3.2.87 \(\int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx\) [187]

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

[Out]

EllipticE(x,I)

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Rubi [A]
time = 0.00, antiderivative size = 4, normalized size of antiderivative = 1.00, 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 = {435} \begin {gather*} E(\text {ArcSin}(x)|-1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x^2]/Sqrt[1 - x^2],x]

[Out]

EllipticE[ArcSin[x], -1]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], 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*}

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Mathematica [A]
time = 0.27, size = 4, normalized size = 1.00 \begin {gather*} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \end {gather*}

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 = 0.10, size = 5, normalized size = 1.25

method	result	size
default	\(\EllipticE \left (x , i\right )\)	\(5\)
elliptic	\(\frac {\sqrt {-x^{4}+1}\, \left (\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \EllipticF \left (x , i\right )}{\sqrt {-x^{4}+1}}-\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (x , i\right )-\EllipticE \left (x , i\right )\right )}{\sqrt {-x^{4}+1}}\right )}{\sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}}\)	\(96\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

EllipticE(x,I)

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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((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 [B] Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (3) = 6\).
time = 0.24, size = 21, normalized size = 5.25 \begin {gather*} -\frac {\sqrt {x^{2} + 1} \sqrt {-x^{2} + 1}}{x} \end {gather*}

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]

-sqrt(x^2 + 1)*sqrt(-x^2 + 1)/x

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 10 vs. \(2 (2) = 4\).
time = 1.31, size = 10, normalized size = 2.50 \begin {gather*} \begin {cases} E\left (\operatorname {asin}{\left (x \right )}\middle | -1\right ) & \text {for}\: x > -1 \wedge x < 1 \end {cases} \end {gather*}

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 = 0.00, 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((x^2+1)^(1/2)/(-x^2+1)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.25 \begin {gather*} \int \frac {\sqrt {x^2+1}}{\sqrt {1-x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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