\(\int \frac {\sin (c+d x)}{x (a+b x^3)^3} \, dx\) [113]

Optimal result

Integrand size = 19, antiderivative size = 1163 \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^3} \, dx=\frac {d \cos (c+d x)}{18 a b^2 x^5}-\frac {d \cos (c+d x)}{18 a^2 b x^2}-\frac {d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}+\frac {4 \sqrt [3]{-1} 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {4 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {4 (-1)^{2/3} 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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac {\operatorname {CosIntegral}(d x) \sin (c)}{a^3}-\frac {\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 )}{3 a^3}+\frac {d^2 \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 )}{54 a^{7/3} b^{2/3}}-\frac {\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 {(-1)^{2/3} d^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 )}{54 a^{7/3} b^{2/3}}-\frac {\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 )}{3 a^3}-\frac {\sqrt [3]{-1} d^2 \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 )}{54 a^{7/3} b^{2/3}}-\frac {\sin (c+d x)}{6 a b^2 x^6}+\frac {\sin (c+d x)}{3 a^2 b x^3}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac {\cos (c) \text {Si}(d x)}{a^3}+\frac {\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 {(-1)^{2/3} d^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 )}{54 a^{7/3} b^{2/3}}+\frac {4 \sqrt [3]{-1} 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {\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 )}{3 a^3}+\frac {d^2 \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 )}{54 a^{7/3} b^{2/3}}+\frac {4 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {\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 )}{3 a^3}-\frac {\sqrt [3]{-1} d^2 \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 )}{54 a^{7/3} b^{2/3}}+\frac {4 (-1)^{2/3} 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 )}{27 a^{8/3} \sqrt [3]{b}} \]

[Out]

-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^2-1/18*d*cos(d*x+c)/b^2/x^5/(b*x^3+a)+1/54*d^2*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)+1/54*d^2*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)+4/27*d*Si(a
^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^(8/3)/b^(1/3)-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*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^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+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)+cos(c)*Si(d*x)/a^3+Ci(d*x)*sin(c)/a^3-1/6*sin(d*x+c)/a/b^2
/x^6+1/3*sin(d*x+c)/a^2/b/x^3-1/6*sin(d*x+c)/b/x^3/(b*x^3+a)^2+1/6*sin(d*x+c)/b^2/x^6/(b*x^3+a)

Rubi [A] (verified)

Time = 2.41 (sec) , antiderivative size = 1163, normalized size of antiderivative = 1.00, number of steps used = 110, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3424, 3426, 3378, 3384, 3380, 3383, 3427, 3415, 3425} \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^3} \, dx=\frac {\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 ) d^2}{54 a^{7/3} b^{2/3}}+\frac {(-1)^{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 ) d^2}{54 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} \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 ) d^2}{54 a^{7/3} b^{2/3}}-\frac {(-1)^{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 ) d^2}{54 a^{7/3} b^{2/3}}+\frac {\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 ) d^2}{54 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} \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 ) d^2}{54 a^{7/3} b^{2/3}}-\frac {\cos (c+d x) d}{18 b^2 x^5 \left (b x^3+a\right )}-\frac {\cos (c+d x) d}{18 a^2 b x^2}+\frac {\cos (c+d x) d}{18 a b^2 x^5}+\frac {4 \sqrt [3]{-1} \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 ) d}{27 a^{8/3} \sqrt [3]{b}}-\frac {4 \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 ) d}{27 a^{8/3} \sqrt [3]{b}}-\frac {4 (-1)^{2/3} \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 ) d}{27 a^{8/3} \sqrt [3]{b}}+\frac {4 \sqrt [3]{-1} \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 ) d}{27 a^{8/3} \sqrt [3]{b}}+\frac {4 \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 ) d}{27 a^{8/3} \sqrt [3]{b}}+\frac {4 (-1)^{2/3} \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 ) d}{27 a^{8/3} \sqrt [3]{b}}+\frac {\operatorname {CosIntegral}(d x) \sin (c)}{a^3}-\frac {\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 {\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 {\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 {\sin (c+d x)}{6 b^2 x^6 \left (b x^3+a\right )}-\frac {\sin (c+d x)}{6 b x^3 \left (b x^3+a\right )^2}+\frac {\sin (c+d x)}{3 a^2 b x^3}-\frac {\sin (c+d x)}{6 a b^2 x^6}+\frac {\cos (c) \text {Si}(d x)}{a^3}+\frac {\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 {\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 {\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} \]

