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

Optimal. Leaf size=58 \[ -\frac{2 i \sqrt{\text{sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt{\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]

[Out]

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

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Rubi [A]  time = 0.0557946, antiderivative size = 58, 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, 2639} \[ -\frac{2 i \sqrt{\text{sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt{\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0703197, size = 58, normalized size = 1. \[ -\frac{2 i E\left (\left .\frac{1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n \sqrt{\text{sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt{\cosh \left (a+b \log \left (c x^n\right )\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.297, size = 183, normalized size = 3.2 \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 EllipticE} \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/sech(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)*EllipticE(cosh(1/2*a+1/2*b*ln(c*x^n)),2^(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)/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{\operatorname{sech}\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/sech(a+b*log(c*x^n))^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

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

[Out]

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