3.220 \(\int \frac{\sin (a+\frac{b}{\sqrt [3]{c+d x}})}{e+f x} \, dx\)

Optimal. Leaf size=434 \[ \frac{\sin \left (a-\frac{b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text{CosIntegral}\left (\frac{b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac{b}{\sqrt [3]{c+d x}}\right )}{f}+\frac{\sin \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}-\frac{b}{\sqrt [3]{c+d x}}\right )}{f}+\frac{\sin \left (a-\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac{b}{\sqrt [3]{c+d x}}\right )}{f}-\frac{3 \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{f}-\frac{\cos \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}-\frac{b}{\sqrt [3]{c+d x}}\right )}{f}+\frac{\cos \left (a-\frac{b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text{Si}\left (\frac{\sqrt [3]{f} b}{\sqrt [3]{d e-c f}}+\frac{b}{\sqrt [3]{c+d x}}\right )}{f}+\frac{\cos \left (a-\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{f} b}{\sqrt [3]{d e-c f}}+\frac{b}{\sqrt [3]{c+d x}}\right )}{f}-\frac{3 \cos (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{f} \]

[Out]

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

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Rubi [A]  time = 1.92008, antiderivative size = 434, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {3431, 3303, 3299, 3302, 3345} \[ \frac{\sin \left (a-\frac{b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text{CosIntegral}\left (\frac{b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac{b}{\sqrt [3]{c+d x}}\right )}{f}+\frac{\sin \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}-\frac{b}{\sqrt [3]{c+d x}}\right )}{f}+\frac{\sin \left (a-\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac{b}{\sqrt [3]{c+d x}}\right )}{f}-\frac{3 \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{f}-\frac{\cos \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}-\frac{b}{\sqrt [3]{c+d x}}\right )}{f}+\frac{\cos \left (a-\frac{b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text{Si}\left (\frac{\sqrt [3]{f} b}{\sqrt [3]{d e-c f}}+\frac{b}{\sqrt [3]{c+d x}}\right )}{f}+\frac{\cos \left (a-\frac{(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{f} b}{\sqrt [3]{d e-c f}}+\frac{b}{\sqrt [3]{c+d x}}\right )}{f}-\frac{3 \cos (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{f} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b/(c + d*x)^(1/3)]/(e + f*x),x]

[Out]

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

Rule 3431

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :
> Dist[1/(n*f), Subst[Int[ExpandIntegrand[(a + b*Sin[c + d*x])^p, x^(1/n - 1)*(g - (e*h)/f + (h*x^(1/n))/f)^m,
x], x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && IntegerQ[1/n]

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 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]

Rubi steps

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

Mathematica [C]  time = 2.74273, size = 170, normalized size = 0.39 \[ \frac{i \left ((\cos (a)-i \sin (a)) \left (\text{RootSum}\left [\text{$\#$1}^3 f-c f+d e\& ,e^{-\frac{i b}{\text{$\#$1}}} \text{Ei}\left (-i b \left (\frac{1}{\sqrt [3]{c+d x}}-\frac{1}{\text{$\#$1}}\right )\right )\& \right ]-3 \text{Ei}\left (-\frac{i b}{\sqrt [3]{c+d x}}\right )\right )+(\cos (a)+i \sin (a)) \left (3 \text{Ei}\left (\frac{i b}{\sqrt [3]{c+d x}}\right )-\text{RootSum}\left [\text{$\#$1}^3 f-c f+d e\& ,e^{\frac{i b}{\text{$\#$1}}} \text{Ei}\left (i b \left (\frac{1}{\sqrt [3]{c+d x}}-\frac{1}{\text{$\#$1}}\right )\right )\& \right ]\right )\right )}{2 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b/(c + d*x)^(1/3)]/(e + f*x),x]

[Out]

((I/2)*((-3*ExpIntegralEi[((-I)*b)/(c + d*x)^(1/3)] + RootSum[d*e - c*f + f*#1^3 & , ExpIntegralEi[(-I)*b*((c
+ d*x)^(-1/3) - #1^(-1))]/E^((I*b)/#1) & ])*(Cos[a] - I*Sin[a]) + (3*ExpIntegralEi[(I*b)/(c + d*x)^(1/3)] - Ro
otSum[d*e - c*f + f*#1^3 & , E^((I*b)/#1)*ExpIntegralEi[I*b*((c + d*x)^(-1/3) - #1^(-1))] & ])*(Cos[a] + I*Sin
[a])))/f

