3.4.53 \(\int x (c \sin ^3(a+b x^n))^{2/3} \, dx\) [353]

Optimal. Leaf size=188 \[ \frac {1}{4} x^2 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}+\frac {4^{-1-\frac {1}{n}} e^{2 i a} x^2 \left (-i b x^n\right )^{-2/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {2}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac {4^{-1-\frac {1}{n}} e^{-2 i a} x^2 \left (i b x^n\right )^{-2/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {2}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n} \]

[Out]

1/4*x^2*csc(a+b*x^n)^2*(c*sin(a+b*x^n)^3)^(2/3)+4^(-1-1/n)*exp(2*I*a)*x^2*csc(a+b*x^n)^2*GAMMA(2/n,-2*I*b*x^n)
*(c*sin(a+b*x^n)^3)^(2/3)/n/((-I*b*x^n)^(2/n))+4^(-1-1/n)*x^2*csc(a+b*x^n)^2*GAMMA(2/n,2*I*b*x^n)*(c*sin(a+b*x
^n)^3)^(2/3)/exp(2*I*a)/n/((I*b*x^n)^(2/n))

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Rubi [A]
time = 0.16, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6852, 3506, 3505, 2250} \begin {gather*} \frac {e^{2 i a} 4^{-\frac {1}{n}-1} x^2 \left (-i b x^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{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} 4^{-\frac {1}{n}-1} x^2 \left (i b x^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{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}{4} x^2 \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*(c*Sin[a + b*x^n]^3)^(2/3),x]

[Out]

(x^2*Csc[a + b*x^n]^2*(c*Sin[a + b*x^n]^3)^(2/3))/4 + (4^(-1 - n^(-1))*E^((2*I)*a)*x^2*Csc[a + b*x^n]^2*Gamma[
2/n, (-2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(n*((-I)*b*x^n)^(2/n)) + (4^(-1 - n^(-1))*x^2*Csc[a + b*x^n]^2*
Gamma[2/n, (2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(E^((2*I)*a)*n*(I*b*x^n)^(2/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 \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 \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}-\frac {1}{2} x \cos \left (2 a+2 b x^n\right )\right ) \, dx\\ &=\frac {1}{4} x^2 \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 \cos \left (2 a+2 b x^n\right ) \, dx\\ &=\frac {1}{4} x^2 \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 \, 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 \, dx\\ &=\frac {1}{4} x^2 \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}+\frac {4^{-1-\frac {1}{n}} e^{2 i a} x^2 \left (-i b x^n\right )^{-2/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {2}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n}+\frac {4^{-1-\frac {1}{n}} e^{-2 i a} x^2 \left (i b x^n\right )^{-2/n} \csc ^2\left (a+b x^n\right ) \Gamma \left (\frac {2}{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 = 160, normalized size = 0.85 \begin {gather*} \frac {4^{-\frac {1+n}{n}} e^{-2 i a} x^2 \left (b^2 x^{2 n}\right )^{-2/n} \csc ^2\left (a+b x^n\right ) \left (4^{\frac {1}{n}} e^{2 i a} n \left (b^2 x^{2 n}\right )^{2/n}+e^{4 i a} \left (i b x^n\right )^{2/n} \Gamma \left (\frac {2}{n},-2 i b x^n\right )+\left (-i b x^n\right )^{2/n} \Gamma \left (\frac {2}{n},2 i b x^n\right )\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-1/8*(x^2 - 2*integrate(x*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*(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, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

int(x*(c*sin(a + b*x^n)^3)^(2/3), x)

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