3.2.26 \(\int \frac {1}{\sqrt {1-\sinh ^2(x)}} \, dx\) [126]

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

[Out]

-I*(cosh(x)^2)^(1/2)/cosh(x)*EllipticF(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 = {3261} \begin {gather*} -i F(i x|-1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 - Sinh[x]^2],x]

[Out]

(-I)*EllipticF[I*x, -1]

Rule 3261

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1-\sinh ^2(x)}} \, dx &=-i F(i x|-1)\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 11, normalized size = 1.00 \begin {gather*} -i F(i x|-1) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 - Sinh[x]^2],x]

[Out]

(-I)*EllipticF[I*x, -1]

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Maple [A]
time = 0.77, size = 41, normalized size = 3.73

method	result	size
default	\(\frac {\sqrt {-\left (-1+\sinh ^{2}\left (x \right )\right ) \left (\cosh ^{2}\left (x \right )\right )}\, \sqrt {\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \EllipticF \left (\sinh \left (x \right ), i\right )}{\sqrt {1-\left (\sinh ^{4}\left (x \right )\right )}\, \cosh \left (x \right )}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate(1/sqrt(-sinh(x)^2 + 1), x)

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Fricas [F]
time = 0.09, size = 1, normalized size = 0.09 \begin {gather*} 0 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(1 - sinh(x)**2), 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/(1-sinh(x)^2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-sinh(x)^2 + 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.09 \begin {gather*} \int \frac {1}{\sqrt {1-{\mathrm {sinh}\left (x\right )}^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1 - sinh(x)^2)^(1/2),x)

[Out]

int(1/(1 - sinh(x)^2)^(1/2), x)

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