sin (c+d x) b}{a^2 x^2}+\frac {\sin (c+d x)}{a x^5}\right )dx}{3 b}-\frac {5 \int \frac {\cos (c+d x)}{x^6 \left (b x^3+a\right )}dx}{3 b}-\frac {\cos (c+d x)}{3 b x^5 \left (a+b x^3\right )}\right )}{6 b}-\frac {-\frac {2 \int \left (-\frac {x^2 \sin (c+d x) b^3}{a^3 \left (b x^3+a\right )}+\frac {\sin (c+d x) b^2}{a^3 x}-\frac {\sin (c+d x) b}{a^2 x^4}+\frac {\sin (c+d x)}{a x^7}\right )dx}{b}+\frac {d \int \frac {\cos (c+d x)}{x^6 \left (b x^3+a\right )}dx}{3 b}-\frac {\sin (c+d x)}{3 b x^6 \left (a+b x^3\right )}}{2 b}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sin (c+d x)}{6 b x^3 \left (b x^3+a\right )^2}+\frac {d \left (-\frac {\cos (c+d x)}{3 b x^5 \left (b x^3+a\right )}-\frac {d \left (\frac {\operatorname {CosIntegral}(d x) \sin (c) d^4}{24 a}+\frac {\cos (c) \text {Si}(d x) d^4}{24 a}+\frac {\cos (c+d x) d^3}{24 a x}+\frac {\sin (c+d x) d^2}{24 a x^2}-\frac {\cos (c+d x) d}{12 a x^3}-\frac {b \cos (c) \operatorname {CosIntegral}(d x) d}{a^2}+\frac {b \sin (c) \text {Si}(d x) d}{a^2}-\frac {b^{4/3} \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^{7/3}}-\frac {(-1)^{2/3} b^{4/3} \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^{7/3}}+\frac {\sqrt [3]{-1} b^{4/3} \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^{7/3}}+\frac {b \sin (c+d x)}{a^2 x}-\frac {\sin (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 ) \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} \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^{7/3}}+\frac {\sqrt [3]{-1} b^{4/3} \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^{7/3}}\right )}{3 b}-\frac {5 \int \frac {\cos (c+d x)}{x^6 \left (b x^3+a\right )}dx}{3 b}\right )}{6 b}-\frac {-\frac {\sin (c+d x)}{3 b x^6 \left (b x^3+a\right )}-\frac {2 \left (-\frac {\operatorname {CosIntegral}(d x) \sin (c) d^6}{720 a}-\frac {\cos (c) \text {Si}(d x) d^6}{720 a}-\frac {\cos (c+d x) d^5}{720 a x}-\frac {\sin (c+d x) d^4}{720 a x^2}+\frac {\cos (c+d x) d^3}{360 a x^3}+\frac {b \cos (c) \operatorname {CosIntegral}(d x) d^3}{6 a^2}-\frac {b \sin (c) \text {Si}(d x) d^3}{6 a^2}-\frac {b \sin (c+d x) d^2}{6 a^2 x}+\frac {\sin (c+d x) d^2}{120 a x^4}+\frac {b \cos (c+d x) d}{6 a^2 x^2}-\frac {\cos (c+d x) d}{30 a x^5}+\frac {b^2 \operatorname {CosIntegral}(d x) \sin (c)}{a^3}-\frac {b^2 \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^3}-\frac {b^2 \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^3}-\frac {b^2 \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^3}+\frac {b \sin (c+d x)}{3 a^2 x^3}-\frac {\sin (c+d x)}{6 a x^6}+\frac {b^2 \cos (c) \text {Si}(d x)}{a^3}+\frac {b^2 \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^3}-\frac {b^2 \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^3}-\frac {b^2 \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^3}\right )}{b}+\frac {d \int \frac {\cos (c+d x)}{x^6 \left (b x^3+a\right )}dx}{3 b}}{2 b}\) |
| \(\Big \downarrow \) 3827 |
| \(\displaystyle -\frac {\sin (c+d x)}{6 b x^3 \left (b x^3+a\right )^2}+\frac {d \left (-\frac {\cos (c+d x)}{3 b x^5 \left (b x^3+a\right )}-\frac {d \left (\frac {\operatorname {CosIntegral}(d x) \sin (c) d^4}{24 a}+\frac {\cos (c) \text {Si}(d x) d^4}{24 a}+\frac {\cos (c+d x) d^3}{24 a x}+\frac {\sin (c+d x) d^2}{24 a x^2}-\frac {\cos (c+d x) d}{12 a x^3}-\frac {b \cos (c) \operatorname {CosIntegral}(d x) d}{a^2}+\frac {b \sin (c) \text {Si}(d x) d}{a^2}-\frac {b^{4/3} \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^{7/3}}-\frac {(-1)^{2/3} b^{4/3} \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^{7/3}}+\frac {\sqrt [3]{-1} b^{4/3} \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^{7/3}}+\frac {b \sin (c+d x)}{a^2 x}-\frac {\sin (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 ) \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} \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^{7/3}}+\frac {\sqrt [3]{-1} b^{4/3} \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^{7/3}}\right )}{3 b}-\frac {5 \int \left (\frac {\cos (c+d x) b^2}{a^2 \left (b x^3+a\right )}-\frac {\cos (c+d x) b}{a^2 x^3}+\frac {\cos (c+d x)}{a x^6}\right )dx}{3 b}\right )}{6 b}-\frac {-\frac {\sin (c+d x)}{3 b x^6 \left (b x^3+a\right )}-\frac {2 \left (-\frac {\operatorname {CosIntegral}(d x) \sin (c) d^6}{720 a}-\frac {\cos (c) \text {Si}(d x) d^6}{720 a}-\frac {\cos (c+d x) d^5}{720 a x}-\frac {\sin (c+d x) d^4}{720 a x^2}+\frac {\cos (c+d x) d^3}{360 a x^3}+\frac {b \cos (c) \operatorname {CosIntegral}(d x) d^3}{6 a^2}-\frac {b \sin (c) \text {Si}(d x) d^3}{6 a^2}-\frac {b \sin (c+d x) d^2}{6 a^2 x}+\frac {\sin (c+d x) d^2}{120 a x^4}+\frac {b \cos (c+d x) d}{6 a^2 x^2}-\frac {\cos (c+d x) d}{30 a x^5}+\frac {b^2 \operatorname {CosIntegral}(d x) \sin (c)}{a^3}-\frac {b^2 \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^3}-\frac {b^2 \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^3}-\frac {b^2 \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^3}+\frac {b \sin (c+d x)}{3 a^2 x^3}-\frac {\sin (c+d x)}{6 a x^6}+\frac {b^2 \cos (c) \text {Si}(d x)}{a^3}+\frac {b^2 \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^3}-\frac {b^2 \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^3}-\frac {b^2 \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^3}\right )}{b}+\frac {d \int \left (\frac {\cos (c+d x) b^2}{a^2 \left (b x^3+a\right )}-\frac {\cos (c+d x) b}{a^2 x^3}+\frac {\cos (c+d x)}{a x^6}\right )dx}{3 b}}{2 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sin (c+d x)}{6 b x^3 \left (b x^3+a\right )^2}+\frac {d \left (-\frac {\cos (c+d x)}{3 b x^5 \left (b x^3+a\right )}-\frac {d \left (\frac {\operatorname {CosIntegral}(d x) \sin (c) d^4}{24 a}+\frac {\cos (c) \text {Si}(d x) d^4}{24 a}+\frac {\cos (c+d x) d^3}{24 a x}+\frac {\sin (c+d x) d^2}{24 a x^2}-\frac {\cos (c+d x) d}{12 a x^3}-\frac {b \cos (c) \operatorname {CosIntegral}(d x) d}{a^2}+\frac {b \sin (c) \text {Si}(d x) d}{a^2}-\frac {b^{4/3} \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^{7/3}}-\frac {(-1)^{2/3} b^{4/3} \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^{7/3}}+\frac {\sqrt [3]{-1} b^{4/3} \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^{7/3}}+\frac {b \sin (c+d x)}{a^2 x}-\frac {\sin (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 ) \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} \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^{7/3}}+\frac {\sqrt [3]{-1} b^{4/3} \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^{7/3}}\right )}{3 b}-\frac {5 \left (-\frac {\operatorname {CosIntegral}(d x) \sin (c) d^5}{120 a}-\frac {\cos (c) \text {Si}(d x) d^5}{120 a}-\frac {\cos (c+d x) d^4}{120 a x}-\frac {\sin (c+d x) d^3}{120 a x^2}+\frac {\cos (c+d x) d^2}{60 a x^3}+\frac {b \cos (c) \operatorname {CosIntegral}(d x) d^2}{2 a^2}-\frac {b \sin (c) \text {Si}(d x) d^2}{2 a^2}-\frac {b \sin (c+d x) d}{2 a^2 x}+\frac {\sin (c+d x) d}{20 a x^4}+\frac {b \cos (c+d x)}{2 a^2 x^2}-\frac {\cos (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 ) \operatorname {CosIntegral}\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 ) \operatorname {CosIntegral}\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 ) \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} \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^{8/3}}-\frac {b^{5/3} \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^{8/3}}-\frac {(-1)^{2/3} b^{5/3} \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^{8/3}}\right )}{3 b}\right )}{6 b}-\frac {-\frac {\sin (c+d x)}{3 b x^6 \left (b x^3+a\right )}-\frac {2 \left (-\frac {\operatorname {CosIntegral}(d x) \sin (c) d^6}{720 a}-\frac {\cos (c) \text {Si}(d x) d^6}{720 a}-\frac {\cos (c+d x) d^5}{720 a x}-\frac {\sin (c+d x) d^4}{720 a x^2}+\frac {\cos (c+d x) d^3}{360 a x^3}+\frac {b \cos (c) \operatorname {CosIntegral}(d x) d^3}{6 a^2}-\frac {b \sin (c) \text {Si}(d x) d^3}{6 a^2}-\frac {b \sin (c+d x) d^2}{6 a^2 x}+\frac {\sin (c+d x) d^2}{120 a x^4}+\frac {b \cos (c+d x) d}{6 a^2 x^2}-\frac {\cos (c+d x) d}{30 a x^5}+\frac {b^2 \operatorname {CosIntegral}(d x) \sin (c)}{a^3}-\frac {b^2 \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^3}-\frac {b^2 \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^3}-\frac {b^2 \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^3}+\frac {b \sin (c+d x)}{3 a^2 x^3}-\frac {\sin (c+d x)}{6 a x^6}+\frac {b^2 \cos (c) \text {Si}(d x)}{a^3}+\frac {b^2 \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^3}-\frac {b^2 \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^3}-\frac {b^2 \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^3}\right )}{b}+\frac {d \left (-\frac {\operatorname {CosIntegral}(d x) \sin (c) d^5}{120 a}-\frac {\cos (c) \text {Si}(d x) d^5}{120 a}-\frac {\cos (c+d x) d^4}{120 a x}-\frac {\sin (c+d x) d^3}{120 a x^2}+\frac {\cos (c+d x) d^2}{60 a x^3}+\frac {b \cos (c) \operatorname {CosIntegral}(d x) d^2}{2 a^2}-\frac {b \sin (c) \text {Si}(d x) d^2}{2 a^2}-\frac {b \sin (c+d x) d}{2 a^2 x}+\frac {\sin (c+d x) d}{20 a x^4}+\frac {b \cos (c+d x)}{2 a^2 x^2}-\frac {\cos (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 ) \operatorname {CosIntegral}\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 ) \operatorname {CosIntegral}\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 ) \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} \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^{8/3}}-\frac {b^{5/3} \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^{8/3}}-\frac {(-1)^{2/3} b^{5/3} \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^{8/3}}\right )}{3 b}}{2 b}\) |
-1/6*Sin[c + d*x]/(b*x^3*(a + b*x^3)^2) + (d*(-1/3*Cos[c + d*x]/(b*x^5*(a + b*x^3)) - (d*(-1/12*(d*Cos[c + d*x])/(a*x^3) + (d^3*Cos[c + d*x])/(24*a* x) - (b*d*Cos[c]*CosIntegral[d*x])/a^2 + (d^4*CosIntegral[d*x]*Sin[c])/(24 *a) - (b^(4/3)*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/ b^(1/3)])/(3*a^(7/3)) - ((-1)^(2/3)*b^(4/3)*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^(7/3)) + ((-1)^(1/3)*b^(4/3)*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^(7/3)) - Sin[c + d*x]/(4*a*x^4) + (d^2*Sin[c + d*x])/(24*a*x^2) + (b*Sin[c + d*x])/(a^2*x) + (d^4*Cos[c]* SinIntegral[d*x])/(24*a) + (b*d*Sin[c]*SinIntegral[d*x])/a^2 + ((-1)^(2/3) *b^(4/3)*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^(7/3)) - (b^(4/3)*Cos[c - (a^(1/3)*d)/b^(1/ 3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(7/3)) + ((-1)^(1/3)*b^(4 /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^(7/3))))/(3*b) - (5*(-1/5*Cos[c + d*x]/(a*x^5) + (d^2*Cos[c + d*x])/(60*a*x^3) + (b*Cos[c + d*x])/(2*a^2*x^2) - (d^4*Cos[c + d*x])/(120*a*x) + (b*d^2*Cos[c]*CosIntegral[d*x])/(2*a^2) - ((-1)^(1/3) *b^(5/3)*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^(8/3)) + (b^(5/3)*Cos[c - (a^(1/3)*d)/b^(1/ 3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^(8/3)) + ((-1)^(2/3)*b...
