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

Optimal. Leaf size=1141 \[ \text{result too large to display} \]

[Out]

(d*Cos[c + d*x])/(18*a*b^2*x^3) - (d*Cos[c + d*x])/(18*b^2*x^3*(a + b*x^3)) - (2*d*Cos[c + ((-1)^(1/3)*a^(1/3)
*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^2*b) - (2*d*Cos[c - (a^(1/3)*d)/b^(1/3)]
*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*b) - (2*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral
[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*b) - (2*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3
)*d)/b^(1/3)])/(27*a^(7/3)*b^(2/3)) + (d^2*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)]
)/(54*a^(5/3)*b^(4/3)) - (2*(-1)^(2/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a
^(1/3)*d)/b^(1/3)])/(27*a^(7/3)*b^(2/3)) - ((-1)^(1/3)*d^2*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*S
in[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(54*a^(5/3)*b^(4/3)) + (2*(-1)^(1/3)*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^(7/3)*b^(2/3)) + ((-1)^(2/3)*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^(5/3)*b^(4/3)) - Sin[c + d
*x]/(18*a*b^2*x^4) + (2*Sin[c + d*x])/(9*a^2*b*x) - Sin[c + d*x]/(6*b*x*(a + b*x^3)^2) + Sin[c + d*x]/(18*b^2*
x^4*(a + b*x^3)) + (2*(-1)^(2/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^
(1/3) - d*x])/(27*a^(7/3)*b^(2/3)) + ((-1)^(1/3)*d^2*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])/(54*a^(5/3)*b^(4/3)) - (2*d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinInteg
ral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^2*b) - (2*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d
)/b^(1/3) + d*x])/(27*a^(7/3)*b^(2/3)) + (d^2*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d
*x])/(54*a^(5/3)*b^(4/3)) + (2*d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*
b) + (2*(-1)^(1/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/
(27*a^(7/3)*b^(2/3)) + ((-1)^(2/3)*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^(5/3)*b^(4/3)) + (2*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3
)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*b)

________________________________________________________________________________________

Rubi [A]  time = 3.1159, antiderivative size = 1141, normalized size of antiderivative = 1., number of steps used = 89, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529, Rules used = {3343, 3345, 3297, 3303, 3299, 3302, 3346, 3344, 3333} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[(x*Sin[c + d*x])/(a + b*x^3)^3,x]

[Out]

(d*Cos[c + d*x])/(18*a*b^2*x^3) - (d*Cos[c + d*x])/(18*b^2*x^3*(a + b*x^3)) - (2*d*Cos[c + ((-1)^(1/3)*a^(1/3)
*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^2*b) - (2*d*Cos[c - (a^(1/3)*d)/b^(1/3)]
*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*b) - (2*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral
[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*b) - (2*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3
)*d)/b^(1/3)])/(27*a^(7/3)*b^(2/3)) + (d^2*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)]
)/(54*a^(5/3)*b^(4/3)) - (2*(-1)^(2/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a
^(1/3)*d)/b^(1/3)])/(27*a^(7/3)*b^(2/3)) - ((-1)^(1/3)*d^2*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*S
in[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(54*a^(5/3)*b^(4/3)) + (2*(-1)^(1/3)*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^(7/3)*b^(2/3)) + ((-1)^(2/3)*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^(5/3)*b^(4/3)) - Sin[c + d
*x]/(18*a*b^2*x^4) + (2*Sin[c + d*x])/(9*a^2*b*x) - Sin[c + d*x]/(6*b*x*(a + b*x^3)^2) + Sin[c + d*x]/(18*b^2*
x^4*(a + b*x^3)) + (2*(-1)^(2/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^
(1/3) - d*x])/(27*a^(7/3)*b^(2/3)) + ((-1)^(1/3)*d^2*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])/(54*a^(5/3)*b^(4/3)) - (2*d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinInteg
ral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(27*a^2*b) - (2*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d
)/b^(1/3) + d*x])/(27*a^(7/3)*b^(2/3)) + (d^2*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d
*x])/(54*a^(5/3)*b^(4/3)) + (2*d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*
b) + (2*(-1)^(1/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/
(27*a^(7/3)*b^(2/3)) + ((-1)^(2/3)*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^(5/3)*b^(4/3)) + (2*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3
)*a^(1/3)*d)/b^(1/3) + d*x])/(27*a^2*b)

