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

Optimal. Leaf size=800 \[ \text{result too large to display} \]

[Out]

-(d*Cos[c + d*x])/(2*a^2*x) - ((-1)^(2/3)*b^(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])/(9*a^(7/3)) - (b^(1/3)*d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)
/b^(1/3) + d*x])/(9*a^(7/3)) + ((-1)^(1/3)*b^(1/3)*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)
^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(7/3)) - (d^2*CosIntegral[d*x]*Sin[c])/(2*a^2) - (5*b^(2/3)*CosIntegral
[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(9*a^(8/3)) + (5*(-1)^(1/3)*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)])/(9*a^(8/3)) - (5*(-1)^(2/3)*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)])/(9*a^(8/3)) + Sin[
c + d*x]/(3*a*b*x^5) - (5*Sin[c + d*x])/(6*a^2*x^2) - Sin[c + d*x]/(3*b*x^5*(a + b*x^3)) - (d^2*Cos[c]*SinInte
gral[d*x])/(2*a^2) - (5*(-1)^(1/3)*b^(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])/(9*a^(8/3)) - ((-1)^(2/3)*b^(1/3)*d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegra
l[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(7/3)) - (5*b^(2/3)*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^
(1/3)*d)/b^(1/3) + d*x])/(9*a^(8/3)) + (b^(1/3)*d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3)
+ d*x])/(9*a^(7/3)) - (5*(-1)^(2/3)*b^(2/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a
^(1/3)*d)/b^(1/3) + d*x])/(9*a^(8/3)) - ((-1)^(1/3)*b^(1/3)*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinInteg
ral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(7/3))

________________________________________________________________________________________

Rubi [A]  time = 1.78803, antiderivative size = 800, normalized size of antiderivative = 1., number of steps used = 51, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {3343, 3345, 3297, 3303, 3299, 3302, 3333, 3346} \[ -\frac{\text{CosIntegral}(d x) \sin (c) d^2}{2 a^2}-\frac{\cos (c) \text{Si}(d x) d^2}{2 a^2}-\frac{\cos (c+d x) d}{2 a^2 x}-\frac{(-1)^{2/3} \sqrt [3]{b} \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) d}{9 a^{7/3}}-\frac{\sqrt [3]{b} \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{9 a^{7/3}}+\frac{\sqrt [3]{-1} \sqrt [3]{b} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{9 a^{7/3}}-\frac{(-1)^{2/3} \sqrt [3]{b} \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) d}{9 a^{7/3}}+\frac{\sqrt [3]{b} \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{9 a^{7/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{b} \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{9 a^{7/3}}-\frac{5 b^{2/3} \text{CosIntegral}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{8/3}}+\frac{5 \sqrt [3]{-1} b^{2/3} \text{CosIntegral}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{8/3}}-\frac{5 (-1)^{2/3} b^{2/3} \text{CosIntegral}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{8/3}}-\frac{\sin (c+d x)}{3 b x^5 \left (b x^3+a\right )}-\frac{5 \sin (c+d x)}{6 a^2 x^2}+\frac{\sin (c+d x)}{3 a b x^5}-\frac{5 \sqrt [3]{-1} b^{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 )}{9 a^{8/3}}-\frac{5 b^{2/3} \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^{8/3}}-\frac{5 (-1)^{2/3} b^{2/3} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{8/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*Cos[c + d*x])/(2*a^2*x) - ((-1)^(2/3)*b^(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])/(9*a^(7/3)) - (b^(1/3)*d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)
/b^(1/3) + d*x])/(9*a^(7/3)) + ((-1)^(1/3)*b^(1/3)*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)
^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(7/3)) - (d^2*CosIntegral[d*x]*Sin[c])/(2*a^2) - (5*b^(2/3)*CosIntegral
[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(9*a^(8/3)) + (5*(-1)^(1/3)*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)])/(9*a^(8/3)) - (5*(-1)^(2/3)*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)])/(9*a^(8/3)) + Sin[
c + d*x]/(3*a*b*x^5) - (5*Sin[c + d*x])/(6*a^2*x^2) - Sin[c + d*x]/(3*b*x^5*(a + b*x^3)) - (d^2*Cos[c]*SinInte
gral[d*x])/(2*a^2) - (5*(-1)^(1/3)*b^(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])/(9*a^(8/3)) - ((-1)^(2/3)*b^(1/3)*d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegra
l[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(7/3)) - (5*b^(2/3)*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^
(1/3)*d)/b^(1/3) + d*x])/(9*a^(8/3)) + (b^(1/3)*d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3)
+ d*x])/(9*a^(7/3)) - (5*(-1)^(2/3)*b^(2/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a
^(1/3)*d)/b^(1/3) + d*x])/(9*a^(8/3)) - ((-1)^(1/3)*b^(1/3)*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinInteg
ral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(7/3))

