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

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

[Out]

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

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Rubi [A]  time = 0.0426943, antiderivative size = 72, normalized size of antiderivative = 1., 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, 2641} \[ -\frac{2 i \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt{\text{csch}\left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac{1}{2} \left (i a+i b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Csch[a + b*Log[c*x^n]]]/x,x]

[Out]

((-2*I)*Sqrt[Csch[a + b*Log[c*x^n]]]*EllipticF[(I*a - Pi/2 + I*b*Log[c*x^n])/2, 2]*Sqrt[I*Sinh[a + b*Log[c*x^n
]]])/(b*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 2641

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

Rubi steps

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

Mathematica [A]  time = 0.0919914, size = 66, normalized size = 0.92 \[ \frac{2 \left (i \sinh \left (a+b \log \left (c x^n\right )\right )\right )^{3/2} \text{csch}^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right ) \text{EllipticF}\left (\frac{1}{4} \left (-2 i a-2 i b \log \left (c x^n\right )+\pi \right ),2\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Csch[a + b*Log[c*x^n]]^(3/2)*EllipticF[((-2*I)*a + Pi - (2*I)*b*Log[c*x^n])/4, 2]*(I*Sinh[a + b*Log[c*x^n]]
)^(3/2))/(b*n)

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Maple [A]  time = 0.118, size = 120, normalized size = 1.7 \begin{align*}{\frac{i\sqrt{2}}{n\cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) b}\sqrt{-i \left ( i+\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }\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 ) }{\it EllipticF} \left ( \sqrt{-i \left ( i+\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) },{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I/n*(-I*(I+sinh(a+b*ln(c*x^n))))^(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)*EllipticF((-I*(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., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{csch}\left (b \log \left (c x^{n}\right ) + a\right )}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out