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 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]

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])

Rubi steps

\begin{align*} \int \frac{x \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx &=-\frac{\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}-\frac{\int \frac{\sin (c+d x)}{x^2 \left (a+b x^3\right )^2} \, dx}{6 b}+\frac{d \int \frac{\cos (c+d x)}{x \left (a+b x^3\right )^2} \, dx}{6 b}\\ &=-\frac{d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac{\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac{2 \int \frac{\sin (c+d x)}{x^5 \left (a+b x^3\right )} \, dx}{9 b^2}-\frac{d \int \frac{\cos (c+d x)}{x^4 \left (a+b x^3\right )} \, dx}{18 b^2}-\frac{d \int \frac{\cos (c+d x)}{x^4 \left (a+b x^3\right )} \, dx}{6 b^2}-\frac{d^2 \int \frac{\sin (c+d x)}{x^3 \left (a+b x^3\right )} \, dx}{18 b^2}\\ &=-\frac{d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac{\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac{2 \int \left (\frac{\sin (c+d x)}{a x^5}-\frac{b \sin (c+d x)}{a^2 x^2}+\frac{b^2 x \sin (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{9 b^2}-\frac{d \int \left (\frac{\cos (c+d x)}{a x^4}-\frac{b \cos (c+d x)}{a^2 x}+\frac{b^2 x^2 \cos (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{18 b^2}-\frac{d \int \left (\frac{\cos (c+d x)}{a x^4}-\frac{b \cos (c+d x)}{a^2 x}+\frac{b^2 x^2 \cos (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{6 b^2}-\frac{d^2 \int \left (\frac{\sin (c+d x)}{a x^3}-\frac{b \sin (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx}{18 b^2}\\ &=-\frac{d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac{\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac{2 \int \frac{x \sin (c+d x)}{a+b x^3} \, dx}{9 a^2}+\frac{2 \int \frac{\sin (c+d x)}{x^5} \, dx}{9 a b^2}-\frac{2 \int \frac{\sin (c+d x)}{x^2} \, dx}{9 a^2 b}-\frac{d \int \frac{x^2 \cos (c+d x)}{a+b x^3} \, dx}{18 a^2}-\frac{d \int \frac{x^2 \cos (c+d x)}{a+b x^3} \, dx}{6 a^2}-\frac{d \int \frac{\cos (c+d x)}{x^4} \, dx}{18 a b^2}-\frac{d \int \frac{\cos (c+d x)}{x^4} \, dx}{6 a b^2}+\frac{d \int \frac{\cos (c+d x)}{x} \, dx}{18 a^2 b}+\frac{d \int \frac{\cos (c+d x)}{x} \, dx}{6 a^2 b}-\frac{d^2 \int \frac{\sin (c+d x)}{x^3} \, dx}{18 a b^2}+\frac{d^2 \int \frac{\sin (c+d x)}{a+b x^3} \, dx}{18 a b}\\ &=\frac{2 d \cos (c+d x)}{27 a b^2 x^3}-\frac{d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac{\sin (c+d x)}{18 a b^2 x^4}+\frac{d^2 \sin (c+d x)}{36 a b^2 x^2}+\frac{2 \sin (c+d x)}{9 a^2 b x}-\frac{\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac{2 \int \left (-\frac{\sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{(-1)^{2/3} \sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac{\sqrt [3]{-1} \sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{9 a^2}-\frac{d \int \left (\frac{\cos (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\cos (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\cos (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 \int \left (\frac{\cos (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\cos (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\cos (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{6 a^2}+\frac{d \int \frac{\cos (c+d x)}{x^4} \, dx}{18 a b^2}-\frac{(2 d) \int \frac{\cos (c+d x)}{x} \, dx}{9 a^2 b}+\frac{d^2 \int \frac{\sin (c+d x)}{x^3} \, dx}{54 a b^2}+\frac{d^2 \int \frac{\sin (c+d x)}{x^3} \, dx}{18 a b^2}+\frac{d^2 \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}{18 a b}-\frac{d^3 \int \frac{\cos (c+d x)}{x^2} \, dx}{36 a b^2}+\frac{(d \cos (c)) \int \frac{\cos (d x)}{x} \, dx}{18 a^2 b}+\frac{(d \cos (c)) \int \frac{\cos (d x)}{x} \, dx}{6 a^2 b}-\frac{(d \sin (c)) \int \frac{\sin (d x)}{x} \, dx}{18 a^2 b}-\frac{(d \sin (c)) \int \frac{\sin (d x)}{x} \, dx}{6 a^2 b}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x^3}+\frac{d^3 \cos (c+d x)}{36 a b^2 x}-\frac{d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}+\frac{2 d \cos (c) \text{Ci}(d x)}{9 a^2 b}-\frac{\sin (c+d x)}{18 a b^2 x^4}-\frac{d^2 \sin (c+d x)}{108 a b^2 x^2}+\frac{2 \sin (c+d x)}{9 a^2 b x}-\frac{\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac{2 d \sin (c) \text{Si}(d x)}{9 a^2 b}-\frac{2 \int \frac{\sin (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac{\left (2 \sqrt [3]{-1}\right ) \int \frac{\sin (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac{\left (2 (-1)^{2/3}\right ) \int \frac{\sin (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac{d \int \frac{\cos (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac{d \int \frac{\cos (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac{d \int \frac{\cos (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac{d \int \frac{\cos (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\frac{d \int \frac{\cos (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\frac{d \int \frac{\cos (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\frac{d^2 \int \frac{\sin (c+d x)}{x^3} \, dx}{54 a b^2}-\frac{d^2 \int \frac{\sin (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{54 a^{5/3} b}-\frac{d^2 \int \frac{\sin (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}-\frac{d^2 \int \frac{\sin (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}+\frac{d^3 \int \frac{\cos (c+d x)}{x^2} \, dx}{108 a b^2}+\frac{d^3 \int \frac{\cos (c+d x)}{x^2} \, dx}{36 a b^2}+\frac{d^4 \int \frac{\sin (c+d x)}{x} \, dx}{36 a b^2}-\frac{(2 d \cos (c)) \int \frac{\cos (d x)}{x} \, dx}{9 a^2 b}+\frac{(2 d \sin (c)) \int \frac{\sin (d x)}{x} \, dx}{9 a^2 b}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x^3}-\frac{d^3 \cos (c+d x)}{108 a b^2 x}-\frac{d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac{\sin (c+d x)}{18 a b^2 x^4}+\frac{2 \sin (c+d x)}{9 a^2 b x}-\frac{\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac{d^3 \int \frac{\cos (c+d x)}{x^2} \, dx}{108 a b^2}-\frac{d^4 \int \frac{\sin (c+d x)}{x} \, dx}{108 a b^2}-\frac{d^4 \int \frac{\sin (c+d x)}{x} \, dx}{36 a b^2}+\frac{\left (d^4 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{36 a b^2}-\frac{\left (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}{27 a^{7/3} \sqrt [3]{b}}-\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}{54 a^2 b^{2/3}}-\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}{18 a^2 b^{2/3}}-\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^{5/3} b}-\frac{\left (2 \sqrt [3]{-1} \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^{7/3} \sqrt [3]{b}}-\frac{\left (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]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac{\left (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]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}+\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]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}-\frac{\left (2 (-1)^{2/3} \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^{7/3} \sqrt [3]{b}}-\frac{\left (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 )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac{\left (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 )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\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 )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}+\frac{\left (d^4 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{36 a b^2}-\frac{\left (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}{27 a^{7/3} \sqrt [3]{b}}+\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}{54 a^2 b^{2/3}}+\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}{18 a^2 b^{2/3}}-\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^{5/3} b}+\frac{\left (2 \sqrt [3]{-1} \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^{7/3} \sqrt [3]{b}}-\frac{\left (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]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac{\left (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]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\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]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}-\frac{\left (2 (-1)^{2/3} \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^{7/3} \sqrt [3]{b}}+\frac{\left (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 )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}+\frac{\left (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 )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{18 a^2 b^{2/3}}-\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 )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{5/3} b}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x^3}-\frac{d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac{2 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 )}{27 a^2 b}-\frac{2 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 )}{27 a^2 b}-\frac{2 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 )}{27 a^2 b}+\frac{d^4 \text{Ci}(d x) \sin (c)}{36 a b^2}-\frac{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 )}{27 a^{7/3} b^{2/3}}+\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^{5/3} b^{4/3}}-\frac{2 (-1)^{2/3} \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^{7/3} b^{2/3}}-\frac{\sqrt [3]{-1} 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^{5/3} b^{4/3}}+\frac{2 \sqrt [3]{-1} \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^{7/3} b^{2/3}}+\frac{(-1)^{2/3} 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^{5/3} b^{4/3}}-\frac{\sin (c+d x)}{18 a b^2 x^4}+\frac{2 \sin (c+d x)}{9 a^2 b