[In]

Int[Sin[c + d*x]/(x*(a + b*x^3)^3),x]

[Out]

(d*Cos[c + d*x])/(18*a*b^2*x^5) - (d*Cos[c + d*x])/(18*a^2*b*x^2) - (d*Cos[c + d*x])/(18*b^2*x^5*(a + b*x^3))
+ (4*(-1)^(1/3)*d*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])/(
27*a^(8/3)*b^(1/3)) - (4*d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(8/3)*b^
(1/3)) - (4*(-1)^(2/3)*d*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])/(27*a^(8/3)*b^(1/3)) + (CosIntegral[d*x]*Sin[c])/a^3 - (CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (
a^(1/3)*d)/b^(1/3)])/(3*a^3) + (d^2*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(54*a
^(7/3)*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^3) + ((-1)^(2/3)*d^2*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)])/(54*a^(7/3)*b^(2/3)) - (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^3) - ((-1)^(1/3)*d^2*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)])/(54*a^(7/3)*b^(2/3)) - Sin[c + d*x]/(6*a*b^2*x^6) + Sin[c + d*x]/(3*a^2*b*x^3) - Sin[c + d*
x]/(6*b*x^3*(a + b*x^3)^2) + Sin[c + d*x]/(6*b^2*x^6*(a + b*x^3)) + (Cos[c]*SinIntegral[d*x])/a^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^3) - ((-1)^(2/3)*d^2*Co
s[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a^(7/3)*b^(2/3))
+ (4*(-1)^(1/3)*d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(
27*a^(8/3)*b^(1/3)) - (Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^3) + (d^2*Cos
[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(54*a^(7/3)*b^(2/3)) + (4*d*Sin[c - (a^(1/3)
*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(8/3)*b^(1/3)) - (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^3) - ((-1)^(1/3)*d^2*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])/(54*a^(7/3)*b^(2/3)) + (4*(-1)^(2/3)*d*Sin[c
- ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(8/3)*b^(1/3))

Rule 3378

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m
+ 1))), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3415

Int[Cos[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cos[c + d*x], (a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])

Rule 3424

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[x^(m - n + 1)*(a + b*x
^n)^(p + 1)*(Sin[c + d*x]/(b*n*(p + 1))), x] + (-Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*(a + b*x^n)^(p
+ 1)*Sin[c + d*x], x], x] - Dist[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]

Rule 3425

Int[Cos[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m - n + 1)*(a + b*x
^n)^(p + 1)*(Cos[c + d*x]/(b*n*(p + 1))), x] + (-Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*(a + b*x^n)^(p
+ 1)*Cos[c + d*x], x], x] + Dist[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]

Rule 3426

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sin[c +
d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ
[p, -1]) && IntegerQ[m]

