Integrand size = 17, antiderivative size = 1141 \[ \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx =\text {Too large to display} \]
-2/27*d*Ci(a^(1/3)*d/b^(1/3)+d*x)*cos(c-a^(1/3)*d/b^(1/3))/a^2/b-2/27*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^2 /b-2/27*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^2/b+1/18*d*cos(d*x+c)/a/b^2/x^3-1/18*d*cos(d*x+c)/b^2/x^3/(b*x^ 3+a)-2/27*(-1)^(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^(7/3)/b^(2/3)-1/54*(-1)^(1/3)*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^(5/3)/b^(4/3)-2 /27*cos(c-a^(1/3)*d/b^(1/3))*Si(a^(1/3)*d/b^(1/3)+d*x)/a^(7/3)/b^(2/3)+1/5 4*d^2*cos(c-a^(1/3)*d/b^(1/3))*Si(a^(1/3)*d/b^(1/3)+d*x)/a^(5/3)/b^(4/3)+2 /27*(-1)^(1/3)*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^(7/3)/b^(2/3)+1/54*(-1)^(2/3)*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^(5/3)/b^(4/3)-2/27*Ci(a ^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^(7/3)/b^(2/3)+1/54*d^2*Ci (a^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^(5/3)/b^(4/3)+2/27*d*Si (a^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^2/b-2/27*(-1)^(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^(7 /3)/b^(2/3)-1/54*(-1)^(1/3)*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^(5/3)/b^(4/3)+2/27*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^2/b+2/27*(-1)^(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/...
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 0.41 (sec) , antiderivative size = 698, normalized size of antiderivative = 0.61 \[ \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx=-\frac {\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {-i a d^2 \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1}))-a d^2 \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-a d^2 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+i a d^2 \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-4 i b \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1})) \text {$\#$1}-4 b \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}-4 b \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}+4 i b \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}+4 b d \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1})) \text {$\#$1}^2-4 i b d \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}^2-4 i b d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2-4 b d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2}{\text {$\#$1}^2}\&\right ]+\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {i a d^2 \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1}))-a d^2 \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-a d^2 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-i a d^2 \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+4 i b \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1})) \text {$\#$1}-4 b \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}-4 b \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}-4 i b \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}+4 b d \cos (c+d \text {$\#$1}) \operatorname {CosIntegral}(d (x-\text {$\#$1})) \text {$\#$1}^2+4 i b d \operatorname {CosIntegral}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}^2+4 i b d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2-4 b d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2}{\text {$\#$1}^2}\&\right ]-\frac {6 b \cos (d x) \left (a d \left (a+b x^3\right ) \cos (c)+b x^2 \left (7 a+4 b x^3\right ) \sin (c)\right )}{\left (a+b x^3\right )^2}-\frac {6 b \left (b x^2 \left (7 a+4 b x^3\right ) \cos (c)-a d \left (a+b x^3\right ) \sin (c)\right ) \sin (d x)}{\left (a+b x^3\right )^2}}{108 a^2 b^2} \]
-1/108*(RootSum[a + b*#1^3 & , ((-I)*a*d^2*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - a*d^2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - a*d^2*Cos[c + d*#1] *SinIntegral[d*(x - #1)] + I*a*d^2*Sin[c + d*#1]*SinIntegral[d*(x - #1)] - (4*I)*b*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - 4*b*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - 4*b*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 + (4 *I)*b*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1 + 4*b*d*Cos[c + d*#1]*CosIn tegral[d*(x - #1)]*#1^2 - (4*I)*b*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]* #1^2 - (4*I)*b*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2 - 4*b*d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2)/#1^2 & ] + RootSum[a + b*#1^3 & , (I *a*d^2*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - a*d^2*CosIntegral[d*(x - #1 )]*Sin[c + d*#1] - a*d^2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - I*a*d^2*S in[c + d*#1]*SinIntegral[d*(x - #1)] + (4*I)*b*Cos[c + d*#1]*CosIntegral[d *(x - #1)]*#1 - 4*b*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - 4*b*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - (4*I)*b*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1 + 4*b*d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1^2 + (4*I)*b*d* CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1^2 + (4*I)*b*d*Cos[c + d*#1]*SinIn tegral[d*(x - #1)]*#1^2 - 4*b*d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2 )/#1^2 & ] - (6*b*Cos[d*x]*(a*d*(a + b*x^3)*Cos[c] + b*x^2*(7*a + 4*b*x^3) *Sin[c]))/(a + b*x^3)^2 - (6*b*(b*x^2*(7*a + 4*b*x^3)*Cos[c] - a*d*(a + b* x^3)*Sin[c])*Sin[d*x])/(a + b*x^3)^2)/(a^2*b^2)
Time = 3.98 (sec) , antiderivative size = 1802, normalized size of antiderivative = 1.58, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, 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.