x}-\frac{\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac{d^4 \cos (c) \text{Si}(d x)}{36 a b^2}+\frac{2 (-1)^{2/3} \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}+\frac{\sqrt [3]{-1} 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^{5/3} b^{4/3}}-\frac{2 d \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^2 b}-\frac{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 )}{27 a^{7/3} b^{2/3}}+\frac{d^2 \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}+\frac{2 d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}+\frac{2 \sqrt [3]{-1} \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^{7/3} b^{2/3}}+\frac{(-1)^{2/3} 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^{5/3} b^{4/3}}+\frac{2 d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}+\frac{d^4 \int \frac{\sin (c+d x)}{x} \, dx}{108 a b^2}-\frac{\left (d^4 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{108 a b^2}-\frac{\left (d^4 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{36 a b^2}-\frac{\left (d^4 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{108 a b^2}-\frac{\left (d^4 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{36 a b^2}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x^3}-\frac{d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac{2 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 )}{27 a^2 b}-\frac{2 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 )}{27 a^2 b}-\frac{2 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 )}{27 a^2 b}-\frac{d^4 \text{Ci}(d x) \sin (c)}{108 a b^2}-\frac{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 )}{27 a^{7/3} b^{2/3}}+\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^{5/3} b^{4/3}}-\frac{2 (-1)^{2/3} \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^{7/3} b^{2/3}}-\frac{\sqrt [3]{-1} 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^{5/3} b^{4/3}}+\frac{2 \sqrt [3]{-1} \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^{7/3} b^{2/3}}+\frac{(-1)^{2/3} 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^{5/3} b^{4/3}}-\frac{\sin (c+d x)}{18 a b^2 x^4}+\frac{2 \sin (c+d x)}{9 a^2 b x}-\frac{\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac{d^4 \cos (c) \text{Si}(d x)}{108 a b^2}+\frac{2 (-1)^{2/3} \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}+\frac{\sqrt [3]{-1} 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^{5/3} b^{4/3}}-\frac{2 d \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^2 b}-\frac{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 )}{27 a^{7/3} b^{2/3}}+\frac{d^2 \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}+\frac{2 d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}+\frac{2 \sqrt [3]{-1} \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^{7/3} b^{2/3}}+\frac{(-1)^{2/3} 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^{5/3} b^{4/3}}+\frac{2 d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}+\frac{\left (d^4 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{108 a b^2}+\frac{\left (d^4 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{108 a b^2}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x^3}-\frac{d \cos (c+d x)}{18 b^2 x^3 \left (a+b x^3\right )}-\frac{2 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 )}{27 a^2 b}-\frac{2 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 )}{27 a^2 b}-\frac{2 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 )}{27 a^2 b}-\frac{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 )}{27 a^{7/3} b^{2/3}}+\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^{5/3} b^{4/3}}-\frac{2 (-1)^{2/3} \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^{7/3} b^{2/3}}-\frac{\sqrt [3]{-1} 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^{5/3} b^{4/3}}+\frac{2 \sqrt [3]{-1} \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^{7/3} b^{2/3}}+\frac{(-1)^{2/3} 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^{5/3} b^{4/3}}-\frac{\sin (c+d x)}{18 a b^2 x^4}+\frac{2 \sin (c+d x)}{9 a^2 b x}-\frac{\sin (c+d x)}{6 b x \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac{2 (-1)^{2/3} \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{7/3} b^{2/3}}+\frac{\sqrt [3]{-1} 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^{5/3} b^{4/3}}-\frac{2 d \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^2 b}-\frac{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 )}{27 a^{7/3} b^{2/3}}+\frac{d^2 \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{5/3} b^{4/3}}+\frac{2 d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}+\frac{2 \sqrt [3]{-1} \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^{7/3} b^{2/3}}+\frac{(-1)^{2/3} 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^{5/3} b^{4/3}}+\frac{2 d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^2 b}\\ \end{align*}

