3.2.12 \(\int \frac {\sin (c+d x)}{(a+b x^3)^3} \, dx\) [112]

Optimal. Leaf size=1161 \[ \frac {d \cos (c+d x)}{18 a b^2 x^4}-\frac {d \cos (c+d x)}{18 a^2 b x}-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {(-1)^{2/3} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\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^2 b}-\frac {5 \sqrt [3]{-1} \text {Ci}\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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\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^2 b}+\frac {5 (-1)^{2/3} \text {Ci}\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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\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^2 b}-\frac {\sin (c+d x)}{9 a b^2 x^5}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {5 \sqrt [3]{-1} \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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac {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^2 b}+\frac {(-1)^{2/3} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \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 )}{27 a^{8/3} \sqrt [3]{b}}-\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^2 b}-\frac {d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 (-1)^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {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^2 b}+\frac {\sqrt [3]{-1} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}} \]

[Out]

1/9*(-1)^(2/3)*d*Ci((-1)^(1/3)*a^(1/3)*d/b^(1/3)-d*x)*cos(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a^(7/3)/b^(2/3)-1/9*
(-1)^(1/3)*d*Ci((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)*cos(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^(7/3)/b^(2/3)+1/9*d*Ci
(a^(1/3)*d/b^(1/3)+d*x)*cos(c-a^(1/3)*d/b^(1/3))/a^(7/3)/b^(2/3)+1/18*d*cos(d*x+c)/a/b^2/x^4-1/18*d*cos(d*x+c)
/a^2/b/x-1/18*d*cos(d*x+c)/b^2/x^4/(b*x^3+a)+5/27*cos(c-a^(1/3)*d/b^(1/3))*Si(a^(1/3)*d/b^(1/3)+d*x)/a^(8/3)/b
^(1/3)+5/27*Ci(a^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^(8/3)/b^(1/3)-1/9*sin(d*x+c)/a/b^2/x^5+5/18*s
in(d*x+c)/a^2/b/x^2-1/6*sin(d*x+c)/b/x^2/(b*x^3+a)^2+1/9*sin(d*x+c)/b^2/x^5/(b*x^3+a)-5/27*(-1)^(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^(8/3)/b^(1/3)-1/54*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^2/b+1/9*(-1)^(1/3)*d*Si((-1)^(2/3)*a^(1/3)*d/b^(1/3)+
d*x)*sin(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^(7/3)/b^(2/3)-1/54*d^2*cos(c-a^(1/3)*d/b^(1/3))*Si(a^(1/3)*d/b^(1/3
)+d*x)/a^2/b+5/27*(-1)^(2/3)*cos(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))*Si((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)/a^(8/3)/
b^(1/3)-1/54*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^2/b-1/54*d^2*Ci(a^
(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^2/b-1/9*d*Si(a^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a
^(7/3)/b^(2/3)-5/27*(-1)^(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^(8/3
)/b^(1/3)-1/54*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^2/b+5/27*(-1)^(2
/3)*Ci((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)*sin(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^(8/3)/b^(1/3)-1/54*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^2/b-1/9*(-1)^(2/3)*d*Si(-(-1)^(1/3)*a^(1/3)
*d/b^(1/3)+d*x)*sin(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a^(7/3)/b^(2/3)

________________________________________________________________________________________

Rubi [A]
time = 2.08, antiderivative size = 1161, normalized size of antiderivative = 1.00, number of steps used = 99, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3412, 3424, 3426, 3378, 3384, 3380, 3383, 3414, 3427, 3425} \begin {gather*} -\frac {\text {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^2 b}-\frac {\text {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^2 b}-\frac {\text {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^2 b}+\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 ) d^2}{54 a^2 b}-\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^2 b}-\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 ) d^2}{54 a^2 b}-\frac {\cos (c+d x) d}{18 a^2 b x}-\frac {\cos (c+d x) d}{18 b^2 x^4 \left (b x^3+a\right )}+\frac {\cos (c+d x) d}{18 a b^2 x^4}+\frac {(-1)^{2/3} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) d}{9 a^{7/3} b^{2/3}}+\frac {\cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {CosIntegral}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{9 