3.173 \(\int \frac {1}{x \sqrt {\text {csch}(a+b \log (c x^n))}} \, dx\)

Optimal. Leaf size=72 \[ -\frac {2 i E\left (\left .\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{b n \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3771, 2639} \[ -\frac {2 i E\left (\left .\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{b n \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[Csch[a + b*Log[c*x^n]]]),x]

[Out]

((-2*I)*EllipticE[(I*a - Pi/2 + I*b*Log[c*x^n])/2, 2])/(b*n*Sqrt[Csch[a + b*Log[c*x^n]]]*Sqrt[I*Sinh[a + b*Log
[c*x^n]]])

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

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Mathematica [A]  time = 0.08, size = 68, normalized size = 0.94 \[ \frac {2 \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (\frac {\pi }{2}-i \left (a+b \log \left (c x^n\right )\right )\right )\right |2\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[Csch[a + b*Log[c*x^n]]]),x]

[Out]

(2*Sqrt[Csch[a + b*Log[c*x^n]]]*EllipticE[(Pi/2 - I*(a + b*Log[c*x^n]))/2, 2]*Sqrt[I*Sinh[a + b*Log[c*x^n]]])/
(b*n)

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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{x \sqrt {\operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(x*sqrt(csch(b*log(c*x^n) + a))), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {\operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x*sqrt(csch(b*log(c*x^n) + a))), x)

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maple [A]  time = 0.59, size = 146, normalized size = 2.03 \[ \frac {\sqrt {-i \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \left (2 \EllipticE \left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )\right )}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/csch(a+b*ln(c*x^n))^(1/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {\operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x*sqrt(csch(b*log(c*x^n) + a))), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x\,\sqrt {\frac {1}{\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(1/sinh(a + b*log(c*x^n)))^(1/2)),x)

[Out]

int(1/(x*(1/sinh(a + b*log(c*x^n)))^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {\operatorname {csch}{\left (a + b \log {\left (c x^{n} \right )} \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/csch(a+b*ln(c*x**n))**(1/2),x)

[Out]

Integral(1/(x*sqrt(csch(a + b*log(c*x**n)))), x)

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