Mathematica [C]  time = 0.552438, size = 698, normalized size = 0.61 \[ -\frac{\text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-4 i \text{$\#$1}^2 b d \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+4 \text{$\#$1}^2 b d \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-4 \text{$\#$1}^2 b d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-4 i \text{$\#$1}^2 b d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-a d^2 \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-i a d^2 \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+i a d^2 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-a d^2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-4 \text{$\#$1} b \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-4 i \text{$\#$1} b \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+4 i \text{$\#$1} b \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-4 \text{$\#$1} b \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]+\text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{4 i \text{$\#$1}^2 b d \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+4 \text{$\#$1}^2 b d \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-4 \text{$\#$1}^2 b d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+4 i \text{$\#$1}^2 b d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-a d^2 \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+i a d^2 \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-i a d^2 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-a d^2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-4 \text{$\#$1} b \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+4 i \text{$\#$1} b \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-4 i \text{$\#$1} b \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-4 \text{$\#$1} b \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]-\frac{6 b \cos (d x) \left (a d \cos (c) \left (a+b x^3\right )+b x^2 \sin (c) \left (7 a+4 b x^3\right )\right )}{\left (a+b x^3\right )^2}-\frac{6 b \sin (d x) \left (b x^2 \cos (c) \left (7 a+4 b x^3\right )-a d \sin (c) \left (a+b x^3\right )\right )}{\left (a+b x^3\right )^2}}{108 a^2 b^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(RootSum[a + b*#1^3 & , ((-I)*a*d^2*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - a*d^2*CosIntegral[d*(x - #1)]*Sin
[c + d*#1] - a*d^2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + I*a*d^2*Sin[c + d*#1]*SinIntegral[d*(x - #1)] - (4*
I)*b*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 - 4*b*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - 4*b*Cos[c + d*#
1]*SinIntegral[d*(x - #1)]*#1 + (4*I)*b*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1 + 4*b*d*Cos[c + d*#1]*CosInte
gral[d*(x - #1)]*#1^2 - (4*I)*b*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1^2 - (4*I)*b*d*Cos[c + d*#1]*SinInte
gral[d*(x - #1)]*#1^2 - 4*b*d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2)/#1^2 & ] + RootSum[a + b*#1^3 & , (I
*a*d^2*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - a*d^2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - a*d^2*Cos[c + d*#
1]*SinIntegral[d*(x - #1)] - I*a*d^2*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + (4*I)*b*Cos[c + d*#1]*CosIntegral
[d*(x - #1)]*#1 - 4*b*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - 4*b*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1
- (4*I)*b*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1 + 4*b*d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1^2 + (4*I)*
b*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1^2 + (4*I)*b*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2 - 4*b*d*
Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2)/#1^2 & ] - (6*b*Cos[d*x]*(a*d*(a + b*x^3)*Cos[c] + b*x^2*(7*a + 4*
b*x^3)*Sin[c]))/(a + b*x^3)^2 - (6*b*(b*x^2*(7*a + 4*b*x^3)*Cos[c] - a*d*(a + b*x^3)*Sin[c])*Sin[d*x])/(a + b*
x^3)^2)/(108*a^2*b^2)

