3.127 \(\int \frac{1}{\sqrt{-1+\sinh ^2(x)}} \, dx\)

Optimal. Leaf size=33 \[ -\frac{i \sqrt{1-\sinh ^2(x)} \text{EllipticF}(i x,-1)}{\sqrt{\sinh ^2(x)-1}} \]

[Out]

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

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Rubi [A]  time = 0.021624, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3183, 3182} \[ -\frac{i \sqrt{1-\sinh ^2(x)} F(i x|-1)}{\sqrt{\sinh ^2(x)-1}} \]

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 3183

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]

Rule 3182

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

Mathematica [A]  time = 0.0414932, size = 33, normalized size = 1. \[ -\frac{i \sqrt{3-\cosh (2 x)} \text{EllipticF}(i x,-1)}{\sqrt{\cosh (2 x)-3}} \]

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 = 0.068, size = 61, normalized size = 1.9 \begin{align*}{\frac{-i{\it EllipticF} \left ( i\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}}\sqrt{1- \left ( \sinh \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{ \left ( \sinh \left ( x \right ) \right ) ^{4}-1}}}{\frac{1}{\sqrt{-1+ \left ( \sinh \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\sinh \left (x\right )^{2} - 1}}\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{\sinh \left (x\right )^{2} - 1}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(sinh(x)^2 - 1), x)

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

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\sinh \left (x\right )^{2} - 1}}\,{d x} \end{align*}

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)