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 3333

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

Rule 3346

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

Rubi steps

\begin{align*} \int \frac{\sin (c+d x)}{x^3 \left (a+b x^3\right )^2} \, dx &=-\frac{\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}-\frac{5 \int \frac{\sin (c+d x)}{x^6 \left (a+b x^3\right )} \, dx}{3 b}+\frac{d \int \frac{\cos (c+d x)}{x^5 \left (a+b x^3\right )} \, dx}{3 b}\\ &=-\frac{\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}-\frac{5 \int \left (\frac{\sin (c+d x)}{a x^6}-\frac{b \sin (c+d x)}{a^2 x^3}+\frac{b^2 \sin (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{3 b}+\frac{d \int \left (\frac{\cos (c+d x)}{a x^5}-\frac{b \cos (c+d x)}{a^2 x^2}+\frac{b^2 x \cos (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{3 b}\\ &=-\frac{\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}+\frac{5 \int \frac{\sin (c+d x)}{x^3} \, dx}{3 a^2}-\frac{5 \int \frac{\sin (c+d x)}{x^6} \, dx}{3 a b}-\frac{(5 b) \int \frac{\sin (c+d x)}{a+b x^3} \, dx}{3 a^2}-\frac{d \int \frac{\cos (c+d x)}{x^2} \, dx}{3 a^2}+\frac{d \int \frac{\cos (c+d x)}{x^5} \, dx}{3 a b}+\frac{(b d) \int \frac{x \cos (c+d x)}{a+b x^3} \, dx}{3 a^2}\\ &=-\frac{d \cos (c+d x)}{12 a b x^4}+\frac{d \cos (c+d x)}{3 a^2 x}+\frac{\sin (c+d x)}{3 a b x^5}-\frac{5 \sin (c+d x)}{6 a^2 x^2}-\frac{\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}-\frac{(5 b) \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}{3 a^2}+\frac{(5 d) \int \frac{\cos (c+d x)}{x^2} \, dx}{6 a^2}-\frac{d \int \frac{\cos (c+d x)}{x^5} \, dx}{3 a b}+\frac{(b d) \int \left (-\frac{\cos (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{(-1)^{2/3} \cos (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac{\sqrt [3]{-1} \cos (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{3 a^2}+\frac{d^2 \int \frac{\sin (c+d x)}{x} \, dx}{3 a^2}-\frac{d^2 \int \frac{\sin (c+d x)}{x^4} \, dx}{12 a b}\\ &=-\frac{d \cos (c+d x)}{2 a^2 x}+\frac{\sin (c+d x)}{3 a b x^5}+\frac{d^2 \sin (c+d x)}{36 a b x^3}-\frac{5 \sin (c+d x)}{6 a^2 x^2}-\frac{\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}+\frac{(5 b) \int \frac{\sin (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{9 a^{8/3}}+\frac{(5 b) \int \frac{\sin (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{8/3}}+\frac{(5 b) \int \frac{\sin (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{8/3}}-\frac{\left (b^{2/3} d\right ) \int \frac{\cos (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{7/3}}+\frac{\left (\sqrt [3]{-1} b^{2/3} d\right ) \int \frac{\cos (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{7/3}}-\frac{\left ((-1)^{2/3} b^{2/3} d\right ) \int \frac{\cos (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{7/3}}-\frac{\left (5 d^2\right ) \int \frac{\sin (c+d x)}{x} \, dx}{6 a^2}+\frac{d^2 \int \frac{\sin (c+d x)}{x^4} \, dx}{12 a b}-\frac{d^3 \int \frac{\cos (c+d x)}{x^3} \, dx}{36 a b}+\frac{\left (d^2 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{3 a^2}+\frac{\left (d^2 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{3 a^2}\\ &=\frac{d^3 \cos (c+d x)}{72 a b x^2}-\frac{d \cos (c+d x)}{2 a^2 x}+\frac{d^2 \text{Ci}(d x) \sin (c)}{3 a^2}+\frac{\sin (c+d x)}{3 a b x^5}-\frac{5 \sin (c+d x)}{6 a^2 x^2}-\frac{\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}+\frac{d^2 \cos (c) \text{Si}(d x)}{3 a^2}+\frac{d^3 \int \frac{\cos (c+d x)}{x^3} \, dx}{36 a b}+\frac{d^4 \int \frac{\sin (c+d x)}{x^2} \, dx}{72 a b}-\frac{\left (5 d^2 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{6 a^2}+\frac{\left (5 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}{9 a^{8/3}}-\frac{\left (b^{2/3} 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^{7/3}}-\frac{\left (5 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]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{8/3}}+\frac{\left (\sqrt [3]{-1} b^{2/3} 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}{9 a^{7/3}}+\frac{\left (5 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 )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{8/3}}-\frac{\left ((-1)^{2/3} b^{2/3} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{7/3}}-\frac{\left (5 d^2 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{6 a^2}+\frac{\left (5 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}{9 a^{8/3}}+\frac{\left (b^{2/3} 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^{7/3}}+\frac{\left (5 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]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{8/3}}+\frac{\left (\sqrt [3]{-1} b^{2/3} 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}{9 a^{7/3}}+\frac{\left (5 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 )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{8/3}}+\frac{\left ((-1)^{2/3} b^{2/3} d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{7/3}}\\ &=-\frac{d \cos (c+d x)}{2 a^2 x}-\frac{(-1)^{2/3} \sqrt [3]{b} d \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3}}-\frac{\sqrt [3]{b} d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}+\frac{\sqrt [3]{-1} \sqrt [3]{b} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}-\frac{d^2 \text{Ci}(d x) \sin (c)}{2 a^2}-\frac{5 