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

Optimal. Leaf size=712 \[ \frac {4 \sqrt [3]{b} \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\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 {Ci}\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 {Ci}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{7/3}}-\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 \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\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 (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^2}+\frac {d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\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 {d \cos (c) \text {Ci}(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 a b x^4}-\frac {\sin (c+d x)}{3 b x^4 \left (a+b x^3\right )} \]

[Out]

d*Ci(d*x)*cos(c)/a^2+1/9*d*Ci(a^(1/3)*d/b^(1/3)+d*x)*cos(c-a^(1/3)*d/b^(1/3))/a^2+1/9*d*Ci((-1)^(1/3)*a^(1/3)*
d/b^(1/3)-d*x)*cos(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a^2+1/9*d*Ci((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)*cos(c-(-1)^(
2/3)*a^(1/3)*d/b^(1/3))/a^2+4/9*(-1)^(2/3)*b^(1/3)*cos(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))*Si(-(-1)^(1/3)*a^(1/3)*
d/b^(1/3)+d*x)/a^(7/3)+4/9*b^(1/3)*cos(c-a^(1/3)*d/b^(1/3))*Si(a^(1/3)*d/b^(1/3)+d*x)/a^(7/3)-4/9*(-1)^(1/3)*b
^(1/3)*cos(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))*Si((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)/a^(7/3)-d*Si(d*x)*sin(c)/a^2+4
/9*b^(1/3)*Ci(a^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^(7/3)-1/9*d*Si(a^(1/3)*d/b^(1/3)+d*x)*sin(c-a^
(1/3)*d/b^(1/3))/a^2+4/9*(-1)^(2/3)*b^(1/3)*Ci((-1)^(1/3)*a^(1/3)*d/b^(1/3)-d*x)*sin(c+(-1)^(1/3)*a^(1/3)*d/b^
(1/3))/a^(7/3)-1/9*d*Si(-(-1)^(1/3)*a^(1/3)*d/b^(1/3)+d*x)*sin(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a^2-4/9*(-1)^(1
/3)*b^(1/3)*Ci((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)*sin(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^(7/3)-1/9*d*Si((-1)^(2/
3)*a^(1/3)*d/b^(1/3)+d*x)*sin(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^2+1/3*sin(d*x+c)/a/b/x^4-4/3*sin(d*x+c)/a^2/x-
1/3*sin(d*x+c)/b/x^4/(b*x^3+a)

________________________________________________________________________________________

Rubi [A]  time = 1.60, antiderivative size = 712, normalized size of antiderivative = 1.00, 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 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 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 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 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 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*}

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Mathematica [C]  time = 1.16, size = 445, normalized size = 0.62 \[ -\frac {-\frac {1}{6} x \left (a+b x^3\right ) \left (\text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\frac {-4 \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-i \text {$\#$1} d \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-4 i \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+\text {$\#$1} d \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+4 i \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-\text {$\#$1} d \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-4 \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-i \text {$\#$1} d \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}}\& \right ]+\text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\frac {-4 \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+i \text {$\#$1} d \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+4 i \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+\text {$\#$1} d \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-4 i \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-\text {$\#$1} d \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-4 \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))+i \text {$\#$1} d \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}}\& \right ]+18 d \cos (c) \text {Ci}(d x)-18 d \sin (c) \text {Si}(d x)\right )+\sin (c) \left (3 a+4 b x^3\right ) \cos (d x)+\cos (c) \left (3 a+4 b x^3\right ) \sin (d x)}{3 a^2 x \left (a+b x^3\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/3*((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]*CosInteg
ral[d*x] + RootSum[a + b*#1^3 & , ((-4*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 4*CosIntegral[d*(x - #1)]*Si
n[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]*SinInteg
ral[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*CosInte
gral[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)/(a^2*x*(a + b*x^3))