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 {CosIntegral}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{9 a^{7/3} b^{2/3}}+\frac {(-1)^{2/3} \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 ) d}{9 a^{7/3} b^{2/3}}-\frac {\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}{9 a^{7/3} b^{2/3}}+\frac {\sqrt [3]{-1} \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}{9 a^{7/3} b^{2/3}}+\frac {5 \text {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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {5 \sqrt [3]{-1} \text {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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac {5 (-1)^{2/3} \text {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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (b x^3+a\right )}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}-\frac {\sin (c+d x)}{6 b x^2 \left (b x^3+a\right )^2}-\frac {\sin (c+d x)}{9 a b^2 x^5}+\frac {5 \sqrt [3]{-1} \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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac {5 \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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac {5 (-1)^{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 )}{27 a^{8/3} \sqrt [3]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(a + b*x^3)^3,x]

[Out]

(d*Cos[c + d*x])/(18*a*b^2*x^4) - (d*Cos[c + d*x])/(18*a^2*b*x) - (d*Cos[c + d*x])/(18*b^2*x^4*(a + b*x^3)) +
((-1)^(2/3)*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])/(9*a^
(7/3)*b^(2/3)) + (d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(7/3)*b^(2/3)) -
((-1)^(1/3)*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])/(9*a
^(7/3)*b^(2/3)) + (5*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(27*a^(8/3)*b^(1/3))
- (d^2*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(54*a^2*b) - (5*(-1)^(1/3)*CosInt
egral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(27*a^(8/3)*b^(1/3)) - (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^2*b) + (5*
(-1)^(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)])/(27*a^(8
/3)*b^(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^2*b) - Sin[c + d*x]/(9*a*b^2*x^5) + (5*Sin[c + d*x])/(18*a^2*b*x^2) - Sin[c + d*x]/(6*b*x^2*(a + b*x^3)
^2) + Sin[c + d*x]/(9*b^2*x^5*(a + b*x^3)) + (5*(-1)^(1/3)*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])/(27*a^(8/3)*b^(1/3)) + (d^2*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*Si
nIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a^2*b) + ((-1)^(2/3)*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])/(9*a^(7/3)*b^(2/3)) + (5*Cos[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)) - (d^2*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[
(a^(1/3)*d)/b^(1/3) + d*x])/(54*a^2*b) - (d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x
])/(9*a^(7/3)*b^(2/3)) + (5*(-1)^(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])/(27*a^(8/3)*b^(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^2*b) + ((-1)^(1/3)*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])/(9*a^(7/3)*b^(2/3))

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 3412

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

Rule 3414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sin[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*} \int \frac {\sin (c+d x)}{\left (a+b x^3\right )^3} \, dx &=-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}-\frac {\int \frac {\sin (c+d x)}{x^3 \left (a+b x^3\right )^2} \, dx}{3 b}+\frac {d \int \frac {\cos (c+d x)}{x^2 \left (a+b x^3\right )^2} \, dx}{6 b}\\ &=-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {5 \int \frac {\sin (c+d x)}{x^6 \left (a+b x^3\right )} \, dx}{9 b^2}-\frac {d \int \frac {\cos (c+d x)}{x^5 \left (a+b x^3\right )} \, dx}{9 b^2}-\frac {(2 d) \int \frac {\cos (c+d x)}{x^5 \left (a+b x^3\right )} \, dx}{9 b^2}-\frac {d^2 \int \frac {\sin (c+d x)}{x^4 \left (a+b x^3\right )} \, dx}{18 b^2}\\ &=-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {5 \int \left (\frac {\sin (c+d x)}{a x^6}-\frac {b \sin (c+d x)}{a^2 x^3}+\frac {b^2 \sin (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{9 b^2}-\frac {d \int \left (\frac {\cos (c+d x)}{a x^5}-\frac {b \cos (c+d x)}{a^2 x^2}+\frac {b^2 x \cos (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{9 b^2}-\frac {(2 d) \int \left (\frac {\cos (c+d x)}{a x^5}-\frac {b \cos (c+d