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

Optimal. Leaf size=184 \[ \frac{e^{2 i a} 4^{\frac{1}{n}-1} \left (-i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},-2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x^2}+\frac{e^{-2 i a} 4^{\frac{1}{n}-1} \left (i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x^2}-\frac{\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{4 x^2} \]

[Out]

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

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Rubi [A]  time = 0.268642, antiderivative size = 184, 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} \[ \frac{e^{2 i a} 4^{\frac{1}{n}-1} \left (-i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},-2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x^2}+\frac{e^{-2 i a} 4^{\frac{1}{n}-1} \left (i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},2 i b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x^2}-\frac{\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{4 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x^n]^3)^(2/3)/x^3,x]

[Out]

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

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

Mathematica [A]  time = 0.352419, size = 129, normalized size = 0.7 \[ \frac{e^{-2 i a} \csc ^2\left (a+b x^n\right ) \left (e^{4 i a} 4^{\frac{1}{n}} \left (-i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},-2 i b x^n\right )+4^{\frac{1}{n}} \left (i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},2 i b x^n\right )-e^{2 i a} n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{4 n x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.115, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \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((c*sin(a+b*x^n)^3)^(2/3)/x^3,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (2 \, x^{2} \int \frac{\cos \left (2 \, b x^{n} + 2 \, a\right )}{x^{3}}\,{d x} + 1\right )} c^{\frac{2}{3}}}{8 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*x^2*integrate(cos(2*b*x^n + 2*a)/x^3, x) + 1)*c^(2/3)/x^2

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\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^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-(c*cos(b*x^n + a)^2 - c)*sin(b*x^n + a))^(2/3)/x^3, 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((c*sin(a+b*x**n)**3)**(2/3)/x**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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