3.287 \(\int \frac{1}{\sinh ^{\frac{3}{2}}(a+\frac{2 \log (c x^n)}{n})} \, dx\)

Optimal. Leaf size=43 \[ -\frac{x \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}{2 \sinh ^{\frac{3}{2}}\left (a+\frac{2 \log \left (c x^n\right )}{n}\right )} \]

[Out]

-(x*(1 - 1/(E^(2*a)*(c*x^n)^(4/n))))/(2*Sinh[a + (2*Log[c*x^n])/n]^(3/2))

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Rubi [A]  time = 0.0528222, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5525, 5533, 264} \[ -\frac{x \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}{2 \sinh ^{\frac{3}{2}}\left (a+\frac{2 \log \left (c x^n\right )}{n}\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + (2*Log[c*x^n])/n]^(-3/2),x]

[Out]

-(x*(1 - 1/(E^(2*a)*(c*x^n)^(4/n))))/(2*Sinh[a + (2*Log[c*x^n])/n]^(3/2))

Rule 5525

Int[Sinh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[
x^(1/n - 1)*Sinh[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n
, 1])

Rule 5533

Int[((e_.)*(x_))^(m_.)*Sinh[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_), x_Symbol] :> Dist[Sinh[d*(a + b*Log[x])]^p/(x
^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p), Int[(e*x)^m*x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p, x], x] /; Fr
eeQ[{a, b, d, e, m, p}, x] &&  !IntegerQ[p]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{\sinh ^{\frac{3}{2}}\left (a+\frac{2 \log \left (c x^n\right )}{n}\right )} \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+\frac{1}{n}}}{\sinh ^{\frac{3}{2}}\left (a+\frac{2 \log (x)}{n}\right )} \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{2/n} \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{x^{-1-\frac{2}{n}}}{\left (1-e^{-2 a} x^{-4/n}\right )^{3/2}} \, dx,x,c x^n\right )}{n \sinh ^{\frac{3}{2}}\left (a+\frac{2 \log \left (c x^n\right )}{n}\right )}\\ &=-\frac{x \left (1-e^{-2 a} \left (c x^n\right )^{-4/n}\right )}{2 \sinh ^{\frac{3}{2}}\left (a+\frac{2 \log \left (c x^n\right )}{n}\right )}\\ \end{align*}

Mathematica [A]  time = 0.14718, size = 61, normalized size = 1.42 \[ \frac{\sinh \left (a+\frac{2 \log \left (c x^n\right )}{n}-2 \log (x)\right )-\cosh \left (a+\frac{2 \log \left (c x^n\right )}{n}-2 \log (x)\right )}{x \sqrt{\sinh \left (a+\frac{2 \log \left (c x^n\right )}{n}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + (2*Log[c*x^n])/n]^(-3/2),x]

[Out]

(-Cosh[a - 2*Log[x] + (2*Log[c*x^n])/n] + Sinh[a - 2*Log[x] + (2*Log[c*x^n])/n])/(x*Sqrt[Sinh[a + (2*Log[c*x^n
])/n]])

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Maple [F]  time = 0.16, size = 0, normalized size = 0. \begin{align*} \int \left ( \sinh \left ( a+2\,{\frac{\ln \left ( c{x}^{n} \right ) }{n}} \right ) \right ) ^{-{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sinh(a+2*ln(c*x^n)/n)^(3/2),x)

[Out]

int(1/sinh(a+2*ln(c*x^n)/n)^(3/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sinh(a+2*log(c*x^n)/n)^(3/2),x, algorithm="maxima")

[Out]

integrate(sinh(a + 2*log(c*x^n)/n)^(-3/2), x)

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Fricas [A]  time = 2.43249, size = 167, normalized size = 3.88 \begin{align*} -\frac{2 \, \sqrt{\frac{1}{2}} x \sqrt{\frac{x^{4} e^{\left (\frac{2 \,{\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 1}{x^{2}}} e^{\left (-\frac{a n + 2 \, \log \left (c\right )}{2 \, n}\right )}}{x^{4} e^{\left (\frac{2 \,{\left (a n + 2 \, \log \left (c\right )\right )}}{n}\right )} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sinh(a+2*log(c*x^n)/n)^(3/2),x, algorithm="fricas")

[Out]

-2*sqrt(1/2)*x*sqrt((x^4*e^(2*(a*n + 2*log(c))/n) - 1)/x^2)*e^(-1/2*(a*n + 2*log(c))/n)/(x^4*e^(2*(a*n + 2*log
(c))/n) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sinh ^{\frac{3}{2}}{\left (a + \frac{2 \log{\left (c x^{n} \right )}}{n} \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sinh(a+2*ln(c*x**n)/n)**(3/2),x)

[Out]

Integral(sinh(a + 2*log(c*x**n)/n)**(-3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sinh(a+2*log(c*x^n)/n)^(3/2),x, algorithm="giac")

[Out]

integrate(sinh(a + 2*log(c*x^n)/n)^(-3/2), x)