Optimal. Leaf size=18 \[ -\frac {\sqrt {1-x}}{\sqrt {1+x}} \]
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Rubi [A]
time = 0.01, 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 = {6264, 37}
\begin {gather*} -\frac {\sqrt {1-x}}{\sqrt {x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 6264
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 \begin {gather*} -\frac {\sqrt {1-x}}{\sqrt {1+x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.02, size = 22, normalized size = 1.22
method | result | size |
gosper | \(\frac {x -1}{\sqrt {-x^{2}+1}}\) | \(14\) |
risch | \(\frac {x -1}{\sqrt {-x^{2}+1}}\) | \(14\) |
trager | \(-\frac {\sqrt {-x^{2}+1}}{1+x}\) | \(17\) |
default | \(-\frac {\sqrt {-\left (1+x \right )^{2}+2+2 x}}{1+x}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 16, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {-x^{2} + 1}}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 19, normalized size = 1.06 \begin {gather*} -\frac {x + \sqrt {-x^{2} + 1} + 1}{x + 1} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 21, normalized size = 1.17 \begin {gather*} \frac {2}{\frac {\sqrt {-x^{2} + 1} - 1}{x} - 1} \end {gather*}
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 \begin {gather*} \frac {x-1}{\sqrt {1-x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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