3.91 \(\int \sqrt{-1+\sinh ^2(x)} \, dx\)

Optimal. Leaf size=33 \[ -\frac{i \sqrt{\sinh ^2(x)-1} E(i x|-1)}{\sqrt{1-\sinh ^2(x)}} \]

[Out]

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

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Rubi [A]  time = 0.0202611, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3178, 3177} \[ -\frac{i \sqrt{\sinh ^2(x)-1} E(i x|-1)}{\sqrt{1-\sinh ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + Sinh[x]^2],x]

[Out]

((-I)*EllipticE[I*x, -1]*Sqrt[-1 + Sinh[x]^2])/Sqrt[1 - Sinh[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+\sinh ^2(x)} \, dx &=\frac{\sqrt{-1+\sinh ^2(x)} \int \sqrt{1-\sinh ^2(x)} \, dx}{\sqrt{1-\sinh ^2(x)}}\\ &=-\frac{i E(i x|-1) \sqrt{-1+\sinh ^2(x)}}{\sqrt{1-\sinh ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0329341, size = 33, normalized size = 1. \[ \frac{i \sqrt{3-\cosh (2 x)} E(i x|-1)}{\sqrt{\cosh (2 x)-3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + Sinh[x]^2],x]

[Out]

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

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Maple [A]  time = 0.064, size = 61, normalized size = 1.9 \begin{align*}{\frac{i{\it EllipticE} \left ( i\sinh \left ( x \right ) ,i \right ) }{\cosh \left ( x \right ) }\sqrt{ \left ( -1+ \left ( \sinh \left ( x \right ) \right ) ^{2} \right ) \left ( \cosh \left ( x \right ) \right ) ^{2}}\sqrt{ \left ( \cosh \left ( x \right ) \right ) ^{2}}\sqrt{1- \left ( \sinh \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{ \left ( \sinh \left ( x \right ) \right ) ^{4}-1}}}{\frac{1}{\sqrt{-1+ \left ( \sinh \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+sinh(x)^2)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sinh(x)^2 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(sinh(x)^2 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+sinh(x)**2)**(1/2),x)

[Out]

Integral(sqrt(sinh(x)**2 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sinh(x)^2 - 1), x)