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Maple [C]  time = 0.059, size = 845, normalized size = 0.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^2*(1/18*sin(d*x+c)*d^3*(4*b*(d*x+c)^5-15*b*c*(d*x+c)^4+20*b*c^2*(d*x+c)^3+7*(d*x+c)^2*a*d^3-10*(d*x+c)^2*b
*c^3-6*(d*x+c)*a*c*d^3-a*c^2*d^3+c^5*b)/a^2/((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*c
os(d*x+c)*d^3*(c*(d*x+c)^2*b-2*(d*x+c)*b*c^2-a*d^3+c^3*b)/a^2/b/((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*d^3/a^2/b^2*sum((_R1^2*b*c-2*_R1*b*c^2-a*d^3+b*c^3-4*_R1*b-6*b*c)/(_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))-1/27*d^3/a^2/
b*sum((2*_RR1+c)/(_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))-d^9*c*(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(-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(x*sin(d*x+c)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

Timed out

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Fricas [C]  time = 3.0455, size = 2934, normalized size = 2.57 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/216*((8*a*b^2*d^3*x^6 + 16*a^2*b*d^3*x^3 + 8*a^3*d^3 - (-4*I*b^3*x^6 - 8*I*a*b^2*x^3 - 4*I*a^2*b - 4*sqrt(3
)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*(I*a*d^3/b)^(2/3) - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3 + sqrt(3)*(I
*a*b^2*d^3*x^6 + 2*I*a^2*b*d^3*x^3 + I*a^3*d^3))*(I*a*d^3/b)^(1/3))*Ei(-I*d*x + 1/2*(I*a*d^3/b)^(1/3)*(-I*sqrt
(3) - 1))*e^(1/2*(I*a*d^3/b)^(1/3)*(I*sqrt(3) + 1) - I*c) + (8*a*b^2*d^3*x^6 + 16*a^2*b*d^3*x^3 + 8*a^3*d^3 -
(4*I*b^3*x^6 + 8*I*a*b^2*x^3 + 4*I*a^2*b + 4*sqrt(3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*(-I*a*d^3/b)^(2/3) - (a*
b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3 + sqrt(3)*(I*a*b^2*d^3*x^6 + 2*I*a^2*b*d^3*x^3 + I*a^3*d^3))*(-I*a*d^3
/b)^(1/3))*Ei(I*d*x + 1/2*(-I*a*d^3/b)^(1/3)*(-I*sqrt(3) - 1))*e^(1/2*(-I*a*d^3/b)^(1/3)*(I*sqrt(3) + 1) + I*c
) + (8*a*b^2*d^3*x^6 + 16*a^2*b*d^3*x^3 + 8*a^3*d^3 - (-4*I*b^3*x^6 - 8*I*a*b^2*x^3 - 4*I*a^2*b + 4*sqrt(3)*(b
^3*x^6 + 2*a*b^2*x^3 + a^2*b))*(I*a*d^3/b)^(2/3) - (a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3 + sqrt(3)*(-I*a*
b^2*d^3*x^6 - 2*I*a^2*b*d^3*x^3 - I*a^3*d^3))*(I*a*d^3/b)^(1/3))*Ei(-I*d*x + 1/2*(I*a*d^3/b)^(1/3)*(I*sqrt(3)
- 1))*e^(1/2*(I*a*d^3/b)^(1/3)*(-I*sqrt(3) + 1) - I*c) + (8*a*b^2*d^3*x^6 + 16*a^2*b*d^3*x^3 + 8*a^3*d^3 - (4*
I*b^3*x^6 + 8*I*a*b^2*x^3 + 4*I*a^2*b - 4*sqrt(3)*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b))*(-I*a*d^3/b)^(2/3) - (a*b^2
*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3 + sqrt(3)*(-I*a*b^2*d^3*x^6 - 2*I*a^2*b*d^3*x^3 - I*a^3*d^3))*(-I*a*d^3/b
)^(1/3))*Ei(I*d*x + 1/2*(-I*a*d^3/b)^(1/3)*(I*sqrt(3) - 1))*e^(1/2*(-I*a*d^3/b)^(1/3)*(-I*sqrt(3) + 1) + I*c)
+ (8*a*b^2*d^3*x^6 + 16*a^2*b*d^3*x^3 + 8*a^3*d^3 - (-8*I*b^3*x^6 - 16*I*a*b^2*x^3 - 8*I*a^2*b)*(-I*a*d^3/b)^(
2/3) + 2*(a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*(-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)) + (8*a*b^2*d^3*x^6 + 16*a^2*b*d^3*x^3 + 8*a^3*d^3 - (8*I*b^3*x^6 + 16*I*a*b^2*x^3 + 8*I
*a^2*b)*(I*a*d^3/b)^(2/3) + 2*(a*b^2*d^3*x^6 + 2*a^2*b*d^3*x^3 + a^3*d^3)*(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)) - 12*(a^2*b*d^3*x^3 + a^3*d^3)*cos(d*x + c) - 12*(4*a*b^2*d^2*x^5 +
7*a^2*b*d^2*x^2)*sin(d*x + c))/(a^3*b^3*d^2*x^6 + 2*a^4*b^2*d^2*x^3 + a^5*b*d^2)

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

[Out]

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