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

Optimal. Leaf size=408 \[ -\frac{b^{2/3} \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{5/3}}+\frac{\sqrt [3]{-1} b^{2/3} \sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac{(-1)^{2/3} b^{2/3} \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{5/3}}-\frac{\sqrt [3]{-1} b^{2/3} \cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac{b^{2/3} \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac{(-1)^{2/3} b^{2/3} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac{d^2 \sin (c) \text{CosIntegral}(d x)}{2 a}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a}-\frac{\sin (c+d x)}{2 a x^2}-\frac{d \cos (c+d x)}{2 a x} \]

[Out]

-(d*Cos[c + d*x])/(2*a*x) - (d^2*CosIntegral[d*x]*Sin[c])/(2*a) - (b^(2/3)*CosIntegral[(a^(1/3)*d)/b^(1/3) + d
*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(3*a^(5/3)) + ((-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)])/(3*a^(5/3)) - ((-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)])/(3*a^(5/3)) - Sin[c + d*x]/(2*a*x^2) - (d^2*Co
s[c]*SinIntegral[d*x])/(2*a) - ((-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])/(3*a^(5/3)) - (b^(2/3)*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^
(1/3) + d*x])/(3*a^(5/3)) - ((-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])/(3*a^(5/3))

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Rubi [A]  time = 0.680667, antiderivative size = 408, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {3345, 3297, 3303, 3299, 3302, 3333} \[ -\frac{b^{2/3} \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{5/3}}+\frac{\sqrt [3]{-1} b^{2/3} \sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac{(-1)^{2/3} b^{2/3} \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 a^{5/3}}-\frac{\sqrt [3]{-1} b^{2/3} \cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 a^{5/3}}-\frac{b^{2/3} \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac{(-1)^{2/3} b^{2/3} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 a^{5/3}}-\frac{d^2 \sin (c) \text{CosIntegral}(d x)}{2 a}-\frac{d^2 \cos (c) \text{Si}(d x)}{2 a}-\frac{\sin (c+d x)}{2 a x^2}-\frac{d \cos (c+d x)}{2 a x} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x^3*(a + b*x^3)),x]

[Out]

-(d*Cos[c + d*x])/(2*a*x) - (d^2*CosIntegral[d*x]*Sin[c])/(2*a) - (b^(2/3)*CosIntegral[(a^(1/3)*d)/b^(1/3) + d
*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(3*a^(5/3)) + ((-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)])/(3*a^(5/3)) - ((-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)])/(3*a^(5/3)) - Sin[c + d*x]/(2*a*x^2) - (d^2*Co
s[c]*SinIntegral[d*x])/(2*a) - ((-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])/(3*a^(5/3)) - (b^(2/3)*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^
(1/3) + d*x])/(3*a^(5/3)) - ((-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])/(3*a^(5/3))

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])

Rubi steps

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

Mathematica [C]  time = 0.492876, size = 253, normalized size = 0.62 \[ \frac{-i x^2 \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-i \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]+i x^2 \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{i \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]-3 \left (d^2 x^2 \sin (c) \text{CosIntegral}(d x)+d^2 x^2 \cos (c) \text{Si}(d x)+\sin (c+d x)+d x \cos (c+d x)\right )}{6 a x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I)*x^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]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] + I*x^2*RootSu
m[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 & ] - 3*(d*x*Cos[c + d*x] + d^2*x^
2*CosIntegral[d*x]*Sin[c] + Sin[c + d*x] + d^2*x^2*Cos[c]*SinIntegral[d*x]))/(6*a*x^2)

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Maple [C]  time = 0.011, size = 136, normalized size = 0.3 \begin{align*}{d}^{2} \left ( -{\frac{1}{3\,a}\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} \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),x)

[Out]

d^2*(-1/3/a*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*(-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 )} 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),x, algorithm="maxima")

[Out]

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

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Fricas [C]  time = 2.39372, size = 1216, normalized size = 2.98 \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),x, algorithm="fricas")

[Out]

1/12*(3*I*a*d^3*x^2*Ei(I*d*x)*e^(I*c) - 3*I*a*d^3*x^2*Ei(-I*d*x)*e^(-I*c) + 2*(-I*a*d^3/b)^(1/3)*b*x^2*Ei(I*d*
x + (-I*a*d^3/b)^(1/3))*e^(I*c - (-I*a*d^3/b)^(1/3)) + 2*(I*a*d^3/b)^(1/3)*b*x^2*Ei(-I*d*x + (I*a*d^3/b)^(1/3)
)*e^(-I*c - (I*a*d^3/b)^(1/3)) - 6*a*d^2*x*cos(d*x + c) + (-I*sqrt(3)*b*x^2 - 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) + (-I*sqrt(3)*b*x
^2 - 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) + (I*sqrt(3)*b*x^2 - 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) + (I*sqrt(3)*b*x^2 - 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) - 6*a*d*sin(d*
x + c))/(a^2*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),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 )} 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),x, algorithm="giac")

[Out]

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