3.2.13.3.1 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[Cos[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Sym bol] :> Simp[x^(m - n + 1)*(a + b*x^n)^(p + 1)*(Cos[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) *Cos[c + d*x], x], x] + Simp[d/(b*n*(p + 1)) Int[x^(m - n + 1)*(a + b*x^n )^(p + 1)*Sin[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 = 1.09 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.31
| method | result | size |
| derivativedivides | \(\frac {\sin \left (d x +c \right ) d^{3} \left (3 a \,d^{3}-2 c^{3} b +6 b \,c^{2} \left (d x +c \right )-6 b c \left (d x +c \right )^{2}+2 b \left (d x +c \right )^{3}\right )}{6 a^{2} \left (a \,d^{3}-c^{3} b +3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}\right )^{2}}-\frac {\cos \left (d x +c \right ) d^{4} x}{18 a^{2} \left (a \,d^{3}-c^{3} b +3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}\right )}+\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{a^{3}}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} b c +3 c^{2} b \textit {\_Z} +a \,d^{3}-c^{3} b \right )}{\sum }\frac {\left (a \,d^{3}+18 \textit {\_R1} b -18 c b \right ) \left (-\operatorname {Si}\left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\operatorname {Ci}\left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )\right )}{-\textit {\_R1} +c}}{54 b \,a^{3}}-\frac {4 d^{3} \left (\munderset {\textit {\_RR1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} b c +3 c^{2} b \textit {\_Z} +a \,d^{3}-c^{3} b \right )}{\sum }\frac {\operatorname {Si}\left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\operatorname {Ci}\left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{\textit {\_RR1}^{2}-2 \textit {\_RR1} c +c^{2}}\right )}{27 a^{2} b}\) | \(363\) |
| default | \(\frac {\sin \left (d x +c \right ) d^{3} \left (3 a \,d^{3}-2 c^{3} b +6 b \,c^{2} \left (d x +c \right )-6 b c \left (d x +c \right )^{2}+2 b \left (d x +c \right )^{3}\right )}{6 a^{2} \left (a \,d^{3}-c^{3} b +3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}\right )^{2}}-\frac {\cos \left (d x +c \right ) d^{4} x}{18 a^{2} \left (a \,d^{3}-c^{3} b +3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}\right )}+\frac {\operatorname {Si}\left (d x \right ) \cos \left (c \right )+\operatorname {Ci}\left (d x \right ) \sin \left (c \right )}{a^{3}}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} b c +3 c^{2} b \textit {\_Z} +a \,d^{3}-c^{3} b \right )}{\sum }\frac {\left (a \,d^{3}+18 \textit {\_R1} b -18 c b \right ) \left (-\operatorname {Si}\left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\operatorname {Ci}\left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )\right )}{-\textit {\_R1} +c}}{54 b \,a^{3}}-\frac {4 d^{3} \left (\munderset {\textit {\_RR1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} b c +3 c^{2} b \textit {\_Z} +a \,d^{3}-c^{3} b \right )}{\sum }\frac {\operatorname {Si}\left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\operatorname {Ci}\left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{\textit {\_RR1}^{2}-2 \textit {\_RR1} c +c^{2}}\right )}{27 a^{2} b}\) | \(363\) |
