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

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

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

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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, 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 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]

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 {\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*}

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Mathematica [A]  time = 0.10, size = 66, normalized size = 0.92 \[ \frac {2 \left (i \sinh \left (a+b \log \left (c x^n\right )\right )\right )^{3/2} \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) F\left (\left .\frac {1}{4} \left (-2 i a-2 i b \log \left (c x^n\right )+\pi \right )\right |2\right )}{b n} \]

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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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )}}{x}, x\right ) \]

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

[Out]

Timed out

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maple [A]  time = 0.34, size = 120, normalized size = 1.67 \[ \frac {i \sqrt {-i \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticF \left (\sqrt {-i \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}, \frac {\sqrt {2}}{2}\right )}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I/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)*EllipticF((-I*(sinh(a+b*ln(c*x^n))+I))^(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 \[ \int \frac {\sqrt {\operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )}}{x}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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