Optimal. Leaf size=42 \[ -\frac {2}{3} (x+1)^{3/2}-2 \sqrt {x+1}+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6129, 80, 50, 63, 206} \[ -\frac {2}{3} (x+1)^{3/2}-2 \sqrt {x+1}+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rule 6129
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(x)} x}{\sqrt {1-x}} \, dx &=\int \frac {x \sqrt {1+x}}{1-x} \, dx\\ &=-\frac {2}{3} (1+x)^{3/2}+\int \frac {\sqrt {1+x}}{1-x} \, dx\\ &=-2 \sqrt {1+x}-\frac {2}{3} (1+x)^{3/2}+2 \int \frac {1}{(1-x) \sqrt {1+x}} \, dx\\ &=-2 \sqrt {1+x}-\frac {2}{3} (1+x)^{3/2}+4 \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+x}\right )\\ &=-2 \sqrt {1+x}-\frac {2}{3} (1+x)^{3/2}+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 36, normalized size = 0.86 \[ 2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {2}}\right )-\frac {2}{3} \sqrt {x+1} (x+4) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 79, normalized size = 1.88 \[ \frac {3 \, \sqrt {2} {\left (x - 1\right )} \log \left (-\frac {x^{2} - 2 \, \sqrt {2} \sqrt {-x^{2} + 1} \sqrt {-x + 1} + 2 \, x - 3}{x^{2} - 2 \, x + 1}\right ) + 2 \, \sqrt {-x^{2} + 1} {\left (x + 4\right )} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 44, normalized size = 1.05 \[ -\frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \sqrt {2} \log \left (\frac {\sqrt {2} - \sqrt {x + 1}}{\sqrt {2} + \sqrt {x + 1}}\right ) - 2 \, \sqrt {x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 61, normalized size = 1.45 \[ -\frac {2 \sqrt {-x^{2}+1}\, \sqrt {1-x}\, \left (3 \arctanh \left (\frac {\sqrt {1+x}\, \sqrt {2}}{2}\right ) \sqrt {2}-\sqrt {1+x}\, x -4 \sqrt {1+x}\right )}{3 \left (-1+x \right ) \sqrt {1+x}} \]
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 \[ \int \frac {{\left (x + 1\right )} x}{\sqrt {-x^{2} + 1} \sqrt {-x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,\left (x+1\right )}{\sqrt {1-x^2}\,\sqrt {1-x}} \,d x \]
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 \[ \int \frac {x \left (x + 1\right )}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {1 - x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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