3.2.27 \(\int \frac {1}{\sqrt {-1+\sinh ^2(x)}} \, dx\) [127]

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

[Out]

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

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Rubi [A]
time = 0.01, 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 = {3262, 3261} \begin {gather*} -\frac {i \sqrt {1-\sinh ^2(x)} F(i x|-1)}{\sqrt {\sinh ^2(x)-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-1 + Sinh[x]^2],x]

[Out]

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

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]

Rule 3262

Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[Sqrt[1 + b*(Sin[e + f*x]^2/a)]/Sqrt[a +
b*Sin[e + f*x]^2], Int[1/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 \frac {1}{\sqrt {-1+\sinh ^2(x)}} \, dx &=\frac {\sqrt {1-\sinh ^2(x)} \int \frac {1}{\sqrt {1-\sinh ^2(x)}} \, dx}{\sqrt {-1+\sinh ^2(x)}}\\ &=-\frac {i F(i x|-1) \sqrt {1-\sinh ^2(x)}}{\sqrt {-1+\sinh ^2(x)}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 33, normalized size = 1.00 \begin {gather*} -\frac {i \sqrt {3-\cosh (2 x)} F(i x|-1)}{\sqrt {-3+\cosh (2 x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-1 + Sinh[x]^2],x]

[Out]

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

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Maple [A]
time = 0.82, size = 61, normalized size = 1.85

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-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)/(sinh(x)^4-1)^(1/2)*EllipticF(I*sinh
(x),I)/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/(-1+sinh(x)^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(sinh(x)^2 - 1), x)

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Fricas [A]
time = 0.12, size = 42, normalized size = 1.27 \begin {gather*} -2 \, \sqrt {2 \, \sqrt {2} + 3} {\left (2 \, \sqrt {2} - 3\right )} F(\arcsin \left (\sqrt {2 \, \sqrt {2} + 3} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}\right )\,|\,-12 \, \sqrt {2} + 17) \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]

-2*sqrt(2*sqrt(2) + 3)*(2*sqrt(2) - 3)*elliptic_f(arcsin(sqrt(2*sqrt(2) + 3)*(cosh(x) + sinh(x))), -12*sqrt(2)
+ 17)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\sinh ^{2}{\left (x \right )} - 1}}\, 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(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/(-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 = -0.03 \begin {gather*} \int \frac {1}{\sqrt {{\mathrm {sinh}\left (x\right )}^2-1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^2 - 1)^(1/2),x)

[Out]

int(1/(sinh(x)^2 - 1)^(1/2), x)

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