3.1.91 \(\int \sqrt {-1+\sinh ^2(x)} \, dx\) [91]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3257, 3256} \begin {gather*} -\frac {i \sqrt {\sinh ^2(x)-1} E(i x|-1)}{\sqrt {1-\sinh ^2(x)}} \end {gather*}

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 3256

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

Rule 3257

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]

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*}

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Mathematica [A]
time = 0.02, size = 33, normalized size = 1.00 \begin {gather*} \frac {i \sqrt {3-\cosh (2 x)} E(i x|-1)}{\sqrt {-3+\cosh (2 x)}} \end {gather*}

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 = 0.77, size = 61, normalized size = 1.85

method	result	size
default	\(\frac {i \sqrt {\left (-1+\sinh ^{2}\left (x \right )\right ) \left (\cosh ^{2}\left (x \right )\right )}\, \sqrt {\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \sqrt {1-\left (\sinh ^{2}\left (x \right )\right )}\, \EllipticE \left (i \sinh \left (x \right ), i\right )}{\sqrt {\sinh ^{4}\left (x \right )-1}\, \cosh \left (x \right ) \sqrt {-1+\sinh ^{2}\left (x \right )}}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.09, size = 10, normalized size = 0.30 \begin {gather*} {\rm integral}\left (\sqrt {\sinh \left (x\right )^{2} - 1}, x\right ) \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\sinh ^{2}{\left (x \right )} - 1}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \sqrt {{\mathrm {sinh}\left (x\right )}^2-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sinh(x)^2 - 1)^(1/2),x)

[Out]

int((sinh(x)^2 - 1)^(1/2), x)

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