x)}{a^2 x^2}+\frac {b^2 x \cos (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{9 b^2}-\frac {d^2 \int \left (\frac {\sin (c+d x)}{a x^4}-\frac {b \sin (c+d x)}{a^2 x}+\frac {b^2 x^2 \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^4 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {5 \int \frac {\sin (c+d x)}{a+b x^3} \, dx}{9 a^2}+\frac {5 \int \frac {\sin (c+d x)}{x^6} \, dx}{9 a b^2}-\frac {5 \int \frac {\sin (c+d x)}{x^3} \, dx}{9 a^2 b}-\frac {d \int \frac {x \cos (c+d x)}{a+b x^3} \, dx}{9 a^2}-\frac {(2 d) \int \frac {x \cos (c+d x)}{a+b x^3} \, dx}{9 a^2}-\frac {d \int \frac {\cos (c+d x)}{x^5} \, dx}{9 a b^2}-\frac {(2 d) \int \frac {\cos (c+d x)}{x^5} \, dx}{9 a b^2}+\frac {d \int \frac {\cos (c+d x)}{x^2} \, dx}{9 a^2 b}+\frac {(2 d) \int \frac {\cos (c+d x)}{x^2} \, dx}{9 a^2 b}-\frac {d^2 \int \frac {x^2 \sin (c+d x)}{a+b x^3} \, dx}{18 a^2}-\frac {d^2 \int \frac {\sin (c+d x)}{x^4} \, dx}{18 a b^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x} \, dx}{18 a^2 b}\\ &=\frac {d \cos (c+d x)}{12 a b^2 x^4}-\frac {d \cos (c+d x)}{3 a^2 b x}-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{9 a b^2 x^5}+\frac {d^2 \sin (c+d x)}{54 a b^2 x^3}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {5 \int \left (-\frac {\sin (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\sin (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac {\sin (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{9 a^2}-\frac {d \int \left (-\frac {\cos (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {(-1)^{2/3} \cos (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} \cos (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{9 a^2}-\frac {(2 d) \int \left (-\frac {\cos (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {(-1)^{2/3} \cos (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} \cos (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{9 a^2}+\frac {d \int \frac {\cos (c+d x)}{x^5} \, dx}{9 a b^2}-\frac {(5 d) \int \frac {\cos (c+d x)}{x^2} \, dx}{18 a^2 b}-\frac {d^2 \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}{18 a^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x^4} \, dx}{36 a b^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x^4} \, dx}{18 a b^2}-\frac {d^2 \int \frac {\sin (c+d x)}{x} \, dx}{9 a^2 b}-\frac {\left (2 d^2\right ) \int \frac {\sin (c+d x)}{x} \, dx}{9 a^2 b}-\frac {d^3 \int \frac {\cos (c+d x)}{x^3} \, dx}{54 a b^2}+\frac {\left (d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{18 a^2 b}+\frac {\left (d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{18 a^2 b}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^4}+\frac {d^3 \cos (c+d x)}{108 a b^2 x^2}-\frac {d \cos (c+d x)}{18 a^2 b x}-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {d^2 \text {Ci}(d x) \sin (c)}{18 a^2 b}-\frac {\sin (c+d x)}{9 a b^2 x^5}-\frac {d^2 \sin (c+d x)}{108 a b^2 x^3}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {d^2 \cos (c) \text {Si}(d x)}{18 a^2 b}-\frac {5 \int \frac {\sin (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{27 a^{8/3}}-\frac {5 \int \frac {\sin (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{8/3}}-\frac {5 \int \frac {\sin (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{8/3}}+\frac {d \int \frac {\cos (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac {(2 d) \int \frac {\cos (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (\sqrt [3]{-1} d\right ) \int \frac {\cos (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (2 \sqrt [3]{-1} d\right ) \int \frac {\cos (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac {\left ((-1)^{2/3} d\right ) \int \frac {\cos (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac {\left (2 (-1)^{2/3} d\right ) \int \frac {\cos (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {d^2 \int \frac {\sin (c+d x)}{x^4} \, dx}{36 a b^2}+\frac {\left (5 d^2\right ) \int \frac {\sin (c+d x)}{x} \, dx}{18 a^2 b}-\frac {d^2 \int \frac {\sin (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {d^2 \int \frac {\sin (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {d^2 \int \frac {\sin (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}+\frac {d^3 \int \frac {\cos (c+d x)}{x^3} \, dx}{108 a b^2}+\frac {d^3 \int \frac {\cos (c+d x)}{x^3} \, dx}{54 a b^2}+\frac {d^4 \int \frac {\sin (c+d x)}{x^2} \, dx}{108 a b^2}-\frac {\left (d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{9 a^2 b}-\frac {\left (2 