3.211 \(\int \frac{\sin (a+b \sqrt [3]{c+d x})}{(e+f x)^2} \, dx\)

Optimal. Leaf size=555 \[ -\frac{\sqrt [3]{-1} b d \cos \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}+\frac{b d \cos \left (a-\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}+\frac{(-1)^{2/3} b d \cos \left (a-\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{\sqrt [3]{-1} b d \sin \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{b d \sin \left (a-\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{\sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{(-1)^{2/3} b d \sin \left (a-\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{\sin \left (a+b \sqrt [3]{c+d x}\right )}{f (e+f x)} \]

[Out]

-((-1)^(1/3)*b*d*Cos[a + ((-1)^(1/3)*b*(d*e - c*f)^(1/3))/f^(1/3)]*CosIntegral[((-1)^(1/3)*b*(d*e - c*f)^(1/3)
)/f^(1/3) - b*(c + d*x)^(1/3)])/(3*f^(4/3)*(d*e - c*f)^(2/3)) + (b*d*Cos[a - (b*(d*e - c*f)^(1/3))/f^(1/3)]*Co
sIntegral[(b*(d*e - c*f)^(1/3))/f^(1/3) + b*(c + d*x)^(1/3)])/(3*f^(4/3)*(d*e - c*f)^(2/3)) + ((-1)^(2/3)*b*d*
Cos[a - ((-1)^(2/3)*b*(d*e - c*f)^(1/3))/f^(1/3)]*CosIntegral[((-1)^(2/3)*b*(d*e - c*f)^(1/3))/f^(1/3) + b*(c
+ d*x)^(1/3)])/(3*f^(4/3)*(d*e - c*f)^(2/3)) - Sin[a + b*(c + d*x)^(1/3)]/(f*(e + f*x)) - ((-1)^(1/3)*b*d*Sin[
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)])/(3*f^(4/3)*(d*e - c*f)^(2/3)) - (b*d*Sin[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)])/(3*f^(4/3)*(d*e - c*f)^(2/3)) - ((-1)^(2/3)*b*d*Sin[a - ((-1)^(2/3)*b
*(d*e - c*f)^(1/3))/f^(1/3)]*SinIntegral[((-1)^(2/3)*b*(d*e - c*f)^(1/3))/f^(1/3) + b*(c + d*x)^(1/3)])/(3*f^(
4/3)*(d*e - c*f)^(2/3))

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Rubi [A]  time = 2.12414, antiderivative size = 555, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3431, 3341, 3334, 3303, 3299, 3302} \[ -\frac{\sqrt [3]{-1} b d \cos \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}+\frac{b d \cos \left (a-\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}+\frac{(-1)^{2/3} b d \cos \left (a-\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{\sqrt [3]{-1} b d \sin \left (a+\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{b d \sin \left (a-\frac{b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{\sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{(-1)^{2/3} b d \sin \left (a-\frac{(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac{\sin \left (a+b \sqrt [3]{c+d x}\right )}{f (e+f x)} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*(c + d*x)^(1/3)]/(e + f*x)^2,x]

[Out]

-((-1)^(1/3)*b*d*Cos[a + ((-1)^(1/3)*b*(d*e - c*f)^(1/3))/f^(1/3)]*CosIntegral[((-1)^(1/3)*b*(d*e - c*f)^(1/3)
)/f^(1/3) - b*(c + d*x)^(1/3)])/(3*f^(4/3)*(d*e - c*f)^(2/3)) + (b*d*Cos[a - (b*(d*e - c*f)^(1/3))/f^(1/3)]*Co
sIntegral[(b*(d*e - c*f)^(1/3))/f^(1/3) + b*(c + d*x)^(1/3)])/(3*f^(4/3)*(d*e - c*f)^(2/3)) + ((-1)^(2/3)*b*d*
Cos[a - ((-1)^(2/3)*b*(d*e - c*f)^(1/3))/f^(1/3)]*CosIntegral[((-1)^(2/3)*b*(d*e - c*f)^(1/3))/f^(1/3) + b*(c
+ d*x)^(1/3)])/(3*f^(4/3)*(d*e - c*f)^(2/3)) - Sin[a + b*(c + d*x)^(1/3)]/(f*(e + f*x)) - ((-1)^(1/3)*b*d*Sin[
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)])/(3*f^(4/3)*(d*e - c*f)^(2/3)) - (b*d*Sin[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)])/(3*f^(4/3)*(d*e - c*f)^(2/3)) - ((-1)^(2/3)*b*d*Sin[a - ((-1)^(2/3)*b
*(d*e - c*f)^(1/3))/f^(1/3)]*SinIntegral[((-1)^(2/3)*b*(d*e - c*f)^(1/3))/f^(1/3) + b*(c + d*x)^(1/3)])/(3*f^(
4/3)*(d*e - c*f)^(2/3))

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 3341

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(e^m*(a + b*x^
n)^(p + 1)*Sin[c + d*x])/(b*n*(p + 1)), x] - Dist[(d*e^m)/(b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*Cos[c + d*x],
x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, -1] && EqQ[m, n - 1] && (IntegerQ[n] || GtQ[e, 0])

