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

Optimal. Leaf size=67 \[ -\cosh (2 \log (c x)) \sqrt{\text{csch}(2 \log (c x))}+\frac{i E\left (\left .\frac{\pi }{4}-i \log (c x)\right |2\right )}{\sqrt{i \sinh (2 \log (c x))} \sqrt{\text{csch}(2 \log (c x))}} \]

[Out]

-(Cosh[2*Log[c*x]]*Sqrt[Csch[2*Log[c*x]]]) + (I*EllipticE[Pi/4 - I*Log[c*x], 2])/(Sqrt[Csch[2*Log[c*x]]]*Sqrt[
I*Sinh[2*Log[c*x]]])

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Rubi [A]  time = 0.0372023, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3768, 3771, 2639} \[ -\cosh (2 \log (c x)) \sqrt{\text{csch}(2 \log (c x))}+\frac{i E\left (\left .\frac{\pi }{4}-i \log (c x)\right |2\right )}{\sqrt{i \sinh (2 \log (c x))} \sqrt{\text{csch}(2 \log (c x))}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cosh[2*Log[c*x]]*Sqrt[Csch[2*Log[c*x]]]) + (I*EllipticE[Pi/4 - I*Log[c*x], 2])/(Sqrt[Csch[2*Log[c*x]]]*Sqrt[
I*Sinh[2*Log[c*x]]])

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int \frac{\text{csch}^{\frac{3}{2}}(2 \log (c x))}{x} \, dx &=\operatorname{Subst}\left (\int \text{csch}^{\frac{3}{2}}(2 x) \, dx,x,\log (c x)\right )\\ &=-\cosh (2 \log (c x)) \sqrt{\text{csch}(2 \log (c x))}+\operatorname{Subst}\left (\int \frac{1}{\sqrt{\text{csch}(2 x)}} \, dx,x,\log (c x)\right )\\ &=-\cosh (2 \log (c x)) \sqrt{\text{csch}(2 \log (c x))}+\frac{\operatorname{Subst}\left (\int \sqrt{i \sinh (2 x)} \, dx,x,\log (c x)\right )}{\sqrt{\text{csch}(2 \log (c x))} \sqrt{i \sinh (2 \log (c x))}}\\ &=-\cosh (2 \log (c x)) \sqrt{\text{csch}(2 \log (c x))}+\frac{i E\left (\left .\frac{\pi }{4}-i \log (c x)\right |2\right )}{\sqrt{\text{csch}(2 \log (c x))} \sqrt{i \sinh (2 \log (c x))}}\\ \end{align*}

Mathematica [A]  time = 0.0935348, size = 54, normalized size = 0.81 \[ \sqrt{\text{csch}(2 \log (c x))} \left (-\cosh (2 \log (c x))+\sqrt{i \sinh (2 \log (c x))} E\left (\left .\frac{\pi }{4}-i \log (c x)\right |2\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[Csch[2*Log[c*x]]]*(-Cosh[2*Log[c*x]] + EllipticE[Pi/4 - I*Log[c*x], 2]*Sqrt[I*Sinh[2*Log[c*x]]])

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Maple [A]  time = 0.168, size = 163, normalized size = 2.4 \begin{align*}{\frac{1}{2\,\cosh \left ( 2\,\ln \left ( cx \right ) \right ) } \left ( 2\,\sqrt{1-i\sinh \left ( 2\,\ln \left ( cx \right ) \right ) }\sqrt{2}\sqrt{1+i\sinh \left ( 2\,\ln \left ( cx \right ) \right ) }\sqrt{i\sinh \left ( 2\,\ln \left ( cx \right ) \right ) }{\it EllipticE} \left ( \sqrt{1-i\sinh \left ( 2\,\ln \left ( cx \right ) \right ) },1/2\,\sqrt{2} \right ) -\sqrt{1-i\sinh \left ( 2\,\ln \left ( cx \right ) \right ) }\sqrt{2}\sqrt{1+i\sinh \left ( 2\,\ln \left ( cx \right ) \right ) }\sqrt{i\sinh \left ( 2\,\ln \left ( cx \right ) \right ) }{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( 2\,\ln \left ( cx \right ) \right ) },{\frac{\sqrt{2}}{2}} \right ) -2\, \left ( \cosh \left ( 2\,\ln \left ( cx \right ) \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{\sinh \left ( 2\,\ln \left ( cx \right ) \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(1-I*sinh(2*ln(c*x)))^(1/2)*2^(1/2)*(1+I*sinh(2*ln(c*x)))^(1/2)*(I*sinh(2*ln(c*x)))^(1/2)*EllipticE((1-
I*sinh(2*ln(c*x)))^(1/2),1/2*2^(1/2))-(1-I*sinh(2*ln(c*x)))^(1/2)*2^(1/2)*(1+I*sinh(2*ln(c*x)))^(1/2)*(I*sinh(
2*ln(c*x)))^(1/2)*EllipticF((1-I*sinh(2*ln(c*x)))^(1/2),1/2*2^(1/2))-2*cosh(2*ln(c*x))^2)/cosh(2*ln(c*x))/sinh
(2*ln(c*x))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csch(2*log(c*x))^(3/2)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(csch(2*log(c*x))^(3/2)/x, 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}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csch(2*log(c*x))^(3/2)/x, x)