b^{2/3} \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^{8/3}}+\frac{5 \sqrt [3]{-1} b^{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 )}{9 a^{8/3}}-\frac{5 (-1)^{2/3} b^{2/3} \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{8/3}}+\frac{\sin (c+d x)}{3 a b x^5}-\frac{5 \sin (c+d x)}{6 a^2 x^2}-\frac{d^4 \sin (c+d x)}{72 a b x}-\frac{\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a^2}-\frac{5 \sqrt [3]{-1} b^{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 )}{9 a^{8/3}}-\frac{(-1)^{2/3} \sqrt [3]{b} d \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3}}-\frac{5 b^{2/3} \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^{8/3}}+\frac{\sqrt [3]{b} d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}-\frac{5 (-1)^{2/3} b^{2/3} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{8/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{b} d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}-\frac{d^4 \int \frac{\sin (c+d x)}{x^2} \, dx}{72 a b}+\frac{d^5 \int \frac{\cos (c+d x)}{x} \, dx}{72 a b}\\ &=-\frac{d \cos (c+d x)}{2 a^2 x}-\frac{(-1)^{2/3} \sqrt [3]{b} d \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3}}-\frac{\sqrt [3]{b} d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}+\frac{\sqrt [3]{-1} \sqrt [3]{b} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}-\frac{d^2 \text{Ci}(d x) \sin (c)}{2 a^2}-\frac{5 b^{2/3} \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^{8/3}}+\frac{5 \sqrt [3]{-1} b^{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 )}{9 a^{8/3}}-\frac{5 (-1)^{2/3} b^{2/3} \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{8/3}}+\frac{\sin (c+d x)}{3 a b x^5}-\frac{5 \sin (c+d x)}{6 a^2 x^2}-\frac{\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a^2}-\frac{5 \sqrt [3]{-1} b^{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 )}{9 a^{8/3}}-\frac{(-1)^{2/3} \sqrt [3]{b} d \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3}}-\frac{5 b^{2/3} \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^{8/3}}+\frac{\sqrt [3]{b} d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}-\frac{5 (-1)^{2/3} b^{2/3} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{8/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{b} d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}-\frac{d^5 \int \frac{\cos (c+d x)}{x} \, dx}{72 a b}+\frac{\left (d^5 \cos (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{72 a b}-\frac{\left (d^5 \sin (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{72 a b}\\ &=-\frac{d \cos (c+d x)}{2 a^2 x}+\frac{d^5 \cos (c) \text{Ci}(d x)}{72 a b}-\frac{(-1)^{2/3} \sqrt [3]{b} d \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3}}-\frac{\sqrt [3]{b} d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}+\frac{\sqrt [3]{-1} \sqrt [3]{b} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}-\frac{d^2 \text{Ci}(d x) \sin (c)}{2 a^2}-\frac{5 b^{2/3} \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^{8/3}}+\frac{5 \sqrt [3]{-1} b^{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 )}{9 a^{8/3}}-\frac{5 (-1)^{2/3} b^{2/3} \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{8/3}}+\frac{\sin (c+d x)}{3 a b x^5}-\frac{5 \sin (c+d x)}{6 a^2 x^2}-\frac{\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a^2}-\frac{d^5 \sin (c) \text{Si}(d x)}{72 a b}-\frac{5 \sqrt [3]{-1} b^{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 )}{9 a^{8/3}}-\frac{(-1)^{2/3} \sqrt [3]{b} d \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3}}-\frac{5 b^{2/3} \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^{8/3}}+\frac{\sqrt [3]{b} d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}-\frac{5 (-1)^{2/3} b^{2/3} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{8/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{b} d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}-\frac{\left (d^5 \cos (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{72 a b}+\frac{\left (d^5 \sin (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{72 a b}\\ &=-\frac{d \cos (c+d x)}{2 a^2 x}-\frac{(-1)^{2/3} \sqrt [3]{b} d \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3}}-\frac{\sqrt [3]{b} d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}+\frac{\sqrt [3]{-1} \sqrt [3]{b} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}-\frac{d^2 \text{Ci}(d x) \sin (c)}{2 a^2}-\frac{5 b^{2/3} \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^{8/3}}+\frac{5 \sqrt [3]{-1} b^{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 )}{9 a^{8/3}}-\frac{5 (-1)^{2/3} b^{2/3} \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{8/3}}+\frac{\sin (c+d x)}{3 a b x^5}-\frac{5 \sin (c+d x)}{6 a^2 x^2}-\frac{\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a^2}-\frac{5 \sqrt [3]{-1} b^{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 )}{9 a^{8/3}}-\frac{(-1)^{2/3} \sqrt [3]{b} d \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3}}-\frac{5 b^{2/3} \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^{8/3}}+\frac{\sqrt [3]{b} d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}-\frac{5 (-1)^{2/3} b^{2/3} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{8/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{b} d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3}}\\ \end{align*}