| \(\displaystyle \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 3824 |
| \(\displaystyle \frac {d \int \frac {\cos (c+d x)}{x \left (b x^3+a\right )^2}dx}{6 b}-\frac {\int \frac {\sin (c+d x)}{x^2 \left (b x^3+a\right )^2}dx}{6 b}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 3824 |
| \(\displaystyle \frac {d \int \frac {\cos (c+d x)}{x \left (b x^3+a\right )^2}dx}{6 b}-\frac {-\frac {4 \int \frac {\sin (c+d x)}{x^5 \left (b x^3+a\right )}dx}{3 b}+\frac {d \int \frac {\cos (c+d x)}{x^4 \left (b x^3+a\right )}dx}{3 b}-\frac {\sin (c+d x)}{3 b x^4 \left (a+b x^3\right )}}{6 b}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 3825 |
| \(\displaystyle \frac {d \left (-\frac {d \int \frac {\sin (c+d x)}{x^3 \left (b x^3+a\right )}dx}{3 b}-\frac {\int \frac {\cos (c+d x)}{x^4 \left (b x^3+a\right )}dx}{b}-\frac {\cos (c+d x)}{3 b x^3 \left (a+b x^3\right )}\right )}{6 b}-\frac {-\frac {4 \int \frac {\sin (c+d x)}{x^5 \left (b x^3+a\right )}dx}{3 b}+\frac {d \int \frac {\cos (c+d x)}{x^4 \left (b x^3+a\right )}dx}{3 b}-\frac {\sin (c+d x)}{3 b x^4 \left (a+b x^3\right )}}{6 b}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 3826 |
| \(\displaystyle -\frac {-\frac {4 \int \left (\frac {x \sin (c+d x) b^2}{a^2 \left (b x^3+a\right )}-\frac {\sin (c+d x) b}{a^2 x^2}+\frac {\sin (c+d x)}{a x^5}\right )dx}{3 b}+\frac {d \int \frac {\cos (c+d x)}{x^4 \left (b x^3+a\right )}dx}{3 b}-\frac {\sin (c+d x)}{3 b x^4 \left (a+b x^3\right )}}{6 b}+\frac {d \left (-\frac {d \int \left (\frac {\sin (c+d x)}{a x^3}-\frac {b \sin (c+d x)}{a \left (b x^3+a\right )}\right )dx}{3 b}-\frac {\int \frac {\cos (c+d x)}{x^4 \left (b x^3+a\right )}dx}{b}-\frac {\cos (c+d x)}{3 b x^3 \left (a+b x^3\right )}\right )}{6 b}-\frac {\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sin (c+d x)}{6 b x \left (b x^3+a\right )^2}+\frac {d \left (-\frac {\cos (c+d x)}{3 b x^3 \left (b x^3+a\right )}-\frac {d \left (-\frac {\operatorname {CosIntegral}(d x) \sin (c) d^2}{2 a}-\frac {\cos (c) \text {Si}(d x) d^2}{2 a}-\frac {\cos (c+d x) d}{2 a x}-\frac {b^{2/3} \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^{5/3}}+\frac {\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 )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \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^{5/3}}-\frac {\sin (c+d x)}{2 a x^2}-\frac {\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 )}{3 a^{5/3}}-\frac {b^{2/3} \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^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \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^{5/3}}\right )}{3 b}-\frac {\int \frac {\cos (c+d x)}{x^4 \left (b x^3+a\right )}dx}{b}\right )}{6 b}-\frac {-\frac {\sin (c+d x)}{3 b x^4 \left (b x^3+a\right )}-\frac {4 \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 {d \int \frac {\cos (c+d x)}{x^4 \left (b x^3+a\right )}dx}{3 b}}{6 b}\) |
| \(\Big \downarrow \) 3827 |
| \(\displaystyle -\frac {\sin (c+d x)}{6 b x \left (b x^3+a\right )^2}+\frac {d \left (-\frac {\cos (c+d x)}{3 b x^3 \left (b x^3+a\right )}-\frac {d \left (-\frac {\operatorname {CosIntegral}(d x) \sin (c) d^2}{2 a}-\frac {\cos (c) \text {Si}(d x) d^2}{2 a}-\frac {\cos (c+d x) d}{2 a x}-\frac {b^{2/3} \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^{5/3}}+\frac {\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 )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \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^{5/3}}-\frac {\sin (c+d x)}{2 a x^2}-\frac {\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 )}{3 a^{5/3}}-\frac {b^{2/3} \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^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \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^{5/3}}\right )}{3 b}-\frac {\int \left (\frac {b^2 \cos (c+d x) x^2}{a^2 \left (b x^3+a\right )}-\frac {b \cos (c+d x)}{a^2 x}+\frac {\cos (c+d x)}{a x^4}\right )dx}{b}\right )}{6 b}-\frac {-\frac {\sin (c+d x)}{3 b x^4 \left (b x^3+a\right )}-\frac {4 \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 {d \int \left (\frac {b^2 \cos (c+d x) x^2}{a^2 \left (b x^3+a\right )}-\frac {b \cos (c+d x)}{a^2 x}+\frac {\cos (c+d x)}{a x^4}\right )dx}{3 b}}{6 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sin (c+d x)}{6 b x \left (b x^3+a\right )^2}+\frac {d \left (-\frac {\cos (c+d x)}{3 b x^3 \left (b x^3+a\right )}-\frac {d \left (-\frac {\operatorname {CosIntegral}(d x) \sin (c) d^2}{2 a}-\frac {\cos (c) \text {Si}(d x) d^2}{2 a}-\frac {\cos (c+d x) d}{2 a x}-\frac {b^{2/3} \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^{5/3}}+\frac {\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 )}{3 a^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \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^{5/3}}-\frac {\sin (c+d x)}{2 a x^2}-\frac {\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 )}{3 a^{5/3}}-\frac {b^{2/3} \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^{5/3}}-\frac {(-1)^{2/3} b^{2/3} \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^{5/3}}\right )}{3 b}-\frac {\frac {\operatorname {CosIntegral}(d x) \sin (c) d^3}{6 a}+\frac {\cos (c) \text {Si}(d x) d^3}{6 a}+\frac {\cos (c+d x) d^2}{6 a x}+\frac {\sin (c+d x) d}{6 a x^2}-\frac {\cos (c+d x)}{3 a x^3}-\frac {b \cos (c) \operatorname {CosIntegral}(d x)}{a^2}+\frac {b \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^2}+\frac {b \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^2}+\frac {b \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^2}+\frac {b \sin (c) \text {Si}(d x)}{a^2}+\frac {b \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^2}-\frac {b \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^2}-\frac {b \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^2}}{b}\right )}{6 b}-\frac {-\frac {\sin (c+d x)}{3 b x^4 \left (b x^3+a\right )}-\frac {4 \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 {d \left (\frac {\operatorname {CosIntegral}(d x) \sin (c) d^3}{6 a}+\frac {\cos (c) \text {Si}(d x) d^3}{6 a}+\frac {\cos (c+d x) d^2}{6 a x}+\frac {\sin (c+d x) d}{6 a x^2}-\frac {\cos (c+d x)}{3 a x^3}-\frac {b \cos (c) \operatorname {CosIntegral}(d x)}{a^2}+\frac {b \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^2}+\frac {b \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^2}+\frac {b \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^2}+\frac {b \sin (c) \text {Si}(d x)}{a^2}+\frac {b \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^2}-\frac {b \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^2}-\frac {b \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^2}\right )}{3 b}}{6 b}\) |
-1/6*Sin[c + d*x]/(b*x*(a + b*x^3)^2) + (d*(-1/3*Cos[c + d*x]/(b*x^3*(a + b*x^3)) - (d*(-1/2*(d*Cos[c + d*x])/(a*x) - (d^2*CosIntegral[d*x]*Sin[c])/ (2*a) - (b^(2/3)*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d )/b^(1/3)])/(3*a^(5/3)) + ((-1)^(1/3)*b^(2/3)*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^(5/3)) - ((-1)^(2/3)*b^(2/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*S in[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(3*a^(5/3)) - Sin[c + d*x]/(2*a*x^ 2) - (d^2*Cos[c]*SinIntegral[d*x])/(2*a) - ((-1)^(1/3)*b^(2/3)*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^(5/3)) - (b^(2/3)*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^( 1/3)*d)/b^(1/3) + d*x])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*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^(5/3))))/(3*b) - (-1/3*Cos[c + d*x]/(a*x^3) + (d^2*Cos[c + d*x])/(6* a*x) - (b*Cos[c]*CosIntegral[d*x])/a^2 + (b*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^2) + (b* Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^ 2) + (b*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^ (1/3)*d)/b^(1/3) + d*x])/(3*a^2) + (d^3*CosIntegral[d*x]*Sin[c])/(6*a) + ( d*Sin[c + d*x])/(6*a*x^2) + (d^3*Cos[c]*SinIntegral[d*x])/(6*a) + (b*Sin[c ]*SinIntegral[d*x])/a^2 + (b*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*Si...
