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

Optimal. Leaf size=1161 \[ \text{result too large to display} \]

[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))

________________________________________________________________________________________

Rubi [A]  time = 3.36789, antiderivative size = 1161, normalized size of antiderivative = 1., 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 = {3331, 3343, 3345, 3297, 3303, 3299, 3302, 3333, 3346, 3344} \[ \text{result too large to display} \]

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 3331

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 3343

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 3345

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 3297

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 3303

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 3299

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 3302

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 3333

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 3346

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]

Rule 3344

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]

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*}

Mathematica [C]  time = 0.431391, size = 675, normalized size = 0.58 \[ \frac{-\frac{i \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-i \text{$\#$1}^2 d^2 \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\text{$\#$1}^2 d^2 \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-\text{$\#$1}^2 d^2 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \text{$\#$1}^2 d^2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-6 \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+10 i \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-6 i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-10 \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+6 i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+10 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-6 \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+10 i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]}{b}+\frac{i \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{i \text{$\#$1}^2 d^2 \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\text{$\#$1}^2 d^2 \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-\text{$\#$1}^2 d^2 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \text{$\#$1}^2 d^2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-6 \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-10 i \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+6 i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-10 \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-6 i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+10 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-6 \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-10 i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]}{b}-\frac{6 x \cos (d x) \left (d x \cos (c) \left (a+b x^3\right )-\sin (c) \left (8 a+5 b x^3\right )\right )}{\left (a+b x^3\right )^2}+\frac{6 x \sin (d x) \left (d x \sin (c) \left (a+b x^3\right )+\cos (c) \left (8 a+5 b x^3\right )\right )}{\left (a+b x^3\right )^2}}{108 a^2} \]

Warning: Unable to verify antiderivative.

[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]  time = 0.031, size = 392, normalized size = 0.3 \begin{align*}{d}^{8} \left ({\frac{\sin \left ( dx+c \right ) \left ( 5\, \left ( dx+c \right ) ^{4}b-20\,c \left ( dx+c \right ) ^{3}b+30\,{c}^{2} \left ( dx+c \right ) ^{2}b+8\, \left ( dx+c \right ) a{d}^{3}-20\, \left ( dx+c \right ) b{c}^{3}-8\,ac{d}^{3}+5\,{c}^{4}b \right ) }{18\,{a}^{2}{d}^{6} \left ( \left ( dx+c \right ) ^{3}b-3\,c \left ( dx+c \right ) ^{2}b+3\, \left ( dx+c \right ) b{c}^{2}+a{d}^{3}-{c}^{3}b \right ) ^{2}}}-{\frac{\cos \left ( dx+c \right ) \left ( \left ( dx+c \right ) ^{2}-2\, \left ( dx+c \right ) c+{c}^{2} \right ) }{18\,{a}^{2}{d}^{6} \left ( \left ( dx+c \right ) ^{3}b-3\,c \left ( dx+c \right ) ^{2}b+3\, \left ( dx+c \right ) b{c}^{2}+a{d}^{3}-{c}^{3}b \right ) }}-{\frac{1}{54\,b{a}^{2}{d}^{6}}\sum _{{\it \_R1}={\it RootOf} \left ({{\it \_Z}}^{3}b-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-{c}^{3}b \right ) }{\frac{ \left ({{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}-10 \right ) \left ( -{\it Si} \left ( -dx+{\it \_R1}-c \right ) \cos \left ({\it \_R1} \right ) +{\it Ci} \left ( dx-{\it \_R1}+c \right ) \sin \left ({\it \_R1} \right ) \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}}}}-{\frac{1}{9\,b{a}^{2}{d}^{6}}\sum _{{\it \_RR1}={\it RootOf} \left ({{\it \_Z}}^{3}b-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-{c}^{3}b \right ) }{\frac{{\it Si} \left ( -dx+{\it \_RR1}-c \right ) \sin \left ({\it \_RR1} \right ) +{\it Ci} \left ( dx-{\it \_RR1}+c \right ) \cos \left ({\it \_RR1} \right ) }{{\it \_RR1}-c}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^8*(1/18*sin(d*x+c)*(5*(d*x+c)^4*b-20*c*(d*x+c)^3*b+30*c^2*(d*x+c)^2*b+8*(d*x+c)*a*d^3-20*(d*x+c)*b*c^3-8*a*c
*d^3+5*c^4*b)/a^2/d^6/((d*x+c)^3*b-3*c*(d*x+c)^2*b+3*(d*x+c)*b*c^2+a*d^3-c^3*b)^2-1/18*cos(d*x+c)*((d*x+c)^2-2
*(d*x+c)*c+c^2)/a^2/d^6/((d*x+c)^3*b-3*c*(d*x+c)^2*b+3*(d*x+c)*b*c^2+a*d^3-c^3*b)-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., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}}\,{d x} \end{align*}

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]  time = 3.03384, size = 2813, normalized size = 2.42 \begin{align*} \text{result too large to display} \end{align*}

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 + 6*a*b^2*x^3 + 3*a^2*b + sqrt(3)*(-3*I*
b^3*x^6 - 6*I*a*b^2*x^3 - 3*I*a^2*b))*(I*a*d^3/b)^(2/3) + (5*b^3*x^6 + 10*a*b^2*x^3 + 5*a^2*b + sqrt(3)*(5*I*b
^3*x^6 + 10*I*a*b^2*x^3 + 5*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
+ 6*a*b^2*x^3 + 3*a^2*b + sqrt(3)*(-3*I*b^3*x^6 - 6*I*a*b^2*x^3 - 3*I*a^2*b))*(-I*a*d^3/b)^(2/3) + (5*b^3*x^6
+ 10*a*b^2*x^3 + 5*a^2*b + sqrt(3)*(5*I*b^3*x^6 + 10*I*a*b^2*x^3 + 5*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 + 6*a*b^2*x^3 + 3*a^2*b + sqrt(3)*(3*I*b^3*x^6 + 6*I*a*b^2*x^3 +
3*I*a^2*b))*(I*a*d^3/b)^(2/3) + (5*b^3*x^6 + 10*a*b^2*x^3 + 5*a^2*b + sqrt(3)*(-5*I*b^3*x^6 - 10*I*a*b^2*x^3
- 5*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 + 6*a*b^2*x^3 + 3*a^2*b
+ sqrt(3)*(3*I*b^3*x^6 + 6*I*a*b^2*x^3 + 3*I*a^2*b))*(-I*a*d^3/b)^(2/3) + (5*b^3*x^6 + 10*a*b^2*x^3 + 5*a^2*b
+ sqrt(3)*(-5*I*b^3*x^6 - 10*I*a*b^2*x^3 - 5*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)*co
s(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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}}\,{d x} \end{align*}

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)