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

Optimal. Leaf size=1163 \[ \text{result too large to display} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 3.89309, antiderivative size = 1163, normalized size of antiderivative = 1., number of steps used = 110, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {3343, 3345, 3297, 3303, 3299, 3302, 3346, 3334, 3344} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x*(a + b*x^3)^3),x]

[Out]

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

Rule 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 3334

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

Rule 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)}{x \left (a+b x^3\right )^3} \, dx &=-\frac{\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}-\frac{\int \frac{\sin (c+d x)}{x^4 \left (a+b x^3\right )^2} \, dx}{2 b}+\frac{d \int \frac{\cos (c+d x)}{x^3 \left (a+b x^3\right )^2} \, dx}{6 b}\\ &=-\frac{d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}-\frac{\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac{\int \frac{\sin (c+d x)}{x^7 \left (a+b x^3\right )} \, dx}{b^2}-\frac{d \int \frac{\cos (c+d x)}{x^6 \left (a+b x^3\right )} \, dx}{6 b^2}-\frac{(5 d) \int \frac{\cos (c+d x)}{x^6 \left (a+b x^3\right )} \, dx}{18 b^2}-\frac{d^2 \int \frac{\sin (c+d x)}{x^5 \left (a+b x^3\right )} \, dx}{18 b^2}\\ &=-\frac{d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}-\frac{\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac{\int \left (\frac{\sin (c+d x)}{a x^7}-\frac{b \sin (c+d x)}{a^2 x^4}+\frac{b^2 \sin (c+d x)}{a^3 x}-\frac{b^3 x^2 \sin (c+d x)}{a^3 \left (a+b x^3\right )}\right ) \, dx}{b^2}-\frac{d \int \left (\frac{\cos (c+d x)}{a x^6}-\frac{b \cos (c+d x)}{a^2 x^3}+\frac{b^2 \cos (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{6 b^2}-\frac{(5 d) \int \left (\frac{\cos (c+d x)}{a x^6}-\frac{b \cos (c+d x)}{a^2 x^3}+\frac{b^2 \cos (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{18 b^2}-\frac{d^2 \int \left (\frac{\sin (c+d x)}{a x^5}-\frac{b \sin (c+d x)}{a^2 x^2}+\frac{b^2 x \sin (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{18 b^2}\\ &=-\frac{d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}-\frac{\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac{\int \frac{\sin (c+d x)}{x} \, dx}{a^3}+\frac{\int \frac{\sin (c+d x)}{x^7} \, dx}{a b^2}-\frac{\int \frac{\sin (c+d x)}{x^4} \, dx}{a^2 b}-\frac{b \int \frac{x^2 \sin (c+d x)}{a+b x^3} \, dx}{a^3}-\frac{d \int \frac{\cos (c+d x)}{a+b x^3} \, dx}{6 a^2}-\frac{(5 d) \int \frac{\cos (c+d x)}{a+b x^3} \, dx}{18 a^2}-\frac{d \int \frac{\cos (c+d x)}{x^6} \, dx}{6 a b^2}-\frac{(5 d) \int \frac{\cos (c+d x)}{x^6} \, dx}{18 a b^2}+\frac{d \int \frac{\cos (c+d x)}{x^3} \, dx}{6 a^2 b}+\frac{(5 d) \int \frac{\cos (c+d x)}{x^3} \, dx}{18 a^2 b}-\frac{d^2 \int \frac{x \sin (c+d x)}{a+b x^3} \, dx}{18 a^2}-\frac{d^2 \int \frac{\sin (c+d x)}{x^5} \, dx}{18 a b^2}+\frac{d^2 \int \frac{\sin (c+d x)}{x^2} \, dx}{18 a^2 b}\\ &=\frac{4 d \cos (c+d x)}{45 a b^2 x^5}-\frac{2 d \cos (c+d x)}{9 a^2 b x^2}-\frac{d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}-\frac{\sin (c+d x)}{6 a b^2 x^6}+\frac{d^2 \sin (c+d x)}{72 a b^2 x^4}+\frac{\sin (c+d x)}{3 a^2 b x^3}-\frac{d^2 \sin (c+d x)}{18 a^2 b x}-\frac{\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}-\frac{b \int \left (\frac{\sin (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\sin (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\sin (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{a^3}-\frac{d \int \left (-\frac{\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac{\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{6 a^2}-\frac{(5 d) \int \left (-\frac{\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac{\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{18 