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

Optimal. Leaf size=217 \[ \frac{e^{2 i a} 2^{-\frac{m+2 n+1}{n}} x^{m+1} \left (-i b x^n\right )^{-\frac{m+1}{n}} \csc ^2\left (a+b x^n\right ) \text{Gamma}\left (\frac{m+1}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{e^{-2 i a} 2^{-\frac{m+2 n+1}{n}} x^{m+1} \left (i b x^n\right )^{-\frac{m+1}{n}} \csc ^2\left (a+b x^n\right ) \text{Gamma}\left (\frac{m+1}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{x^{m+1} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.364997, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6720, 3425, 3424, 2218} \[ \frac{e^{2 i a} 2^{-\frac{m+2 n+1}{n}} x^{m+1} \left (-i b x^n\right )^{-\frac{m+1}{n}} \csc ^2\left (a+b x^n\right ) \text{Gamma}\left (\frac{m+1}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{e^{-2 i a} 2^{-\frac{m+2 n+1}{n}} x^{m+1} \left (i b x^n\right )^{-\frac{m+1}{n}} \csc ^2\left (a+b x^n\right ) \text{Gamma}\left (\frac{m+1}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{x^{m+1} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(c*Sin[a + b*x^n]^3)^(2/3),x]

[Out]

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

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 3425

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 3424

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

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int x^m \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 x^m \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{x^m}{2}-\frac{1}{2} x^m \cos \left (2 a+2 b x^n\right )\right ) \, dx\\ &=\frac{x^{1+m} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 (1+m)}-\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 x^m \cos \left (2 a+2 b x^n\right ) \, dx\\ &=\frac{x^{1+m} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 (1+m)}-\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} x^m \, 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} x^m \, dx\\ &=\frac{x^{1+m} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 (1+m)}+\frac{2^{-\frac{1+m+2 n}{n}} e^{2 i a} x^{1+m} \left (-i b x^n\right )^{-\frac{1+m}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac{1+m}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac{2^{-\frac{1+m+2 n}{n}} e^{-2 i a} x^{1+m} \left (i b x^n\right )^{-\frac{1+m}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac{1+m}{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 = 0.831941, size = 194, normalized size = 0.89 \[ \frac{e^{-2 i a} 2^{-\frac{m+2 n+1}{n}} x^{m+1} \left (b^2 x^{2 n}\right )^{-\frac{m+1}{n}} \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \left (e^{4 i a} (m+1) \left (i b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-2 i b x^n\right )+(m+1) \left (-i b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},2 i b x^n\right )+e^{2 i a} n 2^{\frac{m+n+1}{n}} \left (b^2 x^{2 n}\right )^{\frac{m+1}{n}}\right )}{(m+1) n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.211, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \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(x^m*(c*sin(a+b*x^n)^3)^(2/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(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^m, x)

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

[Out]

Timed out

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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}} x^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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