\(\int \frac {(c \sin ^3(a+b x^n))^{2/3}}{x^2} \, dx\) [356]

Optimal result

Integrand size = 20, antiderivative size = 180 \[ \int \frac {\left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{x^2} \, dx=-\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 x}+\frac {2^{-2+\frac {1}{n}} e^{2 i a} \left (-i b x^n\right )^{\frac {1}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (-\frac {1}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x}+\frac {2^{-2+\frac {1}{n}} e^{-2 i a} \left (i b x^n\right )^{\frac {1}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (-\frac {1}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x} \] Output:

-1/2*csc(a+b*x^n)^2*(c*sin(a+b*x^n)^3)^(2/3)/x+2^(-2+1/n)*exp(2*I*a)*(-I*b 
*x^n)^(1/n)*csc(a+b*x^n)^2*GAMMA(-1/n,-2*I*b*x^n)*(c*sin(a+b*x^n)^3)^(2/3) 
/n/x+2^(-2+1/n)*(I*b*x^n)^(1/n)*csc(a+b*x^n)^2*GAMMA(-1/n,2*I*b*x^n)*(c*si 
n(a+b*x^n)^3)^(2/3)/exp(2*I*a)/n/x
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.69 \[ \int \frac {\left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{x^2} \, dx=\frac {e^{-2 i a} \csc ^2\left (a+b x^n\right ) \left (-2 e^{2 i a} n+2^{\frac {1}{n}} e^{4 i a} \left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-2 i b x^n\right )+2^{\frac {1}{n}} \left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},2 i b x^n\right )\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{4 n x} \] Input:

Integrate[(c*Sin[a + b*x^n]^3)^(2/3)/x^2,x]
 

Output:

(Csc[a + b*x^n]^2*(-2*E^((2*I)*a)*n + 2^n^(-1)*E^((4*I)*a)*((-I)*b*x^n)^n^ 
(-1)*Gamma[-n^(-1), (-2*I)*b*x^n] + 2^n^(-1)*(I*b*x^n)^n^(-1)*Gamma[-n^(-1 
), (2*I)*b*x^n])*(c*Sin[a + b*x^n]^3)^(2/3))/(4*E^((2*I)*a)*n*x)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.72, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {7271, 3906, 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.

\(\displaystyle \int \frac {\left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{x^2} \, dx\)

\(\Big \downarrow \) 7271

\(\displaystyle \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \int \frac {\sin ^2\left (b x^n+a\right )}{x^2}dx\)

\(\Big \downarrow \) 3906

\(\displaystyle \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \int \left (\frac {1}{2 x^2}-\frac {\cos \left (2 b x^n+2 a\right )}{2 x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \csc ^2\left (a+b x^n\right ) \left (\frac {e^{2 i a} 2^{\frac {1}{n}-2} \left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-2 i b x^n\right )}{n x}+\frac {e^{-2 i a} 2^{\frac {1}{n}-2} \left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},2 i b x^n\right )}{n x}-\frac {1}{2 x}\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\)

Input:

Int[(c*Sin[a + b*x^n]^3)^(2/3)/x^2,x]
 

Output:

Csc[a + b*x^n]^2*(-1/2*1/x + (2^(-2 + n^(-1))*E^((2*I)*a)*((-I)*b*x^n)^n^( 
-1)*Gamma[-n^(-1), (-2*I)*b*x^n])/(n*x) + (2^(-2 + n^(-1))*(I*b*x^n)^n^(-1 
)*Gamma[-n^(-1), (2*I)*b*x^n])/(E^((2*I)*a)*n*x))*(c*Sin[a + b*x^n]^3)^(2/ 
3)
 

Defintions of rubi rules used

rule 2009

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

rule 3906

Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_), x 
_Symbol] :> Int[ExpandTrigReduce[(e*x)^m, (a + b*Sin[c + d*x^n])^p, x], x] 
/; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
 

rule 7271

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ 
FracPart[p]/v^(m*FracPart[p]))   Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, 
x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&  !(Eq 
Q[v, x] && EqQ[m, 1])
 

Maple [F]

\[\int \frac {{\left (c \sin \left (a +b \,x^{n}\right )^{3}\right )}^{\frac {2}{3}}}{x^{2}}d x\]

Input:

int((c*sin(a+b*x^n)^3)^(2/3)/x^2,x)
 

Output:

int((c*sin(a+b*x^n)^3)^(2/3)/x^2,x)
 

Fricas [F]

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

integrate((c*sin(a+b*x^n)^3)^(2/3)/x^2,x, algorithm="fricas")
 

Output:

integral((-(c*cos(b*x^n + a)^2 - c)*sin(b*x^n + a))^(2/3)/x^2, x)
 

Sympy [F]

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

integrate((c*sin(a+b*x**n)**3)**(2/3)/x**2,x)
 

Output:

Integral((c*sin(a + b*x**n)**3)**(2/3)/x**2, x)
 

Maxima [F]

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

integrate((c*sin(a+b*x^n)^3)^(2/3)/x^2,x, algorithm="maxima")
 

Output:

1/4*(x*integrate(cos(2*b*x^n + 2*a)/x^2, x) + 1)*c^(2/3)/x
 

Giac [F]

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

integrate((c*sin(a+b*x^n)^3)^(2/3)/x^2,x, algorithm="giac")
 

Output:

integrate((c*sin(b*x^n + a)^3)^(2/3)/x^2, x)
 

Mupad [F(-1)]

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

int((c*sin(a + b*x^n)^3)^(2/3)/x^2,x)
 

Output:

int((c*sin(a + b*x^n)^3)^(2/3)/x^2, x)
 

Reduce [F]

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

int((c*sin(a+b*x^n)^3)^(2/3)/x^2,x)
 

Output:

c**(2/3)*int(sin(x**n*b + a)**2/x**2,x)