a^2}+\frac{d \int \frac{\cos (c+d x)}{x^6} \, dx}{6 a b^2}-\frac{d \int \frac{\cos (c+d x)}{x^3} \, dx}{3 a^2 b}-\frac{d^2 \int \left (-\frac{\sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{(-1)^{2/3} \sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac{\sqrt [3]{-1} \sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{18 a^2}+\frac{d^2 \int \frac{\sin (c+d x)}{x^5} \, dx}{30 a b^2}+\frac{d^2 \int \frac{\sin (c+d x)}{x^5} \, dx}{18 a b^2}-\frac{d^2 \int \frac{\sin (c+d x)}{x^2} \, dx}{12 a^2 b}-\frac{\left (5 d^2\right ) \int \frac{\sin (c+d x)}{x^2} \, dx}{36 a^2 b}-\frac{d^3 \int \frac{\cos (c+d x)}{x^4} \, dx}{72 a b^2}+\frac{d^3 \int \frac{\cos (c+d x)}{x} \, dx}{18 a^2 b}+\frac{\cos (c) \int \frac{\sin (d x)}{x} \, dx}{a^3}+\frac{\sin (c) \int \frac{\cos (d x)}{x} \, dx}{a^3}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x^5}+\frac{d^3 \cos (c+d x)}{216 a b^2 x^3}-\frac{d \cos (c+d x)}{18 a^2 b x^2}-\frac{d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}+\frac{\text{Ci}(d x) \sin (c)}{a^3}-\frac{\sin (c+d x)}{6 a b^2 x^6}-\frac{d^2 \sin (c+d x)}{120 a b^2 x^4}+\frac{\sin (c+d x)}{3 a^2 b x^3}+\frac{d^2 \sin (c+d x)}{6 a^2 b x}-\frac{\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac{\cos (c) \text{Si}(d x)}{a^3}-\frac{\sqrt [3]{b} \int \frac{\sin (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^3}-\frac{\sqrt [3]{b} \int \frac{\sin (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^3}-\frac{\sqrt [3]{b} \int \frac{\sin (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^3}+\frac{d \int \frac{\cos (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{18 a^{8/3}}+\frac{d \int \frac{\cos (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{18 a^{8/3}}+\frac{d \int \frac{\cos (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{18 a^{8/3}}+\frac{(5 d) \int \frac{\cos (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{54 a^{8/3}}+\frac{(5 d) \int \frac{\cos (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{8/3}}+\frac{(5 d) \int \frac{\cos (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{8/3}}-\frac{d^2 \int \frac{\sin (c+d x)}{x^5} \, dx}{30 a b^2}+\frac{d^2 \int \frac{\sin (c+d x)}{x^2} \, dx}{6 a^2 b}+\frac{d^2 \int \frac{\sin (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^{7/3} \sqrt [3]{b}}-\frac{\left (\sqrt [3]{-1} d^2\right ) \int \frac{\sin (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{7/3} \sqrt [3]{b}}+\frac{\left ((-1)^{2/3} d^2\right ) \int \frac{\sin (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{7/3} \sqrt [3]{b}}+\frac{d^3 \int \frac{\cos (c+d x)}{x^4} \, dx}{120 a b^2}+\frac{d^3 \int \frac{\cos (c+d x)}{x^4} \, dx}{72 a b^2}-\frac{d^3 \int \frac{\cos (c+d x)}{x} \, dx}{12 a^2 b}-\frac{\left (5 d^3\right ) \int \frac{\cos (c+d x)}{x} \, dx}{36 a^2 b}+\frac{d^4 \int \frac{\sin (c+d x)}{x^3} \, dx}{216 a b^2}+\frac{\left (d^3 \cos (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{18 a^2 b}-\frac{\left (d^3 \sin (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{18 a^2 b}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x^5}-\frac{d^3 \cos (c+d x)}{360 a b^2 x^3}-\frac{d \cos (c+d x)}{18 a^2 b x^2}-\frac{d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}+\frac{d^3 \cos (c) \text{Ci}(d x)}{18 a^2 b}+\frac{\text{Ci}(d x) \sin (c)}{a^3}-\frac{\sin (c+d x)}{6 a b^2 x^6}+\frac{\sin (c+d x)}{3 a^2 b x^3}-\frac{d^4 \sin (c+d x)}{432 a b^2 x^2}-\frac{\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac{\cos (c) \text{Si}(d x)}{a^3}-\frac{d^3 \sin (c) \text{Si}(d x)}{18 a^2 b}-\frac{d^3 \int \frac{\cos (c+d x)}{x^4} \, dx}{120 a b^2}+\frac{d^3 \int \frac{\cos (c+d x)}{x} \, dx}{6 a^2 b}-\frac{d^4 \int \frac{\sin (c+d x)}{x^3} \, dx}{360 a b^2}-\frac{d^4 \int \frac{\sin (c+d x)}{x^3} \, dx}{216 a b^2}+\frac{d^5 \int \frac{\cos (c+d x)}{x^2} \, dx}{432 a b^2}-\frac{\left (d^3 \cos (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{12 a^2 b}-\frac{\left (5 d^3 \cos (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{36 a^2 b}-\frac{\left (\sqrt [3]{b} \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}{3 a^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^{8/3}}+\frac{\left (5 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^{8/3}}+\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^{7/3} \sqrt [3]{b}}+\frac{\left (\sqrt [3]{b} \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}{3 a^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]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{18 a^{8/3}}+\frac{\left (5 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}{54 a^{8/3}}+\frac{\left (\sqrt [3]{-1} 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^{7/3} \sqrt [3]{b}}-\frac{\left (\sqrt [3]{b} \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}{3 a^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 )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{18 a^{8/3}}+\frac{\left (5 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}{54 a^{8/3}}+\frac{\left ((-1)^{2/3} 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^{7/3} \sqrt [3]{b}}+\frac{\left (d^3 \sin (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{12 a^2 b}+\frac{\left (5 d^3 \sin (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{36 a^2 b}-\frac{\left (\sqrt [3]{b} \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}{3 a^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^{8/3}}-\frac{\left (5 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^{8/3}}+\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^{7/3} \sqrt [3]{b}}-\frac{\left (\sqrt [3]{b} \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}{3 a^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]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{18 a^{8/3}}+\frac{\left (5 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}{54 a^{8/3}}-\frac{\left (\sqrt [3]{-1} 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^{7/3} \sqrt [3]{b}}-\frac{\left (\sqrt [3]{b} \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}{3 a^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 )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{18 a^{8/3}}-\frac{\left (5 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}{54 a^{8/3}}+\frac{\left ((-1)^{2/3} 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^{7/3} \sqrt [3]{b}}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x^5}-\frac{d \cos (c+d x)}{18 a^2 b x^2}-\frac{d^5 \cos (c+d x)}{432 a b^2 x}-\frac{d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}-\frac{d^3 \cos (c) \text{Ci}(d x)}{6 a^2 b}+\frac{4 \sqrt [3]{-1} 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^{8/3} \sqrt [3]{b}}-\frac{4 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^{8/3} \sqrt [3]{b}}-\frac{4 (-1)^{2/3} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac{\text{Ci}(d x) \sin (c)}{a^3}-\frac{\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 )}{3 a^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^{7/3} b^{2/3}}-\frac{\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 )}{3 a^3}+\frac{(-1)^{2/3} 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^{7/3} b^{2/3}}-\frac{\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 )}{3 a^3}-\frac{\sqrt [3]{-1} 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^{7/3} b^{2/3}}-\frac{\sin (c+d x)}{6 a b^2 x^6}+\frac{\sin (c+d x)}{3 a^2 b x^3}+\frac{d^4 \sin (c+d x)}{720 a b^2 x^2}-\frac{\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac{\cos (c) \text{Si}(d x)}{a^3}+\frac{d^3 \sin (c) \text{Si}(d x)}{6 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 )}{3 a^3}-\frac{(-1)^{2/3} d^2 \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{7/3} b^{2/3}}+\frac{4 \sqrt [3]{-1} d \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{\cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^3}+\frac{d^2 \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{7/3} b^{2/3}}+\frac{4 d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{\cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^3}-\frac{\sqrt [3]{-1} d^2 \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{7/3} b^{2/3}}+\frac{4 (-1)^{2/3} d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac{d^4 \int \frac{\sin (c+d x)}{x^3} \, dx}{360 a b^2}-\frac{d^5 \int \frac{\cos (c+d x)}{x^2} \, dx}{720 a b^2}-\frac{d^5 \int \frac{\cos (c+d x)}{x^2} \, dx}{432 a b^2}-\frac{d^6 \int \frac{\sin (c+d x)}{x} \, dx}{432 a b^2}+\frac{\left (d^3 \cos (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{6 a^2 b}-\frac{\left (d^3 \sin (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{6 a^2 b}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x^5}-\frac{d \cos (c+d x)}{18 a^2 b x^2}+\frac{d^5 \cos (c+d x)}{720 a b^2 x}-\frac{d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}+\frac{4 \sqrt [3]{-1} d \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{4 d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{4 (-1)^{2/3} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac{\text{Ci}(d x) \sin (c)}{a^3}-\frac{\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 )}{3 a^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^{7/3} b^{2/3}}-\frac{\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 )}{3 a^3}+\frac{(-1)^{2/3} 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^{7/3} b^{2/3}}-\frac{\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 )}{3 a^3}-\frac{\sqrt [3]{-1} 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^{7/3} b^{2/3}}-\frac{\sin (c+d x)}{6 a b^2 x^6}+\frac{\sin (c+d x)}{3 a^2 b x^3}-\frac{\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac{\cos (c) \text{Si}(d x)}{a^3}+\frac{\cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^3}-\frac{(-1)^{2/3} d^2 \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{7/3} b^{2/3}}+\frac{4 \sqrt [3]{-1} d \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{\cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^3}+\frac{d^2 \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{7/3} b^{2/3}}+\frac{4 d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{\cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^3}-\frac{\sqrt [3]{-1} d^2 \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{7/3} b^{2/3}}+\frac{4 (-1)^{2/3} d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac{d^5 \int \frac{\cos (c+d x)}{x^2} \, dx}{720 a b^2}+\frac{d^6 \int \frac{\sin (c+d x)}{x} \, dx}{720 a b^2}+\frac{d^6 \int \frac{\sin (c+d x)}{x} \, dx}{432 a b^2}-\frac{\left (d^6 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{432 a b^2}-\frac{\left (d^6 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{432 a b^2}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x^5}-\frac{d \cos (c+d x)}{18 a^2 b x^2}-\frac{d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}+\frac{4 \sqrt [3]{-1} d \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{4 d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{4 (-1)^{2/3} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac{\text{Ci}(d x) \sin (c)}{a^3}-\frac{d^6 \text{Ci}(d x) \sin (c)}{432 a b^2}-\frac{\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 )}{3 a^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^{7/3} b^{2/3}}-\frac{\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 )}{3 a^3}+\frac{(-1)^{2/3} 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^{7/3} b^{2/3}}-\frac{\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 )}{3 a^3}-\frac{\sqrt [3]{-1} 