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

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

[Out]

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

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Rubi [A]  time = 0.0210998, 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 = {3178, 3177} \[ -\frac{i \sqrt{-\cosh ^2(x)-1} E\left (\left .i x+\frac{\pi }{2}\right |-1\right )}{\sqrt{\cosh ^2(x)+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 - Cosh[x]^2],x]

[Out]

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

Rule 3178

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

Rule 3177

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

Rubi steps

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

Mathematica [A]  time = 0.0391654, size = 40, normalized size = 1.03 \[ \frac{i \sqrt{2} \sqrt{\cosh (2 x)+3} E\left (i x\left |\frac{1}{2}\right .\right )}{\sqrt{-\cosh (2 x)-3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 - Cosh[x]^2],x]

[Out]

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

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Maple [A]  time = 0.211, size = 62, normalized size = 1.6 \begin{align*} -{\frac{{\it EllipticE} \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-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)*EllipticE(cosh(x),I)/(1-cosh(x)^4)^(1
/2)/sinh(x)/(-1-cosh(x)^2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-cosh(x)^2 - 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (e^{\left (2 \, x\right )} - e^{x}\right )}{\rm integral}\left (\frac{4 \, \sqrt{-e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} - 1}{\left (e^{\left (2 \, x\right )} + 1\right )}}{e^{\left (6 \, x\right )} - 2 \, e^{\left (5 \, x\right )} + 7 \, e^{\left (4 \, x\right )} - 12 \, e^{\left (3 \, x\right )} + 7 \, e^{\left (2 \, x\right )} - 2 \, e^{x} + 1}, x\right ) + \sqrt{-e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} - 1}{\left (e^{x} + 1\right )}}{2 \,{\left (e^{\left (2 \, x\right )} - e^{x}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(e^(2*x) - e^x)*integral(4*sqrt(-e^(4*x) - 6*e^(2*x) - 1)*(e^(2*x) + 1)/(e^(6*x) - 2*e^(5*x) + 7*e^(4*x
) - 12*e^(3*x) + 7*e^(2*x) - 2*e^x + 1), x) + sqrt(-e^(4*x) - 6*e^(2*x) - 1)*(e^x + 1))/(e^(2*x) - e^x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-cosh(x)**2 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-cosh(x)^2 - 1), x)