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

Optimal. Leaf size=11 \[ -i E(i x|-1) \]

[Out]

(-I)*EllipticE[I*x, -1]

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Rubi [A]  time = 0.0102307, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3177} \[ -i E(i x|-1) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - Sinh[x]^2],x]

[Out]

(-I)*EllipticE[I*x, -1]

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 &=-i E(i x|-1)\\ \end{align*}

Mathematica [A]  time = 0.024372, size = 11, normalized size = 1. \[ -i E(i x|-1) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - Sinh[x]^2],x]

[Out]

(-I)*EllipticE[I*x, -1]

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Maple [B]  time = 0.07, size = 51, normalized size = 4.6 \begin{align*}{\frac{2\,{\it EllipticF} \left ( \sinh \left ( x \right ) ,i \right ) -{\it EllipticE} \left ( \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}}{\frac{1}{\sqrt{1- \left ( \sinh \left ( x \right ) \right ) ^{4}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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{1 - \sinh ^{2}{\left (x \right )}}\, 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(1 - sinh(x)**2), 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)