3.2.11.3.1 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 = 1.38 (sec) , antiderivative size = 610, normalized size of antiderivative = 0.53
| method | result | size |
| risch | \(\frac {i d c \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 (-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}+6 i c -6 \textit {\_R1} +10\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 a^{2} b}+\frac {i d \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 (-2 i b \textit {\_R1} \,c^{2}+\textit {\_R1}^{2} b c +4 i b \,\textit {\_R1}^{2}+a \,d^{3}-c^{3} b +2 i b \,c^{2}+2 b c \textit {\_R1} -4 i \textit {\_R1} b +6 c 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 a^{2} b^{2}}-\frac {i d c \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 (-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}-6 i c +6 \textit {\_R1} +10\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 a^{2} b}-\frac {i d \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 (-2 i b \textit {\_R1} \,c^{2}+\textit {\_R1}^{2} b c -4 i b \,\textit {\_R1}^{2}+a \,d^{3}-c^{3} b -2 i b \,c^{2}-2 b c \textit {\_R1} -4 i \textit {\_R1} b +6 c 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 a^{2} b^{2}}+\frac {d^{4} \left (d^{3} x^{3} b +a \,d^{3}\right ) \cos \left (d x +c \right )}{18 a b \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}-\frac {d \left (-4 b \,x^{5} d^{5}-7 a \,d^{5} x^{2}\right ) \sin \left (d x +c \right )}{18 a^{2} \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}\) | \(610\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(847\) |
| default | \(\text {Expression too large to display}\) | \(847\) |
1/108*I*d/a^2/b*c*sum((-2*I*c*_R1+6*I*c+_R1^2-c^2-6*_R1+10)/(-2*I*c*_R1+_R 1^2-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))+1/108*I*d/a^2/b^2*sum((-2*I*b*_R1*c^2+4*I*b*_R1^2 +2*I*b*c^2+_R1^2*b*c+a*d^3-c^3*b-4*I*_R1*b+2*b*c*_R1+6*c*b)/(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))-1/108*I*d/a^2/b*c*sum((-2*I*c*_R1-6*I*c+_R1^2-c^2+ 6*_R1+10)/(-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))-1/108*I*d/a^2/b^2*sum((- 2*I*b*_R1*c^2-4*I*b*_R1^2-2*I*b*c^2+_R1^2*b*c+a*d^3-c^3*b-4*I*_R1*b-2*b*c* _R1+6*c*b)/(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))+1/18*d^4/a*(b*d^3*x^3+a* d^3)/b/(b^2*d^6*x^6+2*a*b*d^6*x^3+a^2*d^6)*cos(d*x+c)-1/18*d*(-4*b*d^5*x^5 -7*a*d^5*x^2)/a^2/(b^2*d^6*x^6+2*a*b*d^6*x^3+a^2*d^6)*sin(d*x+c)
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 1319, normalized size of antiderivative = 1.16 \[ \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]