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^{7/3} b^{2/3}}-\frac{\sin (c+d x)}{6 a b^2 x^6}+\frac{\sin (c+d x)}{3 a^2 b x^3}-\frac{\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac{\cos (c) \text{Si}(d x)}{a^3}-\frac{d^6 \cos (c) \text{Si}(d x)}{432 a b^2}+\frac{\cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^3}-\frac{(-1)^{2/3} d^2 \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{7/3} b^{2/3}}+\frac{4 \sqrt [3]{-1} d \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{\cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^3}+\frac{d^2 \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{7/3} b^{2/3}}+\frac{4 d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{\cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^3}-\frac{\sqrt [3]{-1} d^2 \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{7/3} b^{2/3}}+\frac{4 (-1)^{2/3} d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{d^6 \int \frac{\sin (c+d x)}{x} \, dx}{720 a b^2}+\frac{\left (d^6 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{720 a b^2}+\frac{\left (d^6 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{432 a b^2}+\frac{\left (d^6 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{720 a b^2}+\frac{\left (d^6 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{432 a b^2}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x^5}-\frac{d \cos (c+d x)}{18 a^2 b x^2}-\frac{d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}+\frac{4 \sqrt [3]{-1} d \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{4 d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{4 (-1)^{2/3} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac{\text{Ci}(d x) \sin (c)}{a^3}+\frac{d^6 \text{Ci}(d x) \sin (c)}{720 a b^2}-\frac{\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 )}{3 a^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^{7/3} b^{2/3}}-\frac{\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 )}{3 a^3}+\frac{(-1)^{2/3} 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^{7/3} b^{2/3}}-\frac{\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 )}{3 a^3}-\frac{\sqrt [3]{-1} 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^{7/3} b^{2/3}}-\frac{\sin (c+d x)}{6 a b^2 x^6}+\frac{\sin (c+d x)}{3 a^2 b x^3}-\frac{\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac{\cos (c) \text{Si}(d x)}{a^3}+\frac{d^6 \cos (c) \text{Si}(d x)}{720 a b^2}+\frac{\cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^3}-\frac{(-1)^{2/3} d^2 \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{7/3} b^{2/3}}+\frac{4 \sqrt [3]{-1} d \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{\cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^3}+\frac{d^2 \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{7/3} b^{2/3}}+\frac{4 d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{\cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^3}-\frac{\sqrt [3]{-1} d^2 \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{7/3} b^{2/3}}+\frac{4 (-1)^{2/3} d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{\left (d^6 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{720 a b^2}-\frac{\left (d^6 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{720 a b^2}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x^5}-\frac{d \cos (c+d x)}{18 a^2 b x^2}-\frac{d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}+\frac{4 \sqrt [3]{-1} d \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{4 d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{4 (-1)^{2/3} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac{\text{Ci}(d x) \sin (c)}{a^3}-\frac{\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 )}{3 a^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^{7/3} b^{2/3}}-\frac{\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 )}{3 