3.52 \(\int \frac{1}{\sqrt{1+\cosh ^2(x)}} \, dx\)

Optimal. Leaf size=17 \[ -i \text{EllipticF}\left (\frac{\pi }{2}+i x,-1\right ) \]

[Out]

(-I)*EllipticF[Pi/2 + I*x, -1]

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Rubi [A]  time = 0.0090285, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3182} \[ -i F\left (\left .i x+\frac{\pi }{2}\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 + Cosh[x]^2],x]

[Out]

(-I)*EllipticF[Pi/2 + I*x, -1]

Rule 3182

Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(1*EllipticF[e + f*x, -(b/a)])/(Sqrt[a]*
f), x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1+\cosh ^2(x)}} \, dx &=-i F\left (\left .\frac{\pi }{2}+i x\right |-1\right )\\ \end{align*}

Mathematica [A]  time = 0.0343898, size = 18, normalized size = 1.06 \[ -\frac{i \text{EllipticF}\left (i x,\frac{1}{2}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 + Cosh[x]^2],x]

[Out]

((-I)*EllipticF[I*x, 1/2])/Sqrt[2]

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Maple [B]  time = 0.204, size = 45, normalized size = 2.7 \begin{align*}{\frac{-i{\it EllipticF} \left ( i\cosh \left ( x \right ) ,i \right ) }{\sinh \left ( x \right ) }\sqrt{ \left ( 1+ \left ( \cosh \left ( x \right ) \right ) ^{2} \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}}\sqrt{- \left ( \sinh \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{4}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+cosh(x)^2)^(1/2),x)

[Out]

-I*((1+cosh(x)^2)*sinh(x)^2)^(1/2)*(-sinh(x)^2)^(1/2)/(cosh(x)^4-1)^(1/2)*EllipticF(I*cosh(x),I)/sinh(x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\cosh \left (x\right )^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(x)^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(cosh(x)^2 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{\cosh \left (x\right )^{2} + 1}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(x)^2)^(1/2),x, algorithm="fricas")

[Out]

integral(1/sqrt(cosh(x)^2 + 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\cosh ^{2}{\left (x \right )} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(x)**2)**(1/2),x)

[Out]

Integral(1/sqrt(cosh(x)**2 + 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\cosh \left (x\right )^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+cosh(x)^2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(cosh(x)^2 + 1), x)