| risch | \(-\frac {i \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-3 i \textit {\_Z}^{2} b c -i d^{3} a +i b \,c^{3}+b \,\textit {\_Z}^{3}-3 c^{2} b \textit {\_Z} \right )}{\sum }\frac {\left (i \textit {\_R1} a \,d^{3}+a c \,d^{3}-8 i d^{3} a -36 i \textit {\_R1} b c +18 b \,\textit {\_R1}^{2}-18 c^{2} b \right ) {\mathrm e}^{\textit {\_R1}} \operatorname {Ei}_{1}\left (-i d x -i c +\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{108 b \,a^{3}}+\frac {i {\mathrm e}^{i c} \operatorname {Ei}_{1}\left (-i d x \right )}{2 a^{3}}+\frac {i \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-3 i \textit {\_Z}^{2} b c -i d^{3} a +i b \,c^{3}+b \,\textit {\_Z}^{3}-3 c^{2} b \textit {\_Z} \right )}{\sum }\frac {\left (i \textit {\_R1} a \,d^{3}+a c \,d^{3}+8 i d^{3} a -36 i \textit {\_R1} b c +18 b \,\textit {\_R1}^{2}-18 c^{2} b \right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {Ei}_{1}\left (i d x +i c -\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{108 b \,a^{3}}-\frac {{\mathrm e}^{-i c} \pi \,\operatorname {csgn}\left (d x \right )}{2 a^{3}}+\frac {{\mathrm e}^{-i c} \operatorname {Si}\left (d x \right )}{a^{3}}-\frac {i {\mathrm e}^{-i c} \operatorname {Ei}_{1}\left (-i d x \right )}{2 a^{3}}-\frac {d^{7} \cos \left (d x +c \right ) b \,x^{4}}{18 a^{2} \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}-\frac {d^{7} \cos \left (d x +c \right ) x}{18 a \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}+\frac {d^{6} \sin \left (d x +c \right ) x^{3} b}{3 a^{2} \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}+\frac {d^{6} \sin \left (d x +c \right )}{2 a \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}\) | \(495\) |
1/6*sin(d*x+c)*d^3*(3*a*d^3-2*c^3*b+6*b*c^2*(d*x+c)-6*b*c*(d*x+c)^2+2*b*(d *x+c)^3)/a^2/(a*d^3-c^3*b+3*b*c^2*(d*x+c)-3*b*c*(d*x+c)^2+b*(d*x+c)^3)^2-1 /18*cos(d*x+c)*d^4*x/a^2/(a*d^3-c^3*b+3*b*c^2*(d*x+c)-3*b*c*(d*x+c)^2+b*(d *x+c)^3)+1/a^3*(Si(d*x)*cos(c)+Ci(d*x)*sin(c))+1/54/b/a^3*sum((a*d^3+18*_R 1*b-18*b*c)/(-_R1+c)*(-Si(-d*x+_R1-c)*cos(_R1)+Ci(d*x-_R1+c)*sin(_R1)),_R1 =RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))-4/27*d^3/a^2/b*sum(1/(_ RR1^2-2*_RR1*c+c^2)*(Si(-d*x+_RR1-c)*sin(_RR1)+Ci(d*x-_RR1+c)*cos(_RR1)),_ RR1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 1113, normalized size of antiderivative = 0.96 \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
1/216*((-36*I*b^2*x^6 - 72*I*a*b*x^3 - 36*I*a^2 + (I*b^2*x^6 + 2*I*a*b*x^3 + I*a^2 + sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*(I*a*d^3/b)^(2/3) - 8*(-I* b^2*x^6 - 2*I*a*b*x^3 - I*a^2 + sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^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) + (36*I*b^2*x^6 + 72*I*a*b*x^3 + 3 6*I*a^2 + (-I*b^2*x^6 - 2*I*a*b*x^3 - I*a^2 - sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*(-I*a*d^3/b)^(2/3) - 8*(I*b^2*x^6 + 2*I*a*b*x^3 + I*a^2 - sqrt(3) *(b^2*x^6 + 2*a*b*x^3 + a^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) + (-36*I*b^2*x^6 - 72*I*a*b*x^3 - 36*I*a^2 + (I*b^2*x^6 + 2*I*a*b*x^3 + I*a^2 - sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*(I*a*d^3/b)^(2/3) - 8*(-I*b^ 2*x^6 - 2*I*a*b*x^3 - I*a^2 - sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^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) + (36*I*b^2*x^6 + 72*I*a*b*x^3 + 36* I*a^2 + (-I*b^2*x^6 - 2*I*a*b*x^3 - I*a^2 + sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*(-I*a*d^3/b)^(2/3) - 8*(I*b^2*x^6 + 2*I*a*b*x^3 + I*a^2 + sqrt(3)*( b^2*x^6 + 2*a*b*x^3 + a^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) - 2*(-18*I*b^2*x^6 - 36*I*a*b*x^3 - 18*I*a^2 + (-I*b^2*x^6 - 2*I*a*b*x^3 - I*a^2)*(-I*a*d^3/b)^(2/3) + 8*(-I*b^2*x^6 - 2*I*a*b*x^3 - I*a^2)*(-I*...
Timed out. \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^3} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3} x} \,d x } \]
\[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^3} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3} x} \,d x } \]
Timed out. \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^3} \, dx=\int \frac {\sin \left (c+d\,x\right )}{x\,{\left (b\,x^3+a\right )}^3} \,d x \]