Optimal. Leaf size=11 \[ -2 \sqrt {1-x} \]
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Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6264, 32}
\begin {gather*} -2 \sqrt {1-x} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 6264
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(x)}}{\sqrt {1+x}} \, dx &=\int \frac {1}{\sqrt {1-x}} \, dx\\ &=-2 \sqrt {1-x}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} -2 \sqrt {1-x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.78, size = 17, normalized size = 1.55
method | result | size |
default | \(-\frac {2 \sqrt {-x^{2}+1}}{\sqrt {1+x}}\) | \(17\) |
gosper | \(\frac {2 \left (x -1\right ) \sqrt {1+x}}{\sqrt {-x^{2}+1}}\) | \(20\) |
risch | \(\frac {2 \sqrt {\frac {-x^{2}+1}{1+x}}\, \sqrt {1+x}\, \left (x -1\right )}{\sqrt {-x^{2}+1}\, \sqrt {1-x}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 3 vs. order
2.
time = 0.26, size = 12, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (x - 1\right )}}{\sqrt {-x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order
2.
time = 0.34, size = 16, normalized size = 1.45 \begin {gather*} -\frac {2 \, \sqrt {-x^{2} + 1}}{\sqrt {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 {\sqrt {x + 1}}{\sqrt {- \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 = 0.41, size = 15, normalized size = 1.36 \begin {gather*} 2 \, \sqrt {2} - 2 \, \sqrt {-x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.93, size = 16, normalized size = 1.45 \begin {gather*} -\frac {2\,\sqrt {1-x^2}}{\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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