3.107 \(\int \frac{\sin (c+d x)}{x^2 (a+b x^3)^2} \, dx\)

Optimal. Leaf size=712 \[ \text{result too large to display} \]

[Out]

(d*Cos[c]*CosIntegral[d*x])/a^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])/(9*a^2) + (d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^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])/(9*a^2) + (4*b^(
1/3)*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(9*a^(7/3)) + (4*(-1)^(2/3)*b^(1/3)*
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)])/(9*a^(7/3)) - (4*(-
1)^(1/3)*b^(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)])/(9
*a^(7/3)) + Sin[c + d*x]/(3*a*b*x^4) - (4*Sin[c + d*x])/(3*a^2*x) - Sin[c + d*x]/(3*b*x^4*(a + b*x^3)) - (d*Si
n[c]*SinIntegral[d*x])/a^2 - (4*(-1)^(2/3)*b^(1/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(
1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(7/3)) + (d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/
3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^2) + (4*b^(1/3)*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3
) + d*x])/(9*a^(7/3)) - (d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^2) - (4*(
-1)^(1/3)*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])/(
9*a^(7/3)) - (d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*
a^2)

________________________________________________________________________________________

Rubi [A]  time = 1.60177, antiderivative size = 712, normalized size of antiderivative = 1., number of steps used = 47, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {3343, 3345, 3297, 3303, 3299, 3302, 3346} \[ \frac{4 \sqrt [3]{b} \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}+\frac{4 (-1)^{2/3} \sqrt [3]{b} \sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3}}-\frac{4 \sqrt [3]{-1} \sqrt [3]{b} \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}+\frac{d \cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^2}+\frac{d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^2}+\frac{d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^2}+\frac{d \sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^2}-\frac{d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^2}-\frac{d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^2}-\frac{4 (-1)^{2/3} \sqrt [3]{b} \cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3}}+\frac{4 \sqrt [3]{b} \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{7/3}}-\frac{4 \sqrt [3]{-1} \sqrt [3]{b} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{7/3}}+\frac{d \cos (c) \text{CosIntegral}(d x)}{a^2}-\frac{d \sin (c) \text{Si}(d x)}{a^2}-\frac{4 \sin (c+d x)}{3 a^2 x}-\frac{\sin (c+d x)}{3 b x^4 \left (a+b x^3\right )}+\frac{\sin (c+d x)}{3 a b x^4} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x^2*(a + b*x^3)^2),x]

[Out]

(d*Cos[c]*CosIntegral[d*x])/a^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])/(9*a^2) + (d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^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])/(9*a^2) + (4*b^(
1/3)*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(9*a^(7/3)) + (4*(-1)^(2/3)*b^(1/3)*
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)])/(9*a^(7/3)) - (4*(-
1)^(1/3)*b^(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)])/(9
*a^(7/3)) + Sin[c + d*x]/(3*a*b*x^4) - (4*Sin[c + d*x])/(3*a^2*x) - Sin[c + d*x]/(3*b*x^4*(a + b*x^3)) - (d*Si
n[c]*SinIntegral[d*x])/a^2 - (4*(-1)^(2/3)*b^(1/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(
1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(7/3)) + (d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/
3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^2) + (4*b^(1/3)*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3
) + d*x])/(9*a^(7/3)) - (d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^2) - (4*(
-1)^(1/3)*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])/(
9*a^(7/3)) - (d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*
a^2)

Rule 3343

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

Rule 3345

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

Rule 3297

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

Rule 3303

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

Rule 3299

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

Rule 3302

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

Rule 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]

Rubi steps

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

Mathematica [C]  time = 1.10265, size = 445, normalized size = 0.62 \[ -\frac{-\frac{1}{6} x \left (a+b x^3\right ) \left (\text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-4 \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-4 i \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+4 i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-4 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}}\& \right ]+\text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-4 \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+4 i \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-4 i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-4 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}}\& \right ]+18 d \cos (c) \text{CosIntegral}(d x)-18 d \sin (c) \text{Si}(d x)\right )+\sin (c) \left (3 a+4 b x^3\right ) \cos (d x)+\cos (c) \left (3 a+4 b x^3\right ) \sin (d x)}{3 a^2 x \left (a+b x^3\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sin[c + d*x]/(x^2*(a + b*x^3)^2),x]

[Out]

