Optimal. Leaf size=130 \[ \frac {5 \left (1-\frac {1}{x}\right )^{3/2} \sqrt {1+\frac {1}{x}} x^2}{2 (1-x)^{3/2}}-\frac {\sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{2 (1-x)^{3/2}}-\frac {5 \left (1-\frac {1}{x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {2} (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6311, 6316, 98,
96, 95, 212} \begin {gather*} -\frac {\sqrt {1-\frac {1}{x}} \left (\frac {1}{x}+1\right )^{3/2} x^2}{2 (1-x)^{3/2}}+\frac {5 \left (1-\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{x}+1} x^2}{2 (1-x)^{3/2}}-\frac {5 \left (1-\frac {1}{x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{x}+1}}\right )}{\sqrt {2} (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 98
Rule 212
Rule 6311
Rule 6316
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx &=\frac {\left (\left (1-\frac {1}{x}\right )^{3/2} x^{3/2}\right ) \int \frac {e^{\coth ^{-1}(x)}}{\left (1-\frac {1}{x}\right )^{3/2} \sqrt {x}} \, dx}{(1-x)^{3/2}}\\ &=-\frac {\left (1-\frac {1}{x}\right )^{3/2} \text {Subst}\left (\int \frac {\sqrt {1+x}}{(1-x)^2 x^{3/2}} \, dx,x,\frac {1}{x}\right )}{(1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\\ &=-\frac {\sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{2 (1-x)^{3/2}}-\frac {\left (5 \left (1-\frac {1}{x}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x}}{(1-x) x^{3/2}} \, dx,x,\frac {1}{x}\right )}{4 (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\\ &=\frac {5 \left (1-\frac {1}{x}\right )^{3/2} \sqrt {1+\frac {1}{x}} x^2}{2 (1-x)^{3/2}}-\frac {\sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{2 (1-x)^{3/2}}-\frac {\left (5 \left (1-\frac {1}{x}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {x} \sqrt {1+x}} \, dx,x,\frac {1}{x}\right )}{2 (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\\ &=\frac {5 \left (1-\frac {1}{x}\right )^{3/2} \sqrt {1+\frac {1}{x}} x^2}{2 (1-x)^{3/2}}-\frac {\sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{2 (1-x)^{3/2}}-\frac {\left (5 \left (1-\frac {1}{x}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{(1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\\ &=\frac {5 \left (1-\frac {1}{x}\right )^{3/2} \sqrt {1+\frac {1}{x}} x^2}{2 (1-x)^{3/2}}-\frac {\sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{2 (1-x)^{3/2}}-\frac {5 \left (1-\frac {1}{x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {2} (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 75, normalized size = 0.58 \begin {gather*} -\frac {\sqrt {\frac {-1+x}{x}} x \left (2 \sqrt {1+\frac {1}{x}} (3-2 x)+5 \sqrt {2} (-1+x) \sqrt {\frac {1}{x}} \tanh ^{-1}\left (\sqrt {2} \sqrt {\frac {1}{1+x}}\right )\right )}{2 (1-x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 90, normalized size = 0.69
method | result | size |
default | \(-\frac {\sqrt {1-x}\, \left (5 \sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right ) x -5 \sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right )-4 x \sqrt {-1-x}+6 \sqrt {-1-x}\right )}{2 \sqrt {\frac {-1+x}{1+x}}\, \left (-1+x \right ) \sqrt {-1-x}}\) | \(90\) |
risch | \(-\frac {\left (2 x^{2}-x -3\right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{-1+x}}}{\sqrt {-1-x}\, \sqrt {\frac {-1+x}{1+x}}\, \left (1+x \right ) \sqrt {1-x}}-\frac {5 \sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{-1+x}}\, \left (-1+x \right )}{2 \sqrt {\frac {-1+x}{1+x}}\, \left (1+x \right ) \sqrt {1-x}}\) | \(120\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 84, normalized size = 0.65 \begin {gather*} -\frac {5 \, \sqrt {2} {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{x - 1}\right ) - 2 \, {\left (2 \, x^{2} - x - 3\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 79.46, size = 126, normalized size = 0.97 \begin {gather*} 2 \left (\begin {cases} \sqrt {2} \left (\frac {\sqrt {2} \sqrt {- x - 1}}{2} - \operatorname {acos}{\left (\frac {\sqrt {2}}{\sqrt {1 - x}} \right )}\right ) & \text {for}\: \sqrt {1 - x} > - \sqrt {2} \wedge \sqrt {1 - x} < \sqrt {2} \end {cases}\right ) - 2 \left (\begin {cases} \frac {\sqrt {2} \left (\frac {\operatorname {acos}{\left (\frac {\sqrt {2}}{\sqrt {1 - x}} \right )}}{2} - \frac {\sqrt {2} \sqrt {1 - \frac {2}{1 - x}}}{2 \sqrt {1 - x}}\right )}{2} & \text {for}\: \sqrt {1 - x} > - \sqrt {2} \wedge \sqrt {1 - x} < \sqrt {2} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 41, normalized size = 0.32 \begin {gather*} \frac {5}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-x - 1}\right ) - 2 \, \sqrt {-x - 1} + \frac {\sqrt {-x - 1}}{x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x}{\sqrt {\frac {x-1}{x+1}}\,{\left (1-x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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