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

Optimal. Leaf size=4 \[ E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

EllipticE[ArcSin[x], -1]

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Rubi [A]  time = 0.0052178, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {424} \[ E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x^2]/Sqrt[1 - x^2],x]

[Out]

EllipticE[ArcSin[x], -1]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{1+x^2}}{\sqrt{1-x^2}} \, dx &=E\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end{align*}

Mathematica [A]  time = 0.003282, size = 4, normalized size = 1. \[ E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x^2]/Sqrt[1 - x^2],x]

[Out]

EllipticE[ArcSin[x], -1]

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Maple [A]  time = 0.012, size = 5, normalized size = 1.3 \begin{align*}{\it EllipticE} \left ( x,i \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

EllipticE(x,I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} + 1}}{\sqrt{-x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^2 + 1)/sqrt(-x^2 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{x^{2} + 1} \sqrt{-x^{2} + 1}}{x^{2} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(x^2 + 1)*sqrt(-x^2 + 1)/(x^2 - 1), x)

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Sympy [B]  time = 2.13838, size = 10, normalized size = 2.5 \begin{align*} \begin{cases} E\left (\operatorname{asin}{\left (x \right )}\middle | -1\right ) & \text{for}\: x > -1 \wedge x < 1 \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((elliptic_e(asin(x), -1), (x > -1) & (x < 1)))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} + 1}}{\sqrt{-x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^2 + 1)/sqrt(-x^2 + 1), x)