-((3*a + 4*b*x^3)*Cos[d*x]*Sin[c] + (3*a + 4*b*x^3)*Cos[c]*Sin[d*x] - (x*(a + b*x^3)*(18*d*Cos[c]*CosIntegral[
d*x] + RootSum[a + b*#1^3 & , ((-4*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 4*CosIntegral[d*(x - #1)]*Sin[c
+ d*#1] - 4*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + (4*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*
#1]*CosIntegral[d*(x - #1)]*#1 - I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - I*d*Cos[c + d*#1]*SinIntegral[
d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1 & ] + RootSum[a + b*#1^3 & , ((4*I)*Cos[c + d*
#1]*CosIntegral[d*(x - #1)] - 4*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 4*Cos[c + d*#1]*SinIntegral[d*(x - #1)
] - (4*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 + I*d*CosIntegral
[d*(x - #1)]*Sin[c + d*#1]*#1 + I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(
x - #1)]*#1)/#1 & ] - 18*d*Sin[c]*SinIntegral[d*x]))/6)/(3*a^2*x*(a + b*x^3))

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Maple [C]  time = 0.033, size = 283, normalized size = 0.4 \begin{align*} d \left ({\frac{\sin \left ( dx+c \right ) }{dx \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 ) } \left ( -{\frac{4\, \left ( dx+c \right ) ^{3}b}{3\,{a}^{2}}}+4\,{\frac{c \left ( dx+c \right ) ^{2}b}{{a}^{2}}}-4\,{\frac{ \left ( dx+c \right ) b{c}^{2}}{{a}^{2}}}-{\frac{3\,a{d}^{3}-4\,{c}^{3}b}{3\,{a}^{2}}} \right ) }-{\frac{4}{9\,{a}^{2}}\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{-{\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 ) }{{\it \_R1}-c}}}+{\frac{\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 ) }{\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 ) }{9\,{a}^{2}}}+{\frac{-{\it Si} \left ( dx \right ) \sin \left ( c \right ) +{\it Ci} \left ( dx \right ) \cos \left ( c \right ) }{{a}^{2}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)/x^2/(b*x^3+a)^2,x)

[Out]

d*(sin(d*x+c)*(-4/3*b/a^2*(d*x+c)^3+4*c*b/a^2*(d*x+c)^2-4*c^2*b/a^2*(d*x+c)-1/3*(3*a*d^3-4*b*c^3)/a^2)/x/d/((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)-4/9/a^2*sum(1/(_R1-c)*(-Si(-d*x+_R1-c)*cos(_R1)+Ci(d*x-
_R1+c)*sin(_R1)),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))+1/9/a^2*sum(Si(-d*x+_RR1-c)*sin(_RR1)+C
i(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))+1/a^2*(-Si(d*x)*sin(c)+Ci(d*x)*
cos(c)))

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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 )}^{2} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x^2/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

integrate(sin(d*x + c)/((b*x^3 + a)^2*x^2), x)

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Fricas [C]  time = 2.81415, size = 1713, normalized size = 2.41 \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^2/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

1/18*((a*b*d^3*x^4 + a^2*d^3*x + (2*I*b^2*x^4 + 2*I*a*b*x + 2*sqrt(3)*(b^2*x^4 + a*b*x))*(I*a*d^3/b)^(2/3))*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) + (a*b*d^3*x
^4 + a^2*d^3*x + (-2*I*b^2*x^4 - 2*I*a*b*x - 2*sqrt(3)*(b^2*x^4 + a*b*x))*(-I*a*d^3/b)^(2/3))*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) + (a*b*d^3*x^4 + a^2*d^3*
x + (2*I*b^2*x^4 + 2*I*a*b*x - 2*sqrt(3)*(b^2*x^4 + a*b*x))*(I*a*d^3/b)^(2/3))*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) + (a*b*d^3*x^4 + a^2*d^3*x + (-2*I*b^2*x^
4 - 2*I*a*b*x + 2*sqrt(3)*(b^2*x^4 + a*b*x))*(-I*a*d^3/b)^(2/3))*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) + 9*(a*b*d^3*x^4 + a^2*d^3*x)*Ei(I*d*x)*e^(I*c) + 9*(a
*b*d^3*x^4 + a^2*d^3*x)*Ei(-I*d*x)*e^(-I*c) + (a*b*d^3*x^4 + a^2*d^3*x + (4*I*b^2*x^4 + 4*I*a*b*x)*(-I*a*d^3/b
)^(2/3))*Ei(I*d*x + (-I*a*d^3/b)^(1/3))*e^(I*c - (-I*a*d^3/b)^(1/3)) + (a*b*d^3*x^4 + a^2*d^3*x + (-4*I*b^2*x^
4 - 4*I*a*b*x)*(I*a*d^3/b)^(2/3))*Ei(-I*d*x + (I*a*d^3/b)^(1/3))*e^(-I*c - (I*a*d^3/b)^(1/3)) - 6*(4*a*b*d^2*x
^3 + 3*a^2*d^2)*sin(d*x + c))/(a^3*b*d^2*x^4 + a^4*d^2*x)

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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**2/(b*x**3+a)**2,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 )}^{2} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x^2/(b*x^3+a)^2,x, algorithm="giac")

[Out]

integrate(sin(d*x + c)/((b*x^3 + a)^2*x^2), x)