Optimal. Leaf size=4 \[ E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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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.
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Rule 435
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.
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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.
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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.
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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.
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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.
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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.
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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.
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