3.2.6 \(\int \frac {\sin (c+d x)}{x (a+b x^3)^2} \, dx\) [106]

Optimal. Leaf size=693 \[ \frac {\sqrt [3]{-1} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{5/3} \sqrt [3]{b}}-\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^{5/3} \sqrt [3]{b}}-\frac {(-1)^{2/3} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{5/3} \sqrt [3]{b}}+\frac {\text {Ci}(d x) \sin (c)}{a^2}-\frac {\text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {\text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}-\frac {\text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^2}+\frac {\sin (c+d x)}{3 a b x^3}-\frac {\sin (c+d x)}{3 b x^3 \left (a+b x^3\right )}+\frac {\cos (c) \text {Si}(d x)}{a^2}+\frac {\cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^2}+\frac {\sqrt [3]{-1} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{5/3} \sqrt [3]{b}}-\frac {\cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^2}+\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^{5/3} \sqrt [3]{b}}-\frac {\cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^2}+\frac {(-1)^{2/3} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{5/3} \sqrt [3]{b}} \]

[Out]

-1/9*d*Ci(a^(1/3)*d/b^(1/3)+d*x)*cos(c-a^(1/3)*d/b^(1/3))/a^(5/3)/b^(1/3)+1/9*(-1)^(1/3)*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^(5/3)/b^(1/3)-1/9*(-1)^(2/3)*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^(5/3)/b^(1/3)+cos(c)*Si(d*x)/a^2-1/3*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^2-1/3*cos(c-a^(1/3)*d/b^(1/3))*Si(a^(1/3)*d/b^(1/3)+d*x
)/a^2-1/3*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^2+Ci(d*x)*sin(c)/a^2-1/3*
Ci(a^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^2+1/9*d*Si(a^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3)
)/a^(5/3)/b^(1/3)-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^2-1/9*(-1)^(1
/3)*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^(5/3)/b^(1/3)-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^2+1/9*(-1)^(2/3)*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^(5/3)/b^(1/3)+1/3*sin(d*x+c)/a/b/x^3-1/3*sin(d*x+c)/b/x^3/(b*
x^3+a)

________________________________________________________________________________________

Rubi [A]
time = 0.95, antiderivative size = 693, normalized size of antiderivative = 1.00, number of steps used = 41, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3424, 3426, 3378, 3384, 3380, 3383, 3427, 3415} \begin {gather*} \frac {\sqrt [3]{-1} 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^{5/3} \sqrt [3]{b}}-\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^{5/3} \sqrt [3]{b}}-\frac {(-1)^{2/3} 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^{5/3} \sqrt [3]{b}}+\frac {\sqrt [3]{-1} 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^{5/3} \sqrt [3]{b}}+\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^{5/3} \sqrt [3]{b}}+\frac {(-1)^{2/3} 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^{5/3} \sqrt [3]{b}}-\frac {\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 )}{3 a^2}-\frac {\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 )}{3 a^2}-\frac {\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 )}{3 a^2}+\frac {\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 )}{3 a^2}-\frac {\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 )}{3 a^2}-\frac {\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 )}{3 a^2}+\frac {\sin (c) \text {CosIntegral}(d x)}{a^2}+\frac {\cos (c) \text {Si}(d x)}{a^2}-\frac {\sin (c+d x)}{3 b x^3 \left (a+b x^3\right )}+\frac {\sin (c+d x)}{3 a b x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x*(a + b*x^3)^2),x]

[Out]

((-1)^(1/3)*d*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^
(5/3)*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^(5/3)*b^(1/3)) -
((-1)^(2/3)*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a
^(5/3)*b^(1/3)) + (CosIntegral[d*x]*Sin[c])/a^2 - (CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/
b^(1/3)])/(3*a^2) - (CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]
)/(3*a^2) - (CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)])/(3*a^2
) + Sin[c + d*x]/(3*a*b*x^3) - Sin[c + d*x]/(3*b*x^3*(a + b*x^3)) + (Cos[c]*SinIntegral[d*x])/a^2 + (Cos[c + (
(-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(3*a^2) + ((-1)^(1/3)*d*Sin[
c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(5/3)*b^(1/3)) - (
Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(3*a^2) + (d*Sin[c - (a^(1/3)*d)/b^(1/3)]
*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3)) - (Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*SinInt
egral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(3*a^2) + ((-1)^(2/3)*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*S
inIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(5/3)*b^(1/3))

Rule 3378

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 3380

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 3383

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 3384

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 3415

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

Rule 3424

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 3426

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 3427

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
time = 0.37, size = 446, normalized size = 0.64 \begin {gather*} \frac {-\frac {1}{2} i \text {RootSum}\left [a+b \text {$\#$1}^3\&,\cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))-i \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-\sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))\&\right ]+\frac {1}{2} i \text {RootSum}\left [a+b \text {$\#$1}^3\&,\cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))+i \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})+i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-\sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))\&\right ]-\frac {a d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))-i \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-\sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]}{6 b}-\frac {a d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))+i \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})+i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-\sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]}{6 b}+\frac {a \cos (d x) \sin (c)}{a+b x^3}+3 \text {Ci}(d x) \sin (c)+\frac {a \cos (c) \sin (d x)}{a+b x^3}+3 \cos (c) \text {Si}(d x)}{3 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.18, size = 233, normalized size = 0.34

