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

Optimal. Leaf size=209 \[ e^{2 i a} 2^{-\frac{m}{2}-\frac{7}{2}} x^{m+1} \left (-i b x^2\right )^{\frac{1}{2} (-m-1)} \csc ^2\left (a+b x^2\right ) \text{Gamma}\left (\frac{m+1}{2},-2 i b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+e^{-2 i a} 2^{-\frac{m}{2}-\frac{7}{2}} x^{m+1} \left (i b x^2\right )^{\frac{1}{2} (-m-1)} \csc ^2\left (a+b x^2\right ) \text{Gamma}\left (\frac{m+1}{2},2 i b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac{x^{m+1} \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{2 (m+1)} \]

[Out]

(x^(1 + m)*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/(2*(1 + m)) + 2^(-7/2 - m/2)*E^((2*I)*a)*x^(1 + m)*((-
I)*b*x^2)^((-1 - m)/2)*Csc[a + b*x^2]^2*Gamma[(1 + m)/2, (-2*I)*b*x^2]*(c*Sin[a + b*x^2]^3)^(2/3) + (2^(-7/2 -
m/2)*x^(1 + m)*(I*b*x^2)^((-1 - m)/2)*Csc[a + b*x^2]^2*Gamma[(1 + m)/2, (2*I)*b*x^2]*(c*Sin[a + b*x^2]^3)^(2/
3))/E^((2*I)*a)

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Rubi [A]  time = 0.277051, antiderivative size = 209, 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, 3403, 3390, 2218} \[ e^{2 i a} 2^{-\frac{m}{2}-\frac{7}{2}} x^{m+1} \left (-i b x^2\right )^{\frac{1}{2} (-m-1)} \csc ^2\left (a+b x^2\right ) \text{Gamma}\left (\frac{m+1}{2},-2 i b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+e^{-2 i a} 2^{-\frac{m}{2}-\frac{7}{2}} x^{m+1} \left (i b x^2\right )^{\frac{1}{2} (-m-1)} \csc ^2\left (a+b x^2\right ) \text{Gamma}\left (\frac{m+1}{2},2 i b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac{x^{m+1} \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{2 (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(c*Sin[a + b*x^2]^3)^(2/3),x]

[Out]

(x^(1 + m)*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/(2*(1 + m)) + 2^(-7/2 - m/2)*E^((2*I)*a)*x^(1 + m)*((-
I)*b*x^2)^((-1 - m)/2)*Csc[a + b*x^2]^2*Gamma[(1 + m)/2, (-2*I)*b*x^2]*(c*Sin[a + b*x^2]^3)^(2/3) + (2^(-7/2 -
m/2)*x^(1 + m)*(I*b*x^2)^((-1 - m)/2)*Csc[a + b*x^2]^2*Gamma[(1 + m)/2, (2*I)*b*x^2]*(c*Sin[a + b*x^2]^3)^(2/
3))/E^((2*I)*a)

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 3403

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}, x] && IGtQ[p, 1] && IGtQ[n, 0]

Rule 3390

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}, x] && IGtQ[n, 0]

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^2\right )\right )^{2/3} \, dx &=\left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int x^m \sin ^2\left (a+b x^2\right ) \, dx\\ &=\left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \left (\frac{x^m}{2}-\frac{1}{2} x^m \cos \left (2 a+2 b x^2\right )\right ) \, dx\\ &=\frac{x^{1+m} \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{2 (1+m)}-\frac{1}{2} \left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int x^m \cos \left (2 a+2 b x^2\right ) \, dx\\ &=\frac{x^{1+m} \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{2 (1+m)}-\frac{1}{4} \left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int e^{-2 i a-2 i b x^2} x^m \, dx-\frac{1}{4} \left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int e^{2 i a+2 i b x^2} x^m \, dx\\ &=\frac{x^{1+m} \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{2 (1+m)}+2^{-\frac{7}{2}-\frac{m}{2}} e^{2 i a} x^{1+m} \left (-i b x^2\right )^{\frac{1}{2} (-1-m)} \csc ^2\left (a+b x^2\right ) \Gamma \left (\frac{1+m}{2},-2 i b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+2^{-\frac{7}{2}-\frac{m}{2}} e^{-2 i a} x^{1+m} \left (i b x^2\right )^{\frac{1}{2} (-1-m)} \csc ^2\left (a+b x^2\right ) \Gamma \left (\frac{1+m}{2},2 i b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\\ \end{align*}

