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

Optimal. Leaf size=396 \[ \frac{\sin \left (a-\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{f}+\frac{\sin \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{f}+\frac{\sin \left (a-\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{f}-\frac{\cos \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{f}+\frac{\cos \left (a-\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{\sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{f}+\frac{\cos \left (a-\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{f} \]

[Out]

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

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Rubi [A]  time = 1.38987, antiderivative size = 396, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3431, 3303, 3299, 3302} \[ \frac{\sin \left (a-\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{f}+\frac{\sin \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{f}+\frac{\sin \left (a-\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{f}-\frac{\cos \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{f}+\frac{\cos \left (a-\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{\sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{f}+\frac{\cos \left (a-\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [C]  time = 1.75552, size = 118, normalized size = 0.3 \[ \frac{i \left (\text{RootSum}\left [\text{$\#$1}^3 f-c f+d e\& ,e^{-i \text{$\#$1} b-i a} \text{Ei}\left (-i b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )\& \right ]-\text{RootSum}\left [\text{$\#$1}^3 f-c f+d e\& ,e^{i \text{$\#$1} b+i a} \text{Ei}\left (i b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )\& \right ]\right )}{2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.019, size = 327, normalized size = 0.8 \begin{align*} 3\,{\frac{1}{{b}^{3}} \left ( 1/3\,{\frac{{b}^{3}}{f}\sum _{{\it \_R1}={\it RootOf} \left ( -cf{b}^{3}+de{b}^{3}+f{{\it \_Z}}^{3}-3\,af{{\it \_Z}}^{2}+3\,{a}^{2}f{\it \_Z}-{a}^{3}f \right ) }{\frac{{{\it \_R1}}^{2} \left ( -{\it Si} \left ( -b\sqrt [3]{dx+c}+{\it \_R1}-a \right ) \cos \left ({\it \_R1} \right ) +{\it Ci} \left ( b\sqrt [3]{dx+c}-{\it \_R1}+a \right ) \sin \left ({\it \_R1} \right ) \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,a+{a}^{2}}}}-2/3\,{\frac{{b}^{3}a}{f}\sum _{{\it \_R1}={\it RootOf} \left ( -cf{b}^{3}+de{b}^{3}+f{{\it \_Z}}^{3}-3\,af{{\it \_Z}}^{2}+3\,{a}^{2}f{\it \_Z}-{a}^{3}f \right ) }{\frac{{\it \_R1}\, \left ( -{\it Si} \left ( -b\sqrt [3]{dx+c}+{\it \_R1}-a \right ) \cos \left ({\it \_R1} \right ) +{\it Ci} \left ( b\sqrt [3]{dx+c}-{\it \_R1}+a \right ) \sin \left ({\it \_R1} \right ) \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,a+{a}^{2}}}}+1/3\,{\frac{{a}^{2}{b}^{3}}{f}\sum _{{\it \_R1}={\it RootOf} \left ( -cf{b}^{3}+de{b}^{3}+f{{\it \_Z}}^{3}-3\,af{{\it \_Z}}^{2}+3\,{a}^{2}f{\it \_Z}-{a}^{3}f \right ) }{\frac{-{\it Si} \left ( -b\sqrt [3]{dx+c}+{\it \_R1}-a \right ) \cos \left ({\it \_R1} \right ) +{\it Ci} \left ( b\sqrt [3]{dx+c}-{\it \_R1}+a \right ) \sin \left ({\it \_R1} \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,a+{a}^{2}}}} \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(_R1^2/(_R1^2-2*_R1*a+a^2)*(-Si(-b*(d*x+c)^(1/3)+_R1-a)*cos(_R1)+Ci(b*(d*x+c)^(1/3)-_R1+a)
*sin(_R1)),_R1=RootOf(-b^3*c*f+b^3*d*e+_Z^3*f-3*_Z^2*a*f+3*_Z*a^2*f-a^3*f))-2/3*b^3*a/f*sum(_R1/(_R1^2-2*_R1*a
+a^2)*(-Si(-b*(d*x+c)^(1/3)+_R1-a)*cos(_R1)+Ci(b*(d*x+c)^(1/3)-_R1+a)*sin(_R1)),_R1=RootOf(-b^3*c*f+b^3*d*e+_Z
^3*f-3*_Z^2*a*f+3*_Z*a^2*f-a^3*f))+1/3*a^2*b^3/f*sum(1/(_R1^2-2*_R1*a+a^2)*(-Si(-b*(d*x+c)^(1/3)+_R1-a)*cos(_R
1)+Ci(b*(d*x+c)^(1/3)-_R1+a)*sin(_R1)),_R1=RootOf(-b^3*c*f+b^3*d*e+_Z^3*f-3*_Z^2*a*f+3*_Z*a^2*f-a^3*f)))

