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*ln(c*x))*csch(2*ln(c*x))^(1/2)+I*(sin(1/4*Pi+I*ln(c*x))^2)^(1/2)/sin(1/4*Pi+I*ln(c*x))*EllipticE(cos(1
/4*Pi+I*ln(c*x)),2^(1/2))/csch(2*ln(c*x))^(1/2)/(I*sinh(2*ln(c*x)))^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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]

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]

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*}

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Mathematica [A]  time = 0.10, 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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fricas [F]  time = 1.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x}, x\right ) \]

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 0.41, size = 163, normalized size = 2.43 \[ \frac {2 \sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )+1}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )}\, \EllipticE \left (\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )+1}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\cosh ^{2}\left (2 \ln \left (c x \right )\right )\right )}{2 \cosh \left (2 \ln \left (c x \right )\right ) \sqrt {\sinh \left (2 \ln \left (c x \right )\right )}} \]

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)*(I*sinh(2*ln(c*x))+1)^(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)*(I*sinh(2*ln(c*x))+1)^(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/sinh(2*log(c*x)))^(3/2)/x,x)

[Out]

int((1/sinh(2*log(c*x)))^(3/2)/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x}\, dx \]

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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