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

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

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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]

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 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 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])

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*}

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Mathematica [A]  time = 0.27, size = 149, normalized size = 0.84 \[ \frac {e^{-2 i a} 2^{-\frac {1}{n}-2} x \left (b^2 x^{2 n}\right )^{-1/n} \csc ^2\left (a+b x^n\right ) \left (e^{2 i a} 2^{\frac {1}{n}+1} n \left (b^2 x^{2 n}\right )^{\frac {1}{n}}+e^{4 i a} \left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-2 i b x^n\right )+\left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},2 i b x^n\right )\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n} \]

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))

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fricas [F]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (-{\left (c \cos \left (b x^{n} + a\right )^{2} - c\right )} \sin \left (b x^{n} + a\right )\right )^{\frac {2}{3}}, x\right ) \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {2}{3}}\,{d x} \]

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)

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maple [F]  time = 0.35, size = 0, normalized size = 0.00 \[ \int \left (c \left (\sin ^{3}\left (a +b \,x^{n}\right )\right )\right )^{\frac {2}{3}}\, dx \]

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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{4} \, c^{\frac {2}{3}} {\left (x - \int \cos \left (2 \, b x^{n} + 2 \, a\right )\,{d x}\right )} \]

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))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{2/3} \,d x \]

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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \sin ^{3}{\left (a + b x^{n} \right )}\right )^{\frac {2}{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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