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Maple [C]  time = 0.027, size = 156, normalized size = 0.4 \begin{align*} -3\,{b}^{3} \left ( -1/3\,{\frac{1}{{b}^{3}f}\sum _{{\it \_R1}={\it RootOf} \left ( \left ( cf-de \right ){{\it \_Z}}^{3}+ \left ( -3\,acf+3\,ade \right ){{\it \_Z}}^{2}+ \left ( 3\,{a}^{2}cf-3\,{a}^{2}de \right ){\it \_Z}-{a}^{3}cf+{a}^{3}de-{b}^{3}f \right ) }-{\it Si} \left ( -{\frac{b}{\sqrt [3]{dx+c}}}+{\it \_R1}-a \right ) \cos \left ({\it \_R1} \right ) +{\it Ci} \left ({\frac{b}{\sqrt [3]{dx+c}}}-{\it \_R1}+a \right ) \sin \left ({\it \_R1} \right ) }+{\frac{1}{{b}^{3}f} \left ({\it Si} \left ({\frac{b}{\sqrt [3]{dx+c}}} \right ) \cos \left ( a \right ) +{\it Ci} \left ({\frac{b}{\sqrt [3]{dx+c}}} \right ) \sin \left ( a \right ) \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+b/(d*x+c)^(1/3))/(f*x+e),x)

[Out]

-3*b^3*(-1/3/b^3/f*sum(-Si(-b/(d*x+c)^(1/3)+_R1-a)*cos(_R1)+Ci(b/(d*x+c)^(1/3)-_R1+a)*sin(_R1),_R1=RootOf((c*f
-d*e)*_Z^3+(-3*a*c*f+3*a*d*e)*_Z^2+(3*a^2*c*f-3*a^2*d*e)*_Z-a^3*c*f+a^3*d*e-b^3*f))+1/b^3/f*(Si(b/(d*x+c)^(1/3
))*cos(a)+Ci(b/(d*x+c)^(1/3))*sin(a)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (a + \frac{b}{{\left (d x + c\right )}^{\frac{1}{3}}}\right )}{f x + e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(1/3))/(f*x+e),x, algorithm="maxima")

[Out]

integrate(sin(a + b/(d*x + c)^(1/3))/(f*x + e), x)

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Fricas [C]  time = 2.34048, size = 1359, normalized size = 3.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(1/3))/(f*x+e),x, algorithm="fricas")

[Out]

1/2*(I*Ei(1/2*(-2*I*(d*x + c)^(2/3)*b - (I*b^3*f/(d*e - c*f))^(1/3)*(d*x - sqrt(3)*(-I*d*x - I*c) + c))/(d*x +
c))*e^(1/2*(I*b^3*f/(d*e - c*f))^(1/3)*(I*sqrt(3) + 1) - I*a) - I*Ei(1/2*(2*I*(d*x + c)^(2/3)*b - (-I*b^3*f/(
d*e - c*f))^(1/3)*(d*x - sqrt(3)*(-I*d*x - I*c) + c))/(d*x + c))*e^(1/2*(-I*b^3*f/(d*e - c*f))^(1/3)*(I*sqrt(3
) + 1) + I*a) + I*Ei(1/2*(-2*I*(d*x + c)^(2/3)*b - (I*b^3*f/(d*e - c*f))^(1/3)*(d*x - sqrt(3)*(I*d*x + I*c) +
c))/(d*x + c))*e^(1/2*(I*b^3*f/(d*e - c*f))^(1/3)*(-I*sqrt(3) + 1) - I*a) - I*Ei(1/2*(2*I*(d*x + c)^(2/3)*b -
(-I*b^3*f/(d*e - c*f))^(1/3)*(d*x - sqrt(3)*(I*d*x + I*c) + c))/(d*x + c))*e^(1/2*(-I*b^3*f/(d*e - c*f))^(1/3)
*(-I*sqrt(3) + 1) + I*a) + 3*I*Ei(I*b/(d*x + c)^(1/3))*e^(I*a) - 3*I*Ei(-I*b/(d*x + c)^(1/3))*e^(-I*a) - I*Ei(
(I*(d*x + c)^(2/3)*b + (-I*b^3*f/(d*e - c*f))^(1/3)*(d*x + c))/(d*x + c))*e^(I*a - (-I*b^3*f/(d*e - c*f))^(1/3
)) + I*Ei((-I*(d*x + c)^(2/3)*b + (I*b^3*f/(d*e - c*f))^(1/3)*(d*x + c))/(d*x + c))*e^(-I*a - (I*b^3*f/(d*e -
c*f))^(1/3)))/f

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (a + \frac{b}{\sqrt [3]{c + d x}} \right )}}{e + f x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)**(1/3))/(f*x+e),x)

[Out]

Integral(sin(a + b/(c + d*x)**(1/3))/(e + f*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (a + \frac{b}{{\left (d x + c\right )}^{\frac{1}{3}}}\right )}{f x + e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(1/3))/(f*x+e),x, algorithm="giac")

[Out]

integrate(sin(a + b/(d*x + c)^(1/3))/(f*x + e), x)