Optimal. Leaf size=18 \[ -\frac {\sqrt {1-x}}{\sqrt {x+1}} \]
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Rubi [A] time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6129, 37} \[ -\frac {\sqrt {1-x}}{\sqrt {x+1}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 6129
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(x)}}{(1+x)^2} \, dx &=\int \frac {1}{\sqrt {1-x} (1+x)^{3/2}} \, dx\\ &=-\frac {\sqrt {1-x}}{\sqrt {1+x}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 1.00 \[ -\frac {\sqrt {1-x}}{\sqrt {x+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 19, normalized size = 1.06 \[ -\frac {x + \sqrt {-x^{2} + 1} + 1}{x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 21, normalized size = 1.17 \[ \frac {2}{\frac {\sqrt {-x^{2} + 1} - 1}{x} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 14, normalized size = 0.78 \[ \frac {-1+x}{\sqrt {-x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 16, normalized size = 0.89 \[ -\frac {\sqrt {-x^{2} + 1}}{x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 13, normalized size = 0.72 \[ \frac {x-1}{\sqrt {1-x^2}} \]
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 \[ \int \frac {1}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \left (x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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