Mathematica [C]  time = 1.12044, size = 470, normalized size = 0.59 \[ \frac{\text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-5 \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-5 i \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+5 i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-5 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]+\text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-5 \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+5 i \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-5 i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-5 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]-\frac{3 \left (3 d^2 x^2 \sin (c) \left (a+b x^3\right ) \text{CosIntegral}(d x)+3 d^2 x^2 \cos (c) \left (a+b x^3\right ) \text{Si}(d x)+3 a \sin (c+d x)+3 a d x \cos (c+d x)+5 b x^3 \sin (c+d x)+3 b d x^4 \cos (c+d x)\right )}{x^2 \left (a+b x^3\right )}}{18 a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(RootSum[a + b*#1^3 & , ((-5*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 5*CosIntegral[d*(x - #1)]*Sin[c + d*#1
] - 5*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + (5*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*Co
sIntegral[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)/#1^2 & ] + RootSum[a + b*#1^3 & , ((5*I)*Cos[c + d*#1]*
CosIntegral[d*(x - #1)] - 5*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 5*Cos[c + d*#1]*SinIntegral[d*(x - #1)] -
(5*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)/#1^2 & ] - (3*(3*a*d*x*Cos[c + d*x] + 3*b*d*x^4*Cos[c + d*x] + 3*d^2*x^2*(a + b*x^3)*CosIntegral[d*x]
*Sin[c] + 3*a*Sin[c + d*x] + 5*b*x^3*Sin[c + d*x] + 3*d^2*x^2*(a + b*x^3)*Cos[c]*SinIntegral[d*x]))/(x^2*(a +
b*x^3)))/(18*a^2)

