\(\int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx\) [152]

Optimal result

Integrand size = 15, antiderivative size = 67 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=-\cosh (2 \log (c x)) \sqrt {\text {csch}(2 \log (c x))}+\frac {i E\left (\left .\frac {\pi }{4}-i \log (c x)\right |2\right )}{\sqrt {\text {csch}(2 \log (c x))} \sqrt {i \sinh (2 \log (c x))}} \]

[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)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3853, 3856, 2719} \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=-\cosh (2 \log (c x)) \sqrt {\text {csch}(2 \log (c x))}+\frac {i E\left (\left .\frac {\pi }{4}-i \log (c x)\right |2\right )}{\sqrt {i \sinh (2 \log (c x))} \sqrt {\text {csch}(2 \log (c x))}} \]

[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 2719

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

Rule 3853

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 3856

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*} \text {integral}& = \text {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))}+\text {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 {\text {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] (verified)

Time = 0.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.81 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\sqrt {\text {csch}(2 \log (c x))} \left (-\cosh (2 \log (c x))+E\left (\left .\frac {\pi }{4}-i \log (c x)\right |2\right ) \sqrt {i \sinh (2 \log (c x))}\right ) \]

[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]]])

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.43

method	result	size
derivativedivides	\(\frac {2 \sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )+1}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )+1}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}, \frac {\sqrt {2}}{2}\right )-2 \cosh \left (2 \ln \left (c x \right )\right )^{2}}{2 \cosh \left (2 \ln \left (c x \right )\right ) \sqrt {\sinh \left (2 \ln \left (c x \right )\right )}}\)	\(163\)
default	\(\frac {2 \sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )+1}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )+1}\, \sqrt {i \sinh \left (2 \ln \left (c x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (2 \ln \left (c x \right )\right )}, \frac {\sqrt {2}}{2}\right )-2 \cosh \left (2 \ln \left (c x \right )\right )^{2}}{2 \cosh \left (2 \ln \left (c x \right )\right ) \sqrt {\sinh \left (2 \ln \left (c x \right )\right )}}\)	\(163\)

[In]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.90 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=-\sqrt {2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} c^{2} x^{2} - i \, \sqrt {2} c^{2} E(\arcsin \left (c x\right )\,|\,-1) + i \, \sqrt {2} c^{2} F(\arcsin \left (c x\right )\,|\,-1) \]

[In]

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

[Out]

-sqrt(2)*sqrt(c^2*x^2/(c^4*x^4 - 1))*c^2*x^2 - I*sqrt(2)*c^2*elliptic_e(arcsin(c*x), -1) + I*sqrt(2)*c^2*ellip
tic_f(arcsin(c*x), -1)

Sympy [F]

\[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\int \frac {\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\int { \frac {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x} \,d x } \]

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\int \frac {{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}}{x} \,d x \]

[In]

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

[Out]

int((1/sinh(2*log(c*x)))^(3/2)/x, x)