3.354 \(\int (c \sin ^3(a+b x^n))^{2/3} \, dx\)

Optimal. Leaf size=178 \[ \frac{e^{2 i a} 2^{-\frac{1}{n}-2} x \left (-i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{e^{-2 i a} 2^{-\frac{1}{n}-2} x \left (i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{1}{2} x \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0888038, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6720, 3367, 3366, 2208} \[ \frac{e^{2 i a} 2^{-\frac{1}{n}-2} x \left (-i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{e^{-2 i a} 2^{-\frac{1}{n}-2} x \left (i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{1}{2} x \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x^n]^3)^(2/3),x]

[Out]

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

Rule 6720

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(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]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rule 3367

Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(a +
b*Sin[c + d*(e + f*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[p, 1]

Rule 3366

Int[Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)], x_Symbol] :> Dist[1/2, Int[E^(-(c*I) - d*I*(e + f*x)^n), x],
x] + Dist[1/2, Int[E^(c*I + d*I*(e + f*x)^n), x], x] /; FreeQ[{c, d, e, f, n}, x]

Rule 2208

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)
^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rubi steps

\begin{align*} \int \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \, dx &=\left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int \sin ^2\left (a+b x^n\right ) \, dx\\ &=\left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int \left (\frac{1}{2}-\frac{1}{2} \cos \left (2 a+2 b x^n\right )\right ) \, dx\\ &=\frac{1}{2} x \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}-\frac{1}{2} \left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int \cos \left (2 a+2 b x^n\right ) \, dx\\ &=\frac{1}{2} x \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}-\frac{1}{4} \left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int e^{-2 i a-2 i b x^n} \, dx-\frac{1}{4} \left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int e^{2 i a+2 i b x^n} \, dx\\ &=\frac{1}{2} x \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}+\frac{2^{-2-\frac{1}{n}} e^{2 i a} x \left (-i b x^n\right )^{-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}+\frac{2^{-2-\frac{1}{n}} e^{-2 i a} x \left (i b x^n\right )^{-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}\\ \end{align*}

Mathematica [A]  time = 0.266812, size = 149, normalized size = 0.84 \[ \frac{e^{-2 i a} 2^{-\frac{1}{n}-2} x \left (b^2 x^{2 n}\right )^{-1/n} \csc ^2\left (a+b x^n\right ) \left (e^{4 i a} \left (i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},-2 i b x^n\right )+\left (-i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},2 i b x^n\right )+e^{2 i a} 2^{\frac{1}{n}+1} n \left (b^2 x^{2 n}\right )^{\frac{1}{n}}\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.173, size = 0, normalized size = 0. \begin{align*} \int \left ( c \left ( \sin \left ( a+b{x}^{n} \right ) \right ) ^{3} \right ) ^{{\frac{2}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{4} \, c^{\frac{2}{3}}{\left (x - \int \cos \left (2 \, b x^{n} + 2 \, a\right )\,{d x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (-{\left (c \cos \left (b x^{n} + a\right )^{2} - c\right )} \sin \left (b x^{n} + a\right )\right )^{\frac{2}{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac{2}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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