a^3}+\frac{(-1)^{2/3} 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^{7/3} b^{2/3}}-\frac{\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 )}{3 a^3}-\frac{\sqrt [3]{-1} 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^{7/3} b^{2/3}}-\frac{\sin (c+d x)}{6 a b^2 x^6}+\frac{\sin (c+d x)}{3 a^2 b x^3}-\frac{\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac{\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac{\cos (c) \text{Si}(d x)}{a^3}+\frac{\cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^3}-\frac{(-1)^{2/3} d^2 \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{7/3} b^{2/3}}+\frac{4 \sqrt [3]{-1} d \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{\cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^3}+\frac{d^2 \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{7/3} b^{2/3}}+\frac{4 d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac{\cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^3}-\frac{\sqrt [3]{-1} d^2 \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{7/3} b^{2/3}}+\frac{4 (-1)^{2/3} d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}\\ \end{align*}

Mathematica [B]  time = 11.6903, size = 2929, normalized size = 2.52 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sin[c]*(CosIntegral[d*x]/a^3 - ((-1)^(2/3)*(63 - 64*(-1)^(1/3) + 62*(-1)^(2/3))*(d^2*Cos[(a^(1/3)*d)/b^(1/3)]*
CosIntegral[d*(a^(1/3)/b^(1/3) + x)] + (b^(1/3)*(b^(1/3)*Cos[d*x] - d*(a^(1/3) + b^(1/3)*x)*Sin[d*x]))/(a^(1/3
) + b^(1/3)*x)^2 + d^2*Sin[(a^(1/3)*d)/b^(1/3)]*SinIntegral[d*(a^(1/3)/b^(1/3) + x)]))/(18*(-1 + (-1)^(1/3))*(
1 + (-1)^(1/3))^3*a^(7/3)*b^(2/3)) - ((-1)^(2/3)*(64 - 62*(-1)^(1/3) + 63*(-1)^(2/3))*(d^2*Cos[((-1)^(2/3)*a^(
1/3)*d)/b^(1/3)]*CosIntegral[d*(((-1)^(2/3)*a^(1/3))/b^(1/3) + x)] + (b^(1/3)*(b^(1/3)*Cos[d*x] - d*((-1)^(2/3
)*a^(1/3) + b^(1/3)*x)*Sin[d*x]))/((-1)^(2/3)*a^(1/3) + b^(1/3)*x)^2 + d^2*Sin[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]
*SinIntegral[d*(((-1)^(2/3)*a^(1/3))/b^(1/3) + x)]))/(18*(1 + (-1)^(1/3))^3*a^(7/3)*b^(2/3)) + ((2 - 3*(-1)^(1
/3) + 2*(-1)^(2/3))*(Cos[((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[-(((-1)^(1/3)*a^(1/3)*d)/b^(1/3)) + d*x]
+ Sin[((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]))/((1 + (-1)^(1/3))^2*
a^3) - ((-1)^(2/3)*(64 - 62*(-1)^(1/3) + 63*(-1)^(2/3))*(d^2*Cos[((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[d
*(-(((-1)^(1/3)*a^(1/3))/b^(1/3)) + x)] + (b^(2/3)*Cos[d*x] + b^(1/3)*d*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sin[d
*x])/((-1)^(1/3)*a^(1/3) - b^(1/3)*x)^2 + d^2*Sin[((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1
/3)*d)/b^(1/3) - d*x]))/(18*(-1 + (-1)^(1/3))*(1 + (-1)^(1/3))^3*a^(7/3)*b^(2/3)) - ((-1)^(2/3)*(59 - 67*(-1)^
(1/3) + 63*(-1)^(2/3))*b^(1/3)*(-(Cos[d*x]/(b^(1/3)*(-((-1)^(1/3)*a^(1/3)) + b^(1/3)*x))) + (d*(-(CosIntegral[
-(((-1)^(1/3)*a^(1/3)*d)/b^(1/3)) + d*x]*Sin[((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]) + Cos[((-1)^(1/3)*a^(1/3)*d)/b^(
1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]))/b^(2/3)))/(9*(1 + (-1)^(1/3))^3*a^(8/3)) - ((-1)^(2/
3)*(5*b^(1/3) - 5*(-1)^(1/3)*b^(1/3) + 4*(-1)^(2/3)*b^(1/3))*(Cos[(a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)
/b^(1/3) + d*x] + Sin[(a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x]))/((1 + (-1)^(1/3))^2*a^3*b^
(1/3)) - ((59 - 67*(-1)^(1/3) + 63*(-1)^(2/3))*b^(1/3)*(-(Cos[d*x]/(b^(1/3)*(a^(1/3) + b^(1/3)*x))) + (d*(CosI
ntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[(a^(1/3)*d)/b^(1/3)] - Cos[(a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)
/b^(1/3) + d*x]))/b^(2/3)))/(9*(-1 + (-1)^(1/3))*(1 + (-1)^(1/3))^3*a^(8/3)) + ((-1)^(2/3)*(2*b^(1/3) - 2*(-1)
^(1/3)*b^(1/3) + 3*(-1)^(2/3)*b^(1/3))*(Cos[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)
