\(\int \sin (a+\frac {b}{\sqrt [3]{c+d x}}) \, dx\) [219]

Optimal result

Integrand size = 14, antiderivative size = 136 \[ \int \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=\frac {b (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d}+\frac {b^3 \cos (a) \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d}-\frac {b^2 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d}+\frac {(c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{d}-\frac {b^3 \sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.98 \[ \int \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=\frac {b (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+b^3 \cos (a) \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right )+2 c \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+2 d x \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )-b^2 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )-b^3 \sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.93, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {3842, 3042, 3778, 3042, 3778, 25, 3042, 3778, 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.

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

\(\Big \downarrow \) 3842

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 3783

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

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 3842

Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_S 
ymbol] :> Simp[1/(n*f)   Subst[Int[x^(1/n - 1)*(a + b*Sin[c + d*x])^p, x], 
x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && Intege 
rQ[1/n]
 

Maple [A] (verified)

Time = 1.74 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(-\frac {3 b^{3} \left (-\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )}{3 b^{3}}-\frac {\cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {2}{3}}}{6 b^{2}}+\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {1}{3}}}{6 b}+\frac {\operatorname {Si}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{6}-\frac {\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )}{6}\right )}{d}\)	\(108\)
default	\(-\frac {3 b^{3} \left (-\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )}{3 b^{3}}-\frac {\cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {2}{3}}}{6 b^{2}}+\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {1}{3}}}{6 b}+\frac {\operatorname {Si}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{6}-\frac {\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )}{6}\right )}{d}\)	\(108\)

Input:

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

Output:

-3/d*b^3*(-1/3*sin(a+b/(d*x+c)^(1/3))/b^3*(d*x+c)-1/6*cos(a+b/(d*x+c)^(1/3 
))/b^2*(d*x+c)^(2/3)+1/6*sin(a+b/(d*x+c)^(1/3))/b*(d*x+c)^(1/3)+1/6*Si(b/( 
d*x+c)^(1/3))*sin(a)-1/6*Ci(b/(d*x+c)^(1/3))*cos(a))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.89 \[ \int \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=\frac {b^{3} \cos \left (a\right ) \operatorname {Ci}\left (\frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - b^{3} \sin \left (a\right ) \operatorname {Si}\left (\frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + {\left (d x + c\right )}^{\frac {2}{3}} b \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right ) - {\left ({\left (d x + c\right )}^{\frac {1}{3}} b^{2} - 2 \, d x - 2 \, c\right )} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right )}{2 \, d} \] Input:

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

Output:

1/2*(b^3*cos(a)*cos_integral(b/(d*x + c)^(1/3)) - b^3*sin(a)*sin_integral( 
b/(d*x + c)^(1/3)) + (d*x + c)^(2/3)*b*cos((a*d*x + a*c + (d*x + c)^(2/3)* 
b)/(d*x + c)) - ((d*x + c)^(1/3)*b^2 - 2*d*x - 2*c)*sin((a*d*x + a*c + (d* 
x + c)^(2/3)*b)/(d*x + c)))/d
 

Sympy [F]

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

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

Output:

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.01 \[ \int \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=\frac {{\left ({\left ({\rm Ei}\left (\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + {\rm Ei}\left (-\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \cos \left (a\right ) + {\left (i \, {\rm Ei}\left (\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - i \, {\rm Ei}\left (-\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \sin \left (a\right )\right )} b^{3} + 2 \, {\left (d x + c\right )}^{\frac {2}{3}} b \cos \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b^{2} - 2 \, d x - 2 \, c\right )} \sin \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )}{4 \, d} \] Input:

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

Output:

1/4*(((Ei(I*b/(d*x + c)^(1/3)) + Ei(-I*b/(d*x + c)^(1/3)))*cos(a) + (I*Ei( 
I*b/(d*x + c)^(1/3)) - I*Ei(-I*b/(d*x + c)^(1/3)))*sin(a))*b^3 + 2*(d*x + 
c)^(2/3)*b*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - 2*((d*x + c)^(1/ 
3)*b^2 - 2*d*x - 2*c)*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 663 vs. \(2 (114) = 228\).

Time = 0.20 (sec) , antiderivative size = 663, normalized size of antiderivative = 4.88 \[ \int \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(a^3*b^4*cos(a)*cos_integral(-a + ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1 
/3)) + a^3*b^4*sin(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b)/(d*x + c)^( 
1/3)) - 3*((d*x + c)^(1/3)*a + b)*a^2*b^4*cos(a)*cos_integral(-a + ((d*x + 
 c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(1/3) - 3*((d*x + c)^(1/3)*a + 
 b)*a^2*b^4*sin(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3 
))/(d*x + c)^(1/3) + 3*((d*x + c)^(1/3)*a + b)^2*a*b^4*cos(a)*cos_integral 
(-a + ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(2/3) + 3*((d*x + 
 c)^(1/3)*a + b)^2*a*b^4*sin(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b)/( 
d*x + c)^(1/3))/(d*x + c)^(2/3) - ((d*x + c)^(1/3)*a + b)^3*b^4*cos(a)*cos 
_integral(-a + ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c) - ((d*x 
+ c)^(1/3)*a + b)^3*b^4*sin(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b)/(d 
*x + c)^(1/3))/(d*x + c) + a^2*b^4*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^( 
1/3)) - 2*((d*x + c)^(1/3)*a + b)*a*b^4*sin(((d*x + c)^(1/3)*a + b)/(d*x + 
 c)^(1/3))/(d*x + c)^(1/3) + ((d*x + c)^(1/3)*a + b)^2*b^4*sin(((d*x + c)^ 
(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(2/3) + a*b^4*cos(((d*x + c)^(1/3) 
*a + b)/(d*x + c)^(1/3)) - ((d*x + c)^(1/3)*a + b)*b^4*cos(((d*x + c)^(1/3 
)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(1/3) - 2*b^4*sin(((d*x + c)^(1/3)*a + 
 b)/(d*x + c)^(1/3)))/((a^3 - 3*((d*x + c)^(1/3)*a + b)*a^2/(d*x + c)^(1/3 
) + 3*((d*x + c)^(1/3)*a + b)^2*a/(d*x + c)^(2/3) - ((d*x + c)^(1/3)*a + b 
)^3/(d*x + c))*b*d)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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