3.3.10 \(\int \frac {\sin (a+b \sqrt [3]{c+d x})}{e+f x} \, dx\) [210]

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

[Out]

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

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Rubi [A]
time = 0.96, antiderivative size = 396, normalized size of antiderivative = 1.00, 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 = {3512, 3384, 3380, 3383} \begin {gather*} \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} \end {gather*}

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 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 3512

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]

Rubi steps

\begin {align*} \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{e+f x} \, dx &=\frac {3 \text {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 {\text {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 {\text {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 {\text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
time = 0.36, size = 118, normalized size = 0.30 \begin {gather*} \frac {i \left (\text {RootSum}\left [d e-c f+f \text {$\#$1}^3\&,e^{-i a-i b \text {$\#$1}} \text {Ei}\left (-i b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\&\right ]-\text {RootSum}\left [d e-c f+f \text {$\#$1}^3\&,e^{i a+i b \text {$\#$1}} \text {Ei}\left (i b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\&\right ]\right )}{2 f} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.06, size = 327, normalized size = 0.83

method	result	size
derivativedivides	\(\frac {\frac {b^{3} a^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (-b^{3} c f +b^{3} d e +f \,\textit {\_Z}^{3}-3 a f \,\textit {\_Z}^{2}+3 a^{2} f \textit {\_Z} -a^{3} f \right )}{\sum }\frac {-\sinIntegral \left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )}{f}-\frac {2 b^{3} a \left (\munderset {\textit {\_R1} =\RootOf \left (-b^{3} c f +b^{3} d e +f \,\textit {\_Z}^{3}-3 a f \,\textit {\_Z}^{2}+3 a^{2} f \textit {\_Z} -a^{3} f \right )}{\sum }\frac {\textit {\_R1} \left (-\sinIntegral \left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \sin \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )}{f}+\frac {b^{3} \left (\munderset {\textit {\_R1} =\RootOf \left (-b^{3} c f +b^{3} d e +f \,\textit {\_Z}^{3}-3 a f \,\textit {\_Z}^{2}+3 a^{2} f \textit {\_Z} -a^{3} f \right )}{\sum }\frac {\textit {\_R1}^{2} \left (-\sinIntegral \left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \sin \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )}{f}}{b^{3}}\)	\(327\)
default	\(\frac {\frac {b^{3} a^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (-b^{3} c f +b^{3} d e +f \,\textit {\_Z}^{3}-3 a f \,\textit {\_Z}^{2}+3 a^{2} f \textit {\_Z} -a^{3} f \right )}{\sum }\frac {-\sinIntegral \left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )}{f}-\frac {2 b^{3} a \left (\munderset {\textit {\_R1} =\RootOf \left (-b^{3} c f +b^{3} d e +f \,\textit {\_Z}^{3}-3 a f \,\textit {\_Z}^{2}+3 a^{2} f \textit {\_Z} -a^{3} f \right )}{\sum }\frac {\textit {\_R1} \left (-\sinIntegral \left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \sin \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )}{f}+\frac {b^{3} \left (\munderset {\textit {\_R1} =\RootOf \left (-b^{3} c f +b^{3} d e +f \,\textit {\_Z}^{3}-3 a f \,\textit {\_Z}^{2}+3 a^{2} f \textit {\_Z} -a^{3} f \right )}{\sum }\frac {\textit {\_R1}^{2} \left (-\sinIntegral \left (-b \left (d x +c \right )^{\frac {1}{3}}+\textit {\_R1} -a \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (b \left (d x +c \right )^{\frac {1}{3}}-\textit {\_R1} +a \right ) \sin \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}}\right )}{f}}{b^{3}}\)	\(327\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/b^3*(1/3*b^3*a^2/f*sum(1/(_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*b^3/f*sum(_R1^2/(_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 = 0.00, 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(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] Result contains complex when optimal does not.
time = 0.42, size = 460, normalized size = 1.16 \begin {gather*} \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} c f - i \, b^{3} d e}{f}\right )^{\frac {1}{3}}\right ) e^{\left (\frac {1}{2} \, {\left (i \, \sqrt {3} + 1\right )} \left (\frac {i \, b^{3} c f - i \, b^{3} d e}{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} c f - i \, b^{3} d e}{f}\right )^{\frac {1}{3}}\right ) e^{\left (\frac {1}{2} \, {\left (-i \, \sqrt {3} + 1\right )} \left (\frac {i \, b^{3} c f - i \, b^{3} d e}{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} c f + i \, b^{3} d e}{f}\right )^{\frac {1}{3}}\right ) e^{\left (\frac {1}{2} \, {\left (i \, \sqrt {3} + 1\right )} \left (\frac {-i \, b^{3} c f + i \, b^{3} d e}{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} c f + i \, b^{3} d e}{f}\right )^{\frac {1}{3}}\right ) e^{\left (\frac {1}{2} \, {\left (-i \, \sqrt {3} + 1\right )} \left (\frac {-i \, b^{3} c f + i \, b^{3} d e}{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} c f - i \, b^{3} d e}{f}\right )^{\frac {1}{3}}\right ) e^{\left (i \, a - \left (\frac {i \, b^{3} c f - i \, b^{3} d e}{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} c f + i \, b^{3} d e}{f}\right )^{\frac {1}{3}}\right ) e^{\left (-i \, a - \left (\frac {-i \, b^{3} c f + i \, b^{3} d e}{f}\right )^{\frac {1}{3}}\right )}}{2 \, f} \end {gather*}

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

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

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 = 0.00, 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(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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