Optimal. Leaf size=51 \[ \frac {(x+1)^{3/2}}{2 (1-x)}+\frac {5 \sqrt {x+1}}{2}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {2}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.05, antiderivative size = 51, 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, 78, 50, 63, 206} \[ \frac {(x+1)^{3/2}}{2 (1-x)}+\frac {5 \sqrt {x+1}}{2}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
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)^2} \, dx\\ &=\frac {(1+x)^{3/2}}{2 (1-x)}-\frac {5}{4} \int \frac {\sqrt {1+x}}{1-x} \, dx\\ &=\frac {5 \sqrt {1+x}}{2}+\frac {(1+x)^{3/2}}{2 (1-x)}-\frac {5}{2} \int \frac {1}{(1-x) \sqrt {1+x}} \, dx\\ &=\frac {5 \sqrt {1+x}}{2}+\frac {(1+x)^{3/2}}{2 (1-x)}-5 \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {5 \sqrt {1+x}}{2}+\frac {(1+x)^{3/2}}{2 (1-x)}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {2}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 0.78 \[ \frac {\sqrt {x+1} (2 x-3)}{x-1}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 91, normalized size = 1.78 \[ \frac {5 \, \sqrt {2} {\left (x^{2} - 2 \, x + 1\right )} \log \left (-\frac {x^{2} + 2 \, \sqrt {2} \sqrt {-x^{2} + 1} \sqrt {-x + 1} + 2 \, x - 3}{x^{2} - 2 \, x + 1}\right ) - 4 \, \sqrt {-x^{2} + 1} {\left (2 \, x - 3\right )} \sqrt {-x + 1}}{4 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 49, normalized size = 0.96 \[ \frac {5}{4} \, \sqrt {2} \log \left (\frac {\sqrt {2} - \sqrt {x + 1}}{\sqrt {2} + \sqrt {x + 1}}\right ) + 2 \, \sqrt {x + 1} - \frac {\sqrt {x + 1}}{x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 78, normalized size = 1.53 \[ \frac {\sqrt {-x^{2}+1}\, \sqrt {1-x}\, \left (5 \sqrt {2}\, \arctanh \left (\frac {\sqrt {1+x}\, \sqrt {2}}{2}\right ) x -5 \arctanh \left (\frac {\sqrt {1+x}\, \sqrt {2}}{2}\right ) \sqrt {2}-4 \sqrt {1+x}\, x +6 \sqrt {1+x}\right )}{2 \left (-1+x \right )^{2} \sqrt {1+x}} \]
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 \[ \int \frac {{\left (x + 1\right )} x}{\sqrt {-x^{2} + 1} {\left (-x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x\,\left (x+1\right )}{\sqrt {1-x^2}\,{\left (1-x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \left (x + 1\right )}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \left (1 - x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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