3.55 \(\int \frac{1}{\sqrt{-1-\cosh ^2(x)}} \, dx\)

Optimal. Leaf size=39 \[ -\frac{i \sqrt{\cosh ^2(x)+1} \text{EllipticF}\left (\frac{\pi }{2}+i x,-1\right )}{\sqrt{-\cosh ^2(x)-1}} \]

[Out]

((-I)*Sqrt[1 + Cosh[x]^2]*EllipticF[Pi/2 + I*x, -1])/Sqrt[-1 - Cosh[x]^2]

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Rubi [A]  time = 0.0202647, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3183, 3182} \[ -\frac{i \sqrt{\cosh ^2(x)+1} F\left (\left .i x+\frac{\pi }{2}\right |-1\right )}{\sqrt{-\cosh ^2(x)-1}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-1 - Cosh[x]^2],x]

[Out]

((-I)*Sqrt[1 + Cosh[x]^2]*EllipticF[Pi/2 + I*x, -1])/Sqrt[-1 - Cosh[x]^2]

Rule 3183

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

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

Mathematica [A]  time = 0.0439794, size = 40, normalized size = 1.03 \[ -\frac{i \sqrt{\cosh (2 x)+3} \text{EllipticF}\left (i x,\frac{1}{2}\right )}{\sqrt{2} \sqrt{-\cosh (2 x)-3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-1 - Cosh[x]^2],x]

[Out]

((-I)*Sqrt[3 + Cosh[2*x]]*EllipticF[I*x, 1/2])/(Sqrt[2]*Sqrt[-3 - Cosh[2*x]])

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Maple [A]  time = 0.224, size = 61, normalized size = 1.6 \begin{align*}{\frac{{\it EllipticF} \left ( \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}}\sqrt{1+ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{1- \left ( \cosh \left ( x \right ) \right ) ^{4}}}}{\frac{1}{\sqrt{-1- \left ( \cosh \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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{2 \, \sqrt{-e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} - 1}}{e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 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(-2*sqrt(-e^(4*x) - 6*e^(2*x) - 1)/(e^(4*x) + 6*e^(2*x) + 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)