Optimal. Leaf size=13 \[ 2 \operatorname {EllipticF}\left (\sin ^{-1}(x),-1\right )-E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {423, 424, 248, 221} \[ 2 F\left (\left .\sin ^{-1}(x)\right |-1\right )-E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 221
Rule 248
Rule 423
Rule 424
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^2}}{\sqrt {1+x^2}} \, dx &=2 \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2}} \, dx-\int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx\\ &=-E\left (\left .\sin ^{-1}(x)\right |-1\right )+2 \int \frac {1}{\sqrt {1-x^4}} \, dx\\ &=-E\left (\left .\sin ^{-1}(x)\right |-1\right )+2 F\left (\left .\sin ^{-1}(x)\right |-1\right )\\ \end {align*}
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Mathematica [C] time = 0.00, size = 12, normalized size = 0.92 \[ -i E\left (\left .i \sinh ^{-1}(x)\right |-1\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{2} + 1}}{\sqrt {x^{2} + 1}}, x\right ) \]
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 \[ \int \frac {\sqrt {-x^{2} + 1}}{\sqrt {x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 14, normalized size = 1.08 \[ -\EllipticE \left (x , i\right )+2 \EllipticF \left (x , i\right ) \]
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 \[ \int \frac {\sqrt {-x^{2} + 1}}{\sqrt {x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.08 \[ \int \frac {\sqrt {1-x^2}}{\sqrt {x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt {x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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