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

Optimal result

Integrand size = 22, antiderivative size = 396 \[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{e+f x} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right ) \sin \left (a-\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right )}{f}+\frac {\operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right ) \sin \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right )}{f}+\frac {\operatorname {CosIntegral}\left (\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right ) \sin \left (a-\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right )}{f}-\frac {\cos \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{f}+\frac {\cos \left (a-\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{f}+\frac {\cos \left (a-\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{f} \] Output:

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+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
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.

Time = 25.34 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.30 \[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{e+f x} \, dx=\frac {i \left (\text {RootSum}\left [d e-c f+f \text {$\#$1}^3\&,e^{-i a-i b \text {$\#$1}} \operatorname {ExpIntegralEi}\left (-i b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\&\right ]-\text {RootSum}\left [d e-c f+f \text {$\#$1}^3\&,e^{i a+i b \text {$\#$1}} \operatorname {ExpIntegralEi}\left (i b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )\&\right ]\right )}{2 f} \] Input:

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

Output:

((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
 

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3912, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3912

\(\displaystyle \frac {3 \int \left (-\frac {d \sin \left (a+b \sqrt [3]{c+d x}\right )}{3 f^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{d e-c f}-\sqrt [3]{f} \sqrt [3]{c+d x}\right )}+\frac {d \sin \left (a+b \sqrt [3]{c+d x}\right )}{3 f^{2/3} \left (\sqrt [3]{d e-c f}+\sqrt [3]{f} \sqrt [3]{c+d x}\right )}+\frac {d \sin \left (a+b \sqrt [3]{c+d x}\right )}{3 f^{2/3} \left ((-1)^{2/3} \sqrt [3]{d e-c f}+\sqrt [3]{f} \sqrt [3]{c+d x}\right )}\right )d\sqrt [3]{c+d x}}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \left (\frac {d \sin \left (a-\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 f}+\frac {d \sin \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f}+\frac {d \sin \left (a-\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \operatorname {CosIntegral}\left (\frac {(-1)^{2/3} \sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 f}-\frac {d \cos \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f}+\frac {d \cos \left (a-\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {\sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 f}+\frac {d \cos \left (a-\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 f}\right )}{d}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3912

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f 
_.)*(x_))^(n_)])^(p_.), x_Symbol] :> Simp[1/(n*f)   Subst[Int[ExpandIntegra 
nd[(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]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.93 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.83

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

Input:

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

Output:

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(_R 
1)),_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(_R1)+C 
i(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)))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.13 \[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{e+f x} \, dx=\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} \] Input:

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

Output:

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
 

Sympy [F]

\[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{e+f x} \, dx=\int \frac {\sin {\left (a + b \sqrt [3]{c + d x} \right )}}{e + f x}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{e+f x} \, dx=\int { \frac {\sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{f x + e} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{e+f x} \, dx=\int { \frac {\sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{f x + e} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{e+f x} \, dx=\int \frac {\sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{e+f\,x} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{e+f x} \, dx=\int \frac {\sin \left (\left (d x +c \right )^{\frac {1}{3}} b +a \right )}{f x +e}d x \] Input:

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

Output:

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