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Maple [C]  time = 0.024, size = 388, normalized size = 0.5 \begin{align*}{d}^{2} \left ( -{\frac{b{d}^{3}}{a} \left ({\frac{\sin \left ( dx+c \right ) }{ \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} \left ({\frac{dx+c}{3\,a{d}^{3}}}-{\frac{c}{3\,a{d}^{3}}} \right ) }+{\frac{2}{9\,ab{d}^{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{-{\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}}^{2}-2\,{\it \_R1}\,c+{c}^{2}}}}-{\frac{1}{9\,ab{d}^{3}}\sum _{{\it \_RR1}={\it RootOf} \left ({{\it \_Z}}^{3}b-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-{c}^{3}b \right ) }{\frac{{\it Si} \left ( -dx+{\it \_RR1}-c \right ) \sin \left ({\it \_RR1} \right ) +{\it Ci} \left ( dx-{\it \_RR1}+c \right ) \cos \left ({\it \_RR1} \right ) }{{\it \_RR1}-c}}} \right ) }-{\frac{1}{3\,{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}}^{2}-2\,{\it \_R1}\,c+{c}^{2}}}}+{\frac{1}{{a}^{2}} \left ( -{\frac{\sin \left ( dx+c \right ) }{2\,{d}^{2}{x}^{2}}}-{\frac{\cos \left ( dx+c \right ) }{2\,dx}}-{\frac{{\it Si} \left ( dx \right ) \cos \left ( c \right ) }{2}}-{\frac{{\it Ci} \left ( dx \right ) \sin \left ( c \right ) }{2}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^2*(-1/a*b*d^3*(sin(d*x+c)*(1/3/a/d^3*(d*x+c)-1/3*c/a/d^3)/((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)+2/9/a/d^3/b*sum(1/(_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/d^3/b*sum(1/(_RR1-c)*(Si(-d*x+_RR1-c)*sin(_RR1)+Ci(d*x-_RR1+c)*c
os(_RR1)),_RR1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3)))-1/3/a^2*sum(1/(_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/a^2*(-1/2*sin(
d*x+c)/x^2/d^2-1/2*cos(d*x+c)/x/d-1/2*Si(d*x)*cos(c)-1/2*Ci(d*x)*sin(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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 2.86145, size = 2118, normalized size = 2.65 \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^3/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

-1/36*(((b^2*x^5 + a*b*x^2 - sqrt(3)*(I*b^2*x^5 + I*a*b*x^2))*(I*a*d^3/b)^(2/3) + (5*b^2*x^5 + 5*a*b*x^2 - sqr
t(3)*(-5*I*b^2*x^5 - 5*I*a*b*x^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) + ((b^2*x^5 + a*b*x^2 - sqrt(3)*(I*b^2*x^5 + I*a*b*x^2))*(-I*a*d^3
/b)^(2/3) + (5*b^2*x^5 + 5*a*b*x^2 - sqrt(3)*(-5*I*b^2*x^5 - 5*I*a*b*x^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) + ((b^2*x^5 + a*b*x^2 -
sqrt(3)*(-I*b^2*x^5 - I*a*b*x^2))*(I*a*d^3/b)^(2/3) + (5*b^2*x^5 + 5*a*b*x^2 - sqrt(3)*(5*I*b^2*x^5 + 5*I*a*b*
x^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) + ((b^2*x^5 + a*b*x^2 - sqrt(3)*(-I*b^2*x^5 - I*a*b*x^2))*(-I*a*d^3/b)^(2/3) + (5*b^2*x^5 + 5*a
*b*x^2 - sqrt(3)*(5*I*b^2*x^5 + 5*I*a*b*x^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) - (9*I*a*b*d^3*x^5 + 9*I*a^2*d^3*x^2)*Ei(I*d*x)*e^(I*
c) - (-9*I*a*b*d^3*x^5 - 9*I*a^2*d^3*x^2)*Ei(-I*d*x)*e^(-I*c) - 2*((b^2*x^5 + a*b*x^2)*(-I*a*d^3/b)^(2/3) + 5*
(b^2*x^5 + a*b*x^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)) - 2*((b^2*
x^5 + a*b*x^2)*(I*a*d^3/b)^(2/3) + 5*(b^2*x^5 + a*b*x^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)) + 18*(a*b*d^2*x^4 + a^2*d^2*x)*cos(d*x + c) + 6*(5*a*b*d*x^3 + 3*a^2*d)*sin(d*x + c)
)/(a^3*b*d*x^5 + a^4*d*x^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(sin(d*x+c)/x**3/(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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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