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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, size = 11, normalized size = 1.00 \[ -i E(i x|-1) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - Sinh[x]^2],x]

[Out]

(-I)*EllipticE[I*x, -1]

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fricas [F]  time = 1.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-\sinh \relax (x)^{2} + 1}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-\sinh \relax (x)^{2} + 1}\,{d x} \]

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)

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maple [B]  time = 0.12, size = 51, normalized size = 4.64 \[ \frac {\sqrt {-\left (-1+\sinh ^{2}\relax (x )\right ) \left (\cosh ^{2}\relax (x )\right )}\, \sqrt {\frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}}\, \left (2 \EllipticF \left (\sinh \relax (x ), i\right )-\EllipticE \left (\sinh \relax (x ), i\right )\right )}{\sqrt {1-\left (\sinh ^{4}\relax (x )\right )}\, \cosh \relax (x )} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-\sinh \relax (x)^{2} + 1}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.09 \[ \int \sqrt {1-{\mathrm {sinh}\relax (x)}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - sinh(x)^2)^(1/2),x)

[Out]

int((1 - sinh(x)^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {1 - \sinh ^{2}{\relax (x )}}\, dx \]

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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