method	result	size
derivativedivides	\(\frac {\sin \left (d x +c \right ) d^{3}}{3 a \left (a \,d^{3}-b \,c^{3}+3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}\right )}-\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (-\sinIntegral \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )\right )}{3 a^{2}}+\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )+\cosineIntegral \left (d x \right ) \sin \left (c \right )}{a^{2}}-\frac {d^{3} \left (\munderset {\textit {\_RR1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\sinIntegral \left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\cosineIntegral \left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{\textit {\_RR1}^{2}-2 \textit {\_RR1} c +c^{2}}\right )}{9 a b}\)	\(233\)
default	\(\frac {\sin \left (d x +c \right ) d^{3}}{3 a \left (a \,d^{3}-b \,c^{3}+3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}\right )}-\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (-\sinIntegral \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )\right )}{3 a^{2}}+\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )+\cosineIntegral \left (d x \right ) \sin \left (c \right )}{a^{2}}-\frac {d^{3} \left (\munderset {\textit {\_RR1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\sinIntegral \left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\cosineIntegral \left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{\textit {\_RR1}^{2}-2 \textit {\_RR1} c +c^{2}}\right )}{9 a b}\)	\(233\)
risch	\(-\frac {i \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (-i a \,d^{3}-6 i \textit {\_R1} b c +3 \textit {\_R1}^{2} b -3 b \,c^{2}\right ) {\mathrm e}^{\textit {\_R1}} \expIntegral \left (1, -i d x -i c +\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{18 a^{2} b}+\frac {i {\mathrm e}^{i c} \expIntegral \left (1, -i d x \right )}{2 a^{2}}+\frac {i \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (i a \,d^{3}-6 i \textit {\_R1} b c +3 \textit {\_R1}^{2} b -3 b \,c^{2}\right ) {\mathrm e}^{-\textit {\_R1}} \expIntegral \left (1, i d x +i c -\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{18 a^{2} b}-\frac {{\mathrm e}^{-i c} \pi \,\mathrm {csgn}\left (d x \right )}{2 a^{2}}+\frac {{\mathrm e}^{-i c} \sinIntegral \left (d x \right )}{a^{2}}-\frac {i {\mathrm e}^{-i c} \expIntegral \left (1, -i d x \right )}{2 a^{2}}+\frac {d^{3} \sin \left (d x +c \right )}{3 a \left (d^{3} x^{3} b +a \,d^{3}\right )}\)	\(312\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sin(d*x+c)*d^3/a/(a*d^3-b*c^3+3*b*c^2*(d*x+c)-3*b*c*(d*x+c)^2+b*(d*x+c)^3)-1/3/a^2*sum(-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)*cos(c)+Ci(d*
x)*sin(c))-1/9*d^3/a/b*sum(1/(_RR1^2-2*_RR1*c+c^2)*(Si(-d*x+_RR1-c)*sin(_RR1)+Ci(d*x-_RR1+c)*cos(_RR1)),_RR1=R
ootOf(_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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 0.42, size = 588, normalized size = 0.85 \begin {gather*} \frac {{\left (-6 i \, b x^{3} + {\left (i \, b x^{3} - \sqrt {3} {\left (b x^{3} + a\right )} + i \, a\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} - 6 i \, a\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 (6 i \, b x^{3} + {\left (-i \, b x^{3} + \sqrt {3} {\left (b x^{3} + a\right )} - i \, a\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} + 6 i \, a\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 (-6 i \, b x^{3} + {\left (i \, b x^{3} + \sqrt {3} {\left (b x^{3} + a\right )} + i \, a\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} - 6 i \, a\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 (6 i \, b x^{3} + {\left (-i \, b x^{3} - \sqrt {3} {\left (b x^{3} + a\right )} - i \, a\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} + 6 i \, a\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 )} - 18 \, {\left (i \, b x^{3} + i \, a\right )} {\rm Ei}\left (i \, d x\right ) e^{\left (i \, c\right )} - 18 \, {\left (-i \, b x^{3} - i \, a\right )} {\rm Ei}\left (-i \, d x\right ) e^{\left (-i \, c\right )} - 2 \, {\left (-3 i \, b x^{3} + {\left (-i \, b x^{3} - i \, a\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} - 3 i \, a\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 )} - 2 \, {\left (3 i \, b x^{3} + {\left (i \, b x^{3} + i \, a\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} + 3 i \, a\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 )} + 12 \, a \sin \left (d x + c\right )}{36 \, {\left (a^{2} b x^{3} + a^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*((-6*I*b*x^3 + (I*b*x^3 - sqrt(3)*(b*x^3 + a) + I*a)*(I*a*d^3/b)^(1/3) - 6*I*a)*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) + (6*I*b*x^3 + (-I*b*x^3 + sqrt(3)*
(b*x^3 + a) - I*a)*(-I*a*d^3/b)^(1/3) + 6*I*a)*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) + (-6*I*b*x^3 + (I*b*x^3 + sqrt(3)*(b*x^3 + a) + I*a)*(I*a*d^3/b)^(1/3)
- 6*I*a)*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) +
(6*I*b*x^3 + (-I*b*x^3 - sqrt(3)*(b*x^3 + a) - I*a)*(-I*a*d^3/b)^(1/3) + 6*I*a)*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) - 18*(I*b*x^3 + I*a)*Ei(I*d*x)*e^(I*c)
- 18*(-I*b*x^3 - I*a)*Ei(-I*d*x)*e^(-I*c) - 2*(-3*I*b*x^3 + (-I*b*x^3 - I*a)*(-I*a*d^3/b)^(1/3) - 3*I*a)*Ei(I
*d*x + (-I*a*d^3/b)^(1/3))*e^(I*c - (-I*a*d^3/b)^(1/3)) - 2*(3*I*b*x^3 + (I*b*x^3 + I*a)*(I*a*d^3/b)^(1/3) + 3
*I*a)*Ei(-I*d*x + (I*a*d^3/b)^(1/3))*e^(-I*c - (I*a*d^3/b)^(1/3)) + 12*a*sin(d*x + c))/(a^2*b*x^3 + a^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)/(x*(a + b*x^3)^2),x)

[Out]

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

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