d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{9 a^2 b}-\frac {\left (d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{9 a^2 b}-\frac {\left (2 d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{9 a^2 b}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^4}-\frac {d^3 \cos (c+d x)}{216 a b^2 x^2}-\frac {d \cos (c+d x)}{18 a^2 b x}-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac {5 d^2 \text {Ci}(d x) \sin (c)}{18 a^2 b}-\frac {\sin (c+d x)}{9 a b^2 x^5}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}-\frac {d^4 \sin (c+d x)}{108 a b^2 x}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}-\frac {5 d^2 \cos (c) \text {Si}(d x)}{18 a^2 b}-\frac {d^3 \int \frac {\cos (c+d x)}{x^3} \, dx}{108 a b^2}-\frac {d^4 \int \frac {\sin (c+d x)}{x^2} \, dx}{216 a b^2}-\frac {d^4 \int \frac {\sin (c+d x)}{x^2} \, dx}{108 a b^2}+\frac {d^5 \int \frac {\cos (c+d x)}{x} \, dx}{108 a b^2}+\frac {\left (5 d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{18 a^2 b}-\frac {\left (5 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{27 a^{8/3}}+\frac {\left (d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac {\left (2 d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (d^2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}+\frac {\left (5 \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{8/3}}-\frac {\left (\sqrt [3]{-1} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (2 \sqrt [3]{-1} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac {\left (d^2 \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {\left (5 \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{8/3}}+\frac {\left ((-1)^{2/3} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac {\left (2 (-1)^{2/3} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (d^2 \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}+\frac {\left (5 d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{18 a^2 b}-\frac {\left (5 \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{27 a^{8/3}}-\frac {\left (d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (2 d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (d^2 \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {\left (5 \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{8/3}}-\frac {\left (\sqrt [3]{-1} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (2 \sqrt [3]{-1} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (d^2 \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {\left (5 \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{8/3}}-\frac {\left ((-1)^{2/3} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (2 (-1)^{2/3} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (d^2 \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^4}-\frac {d \cos (c+d x)}{18 a^2 b x}-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {(-1)^{2/3} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\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^2 b}-\frac {5 \sqrt [3]{-1} \text {Ci}\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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\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^2 b}+\frac {5 (-1)^{2/3} \text {Ci}\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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\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^2 b}-\frac {\sin (c+d x)}{9 a b^2 x^5}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}+\frac {d^4 \sin (c+d x)}{216 a b^2 x}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {5 \sqrt [3]{-1} \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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac {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^2 b}+\frac {(-1)^{2/3} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \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 )}{27 a^{8/3} \sqrt [3]{b}}-\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^2 b}-\frac {d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 (-1)^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {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^2 b}+\frac {\sqrt [3]{-1} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {d^4 \int \frac {\sin (c+d x)}{x^2} \, dx}{216 a b^2}-\frac {d^5 \int \frac {\cos (c+d x)}{x} \, dx}{216 a b^2}-\frac {d^5 \int \frac {\cos (c+d x)}{x} \, dx}{108 a b^2}+\frac {\left (d^5 \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{108 a b^2}-\frac {\left (d^5 \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{108 a b^2}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^4}-\frac {d \cos (c+d x)}{18 a^2 b x}-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {d^5 \cos (c) \text {Ci}(d x)}{108 a b^2}+\frac {(-1)^{2/3} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\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^2 b}-\frac {5 \sqrt [3]{-1} \text {Ci}\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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\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^2 b}+\frac {5 (-1)^{2/3} \text {Ci}\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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\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^2 b}-\frac {\sin (c+d x)}{9 a b^2 x^5}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}-\frac {d^5 \sin (c) \text {Si}(d x)}{108 a b^2}+\frac {5 \sqrt [3]{-1} \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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac {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^2 b}+\frac {(-1)^{2/3} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \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 )}{27 a^{8/3} \sqrt [3]{b}}-\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^2 b}-\frac {d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 (-1)^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {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^2 b}+\frac {\sqrt [3]{-1} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {d^5 \int \frac {\cos (c+d x)}{x} \, dx}{216 a b^2}-\frac {\left (d^5 \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{216 a b^2}-\frac {\left (d^5 \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{108 a b^2}+\frac {\left (d^5 \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{216 a b^2}+\frac {\left (d^5 \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{108 a b^2}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^4}-\frac {d \cos (c+d x)}{18 a^2 b x}-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac {d^5 \cos (c) \text {Ci}(d x)}{216 a b^2}+\frac {(-1)^{2/3} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\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^2 b}-\frac {5 \sqrt [3]{-1} \text {Ci}\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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\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^2 b}+\frac {5 (-1)^{2/3} \text {Ci}\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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\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^2 b}-\frac {\sin (c+d x)}{9 a b^2 x^5}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {d^5 \sin (c) \text {Si}(d x)}{216 a b^2}+\frac {5 \sqrt [3]{-1} \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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac {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^2 b}+\frac {(-1)^{2/3} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \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 )}{27 a^{8/3} \sqrt [3]{b}}-\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^2 b}-\frac {d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 (-1)^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {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^2 b}+\frac {\sqrt [3]{-1} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {\left (d^5 \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{216 a b^2}-\frac {\left (d^5 \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{216 a b^2}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^4}-\frac {d \cos (c+d x)}{18 a^2 b x}-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {(-1)^{2/3} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\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^2 b}-\frac {5 \sqrt [3]{-1} \text {Ci}\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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\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^2 b}+\frac {5 (-1)^{2/3} \text {Ci}\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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\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^2 b}-\frac {\sin (c+d x)}{9 a b^2 x^5}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {5 \sqrt [3]{-1} \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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac {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^2 b}+\frac {(-1)^{2/3} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \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 )}{27 a^{8/3} \sqrt [3]{b}}-\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^2 b}-\frac {d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 (-1)^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {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^2 b}+\frac {\sqrt [3]{-1} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