Rule 3427

Int[Cos[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cos[c +
d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ
[p, -1]) && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}-\frac {\int \frac {\sin (c+d x)}{x^4 \left (a+b x^3\right )^2} \, dx}{2 b}+\frac {d \int \frac {\cos (c+d x)}{x^3 \left (a+b x^3\right )^2} \, dx}{6 b} \\ & = -\frac {d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac {\int \frac {\sin (c+d x)}{x^7 \left (a+b x^3\right )} \, dx}{b^2}-\frac {d \int \frac {\cos (c+d x)}{x^6 \left (a+b x^3\right )} \, dx}{6 b^2}-\frac {(5 d) \int \frac {\cos (c+d x)}{x^6 \left (a+b x^3\right )} \, dx}{18 b^2}-\frac {d^2 \int \frac {\sin (c+d x)}{x^5 \left (a+b x^3\right )} \, dx}{18 b^2} \\ & = -\frac {d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac {\int \left (\frac {\sin (c+d x)}{a x^7}-\frac {b \sin (c+d x)}{a^2 x^4}+\frac {b^2 \sin (c+d x)}{a^3 x}-\frac {b^3 x^2 \sin (c+d x)}{a^3 \left (a+b x^3\right )}\right ) \, dx}{b^2}-\frac {d \int \left (\frac {\cos (c+d x)}{a x^6}-\frac {b \cos (c+d x)}{a^2 x^3}+\frac {b^2 \cos (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{6 b^2}-\frac {(5 d) \int \left (\frac {\cos (c+d x)}{a x^6}-\frac {b \cos (c+d x)}{a^2 x^3}+\frac {b^2 \cos (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{18 b^2}-\frac {d^2 \int \left (\frac {\sin (c+d x)}{a x^5}-\frac {b \sin (c+d x)}{a^2 x^2}+\frac {b^2 x \sin (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{18 b^2} \\ & = -\frac {d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac {\int \frac {\sin (c+d x)}{x} \, dx}{a^3}+\frac {\int \frac {\sin (c+d x)}{x^7} \, dx}{a b^2}-\frac {\int \frac {\sin (c+d x)}{x^4} \, dx}{a^2 b}-\frac {b \int \frac {x^2 \sin (c+d x)}{a+b x^3} \, dx}{a^3}-\frac {d \int \frac {\cos (c+d x)}{a+b x^3} \, dx}{6 a^2}-\frac {(5 d) \int \frac {\cos (c+d x)}{a+b x^3} \, dx}{18 a^2}-\frac {d \int \frac {\cos (c+d x)}{x^6} \, dx}{6 a b^2}-\frac {(5 d) \int \frac {\cos (c+d x)}{x^6} \, dx}{18 a b^2}+\frac {d \int \frac {\cos (c+d x)}{x^3} \, dx}{6 a^2 b}+\frac {(5 d) \int \frac {\cos (c+d x)}{x^3} \, dx}{18 a^2 b}-\frac {d^2 \int \frac {x \sin (c+d x)}{a+b x^3} \, dx}{18 a^2}-\frac {d^2 \int \frac {\sin (c+d x)}{x^5} \, dx}{18 a b^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x^2} \, dx}{18 a^2 b} \\ & = \frac {4 d \cos (c+d x)}{45 a b^2 x^5}-\frac {2 d \cos (c+d x)}{9 a^2 b x^2}-\frac {d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 a b^2 x^6}+\frac {d^2 \sin (c+d x)}{72 a b^2 x^4}+\frac {\sin (c+d x)}{3 a^2 b x^3}-\frac {d^2 \sin (c+d x)}{18 a^2 b x}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}-\frac {b \int \left (\frac {\sin (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\sin (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\sin (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{a^3}-\frac {d \int \left (-\frac {\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac {\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{6 a^2}-\frac {(5 d) \int \left (-\frac {\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac {\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{18 a^2}+\frac {d \int \frac {\cos (c+d x)}{x^6} \, dx}{6 a b^2}-\frac {d \int \frac {\cos (c+d x)}{x^3} \, dx}{3 a^2 b}-\frac {d^2 \int \left (-\frac {\sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {(-1)^{2/3} \sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac {\sqrt [3]{-1} \sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{18 a^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x^5} \, dx}{30 a b^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x^5} \, dx}{18 a b^2}-\frac {d^2 \int \frac {\sin (c+d x)}{x^2} \, dx}{12 a^2 b}-\frac {\left (5 d^2\right ) \int \frac {\sin (c+d x)}{x^2} \, dx}{36 a^2 b}-\frac {d^3 \int \frac {\cos (c+d x)}{x^4} \, dx}{72 a b^2}+\frac {d^3 \int \frac {\cos (c+d x)}{x} \, dx}{18 a^2 b}+\frac {\cos (c) \int \frac {\sin (d x)}{x} \, dx}{a^3}+\frac {\sin (c) \int \frac {\cos (d x)}{x} \, dx}{a^3} \\ & = \frac {d \cos (c+d x)}{18 a b^2 x^5}+\frac {d^3 \cos (c+d x)}{216 a b^2 x^3}-\frac {d \cos (c+d x)}{18 a^2 b x^2}-\frac {d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}+\frac {\operatorname {CosIntegral}(d x) \sin (c)}{a^3}-\frac {\sin (c+d x)}{6 a b^2 x^6}-\frac {d^2 \sin (c+d x)}{120 a b^2 x^4}+\frac {\sin (c+d x)}{3 a^2 b x^3}+\frac {d^2 \sin (c+d x)}{6 a^2 b x}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac {\cos (c) \text {Si}(d x)}{a^3}-\frac {\sqrt [3]{b} \int \frac {\sin (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^3}-\frac {\sqrt [3]{b} \int \frac {\sin (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^3}-\frac {\sqrt [3]{b} \int \frac {\sin (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^3}+\frac {d \int \frac {\cos (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{18 a^{8/3}}+\frac {d \int \frac {\cos (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{18 a^{8/3}}+\frac {d \int \frac {\cos (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{18 a^{8/3}}+\frac {(5 d) \int \frac {\cos (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{54 a^{8/3}}+\frac {(5 d) \int \frac {\cos (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{8/3}}+\frac {(5 d) \int \frac {\cos (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{8/3}}-\frac {d^2 \int \frac {\sin (c+d x)}{x^5} \, dx}{30 a b^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x^2} \, dx}{6 a^2 b}+\frac {d^2 \int \frac {\sin (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^{7/3} \sqrt [3]{b}}-\frac {\left (\sqrt [3]{-1} d^2\right ) \int \frac {\sin (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{7/3} \sqrt [3]{b}}+\frac {\left ((-1)^{2/3} d^2\right ) \int \frac {\sin (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{7/3} \sqrt [3]{b}}+\frac {d^3 \int \frac {\cos (c+d x)}{x^4} \, dx}{120 a b^2}+\frac {d^3 \int \frac {\cos (c+d x)}{x^4} \, dx}{72 a b^2}-\frac {d^3 \int \frac {\cos (c+d x)}{x} \, dx}{12 a^2 b}-\frac {\left (5 d^3\right ) \int \frac {\cos (c+d x)}{x} \, dx}{36 a^2 b}+\frac {d^4 \int \frac {\sin (c+d x)}{x^3} \, dx}{216 a b^2}+\frac {\left (d^3 \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{18 a^2 b}-\frac {\left (d^3 \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{18 a^2 b} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (verified)

