3.4.52 \(\int x^2 (c \sin ^3(a+b x^n))^{2/3} \, dx\) [352]

Optimal. Leaf size=188 \[ \frac {1}{6} x^3 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}+\frac {2^{-2-\frac {3}{n}} e^{2 i a} x^3 \left (-i b x^n\right )^{-3/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {3}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac {2^{-2-\frac {3}{n}} e^{-2 i a} x^3 \left (i b x^n\right )^{-3/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {3}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n} \]

[Out]

1/6*x^3*csc(a+b*x^n)^2*(c*sin(a+b*x^n)^3)^(2/3)+2^(-2-3/n)*exp(2*I*a)*x^3*csc(a+b*x^n)^2*GAMMA(3/n,-2*I*b*x^n)
*(c*sin(a+b*x^n)^3)^(2/3)/n/((-I*b*x^n)^(3/n))+2^(-2-3/n)*x^3*csc(a+b*x^n)^2*GAMMA(3/n,2*I*b*x^n)*(c*sin(a+b*x
^n)^3)^(2/3)/exp(2*I*a)/n/((I*b*x^n)^(3/n))

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Rubi [A]
time = 0.21, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6852, 3506, 3505, 2250} \begin {gather*} \frac {e^{2 i a} 2^{-\frac {3}{n}-2} x^3 \left (-i b x^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{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 {3}{n}-2} x^3 \left (i b x^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{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}{6} x^3 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(c*Sin[a + b*x^n]^3)^(2/3),x]

[Out]

(x^3*Csc[a + b*x^n]^2*(c*Sin[a + b*x^n]^3)^(2/3))/6 + (2^(-2 - 3/n)*E^((2*I)*a)*x^3*Csc[a + b*x^n]^2*Gamma[3/n
, (-2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(n*((-I)*b*x^n)^(3/n)) + (2^(-2 - 3/n)*x^3*Csc[a + b*x^n]^2*Gamma[
3/n, (2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(E^((2*I)*a)*n*(I*b*x^n)^(3/n))

Rule 2250

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

Rule 3505

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 3506

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 6852

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 x^2 \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^2 \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^2}{2}-\frac {1}{2} x^2 \cos \left (2 a+2 b x^n\right )\right ) \, dx\\ &=\frac {1}{6} x^3 \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 x^2 \cos \left (2 a+2 b x^n\right ) \, dx\\ &=\frac {1}{6} x^3 \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} x^2 \, 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^2 \, dx\\ &=\frac {1}{6} x^3 \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 {3}{n}} e^{2 i a} x^3 \left (-i b x^n\right )^{-3/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {3}{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 {3}{n}} e^{-2 i a} x^3 \left (i b x^n\right )^{-3/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {3}{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 = 0.38, size = 168, normalized size = 0.89 \begin {gather*} \frac {2^{-2-\frac {3}{n}} e^{-2 i a} x^3 \left (b^2 x^{2 n}\right )^{-3/n} \csc ^2\left (a+b x^n\right ) \left (2^{\frac {3+n}{n}} e^{2 i a} n \left (b^2 x^{2 n}\right )^{3/n}+3 e^{4 i a} \left (i b x^n\right )^{3/n} \Gamma \left (\frac {3}{n},-2 i b x^n\right )+3 \left (-i b x^n\right )^{3/n} \Gamma \left (\frac {3}{n},2 i b x^n\right )\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{3 n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int x^{2} \left (c \left (\sin ^{3}\left (a +b \,x^{n}\right )\right )\right )^{\frac {2}{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(x^3 - 3*integrate(x^2*cos(2*b*x^n + 2*a), x))*c^(2/3)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(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^2, x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(c*sin(a + b*x^n)^3)^(2/3),x)

[Out]

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

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