time = 0.32, size = 675, normalized size = 0.58 \begin {gather*} \frac {-\frac {i \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {-10 \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))+10 i \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})+10 i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+10 \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-6 i d \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1})) \text {$\#$1}-6 d \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}-6 d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}+6 i d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}+d^2 \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1})) \text {$\#$1}^2-i d^2 \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}^2-i d^2 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2-d^2 \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2}{\text {$\#$1}^2}\&\right ]}{b}+\frac {i \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {-10 \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))-10 i \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-10 i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+10 \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+6 i d \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1})) \text {$\#$1}-6 d \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}-6 d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}-6 i d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}+d^2 \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1})) \text {$\#$1}^2+i d^2 \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}^2+i d^2 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2-d^2 \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2}{\text {$\#$1}^2}\&\right ]}{b}-\frac {6 x \cos (d x) \left (d x \left (a+b x^3\right ) \cos (c)-\left (8 a+5 b x^3\right ) \sin (c)\right )}{\left (a+b x^3\right )^2}+\frac {6 x \left (\left (8 a+5 b x^3\right ) \cos (c)+d x \left (a+b x^3\right ) \sin (c)\right ) \sin (d x)}{\left (a+b x^3\right )^2}}{108 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((-I)*RootSum[a + b*#1^3 & , (-10*Cos[c + d*#1]*CosIntegral[d*(x - #1)] + (10*I)*CosIntegral[d*(x - #1)]*Sin[
c + d*#1] + (10*I)*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + 10*Sin[c + d*#1]*SinIntegral[d*(x - #1)] - (6*I)*d*
Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - 6*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - 6*d*Cos[c + d*#1]*Si
nIntegral[d*(x - #1)]*#1 + (6*I)*d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1 + d^2*Cos[c + d*#1]*CosIntegral[d*
(x - #1)]*#1^2 - I*d^2*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1^2 - I*d^2*Cos[c + d*#1]*SinIntegral[d*(x - #1)
]*#1^2 - d^2*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2)/#1^2 & ])/b + (I*RootSum[a + b*#1^3 & , (-10*Cos[c +
d*#1]*CosIntegral[d*(x - #1)] - (10*I)*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - (10*I)*Cos[c + d*#1]*SinIntegra
l[d*(x - #1)] + 10*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + (6*I)*d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 -
6*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - 6*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - (6*I)*d*Sin[c +
d*#1]*SinIntegral[d*(x - #1)]*#1 + d^2*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1^2 + I*d^2*CosIntegral[d*(x - #
1)]*Sin[c + d*#1]*#1^2 + I*d^2*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2 - d^2*Sin[c + d*#1]*SinIntegral[d*(x
- #1)]*#1^2)/#1^2 & ])/b - (6*x*Cos[d*x]*(d*x*(a + b*x^3)*Cos[c] - (8*a + 5*b*x^3)*Sin[c]))/(a + b*x^3)^2 + (
6*x*((8*a + 5*b*x^3)*Cos[c] + d*x*(a + b*x^3)*Sin[c])*Sin[d*x])/(a + b*x^3)^2)/(108*a^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.11, size = 392, normalized size = 0.34

method	result	size
