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

Optimal. Leaf size=28 \[ -\frac{2 i \text{EllipticF}\left (\frac{1}{2} i \left (a+b \log \left (c x^n\right )\right ),2\right )}{b n} \]

[Out]

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

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Rubi [A]  time = 0.0277238, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2641} \[ -\frac{2 i F\left (\left .\frac{1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0203363, size = 28, normalized size = 1. \[ -\frac{2 i \text{EllipticF}\left (\frac{1}{2} i \left (a+b \log \left (c x^n\right )\right ),2\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.069, size = 183, normalized size = 6.5 \begin{align*} 2\,{\frac{\sqrt{ \left ( 2\, \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}\sqrt{- \left ( \sinh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) ,\sqrt{2} \right ) }{n\sqrt{2\, \left ( \sinh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}+ \left ( \sinh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}\sinh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \sqrt{2\, \left ( \cosh \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1}b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/n*((2*cosh(1/2*a+1/2*b*ln(c*x^n))^2-1)*sinh(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)*(-sinh(1/2*a+1/2*b*ln(c*x^n))^2)
^(1/2)*(-2*cosh(1/2*a+1/2*b*ln(c*x^n))^2+1)^(1/2)/(2*sinh(1/2*a+1/2*b*ln(c*x^n))^4+sinh(1/2*a+1/2*b*ln(c*x^n))
^2)^(1/2)*EllipticF(cosh(1/2*a+1/2*b*ln(c*x^n)),2^(1/2))/sinh(1/2*a+1/2*b*ln(c*x^n))/(2*cosh(1/2*a+1/2*b*ln(c*
x^n))^2-1)^(1/2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\cosh \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/cosh(a+b*log(c*x^n))^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

integral(1/(x*sqrt(cosh(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{\cosh{\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/cosh(a+b*ln(c*x**n))**(1/2),x)

[Out]

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

[Out]

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