\(\int \sqrt [3]{c e+d e x} \sin (a+\frac {b}{\sqrt [3]{c+d x}}) \, dx\) [242]

Optimal result

Integrand size = 27, antiderivative size = 247 \[ \int \sqrt [3]{c e+d e x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=-\frac {b^3 \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{8 d}+\frac {b (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{4 d}-\frac {b^4 \sqrt [3]{e (c+d x)} \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{8 d \sqrt [3]{c+d x}}-\frac {b^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{8 d}+\frac {3 (c+d x) \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{4 d}-\frac {b^4 \sqrt [3]{e (c+d x)} \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{8 d \sqrt [3]{c+d x}} \] Output:

-1/8*b^3*(e*(d*x+c))^(1/3)*cos(a+b/(d*x+c)^(1/3))/d+1/4*b*(d*x+c)^(2/3)*(e 
*(d*x+c))^(1/3)*cos(a+b/(d*x+c)^(1/3))/d-1/8*b^4*(e*(d*x+c))^(1/3)*Ci(b/(d 
*x+c)^(1/3))*sin(a)/d/(d*x+c)^(1/3)-1/8*b^2*(d*x+c)^(1/3)*(e*(d*x+c))^(1/3 
)*sin(a+b/(d*x+c)^(1/3))/d+3/4*(d*x+c)*(e*(d*x+c))^(1/3)*sin(a+b/(d*x+c)^( 
1/3))/d-1/8*b^4*(e*(d*x+c))^(1/3)*cos(a)*Si(b/(d*x+c)^(1/3))/d/(d*x+c)^(1/ 
3)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.84 \[ \int \sqrt [3]{c e+d e x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=-\frac {\sqrt [3]{e (c+d x)} \left (-2 b c \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )-2 b d x \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+b^3 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+b^4 \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a)-6 c \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )-6 d x \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+b^2 (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+b^4 \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{8 d \sqrt [3]{c+d x}} \] Input:

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

Output:

-1/8*((e*(c + d*x))^(1/3)*(-2*b*c*Cos[a + b/(c + d*x)^(1/3)] - 2*b*d*x*Cos 
[a + b/(c + d*x)^(1/3)] + b^3*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(1/3)] + 
 b^4*CosIntegral[b/(c + d*x)^(1/3)]*Sin[a] - 6*c*(c + d*x)^(1/3)*Sin[a + b 
/(c + d*x)^(1/3)] - 6*d*x*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(1/3)] + b^2 
*(c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(1/3)] + b^4*Cos[a]*SinIntegral[b/(c 
+ d*x)^(1/3)]))/(d*(c + d*x)^(1/3))
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.73, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {3912, 30, 3042, 3778, 3042, 3778, 25, 3042, 3778, 3042, 3778, 25, 3042, 3784, 3042, 3780, 3783}

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.

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

\(\Big \downarrow \) 3912

\(\displaystyle -\frac {3 \int (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}}{d}\)

\(\Big \downarrow \) 30

\(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \int (c+d x)^{5/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \int (c+d x)^{5/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{4} b \int (c+d x)^{4/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}-\frac {1}{4} (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{4} b \int (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}+\frac {\pi }{2}\right )d\frac {1}{\sqrt [3]{c+d x}}-\frac {1}{4} (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{4} b \left (\frac {1}{3} b \int -\left ((c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )d\frac {1}{\sqrt [3]{c+d x}}-\frac {1}{3} (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{4} (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{4} b \left (-\frac {1}{3} b \int (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}-\frac {1}{3} (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{4} (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{4} b \left (-\frac {1}{3} b \int (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}-\frac {1}{3} (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{4} (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \int (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{3} (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{4} (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \int (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}+\frac {\pi }{2}\right )d\frac {1}{\sqrt [3]{c+d x}}-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{3} (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{4} (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \left (b \int -\sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}-\sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{3} (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{4} (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \left (-b \int \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}-\sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{3} (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{4} (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \left (-b \int \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}-\sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{3} (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{4} (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 3784

\(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \left (-b \left (\sin (a) \int \sqrt [3]{c+d x} \cos \left (\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}+\cos (a) \int \sqrt [3]{c+d x} \sin \left (\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}\right )-\sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{3} (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{4} (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \left (-b \left (\sin (a) \int \sqrt [3]{c+d x} \sin \left (\frac {b}{\sqrt [3]{c+d x}}+\frac {\pi }{2}\right )d\frac {1}{\sqrt [3]{c+d x}}+\cos (a) \int \sqrt [3]{c+d x} \sin \left (\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}\right )-\sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{3} (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{4} (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 3780

\(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \left (-b \left (\sin (a) \int \sqrt [3]{c+d x} \sin \left (\frac {b}{\sqrt [3]{c+d x}}+\frac {\pi }{2}\right )d\frac {1}{\sqrt [3]{c+d x}}+\cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{3} (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{4} (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{c+d x}}\)

\(\Big \downarrow \) 3783

\(\displaystyle -\frac {3 \sqrt [3]{e (c+d x)} \left (\frac {1}{4} b \left (-\frac {1}{3} b \left (\frac {1}{2} b \left (-b \left (\sin (a) \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right )+\cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{3} (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{4} (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{c+d x}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 30

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I 
ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) 
Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & 
&  !IntegerQ[p]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3778

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c 
 + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1))   Int[( 
c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 
1]
 

rule 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

rule 3783

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte 
gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - 
c*f, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[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 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 [F]

\[\int \left (d e x +c e \right )^{\frac {1}{3}} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right )d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [C] (verification not implemented)

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

Time = 0.32 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.52 \[ \int \sqrt [3]{c e+d e x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=-\frac {3 \, {\left ({\left (-i \, \Gamma \left (-4, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) + i \, \Gamma \left (-4, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) - i \, \Gamma \left (-4, \frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + i \, \Gamma \left (-4, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \cos \left (a\right ) - {\left (\Gamma \left (-4, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) + \Gamma \left (-4, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) + \Gamma \left (-4, \frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (-4, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \sin \left (a\right )\right )} b^{4} e^{\frac {1}{3}}}{4 \, d} \] Input:

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

Output:

-3/4*((-I*gamma(-4, I*b*conjugate((d*x + c)^(-1/3))) + I*gamma(-4, -I*b*co 
njugate((d*x + c)^(-1/3))) - I*gamma(-4, I*b/(d*x + c)^(1/3)) + I*gamma(-4 
, -I*b/(d*x + c)^(1/3)))*cos(a) - (gamma(-4, I*b*conjugate((d*x + c)^(-1/3 
))) + gamma(-4, -I*b*conjugate((d*x + c)^(-1/3))) + gamma(-4, I*b/(d*x + c 
)^(1/3)) + gamma(-4, -I*b/(d*x + c)^(1/3)))*sin(a))*b^4*e^(1/3)/d
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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