risch	\(-\frac {i d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}+6 i c -6 \textit {\_R1} +10\right ) {\mathrm e}^{\textit {\_R1}} \expIntegral \left (1, -i d x -i c +\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{108 a^{2} b}+\frac {i d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}-6 i c +6 \textit {\_R1} +10\right ) {\mathrm e}^{-\textit {\_R1}} \expIntegral \left (1, i d x +i c -\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{108 a^{2} b}+\frac {d^{2} \left (-b \,x^{5} d^{5}-a \,d^{5} x^{2}\right ) \cos \left (d x +c \right )}{18 a^{2} \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}+\frac {d^{2} \left (5 b \,x^{4} d^{4}+8 a \,d^{4} x \right ) \sin \left (d x +c \right )}{18 a^{2} \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}\)	\(340\)
derivativedivides	\(d^{8} \left (-\frac {\sin \left (d x +c \right ) \left (8 a c \,d^{3}-8 a \,d^{3} \left (d x +c \right )-5 b \,c^{4}+20 b \,c^{3} \left (d x +c \right )-30 b \,c^{2} \left (d x +c \right )^{2}+20 b c \left (d x +c \right )^{3}-5 b \left (d x +c \right )^{4}\right )}{18 a^{2} d^{6} \left (a \,d^{3}-b \,c^{3}+3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}\right )^{2}}-\frac {\cos \left (d x +c \right ) \left (c^{2}-2 \left (d x +c \right ) c +\left (d x +c \right )^{2}\right )}{18 a^{2} d^{6} \left (a \,d^{3}-b \,c^{3}+3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}\right )}-\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}-10\right ) \left (-\sinIntegral \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{54 a^{2} d^{6} b}+\frac {\munderset {\textit {\_RR1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\sinIntegral \left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\cosineIntegral \left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{-\textit {\_RR1} +c}}{9 a^{2} d^{6} b}\right )\)	\(392\)
default	\(d^{8} \left (-\frac {\sin \left (d x +c \right ) \left (8 a c \,d^{3}-8 a \,d^{3} \left (d x +c \right )-5 b \,c^{4}+20 b \,c^{3} \left (d x +c \right )-30 b \,c^{2} \left (d x +c \right )^{2}+20 b c \left (d x +c \right )^{3}-5 b \left (d x +c \right )^{4}\right )}{18 a^{2} d^{6} \left (a \,d^{3}-b \,c^{3}+3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}\right )^{2}}-\frac {\cos \left (d x +c \right ) \left (c^{2}-2 \left (d x +c \right ) c +\left (d x +c \right )^{2}\right )}{18 a^{2} d^{6} \left (a \,d^{3}-b \,c^{3}+3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}\right )}-\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}-10\right ) \left (-\sinIntegral \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{54 a^{2} d^{6} b}+\frac {\munderset {\textit {\_RR1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\sinIntegral \left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\cosineIntegral \left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{-\textit {\_RR1} +c}}{9 a^{2} d^{6} b}\right )\)	\(392\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^8*(-1/18*sin(d*x+c)*(8*a*c*d^3-8*a*d^3*(d*x+c)-5*b*c^4+20*b*c^3*(d*x+c)-30*b*c^2*(d*x+c)^2+20*b*c*(d*x+c)^3-
5*b*(d*x+c)^4)/a^2/d^6/(a*d^3-b*c^3+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)*(c^2-2*(d*x
+c)*c+(d*x+c)^2)/a^2/d^6/(a*d^3-b*c^3+3*b*c^2*(d*x+c)-3*b*c*(d*x+c)^2+b*(d*x+c)^3)-1/54/a^2/d^6/b*sum((_R1^2-2
*_R1*c+c^2-10)/(_R1^2-2*_R1*c+c^2)*(-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))+1/9/a^2/d^6/b*sum(1/(-_RR1+c)*(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)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 0.44, size = 1223, normalized size = 1.05 \begin {gather*} \frac {{\left (-i \, a b^{2} d^{3} x^{6} - 2 i \, a^{2} b d^{3} x^{3} - i \, a^{3} d^{3} + 3 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b - \sqrt {3} {\left (i \, b^{3} x^{6} + 2 i \, a b^{2} x^{3} + i \, a^{2} b\right )}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} + 5 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b - \sqrt {3} {\left (-i \, b^{3} x^{6} - 2 i \, a b^{2} x^{3} - i \, a^{2} b\right )}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (-i \, d x + \frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} - i \, c\right )} + {\left (i \, a b^{2} d^{3} x^{6} + 2 i \, a^{2} b d^{3} x^{3} + i \, a^{3} d^{3} + 3 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b - \sqrt {3} {\left (i \, b^{3} x^{6} + 2 i \, a b^{2} x^{3} + i \, a^{2} b\right )}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} + 5 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b - \sqrt {3} {\left (-i \, b^{3} x^{6} - 2 i \, a b^{2} x^{3} - i \, a^{2} b\right )}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (i \, d x + \frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} + i \, c\right )} + {\left (-i \, a b^{2} d^{3} x^{6} - 2 i \, a^{2} b d^{3} x^{3} - i \, a^{3} d^{3} + 3 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b - \sqrt {3} {\left (-i \, b^{3} x^{6} - 2 i \, a b^{2} x^{3} - i \, a^{2} b\right )}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} + 5 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b - \sqrt {3} {\left (i \, b^{3} x^{6} + 2 i \, a b^{2} x^{3} + i \, a^{2} b\right )}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (-i \, d x + \frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} - i \, c\right )} + {\left (i \, a b^{2} d^{3} x^{6} + 2 i \, a^{2} b d^{3} x^{3} + i \, a^{3} d^{3} + 3 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b - \sqrt {3} {\left (-i \, b^{3} x^{6} - 2 i \, a b^{2} x^{3} - i \, a^{2} b\right )}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} + 5 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b - \sqrt {3} {\left (i \, b^{3} x^{6} + 2 i \, a b^{2} x^{3} + i \, a^{2} b\right )}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (i \, d x + \frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} + i \, c\right )} + {\left (i \, a b^{2} d^{3} x^{6} + 2 i \, a^{2} b d^{3} x^{3} + i \, a^{3} d^{3} - 6 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} - 10 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (i \, d x + \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right ) e^{\left (i \, c - \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} + {\left (-i \, a b^{2} d^{3} x^{6} - 2 i \, a^{2} b d^{3} x^{3} - i \, a^{3} d^{3} - 6 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} - 10 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (-i \, d x + \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right ) e^{\left (-i \, c - \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} - 6 \, {\left (a b^{2} d^{2} x^{5} + a^{2} b d^{2} x^{2}\right )} \cos \left (d x + c\right ) + 6 \, {\left (5 \, a b^{2} d x^{4} + 8 \, a^{2} b d x\right )} \sin \left (d x + c\right )}{108 \, {\left (a^{3} b^{3} d x^{6} + 2 \, a^{4} b^{2} d x^{3} + a^{5} b d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/108*((-I*a*b^2*d^3*x^6 - 2*I*a^2*b*d^3*x^3 - I*a^3*d^3 + 3*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b - sqrt(3)*(I*b^3*x
^6 + 2*I*a*b^2*x^3 + I*a^2*b))*(I*a*d^3/b)^(2/3) + 5*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b - sqrt(3)*(-I*b^3*x^6 - 2*
I*a*b^2*x^3 - I*a^2*b))*(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) + (I*a*b^2*d^3*x^6 + 2*I*a^2*b*d^3*x^3 + I*a^3*d^3 + 3*(b^3*x^6 + 2*a*b^2*x^3
+ a^2*b - sqrt(3)*(I*b^3*x^6 + 2*I*a*b^2*x^3 + I*a^2*b))*(-I*a*d^3/b)^(2/3) + 5*(b^3*x^6 + 2*a*b^2*x^3 + a^2*
b - sqrt(3)*(-I*b^3*x^6 - 2*I*a*b^2*x^3 - I*a^2*b))*(-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) + (-I*a*b^2*d^3*x^6 - 2*I*a^2*b*d^3*x^3 - I*a^3
*d^3 + 3*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b - sqrt(3)*(-I*b^3*x^6 - 2*I*a*b^2*x^3 - I*a^2*b))*(I*a*d^3/b)^(2/3) +
5*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b - sqrt(3)*(I*b^3*x^6 + 2*I*a*b^2*x^3 + I*a^2*b))*(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) + (I*a*b^2*d^3*x^6
+ 2*I*a^2*b*d^3*x^3 + I*a^3*d^3 + 3*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b - sqrt(3)*(-I*b^3*x^6 - 2*I*a*b^2*x^3 - I*a
^2*b))*(-I*a*d^3/b)^(2/3) + 5*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b - sqrt(3)*(I*b^3*x^6 + 2*I*a*b^2*x^3 + I*a^2*b))*
(-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) + (I*a*b^2*d^3*x^6 + 2*I*a^2*b*d^3*x^3 + I*a^3*d^3 - 6*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b)*(-I*a*d^3/b)
^(2/3) - 10*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b)*(-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)) + (-I*a*b^2*d^3*x^6 - 2*I*a^2*b*d^3*x^3 - I*a^3*d^3 - 6*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b)*(I*a*d^3
/b)^(2/3) - 10*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b)*(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)) - 6*(a*b^2*d^2*x^5 + a^2*b*d^2*x^2)*cos(d*x + c) + 6*(5*a*b^2*d*x^4 + 8*a^2*b*d*x)*sin(d*x + c
))/(a^3*b^3*d*x^6 + 2*a^4*b^2*d*x^3 + a^5*b*d)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sin \left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)/(a + b*x^3)^3,x)

[Out]

int(sin(c + d*x)/(a + b*x^3)^3, x)

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