Mathematica [A]  time = 0.895489, size = 189, normalized size = 0.9 \[ \frac{2^{\frac{1}{2} (-m-7)} x^{m+1} \left (b^2 x^4\right )^{\frac{1}{2} (-m-1)} \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \left ((m+1) (\cos (2 a)-i \sin (2 a)) \left (-i b x^2\right )^{\frac{m+1}{2}} \text{Gamma}\left (\frac{m+1}{2},2 i b x^2\right )+(m+1) (\cos (2 a)+i \sin (2 a)) \left (i b x^2\right )^{\frac{m+1}{2}} \text{Gamma}\left (\frac{m+1}{2},-2 i b x^2\right )+2^{\frac{m+5}{2}} \left (b^2 x^4\right )^{\frac{m+1}{2}}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2^((-7 - m)/2)*x^(1 + m)*(b^2*x^4)^((-1 - m)/2)*Csc[a + b*x^2]^2*(2^((5 + m)/2)*(b^2*x^4)^((1 + m)/2) + (1 +
m)*((-I)*b*x^2)^((1 + m)/2)*Gamma[(1 + m)/2, (2*I)*b*x^2]*(Cos[2*a] - I*Sin[2*a]) + (1 + m)*(I*b*x^2)^((1 + m)
/2)*Gamma[(1 + m)/2, (-2*I)*b*x^2]*(Cos[2*a] + I*Sin[2*a]))*(c*Sin[a + b*x^2]^3)^(2/3))/(1 + m)

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Maple [F]  time = 0.132, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( c \left ( \sin \left ( b{x}^{2}+a \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(b*x^2+a)^3)^(2/3),x)

[Out]

int(x^m*(c*sin(b*x^2+a)^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^{2} + 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(b*x^2+a)^3)^(2/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.80256, size = 359, normalized size = 1.72 \begin{align*} -\frac{{\left (8 \, b x x^{m} -{\left (i \, m + i\right )} e^{\left (-\frac{1}{2} \,{\left (m - 1\right )} \log \left (2 i \, b\right ) - 2 i \, a\right )} \Gamma \left (\frac{1}{2} \, m + \frac{1}{2}, 2 i \, b x^{2}\right ) -{\left (-i \, m - i\right )} e^{\left (-\frac{1}{2} \,{\left (m - 1\right )} \log \left (-2 i \, b\right ) + 2 i \, a\right )} \Gamma \left (\frac{1}{2} \, m + \frac{1}{2}, -2 i \, b x^{2}\right )\right )} \left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac{2}{3}}}{16 \,{\left ({\left (b m + b\right )} \cos \left (b x^{2} + a\right )^{2} - b m - b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(8*b*x*x^m - (I*m + I)*e^(-1/2*(m - 1)*log(2*I*b) - 2*I*a)*gamma(1/2*m + 1/2, 2*I*b*x^2) - (-I*m - I)*e^
(-1/2*(m - 1)*log(-2*I*b) + 2*I*a)*gamma(1/2*m + 1/2, -2*I*b*x^2))*(-(c*cos(b*x^2 + a)^2 - c)*sin(b*x^2 + a))^
(2/3)/((b*m + b)*cos(b*x^2 + a)^2 - b*m - b)

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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(b*x**2+a)**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^{2} + 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(b*x^2+a)^3)^(2/3),x, algorithm="giac")

[Out]

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