Rule 3334

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

Mathematica [C]  time = 1.11035, size = 180, normalized size = 0.32 \[ \frac{b d \text{RootSum}\left [\text{$\#$1}^3 f-c f+d e\& ,\frac{e^{-i \text{$\#$1} b-i a} \text{Ei}\left (-i b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )}{\text{$\#$1}^2}\& \right ]+b d \text{RootSum}\left [\text{$\#$1}^3 f-c f+d e\& ,\frac{e^{i \text{$\#$1} b+i a} \text{Ei}\left (i b \left (\sqrt [3]{c+d x}-\text{$\#$1}\right )\right )}{\text{$\#$1}^2}\& \right ]+\frac{3 i f e^{-i \left (a+b \sqrt [3]{c+d x}\right )} \left (-1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right )}{e+f x}}{6 f^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*(c + d*x)^(1/3)]/(e + f*x)^2,x]

[Out]

(((3*I)*(-1 + E^((2*I)*(a + b*(c + d*x)^(1/3))))*f)/(E^(I*(a + b*(c + d*x)^(1/3)))*(e + f*x)) + b*d*RootSum[d*
e - c*f + f*#1^3 & , (E^((-I)*a - I*b*#1)*ExpIntegralEi[(-I)*b*((c + d*x)^(1/3) - #1)])/#1^2 & ] + b*d*RootSum
[d*e - c*f + f*#1^3 & , (E^(I*a + I*b*#1)*ExpIntegralEi[I*b*((c + d*x)^(1/3) - #1)])/#1^2 & ])/(6*f^2)

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Maple [C]  time = 0.067, size = 1175, normalized size = 2.1 \begin{align*} \text{result too large to display} \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)^2,x)

[Out]

3*d/b^3*(sin(a+b*(d*x+c)^(1/3))*(-2/3*a*b^3/(c*f-d*e)*(a+b*(d*x+c)^(1/3))^2+a^2*b^3/(c*f-d*e)*(a+b*(d*x+c)^(1/
3))-1/3*b^3*(b^3*c*f-b^3*d*e+a^3*f)/(c*f-d*e)/f)/(-c*f*b^3+d*e*b^3+(a+b*(d*x+c)^(1/3))^3*f-3*(a+b*(d*x+c)^(1/3
))^2*a*f+3*(a+b*(d*x+c)^(1/3))*a^2*f-a^3*f)-2/9*a*b^3/f*sum(_R1/(c*f-d*e)/(_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))+1/9*b^3/f^2*sum((b^3*c*f-b^3*d*e+2*_RR1^2*a*f-3*_RR1*a^2*f+a^3*f)/(c*f-d*e)/(_RR1^2-2*_RR1*a+a^2)
*(Si(-b*(d*x+c)^(1/3)+_RR1-a)*sin(_RR1)+Ci(b*(d*x+c)^(1/3)-_RR1+a)*cos(_RR1)),_RR1=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))+sin(a+b*(d*x+c)^(1/3))*(2/3*a*b^3/(c*f-d*e)*(a+b*(d*x+c)^(1/3))^2-2/3*a^2*b^
3/(c*f-d*e)*(a+b*(d*x+c)^(1/3)))/(-c*f*b^3+d*e*b^3+(a+b*(d*x+c)^(1/3))^3*f-3*(a+b*(d*x+c)^(1/3))^2*a*f+3*(a+b*
(d*x+c)^(1/3))*a^2*f-a^3*f)+2/9*a*b^3/f*sum((_R1+a)/(c*f-d*e)/(_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))
-2/9*a*b^3/f*sum(_RR1/(_RR1-a)/(c*f-d*e)*(Si(-b*(d*x+c)^(1/3)+_RR1-a)*sin(_RR1)+Ci(b*(d*x+c)^(1/3)-_RR1+a)*cos
(_RR1)),_RR1=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))+b^6*a^2*(sin(a+b*(d*x+c)^(1/3))*(-1/
3/b^3/(c*f-d*e)*(a+b*(d*x+c)^(1/3))+1/3*a/b^3/(c*f-d*e))/(-c*f*b^3+d*e*b^3+(a+b*(d*x+c)^(1/3))^3*f-3*(a+b*(d*x
+c)^(1/3))^2*a*f+3*(a+b*(d*x+c)^(1/3))*a^2*f-a^3*f)-2/9/b^3/f*sum(1/(c*f-d*e)/(_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))+1/9/b^3/f*sum(1/(_RR1-a)/(c*f-d*e)*(Si(-b*(d*x+c)^(1/3)+_RR1-a)*sin(_RR1)+Ci(b*(d*x+c)^(1/3)-
_RR1+a)*cos(_RR1)),_RR1=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(-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(a+b*(d*x+c)^(1/3))/(f*x+e)^2,x, algorithm="maxima")

[Out]

Timed out

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Fricas [C]  time = 2.12379, size = 1789, normalized size = 3.22 \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)^2,x, algorithm="fricas")

[Out]

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

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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 )}}{\left (e + f x\right )^{2}}\, 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)**2,x)

[Out]

Integral(sin(a + b*(c + d*x)**(1/3))/(e + f*x)**2, 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 )}{{\left (f x + e\right )}^{2}}\,{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)^2,x, algorithm="giac")

[Out]

integrate(sin((d*x + c)^(1/3)*b + a)/(f*x + e)^2, x)