3.153 \(\int \frac{\text{csch}^{\frac{3}{2}}(2 \log (c x))}{x^2} \, dx\)

Optimal. Leaf size=27 \[ -\frac{1}{2} x^3 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x)) \]

[Out]

-((c^4 - x^(-4))*x^3*Csch[2*Log[c*x]]^(3/2))/2

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Rubi [A]  time = 0.039279, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5552, 5550, 261} \[ -\frac{1}{2} x^3 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x)) \]

Antiderivative was successfully verified.

[In]

Int[Csch[2*Log[c*x]]^(3/2)/x^2,x]

[Out]

-((c^4 - x^(-4))*x^3*Csch[2*Log[c*x]]^(3/2))/2

Rule 5552

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

Rule 5550

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

Rule 261

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

Rubi steps

\begin{align*} \int \frac{\text{csch}^{\frac{3}{2}}(2 \log (c x))}{x^2} \, dx &=c \operatorname{Subst}\left (\int \frac{\text{csch}^{\frac{3}{2}}(2 \log (x))}{x^2} \, dx,x,c x\right )\\ &=\left (c^4 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{1}{x^4}\right )^{3/2} x^5} \, dx,x,c x\right )\\ &=-\frac{1}{2} \left (c^4-\frac{1}{x^4}\right ) x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))\\ \end{align*}

Mathematica [A]  time = 0.033557, size = 33, normalized size = 1.22 \[ -\sqrt{2} c^2 x \sqrt{\frac{c^2 x^2}{c^4 x^4-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[2*Log[c*x]]^(3/2)/x^2,x]

[Out]

-(Sqrt[2]*c^2*x*Sqrt[(c^2*x^2)/(-1 + c^4*x^4)])

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ({\rm csch} \left (2\,\ln \left ( cx \right ) \right ) \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(2*ln(c*x))^(3/2)/x^2,x)

[Out]

int(csch(2*ln(c*x))^(3/2)/x^2,x)

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Maxima [B]  time = 1.58188, size = 117, normalized size = 4.33 \begin{align*} -c{\left (\frac{\sqrt{2}}{{\left (\frac{1}{c x} + 1\right )}^{\frac{3}{2}}{\left (-\frac{1}{c x} + 1\right )}^{\frac{3}{2}}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}}} - \frac{\sqrt{2}}{c^{4} x^{4}{\left (\frac{1}{c x} + 1\right )}^{\frac{3}{2}}{\left (-\frac{1}{c x} + 1\right )}^{\frac{3}{2}}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*log(c*x))^(3/2)/x^2,x, algorithm="maxima")

[Out]

-c*(sqrt(2)/((1/(c*x) + 1)^(3/2)*(-1/(c*x) + 1)^(3/2)*(1/(c^2*x^2) + 1)^(3/2)) - sqrt(2)/(c^4*x^4*(1/(c*x) + 1
)^(3/2)*(-1/(c*x) + 1)^(3/2)*(1/(c^2*x^2) + 1)^(3/2)))

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Fricas [A]  time = 1.55671, size = 59, normalized size = 2.19 \begin{align*} -\sqrt{2} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} - 1}} c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*log(c*x))^(3/2)/x^2,x, algorithm="fricas")

[Out]

-sqrt(2)*sqrt(c^2*x^2/(c^4*x^4 - 1))*c^2*x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*ln(c*x))**(3/2)/x**2,x)

[Out]

Integral(csch(2*log(c*x))**(3/2)/x**2, x)

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Giac [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(csch(2*log(c*x))^(3/2)/x^2,x, algorithm="giac")

[Out]

Timed out