Optimal. Leaf size=34 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )-2 \sqrt {1-x} \]
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Rubi [A] time = 0.05, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6129, 80, 63, 206} \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )-2 \sqrt {1-x} \]
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 206
Rule 6129
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(x)} x}{(1+x)^{3/2}} \, dx &=\int \frac {x}{\sqrt {1-x} (1+x)} \, dx\\ &=-2 \sqrt {1-x}-\int \frac {1}{\sqrt {1-x} (1+x)} \, dx\\ &=-2 \sqrt {1-x}+2 \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1-x}\right )\\ &=-2 \sqrt {1-x}+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 1.00 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )-2 \sqrt {1-x} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 71, normalized size = 2.09 \[ \frac {\sqrt {2} {\left (x + 1\right )} \log \left (-\frac {x^{2} - 2 \, \sqrt {2} \sqrt {-x^{2} + 1} \sqrt {x + 1} - 2 \, x - 3}{x^{2} + 2 \, x + 1}\right ) - 4 \, \sqrt {-x^{2} + 1} \sqrt {x + 1}}{2 \, {\left (x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 43, normalized size = 1.26 \[ -\frac {1}{2} \, \sqrt {2} \log \left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {2} + \sqrt {-x + 1}}\right ) - 2 \, \sqrt {-x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 50, normalized size = 1.47 \[ \frac {\sqrt {-x^{2}+1}\, \left (\arctanh \left (\frac {\sqrt {1-x}\, \sqrt {2}}{2}\right ) \sqrt {2}-2 \sqrt {1-x}\right )}{\sqrt {1+x}\, \sqrt {1-x}} \]
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 \[ \int \frac {x}{\sqrt {-x^{2} + 1} \sqrt {x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x}{\sqrt {1-x^2}\,\sqrt {x+1}} \,d x \]
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 \[ \int \frac {x}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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