/b^(1/3) + d*x] + Sin[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]))/((1
+ (-1)^(1/3))^2*a^3*b^(1/3)) - ((-1)^(2/3)*(59*b^(1/3) - 67*(-1)^(1/3)*b^(1/3) + 63*(-1)^(2/3)*b^(1/3))*(-(Cos
[d*x]/(b^(1/3)*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))) + (d*(CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[
((-1)^(2/3)*a^(1/3)*d)/b^(1/3)] - Cos[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/
3) + d*x]))/b^(2/3)))/(9*(-1 + (-1)^(1/3))*(1 + (-1)^(1/3))^3*a^(8/3))) + Cos[c]*(SinIntegral[d*x]/a^3 - ((-1)
^(2/3)*(63 - 64*(-1)^(1/3) + 62*(-1)^(2/3))*(-(d^2*CosIntegral[d*(a^(1/3)/b^(1/3) + x)]*Sin[(a^(1/3)*d)/b^(1/3
)]) + (b^(1/3)*(d*(a^(1/3) + b^(1/3)*x)*Cos[d*x] + b^(1/3)*Sin[d*x]))/(a^(1/3) + b^(1/3)*x)^2 + d^2*Cos[(a^(1/
3)*d)/b^(1/3)]*SinIntegral[d*(a^(1/3)/b^(1/3) + x)]))/(18*(-1 + (-1)^(1/3))*(1 + (-1)^(1/3))^3*a^(7/3)*b^(2/3)
) - ((-1)^(2/3)*(64 - 62*(-1)^(1/3) + 63*(-1)^(2/3))*(-(d^2*CosIntegral[d*(((-1)^(2/3)*a^(1/3))/b^(1/3) + x)]*
Sin[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]) + (b^(1/3)*(d*((-1)^(2/3)*a^(1/3) + b^(1/3)*x)*Cos[d*x] + b^(1/3)*Sin[d*x
]))/((-1)^(2/3)*a^(1/3) + b^(1/3)*x)^2 + d^2*Cos[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[d*(((-1)^(2/3)*a^
(1/3))/b^(1/3) + x)]))/(18*(1 + (-1)^(1/3))^3*a^(7/3)*b^(2/3)) + ((2 - 3*(-1)^(1/3) + 2*(-1)^(2/3))*(CosIntegr
al[-(((-1)^(1/3)*a^(1/3)*d)/b^(1/3)) + d*x]*Sin[((-1)^(1/3)*a^(1/3)*d)/b^(1/3)] - Cos[((-1)^(1/3)*a^(1/3)*d)/b
^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]))/((1 + (-1)^(1/3))^2*a^3) + ((-1)^(2/3)*(64 - 62*(-
1)^(1/3) + 63*(-1)^(2/3))*(-(d^2*CosIntegral[d*(-(((-1)^(1/3)*a^(1/3))/b^(1/3)) + x)]*Sin[((-1)^(1/3)*a^(1/3)*
d)/b^(1/3)]) + (b^(1/3)*d*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Cos[d*x] - b^(2/3)*Sin[d*x])/((-1)^(1/3)*a^(1/3) -
b^(1/3)*x)^2 + d^2*Cos[((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]))/(18
*(-1 + (-1)^(1/3))*(1 + (-1)^(1/3))^3*a^(7/3)*b^(2/3)) - ((-1)^(2/3)*(59 - 67*(-1)^(1/3) + 63*(-1)^(2/3))*b^(1
/3)*(-(Sin[d*x]/(b^(1/3)*(-((-1)^(1/3)*a^(1/3)) + b^(1/3)*x))) + (d*(Cos[((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIn
tegral[-(((-1)^(1/3)*a^(1/3)*d)/b^(1/3)) + d*x] + Sin[((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*
a^(1/3)*d)/b^(1/3) - d*x]))/b^(2/3)))/(9*(1 + (-1)^(1/3))^3*a^(8/3)) - ((-1)^(2/3)*(5*b^(1/3) - 5*(-1)^(1/3)*b
^(1/3) + 4*(-1)^(2/3)*b^(1/3))*(-(CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[(a^(1/3)*d)/b^(1/3)]) + Cos[(a^(1
/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x]))/((1 + (-1)^(1/3))^2*a^3*b^(1/3)) - ((59 - 67*(-1)^(1/
3) + 63*(-1)^(2/3))*b^(1/3)*(-(Sin[d*x]/(b^(1/3)*(a^(1/3) + b^(1/3)*x))) + (d*(Cos[(a^(1/3)*d)/b^(1/3)]*CosInt
egral[(a^(1/3)*d)/b^(1/3) + d*x] + Sin[(a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x]))/b^(2/3)))
/(9*(-1 + (-1)^(1/3))*(1 + (-1)^(1/3))^3*a^(8/3)) + ((-1)^(2/3)*(2*b^(1/3) - 2*(-1)^(1/3)*b^(1/3) + 3*(-1)^(2/
3)*b^(1/3))*(-(CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]) + Cos[((
-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]))/((1 + (-1)^(1/3))^2*a^3*b^(1
/3)) - ((-1)^(2/3)*(59*b^(1/3) - 67*(-1)^(1/3)*b^(1/3) + 63*(-1)^(2/3)*b^(1/3))*(-(Sin[d*x]/(b^(1/3)*((-1)^(2/
3)*a^(1/3) + b^(1/3)*x))) + (d*(Cos[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)
+ d*x] + Sin[((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]))/b^(2/3)))/(9
*(-1 + (-1)^(1/3))*(1 + (-1)^(1/3))^3*a^(8/3)))

