\(\int \frac {\sin (a+\frac {b}{\sqrt [3]{c+d x}})}{(e+f x)^2} \, dx\) [221]

Optimal result

Integrand size = 22, antiderivative size = 566 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(e+f x)^2} \, dx=-\frac {b d \cos \left (a+\frac {b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \operatorname {CosIntegral}\left (\frac {b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}}-\frac {(-1)^{2/3} b d \cos \left (a+\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \operatorname {CosIntegral}\left (\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}}+\frac {\sqrt [3]{-1} b d \cos \left (a-\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}}+\frac {(c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(d e-c f) (e+f x)}-\frac {b d \sin \left (a+\frac {b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \text {Si}\left (\frac {b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}}-\frac {(-1)^{2/3} b d \sin \left (a+\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \text {Si}\left (\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}}-\frac {\sqrt [3]{-1} b d \sin \left (a-\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{-d e+c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f^{2/3} (-d e+c f)^{4/3}} \] Output:

-1/3*b*d*cos(a+b*f^(1/3)/(c*f-d*e)^(1/3))*Ci(b*f^(1/3)/(c*f-d*e)^(1/3)-b/( 
d*x+c)^(1/3))/f^(2/3)/(c*f-d*e)^(4/3)-1/3*(-1)^(2/3)*b*d*cos(a+(-1)^(2/3)* 
b*f^(1/3)/(c*f-d*e)^(1/3))*Ci((-1)^(2/3)*b*f^(1/3)/(c*f-d*e)^(1/3)-b/(d*x+ 
c)^(1/3))/f^(2/3)/(c*f-d*e)^(4/3)+1/3*(-1)^(1/3)*b*d*cos(a-(-1)^(1/3)*b*f^ 
(1/3)/(c*f-d*e)^(1/3))*Ci((-1)^(1/3)*b*f^(1/3)/(c*f-d*e)^(1/3)+b/(d*x+c)^( 
1/3))/f^(2/3)/(c*f-d*e)^(4/3)+(d*x+c)*sin(a+b/(d*x+c)^(1/3))/(-c*f+d*e)/(f 
*x+e)-1/3*b*d*sin(a+b*f^(1/3)/(c*f-d*e)^(1/3))*Si(b*f^(1/3)/(c*f-d*e)^(1/3 
)-b/(d*x+c)^(1/3))/f^(2/3)/(c*f-d*e)^(4/3)-1/3*(-1)^(2/3)*b*d*sin(a+(-1)^( 
2/3)*b*f^(1/3)/(c*f-d*e)^(1/3))*Si((-1)^(2/3)*b*f^(1/3)/(c*f-d*e)^(1/3)-b/ 
(d*x+c)^(1/3))/f^(2/3)/(c*f-d*e)^(4/3)-1/3*(-1)^(1/3)*b*d*sin(a-(-1)^(1/3) 
*b*f^(1/3)/(c*f-d*e)^(1/3))*Si((-1)^(1/3)*b*f^(1/3)/(c*f-d*e)^(1/3)+b/(d*x 
+c)^(1/3))/f^(2/3)/(c*f-d*e)^(4/3)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

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

Time = 1.61 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.55 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(e+f x)^2} \, dx=\frac {(\cos (a)+i \sin (a)) \left (b d (e+f x) \text {RootSum}\left [d e-c f+f \text {$\#$1}^3\&,\frac {\operatorname {ExpIntegralEi}\left (\frac {i b}{\sqrt [3]{c+d x}}\right )-e^{\frac {i b}{\text {$\#$1}}} \operatorname {ExpIntegralEi}\left (i b \left (\frac {1}{\sqrt [3]{c+d x}}-\frac {1}{\text {$\#$1}}\right )\right )}{\text {$\#$1}}\&\right ]+(c+d x) \left (3 i f \cos \left (\frac {b}{\sqrt [3]{c+d x}}\right )-3 f \sin \left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )\right )+i \left (-3 c f-3 d f x+b d (e+f x) \text {RootSum}\left [d e-c f+f \text {$\#$1}^3\&,\frac {\operatorname {ExpIntegralEi}\left (-\frac {i b}{\sqrt [3]{c+d x}}\right )-e^{-\frac {i b}{\text {$\#$1}}} \operatorname {ExpIntegralEi}\left (-i b \left (\frac {1}{\sqrt [3]{c+d x}}-\frac {1}{\text {$\#$1}}\right )\right )}{\text {$\#$1}}\&\right ] \left (-i \cos \left (\frac {b}{\sqrt [3]{c+d x}}\right )+\sin \left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )\right ) \left (\cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )-i \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{6 f (-d e+c f) (e+f x)} \] Input:

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

Output:

((Cos[a] + I*Sin[a])*(b*d*(e + f*x)*RootSum[d*e - c*f + f*#1^3 & , (ExpInt 
egralEi[(I*b)/(c + d*x)^(1/3)] - E^((I*b)/#1)*ExpIntegralEi[I*b*((c + d*x) 
^(-1/3) - #1^(-1))])/#1 & ] + (c + d*x)*((3*I)*f*Cos[b/(c + d*x)^(1/3)] - 
3*f*Sin[b/(c + d*x)^(1/3)])) + I*(-3*c*f - 3*d*f*x + b*d*(e + f*x)*RootSum 
[d*e - c*f + f*#1^3 & , (ExpIntegralEi[((-I)*b)/(c + d*x)^(1/3)] - ExpInte 
gralEi[(-I)*b*((c + d*x)^(-1/3) - #1^(-1))]/E^((I*b)/#1))/#1 & ]*((-I)*Cos 
[b/(c + d*x)^(1/3)] + Sin[b/(c + d*x)^(1/3)]))*(Cos[a + b/(c + d*x)^(1/3)] 
 - I*Sin[a + b/(c + d*x)^(1/3)]))/(6*f*(-(d*e) + c*f)*(e + f*x))
 

Rubi [A] (verified)

Time = 1.84 (sec) , antiderivative size = 584, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {3912, 27, 3822, 3815, 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.

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

\(\Big \downarrow \) 3912

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3822

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

\(\Big \downarrow \) 3815

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 3815

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 3822

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] - Simp[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 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] (warning: unable to verify)

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

Time = 2.48 (sec) , antiderivative size = 1554, normalized size of antiderivative = 2.75

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1554\)
default	\(\text {Expression too large to display}\)	\(1554\)

Input:

int(sin(a+b/(d*x+c)^(1/3))/(f*x+e)^2,x,method=_RETURNVERBOSE)
 

Output:

-3*d*b^3*(a^2*(sin(a+b/(d*x+c)^(1/3))*(1/3/f/b^3*(a+b/(d*x+c)^(1/3))-1/3*a 
/f/b^3)/(a^3*c*f-a^3*d*e-3*a^2*c*f*(a+b/(d*x+c)^(1/3))+3*a^2*d*e*(a+b/(d*x 
+c)^(1/3))+3*a*c*f*(a+b/(d*x+c)^(1/3))^2-3*a*d*e*(a+b/(d*x+c)^(1/3))^2-c*f 
*(a+b/(d*x+c)^(1/3))^3+d*e*(a+b/(d*x+c)^(1/3))^3+f*b^3)-2/9/f/b^3*sum(1/(_ 
R1^2*c*f-_R1^2*d*e-2*_R1*a*c*f+2*_R1*a*d*e+a^2*c*f-a^2*d*e)*(-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-f*b^3))-1/9/f/b^3*sum(1/(-_RR1*c*f+_RR1*d*e+a*c*f-a*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 
((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-f*b^3)))+sin(a+b/(d*x+c)^(1/3))*(-2/3*a/f/b^3*(a+b/(d*x+c)^(1/3))^2 
+2/3*a^2/f/b^3*(a+b/(d*x+c)^(1/3)))/(a^3*c*f-a^3*d*e-3*a^2*c*f*(a+b/(d*x+c 
)^(1/3))+3*a^2*d*e*(a+b/(d*x+c)^(1/3))+3*a*c*f*(a+b/(d*x+c)^(1/3))^2-3*a*d 
*e*(a+b/(d*x+c)^(1/3))^2-c*f*(a+b/(d*x+c)^(1/3))^3+d*e*(a+b/(d*x+c)^(1/3)) 
^3+f*b^3)+2/9*a/f/b^3*sum((_R1+a)/(_R1^2*c*f-_R1^2*d*e-2*_R1*a*c*f+2*_R1*a 
*d*e+a^2*c*f-a^2*d*e)*(-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-f*b^3))+2/9*a/f/b^3*sum(_RR1/(-_RR1 
*c*f+_RR1*d*e+a*c*f-a*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((c*f-d*e)*_Z^3+(-3*a*c*f+3*a*...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 798, normalized size of antiderivative = 1.41 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(e+f x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

\[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(e+f x)^2} \, dx=\int { \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )}{{\left (f x + e\right )}^{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(e+f x)^2} \, dx=\int { \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )}{{\left (f x + e\right )}^{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(e+f x)^2} \, dx=\int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{1/3}}\right )}{{\left (e+f\,x\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(e+f x)^2} \, dx=\int \frac {\sin \left (\frac {\left (d x +c \right )^{\frac {1}{3}} a +b}{\left (d x +c \right )^{\frac {1}{3}}}\right )}{f^{2} x^{2}+2 e f x +e^{2}}d x \] Input:

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

Output:

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