3.282 \(\int \frac{1}{x \sqrt{\sinh (a+b \log (c x^n))}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0439565, 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 = {2642, 2641} \[ -\frac{2 i \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac{1}{2} \left (i a+i b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{b n \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

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

Mathematica [A]  time = 0.0437464, size = 66, normalized size = 0.92 \[ -\frac{2 \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )} \text{EllipticF}\left (\frac{1}{4} \left (-2 i a-2 i b \log \left (c x^n\right )+\pi \right ),2\right )}{b n \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 120, normalized size = 1.7 \begin{align*}{\frac{i\sqrt{2}}{bn\cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }\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(1/x/sinh(a+b*ln(c*x^n))^(1/2),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{1}{x \sqrt{\sinh \left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{x \sqrt{\sinh \left (b \log \left (c x^{n}\right ) + a\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\sinh{\left (a + b \log{\left (c x^{n} \right )} \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\sinh \left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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