-1/216*((8*a*b^2*d^3*x^6 + 16*a^2*b*d^3*x^3 + 8*a^3*d^3 + 4*(I*b^3*x^6 + 2 *I*a*b^2*x^3 + I*a^2*b + sqrt(3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*(I*a*d^3 /b)^(2/3) - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3 + sqrt(3)*(I*a*b^2* d^3*x^6 + 2*I*a^2*b*d^3*x^3 + I*a^3*d^3))*(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) + (8*a*b^2*d^3*x^6 + 16*a^2*b*d^3*x^3 + 8*a^3*d^3 + 4*(-I*b^3 *x^6 - 2*I*a*b^2*x^3 - I*a^2*b - sqrt(3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))* (-I*a*d^3/b)^(2/3) - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3 + sqrt(3)* (I*a*b^2*d^3*x^6 + 2*I*a^2*b*d^3*x^3 + I*a^3*d^3))*(-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) + (8*a*b^2*d^3*x^6 + 16*a^2*b*d^3*x^3 + 8*a^3*d^3 + 4*(I*b^3*x^6 + 2*I*a*b^2*x^3 + I*a^2*b - sqrt(3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*(I*a*d^3/b)^(2/3) - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3 + sqrt(3)*(-I*a*b^2*d^3*x^6 - 2*I*a^2*b*d^3*x^3 - I*a^3*d^3))*(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) + (8*a*b^2*d^3*x^6 + 16*a^2*b*d^3*x^3 + 8* a^3*d^3 + 4*(-I*b^3*x^6 - 2*I*a*b^2*x^3 - I*a^2*b + sqrt(3)*(b^3*x^6 + 2*a *b^2*x^3 + a^2*b))*(-I*a*d^3/b)^(2/3) - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3 + sqrt(3)*(-I*a*b^2*d^3*x^6 - 2*I*a^2*b*d^3*x^3 - I*a^3*d^3))*(-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^(...
Timed out. \[ \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int { \frac {x \sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}} \,d x } \]
-1/2*((cos(c)^2 + sin(c)^2)*x*cos(d*x + c) + (x*cos(d*x + c)^2*cos(c) + x* cos(c)*sin(d*x + c)^2)*cos(d*x + 2*c) + 2*(((b^3*cos(c)^2 + b^3*sin(c)^2)* d*x^9 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d*x^6 + 3*(a^2*b*cos(c)^2 + a^ 2*b*sin(c)^2)*d*x^3 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d)*cos(d*x + c)^2 + (( b^3*cos(c)^2 + b^3*sin(c)^2)*d*x^9 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d *x^6 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d*x^3 + (a^3*cos(c)^2 + a^3*sin (c)^2)*d)*sin(d*x + c)^2)*integrate(1/2*(8*b*x^3 - a)*cos(d*x + c)/(b^4*d* x^12 + 4*a*b^3*d*x^9 + 6*a^2*b^2*d*x^6 + 4*a^3*b*d*x^3 + a^4*d), x) + 2*(( (b^3*cos(c)^2 + b^3*sin(c)^2)*d*x^9 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)* d*x^6 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d*x^3 + (a^3*cos(c)^2 + a^3*si n(c)^2)*d)*cos(d*x + c)^2 + ((b^3*cos(c)^2 + b^3*sin(c)^2)*d*x^9 + 3*(a*b^ 2*cos(c)^2 + a*b^2*sin(c)^2)*d*x^6 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d *x^3 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d)*sin(d*x + c)^2)*integrate(1/2*(8*b *x^3 - a)*cos(d*x + c)/((b^4*d*x^12 + 4*a*b^3*d*x^9 + 6*a^2*b^2*d*x^6 + 4* a^3*b*d*x^3 + a^4*d)*cos(d*x + c)^2 + (b^4*d*x^12 + 4*a*b^3*d*x^9 + 6*a^2* b^2*d*x^6 + 4*a^3*b*d*x^3 + a^4*d)*sin(d*x + c)^2), x) + (x*cos(d*x + c)^2 *sin(c) + x*sin(d*x + c)^2*sin(c))*sin(d*x + 2*c))/(((b^3*cos(c)^2 + b^3*s in(c)^2)*d*x^9 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d*x^6 + 3*(a^2*b*cos( c)^2 + a^2*b*sin(c)^2)*d*x^3 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d)*cos(d*x + c)^2 + ((b^3*cos(c)^2 + b^3*sin(c)^2)*d*x^9 + 3*(a*b^2*cos(c)^2 + a*b^2...
\[ \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int { \frac {x \sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx=\int \frac {x\,\sin \left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^3} \,d x \]