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Maple [C]  time = 0.051, size = 363, normalized size = 0.3 \begin{align*}{\frac{\sin \left ( dx+c \right ){d}^{3} \left ( 2\, \left ( dx+c \right ) ^{3}b-6\,c \left ( dx+c \right ) ^{2}b+6\, \left ( dx+c \right ) b{c}^{2}+3\,a{d}^{3}-2\,{c}^{3}b \right ) }{6\,{a}^{2} \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 ){d}^{4}x}{ \left ( 18\, \left ( dx+c \right ) ^{3}b-54\,c \left ( dx+c \right ) ^{2}b+54\, \left ( dx+c \right ) b{c}^{2}+18\,a{d}^{3}-18\,{c}^{3}b \right ){a}^{2}}}-{\frac{1}{54\,b{a}^{3}}\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 ( a{d}^{3}+18\,{\it \_R1}\,b-18\,cb \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}-c}}}+{\frac{{\it Si} \left ( dx \right ) \cos \left ( c \right ) +{\it Ci} \left ( dx \right ) \sin \left ( c \right ) }{{a}^{3}}}-{\frac{4\,{d}^{3}}{27\,{a}^{2}b}\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}}^{2}-2\,{\it \_RR1}\,c+{c}^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sin(d*x+c)*d^3*(2*(d*x+c)^3*b-6*c*(d*x+c)^2*b+6*(d*x+c)*b*c^2+3*a*d^3-2*c^3*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*cos(d*x+c)*d^4*x/((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)/a^2-1/54/b/a^3*sum((a*d^3+18*_R1*b-18*b*c)/(_R1-c)*(-Si(-d*x+_R1-c)*cos(_R1)+Ci(d*x-_R1+c)*sin(_R1)),_R
1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))+1/a^3*(Si(d*x)*cos(c)+Ci(d*x)*sin(c))-4/27*d^3/a^2/b*sum(1
/(_RR1^2-2*_RR1*c+c^2)*(Si(-d*x+_RR1-c)*sin(_RR1)+Ci(d*x-_RR1+c)*cos(_RR1)),_RR1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z
*b*c^2+a*d^3-b*c^3))

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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} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 2.98755, size = 2763, normalized size = 2.38 \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)/x/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/216*((-36*I*b^2*x^6 - 72*I*a*b*x^3 - 36*I*a^2 + (I*b^2*x^6 + 2*I*a*b*x^3 + I*a^2 + sqrt(3)*(b^2*x^6 + 2*a*b*
x^3 + a^2))*(I*a*d^3/b)^(2/3) + (8*I*b^2*x^6 + 16*I*a*b*x^3 + 8*I*a^2 - 8*sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2))
*(I*a*d^3/b)^(1/3))*Ei(-I*d*x + 1/2*(I*a*d^3/b)^(1/3)*(-I*sqrt(3) - 1))*e^(1/2*(I*a*d^3/b)^(1/3)*(I*sqrt(3) +
1) - I*c) + (36*I*b^2*x^6 + 72*I*a*b*x^3 + 36*I*a^2 + (-I*b^2*x^6 - 2*I*a*b*x^3 - I*a^2 - sqrt(3)*(b^2*x^6 + 2
*a*b*x^3 + a^2))*(-I*a*d^3/b)^(2/3) + (-8*I*b^2*x^6 - 16*I*a*b*x^3 - 8*I*a^2 + 8*sqrt(3)*(b^2*x^6 + 2*a*b*x^3
+ a^2))*(-I*a*d^3/b)^(1/3))*Ei(I*d*x + 1/2*(-I*a*d^3/b)^(1/3)*(-I*sqrt(3) - 1))*e^(1/2*(-I*a*d^3/b)^(1/3)*(I*s
qrt(3) + 1) + I*c) + (-36*I*b^2*x^6 - 72*I*a*b*x^3 - 36*I*a^2 + (I*b^2*x^6 + 2*I*a*b*x^3 + I*a^2 - sqrt(3)*(b^
2*x^6 + 2*a*b*x^3 + a^2))*(I*a*d^3/b)^(2/3) + (8*I*b^2*x^6 + 16*I*a*b*x^3 + 8*I*a^2 + 8*sqrt(3)*(b^2*x^6 + 2*a
*b*x^3 + a^2))*(I*a*d^3/b)^(1/3))*Ei(-I*d*x + 1/2*(I*a*d^3/b)^(1/3)*(I*sqrt(3) - 1))*e^(1/2*(I*a*d^3/b)^(1/3)*
(-I*sqrt(3) + 1) - I*c) + (36*I*b^2*x^6 + 72*I*a*b*x^3 + 36*I*a^2 + (-I*b^2*x^6 - 2*I*a*b*x^3 - I*a^2 + sqrt(3
)*(b^2*x^6 + 2*a*b*x^3 + a^2))*(-I*a*d^3/b)^(2/3) + (-8*I*b^2*x^6 - 16*I*a*b*x^3 - 8*I*a^2 - 8*sqrt(3)*(b^2*x^
6 + 2*a*b*x^3 + a^2))*(-I*a*d^3/b)^(1/3))*Ei(I*d*x + 1/2*(-I*a*d^3/b)^(1/3)*(I*sqrt(3) - 1))*e^(1/2*(-I*a*d^3/
b)^(1/3)*(-I*sqrt(3) + 1) + I*c) + (-108*I*b^2*x^6 - 216*I*a*b*x^3 - 108*I*a^2)*Ei(I*d*x)*e^(I*c) + (108*I*b^2
*x^6 + 216*I*a*b*x^3 + 108*I*a^2)*Ei(-I*d*x)*e^(-I*c) + (36*I*b^2*x^6 + 72*I*a*b*x^3 + 36*I*a^2 + (2*I*b^2*x^6
+ 4*I*a*b*x^3 + 2*I*a^2)*(-I*a*d^3/b)^(2/3) + (16*I*b^2*x^6 + 32*I*a*b*x^3 + 16*I*a^2)*(-I*a*d^3/b)^(1/3))*Ei
(I*d*x + (-I*a*d^3/b)^(1/3))*e^(I*c - (-I*a*d^3/b)^(1/3)) + (-36*I*b^2*x^6 - 72*I*a*b*x^3 - 36*I*a^2 + (-2*I*b
^2*x^6 - 4*I*a*b*x^3 - 2*I*a^2)*(I*a*d^3/b)^(2/3) + (-16*I*b^2*x^6 - 32*I*a*b*x^3 - 16*I*a^2)*(I*a*d^3/b)^(1/3
))*Ei(-I*d*x + (I*a*d^3/b)^(1/3))*e^(-I*c - (I*a*d^3/b)^(1/3)) - 12*(a*b*d*x^4 + a^2*d*x)*cos(d*x + c) + 36*(2
*a*b*x^3 + 3*a^2)*sin(d*x + c))/(a^3*b^2*x^6 + 2*a^4*b*x^3 + a^5)

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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)/x/(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} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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