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 \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^3} \, dx=\text {Result too large to show} \]

[In]

Integrate[Sin[c + d*x]/(x*(a + b*x^3)^3),x]

[Out]

(-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*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)] & ] + (18*I)*b*(a + b*x^3)^2*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)] & ] - 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]*SinI
ntegral[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]*CosIntegral[d*(x - #1)] - I*CosIn
tegral[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*CosIntegral[d*(x - #1)]*Si
n[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]*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 & ] - I*a^3*d*Root
Sum[a + b*#1^3 & , ((-2*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 2
*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + (2*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosInte
gral[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 & ] - (2*I)*a^2*b*d*x^3*RootSum[a + b*#1^3 & , ((-2*I)*
Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 2*Cos[c + d*#1]*SinIntegral[
d*(x - #1)] + (2*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - I*d*C
osIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIn
tegral[d*(x - #1)]*#1)/#1^2 & ] - I*a*b^2*d*x^6*RootSum[a + b*#1^3 & , ((-2*I)*Cos[c + d*#1]*CosIntegral[d*(x
- #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + (2*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 & ]
+ I*a^3*d*RootSum[a + b*#1^3 & , ((2*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin
[c + d*#1] - 2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - (2*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]*SinIntegr
al[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] + (2*I)*a^2*b*d*x^3*RootSum[a + b*#1^
3 & , ((2*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 2*Cos[c + d*#1]
*SinIntegral[d*(x - #1)] - (2*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 & ] + I*a*b^2*d*x^6*RootSum[a + b*#1^3 & , ((2*I)*Cos[c + d*#1]*CosIn
tegral[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - (2*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 & ] + 108*a^2*b*CosIntegral[d*x]*Sin[c] + 216*a*b^2*x^3*CosIntegral[d*x]*Sin[c] + 108*b^3*x^6*CosInte
gral[d*x]*Sin[c] + 54*a^2*b*Sin[c + d*x] + 36*a*b^2*x^3*Sin[c + d*x] + 108*a^2*b*Cos[c]*SinIntegral[d*x] + 216
*a*b^2*x^3*Cos[c]*SinIntegral[d*x] + 108*b^3*x^6*Cos[c]*SinIntegral[d*x])/(108*a^3*b*(a + b*x^3)^2)