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fricas [C]  time = 0.71, size = 722, normalized size = 1.01 \[ \frac {{\left (a b d^{3} x^{4} + a^{2} d^{3} x + {\left (2 i \, b^{2} x^{4} + 2 i \, a b x + 2 \, \sqrt {3} {\left (b^{2} x^{4} + a b x\right )}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}}\right )} {\rm Ei}\left (-i \, d x + \frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} - i \, c\right )} + {\left (a b d^{3} x^{4} + a^{2} d^{3} x + {\left (-2 i \, b^{2} x^{4} - 2 i \, a b x - 2 \, \sqrt {3} {\left (b^{2} x^{4} + a b x\right )}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}}\right )} {\rm Ei}\left (i \, d x + \frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} + i \, c\right )} + {\left (a b d^{3} x^{4} + a^{2} d^{3} x + {\left (2 i \, b^{2} x^{4} + 2 i \, a b x - 2 \, \sqrt {3} {\left (b^{2} x^{4} + a b x\right )}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}}\right )} {\rm Ei}\left (-i \, d x + \frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} - i \, c\right )} + {\left (a b d^{3} x^{4} + a^{2} d^{3} x + {\left (-2 i \, b^{2} x^{4} - 2 i \, a b x + 2 \, \sqrt {3} {\left (b^{2} x^{4} + a b x\right )}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}}\right )} {\rm Ei}\left (i \, d x + \frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} + i \, c\right )} + 9 \, {\left (a b d^{3} x^{4} + a^{2} d^{3} x\right )} {\rm Ei}\left (i \, d x\right ) e^{\left (i \, c\right )} + 9 \, {\left (a b d^{3} x^{4} + a^{2} d^{3} x\right )} {\rm Ei}\left (-i \, d x\right ) e^{\left (-i \, c\right )} + {\left (a b d^{3} x^{4} + a^{2} d^{3} x + {\left (4 i \, b^{2} x^{4} + 4 i \, a b x\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}}\right )} {\rm Ei}\left (i \, d x + \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right ) e^{\left (i \, c - \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} + {\left (a b d^{3} x^{4} + a^{2} d^{3} x + {\left (-4 i \, b^{2} x^{4} - 4 i \, a b x\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}}\right )} {\rm Ei}\left (-i \, d x + \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right ) e^{\left (-i \, c - \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} - 6 \, {\left (4 \, a b d^{2} x^{3} + 3 \, a^{2} d^{2}\right )} \sin \left (d x + c\right )}{18 \, {\left (a^{3} b d^{2} x^{4} + a^{4} d^{2} x\right )}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2} x^{2}}\,{d x} \]

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)

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maple [C]  time = 0.07, size = 283, normalized size = 0.40 \[ d \left (\frac {\sin \left (d x +c \right ) \left (-\frac {4 b \left (d x +c \right )^{3}}{3 a^{2}}+\frac {4 c b \left (d x +c \right )^{2}}{a^{2}}-\frac {4 c^{2} b \left (d x +c \right )}{a^{2}}-\frac {3 a \,d^{3}-4 b \,c^{3}}{3 a^{2}}\right )}{x d \left (\left (d x +c \right )^{3} b -3 c \left (d x +c \right )^{2} b +3 \left (d x +c \right ) b \,c^{2}+a \,d^{3}-b \,c^{3}\right )}-\frac {4 \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {-\Si \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\Ci \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1} -c}\right )}{9 a^{2}}+\frac {-\Si \left (d x \right ) \sin \relax (c )+\Ci \left (d x \right ) \cos \relax (c )}{a^{2}}+\frac {\munderset {\textit {\_RR1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\Si \left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\Ci \left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )\right )}{9 a^{2}}\right ) \]

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/a^2*b*(d*x+c)^3+4*c/a^2*b*(d*x+c)^2-4*c^2/a^2*b*(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-b*c^3)-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/a^2*(-Si(d*x)*sin(c)+Ci(d*x)*cos(c))+
1/9/a^2*sum(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2} x^{2}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sin \left (c+d\,x\right )}{x^2\,{\left (b\,x^3+a\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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