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

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Fricas [C]  time = 1.93127, size = 1119, normalized size = 2.83 \begin{align*} \frac{i \,{\rm Ei}\left (-i \,{\left (d x + c\right )}^{\frac{1}{3}} b + \frac{1}{2} \,{\left (-i \, \sqrt{3} - 1\right )} \left (\frac{i \, b^{3} d e - i \, b^{3} c f}{f}\right )^{\frac{1}{3}}\right ) e^{\left (\frac{1}{2} \,{\left (i \, \sqrt{3} + 1\right )} \left (\frac{i \, b^{3} d e - i \, b^{3} c f}{f}\right )^{\frac{1}{3}} - i \, a\right )} + i \,{\rm Ei}\left (-i \,{\left (d x + c\right )}^{\frac{1}{3}} b + \frac{1}{2} \,{\left (i \, \sqrt{3} - 1\right )} \left (\frac{i \, b^{3} d e - i \, b^{3} c f}{f}\right )^{\frac{1}{3}}\right ) e^{\left (\frac{1}{2} \,{\left (-i \, \sqrt{3} + 1\right )} \left (\frac{i \, b^{3} d e - i \, b^{3} c f}{f}\right )^{\frac{1}{3}} - i \, a\right )} - i \,{\rm Ei}\left (i \,{\left (d x + c\right )}^{\frac{1}{3}} b + \frac{1}{2} \,{\left (-i \, \sqrt{3} - 1\right )} \left (\frac{-i \, b^{3} d e + i \, b^{3} c f}{f}\right )^{\frac{1}{3}}\right ) e^{\left (\frac{1}{2} \,{\left (i \, \sqrt{3} + 1\right )} \left (\frac{-i \, b^{3} d e + i \, b^{3} c f}{f}\right )^{\frac{1}{3}} + i \, a\right )} - i \,{\rm Ei}\left (i \,{\left (d x + c\right )}^{\frac{1}{3}} b + \frac{1}{2} \,{\left (i \, \sqrt{3} - 1\right )} \left (\frac{-i \, b^{3} d e + i \, b^{3} c f}{f}\right )^{\frac{1}{3}}\right ) e^{\left (\frac{1}{2} \,{\left (-i \, \sqrt{3} + 1\right )} \left (\frac{-i \, b^{3} d e + i \, b^{3} c f}{f}\right )^{\frac{1}{3}} + i \, a\right )} - i \,{\rm Ei}\left (i \,{\left (d x + c\right )}^{\frac{1}{3}} b + \left (\frac{-i \, b^{3} d e + i \, b^{3} c f}{f}\right )^{\frac{1}{3}}\right ) e^{\left (i \, a - \left (\frac{-i \, b^{3} d e + i \, b^{3} c f}{f}\right )^{\frac{1}{3}}\right )} + i \,{\rm Ei}\left (-i \,{\left (d x + c\right )}^{\frac{1}{3}} b + \left (\frac{i \, b^{3} d e - i \, b^{3} c f}{f}\right )^{\frac{1}{3}}\right ) e^{\left (-i \, a - \left (\frac{i \, b^{3} d e - i \, b^{3} c f}{f}\right )^{\frac{1}{3}}\right )}}{2 \, f} \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(-I*(d*x + c)^(1/3)*b + 1/2*(-I*sqrt(3) - 1)*((I*b^3*d*e - I*b^3*c*f)/f)^(1/3))*e^(1/2*(I*sqrt(3) + 1
)*((I*b^3*d*e - I*b^3*c*f)/f)^(1/3) - I*a) + I*Ei(-I*(d*x + c)^(1/3)*b + 1/2*(I*sqrt(3) - 1)*((I*b^3*d*e - I*b
^3*c*f)/f)^(1/3))*e^(1/2*(-I*sqrt(3) + 1)*((I*b^3*d*e - I*b^3*c*f)/f)^(1/3) - I*a) - I*Ei(I*(d*x + c)^(1/3)*b
+ 1/2*(-I*sqrt(3) - 1)*((-I*b^3*d*e + I*b^3*c*f)/f)^(1/3))*e^(1/2*(I*sqrt(3) + 1)*((-I*b^3*d*e + I*b^3*c*f)/f)
^(1/3) + I*a) - I*Ei(I*(d*x + c)^(1/3)*b + 1/2*(I*sqrt(3) - 1)*((-I*b^3*d*e + I*b^3*c*f)/f)^(1/3))*e^(1/2*(-I*
sqrt(3) + 1)*((-I*b^3*d*e + I*b^3*c*f)/f)^(1/3) + I*a) - I*Ei(I*(d*x + c)^(1/3)*b + ((-I*b^3*d*e + I*b^3*c*f)/
f)^(1/3))*e^(I*a - ((-I*b^3*d*e + I*b^3*c*f)/f)^(1/3)) + I*Ei(-I*(d*x + c)^(1/3)*b + ((I*b^3*d*e - I*b^3*c*f)/
f)^(1/3))*e^(-I*a - ((I*b^3*d*e - I*b^3*c*f)/f)^(1/3)))/f

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