Maple [C] (verified)

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	\(-\frac {i \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-3 i \textit {\_Z}^{2} b c -i d^{3} a +i b \,c^{3}+b \,\textit {\_Z}^{3}-3 c^{2} b \textit {\_Z} \right )}{\sum }\frac {\left (i \textit {\_R1} a \,d^{3}+a c \,d^{3}-8 i d^{3} a -36 i \textit {\_R1} b c +18 b \,\textit {\_R1}^{2}-18 c^{2} b \right ) {\mathrm e}^{\textit {\_R1}} \operatorname {Ei}_{1}\left (-i d x -i c +\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{108 b \,a^{3}}+\frac {i {\mathrm e}^{i c} \operatorname {Ei}_{1}\left (-i d x \right )}{2 a^{3}}+\frac {i \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (-3 i \textit {\_Z}^{2} b c -i d^{3} a +i b \,c^{3}+b \,\textit {\_Z}^{3}-3 c^{2} b \textit {\_Z} \right )}{\sum }\frac {\left (i \textit {\_R1} a \,d^{3}+a c \,d^{3}+8 i d^{3} a -36 i \textit {\_R1} b c +18 b \,\textit {\_R1}^{2}-18 c^{2} b \right ) {\mathrm e}^{-\textit {\_R1}} \operatorname {Ei}_{1}\left (i d x +i c -\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{108 b \,a^{3}}-\frac {{\mathrm e}^{-i c} \pi \,\operatorname {csgn}\left (d x \right )}{2 a^{3}}+\frac {{\mathrm e}^{-i c} \operatorname {Si}\left (d x \right )}{a^{3}}-\frac {i {\mathrm e}^{-i c} \operatorname {Ei}_{1}\left (-i d x \right )}{2 a^{3}}-\frac {d^{7} \cos \left (d x +c \right ) b \,x^{4}}{18 a^{2} \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}-\frac {d^{7} \cos \left (d x +c \right ) x}{18 a \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}+\frac {d^{6} \sin \left (d x +c \right ) x^{3} b}{3 a^{2} \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}+\frac {d^{6} \sin \left (d x +c \right )}{2 a \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}\)	\(495\)

[In]

int(sin(d*x+c)/x/(b*x^3+a)^3,x,method=_RETURNVERBOSE)

[Out]

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*_R1*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))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.35 (sec) , antiderivative size = 1113, normalized size of antiderivative = 0.96 \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(sin(d*x+c)/x/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

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 + 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) + (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*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*(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*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)) + 216*(b^2*x^6 + 2*a*b*x^3 + a^2)*cos_integral(d*x)*sin(c) + 216*(b^2*x^6 + 2*a*b*x^3 + a^2)*cos(c)*si
n_integral(d*x) - 12*(a*b*d*x^4 + a^2*d*x)*cos(d*x + c) + 36*(2*a*b*x^3 + 3*a^2)*sin(d*x + c))/(a^3*b^2*x^6 +
2*a^4*b*x^3 + a^5)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(sin(d*x+c)/x/(b*x**3+a)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^3} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3} x} \,d x } \]

[In]

integrate(sin(d*x+c)/x/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

integrate(sin(d*x + c)/((b*x^3 + a)^3*x), x)

Giac [F]

\[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^3} \, dx=\int { \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3} x} \,d x } \]

[In]

integrate(sin(d*x+c)/x/(b*x^3+a)^3,x, algorithm="giac")

[Out]

integrate(sin(d*x + c)/((b*x^3 + a)^3*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^3} \, dx=\int \frac {\sin \left (c+d\,x\right )}{x\,{\left (b\,x^3+a\right )}^3} \,d x \]

[In]

int(sin(c + d*x)/(x*(a + b*x^3)^3),x)

[Out]

int(sin(c + d*x)/(x*(a + b*x^3)^3), x)