Optimal. Leaf size=18 \[ -\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3 \]
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Rubi [A]
time = 0.04, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6310, 6313,
270} \begin {gather*} -\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 6310
Rule 6313
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(x)} (1-x) x \, dx &=-\int e^{\coth ^{-1}(x)} \left (1-\frac {1}{x}\right ) x^2 \, dx\\ &=\text {Subst}\left (\int \frac {\sqrt {1-x^2}}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 21, normalized size = 1.17 \begin {gather*} -\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x \left (-1+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 22, normalized size = 1.22
method | result | size |
gosper | \(-\frac {\left (1+x \right ) \left (-1+x \right )^{2}}{3 \sqrt {\frac {-1+x}{1+x}}}\) | \(22\) |
default | \(-\frac {\left (1+x \right ) \left (-1+x \right )^{2}}{3 \sqrt {\frac {-1+x}{1+x}}}\) | \(22\) |
risch | \(-\frac {\left (x^{2}-1\right ) \left (-1+x \right )}{3 \sqrt {\frac {-1+x}{1+x}}}\) | \(22\) |
trager | \(-\frac {\left (1+x \right ) \left (x^{2}-1\right ) \sqrt {-\frac {1-x}{1+x}}}{3}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 50 vs.
\(2 (14) = 28\).
time = 0.27, size = 50, normalized size = 2.78 \begin {gather*} \frac {8 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {3}{2}}}{3 \, {\left (\frac {3 \, {\left (x - 1\right )}}{x + 1} - \frac {3 \, {\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 24, normalized size = 1.33 \begin {gather*} -\frac {1}{3} \, {\left (x^{3} + x^{2} - x - 1\right )} \sqrt {\frac {x - 1}{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 \left (- \frac {x}{\sqrt {\frac {x}{x + 1} - \frac {1}{x + 1}}}\right )\, dx - \int \frac {x^{2}}{\sqrt {\frac {x}{x + 1} - \frac {1}{x + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 15, normalized size = 0.83 \begin {gather*} -\frac {{\left (x^{2} - 1\right )}^{\frac {3}{2}}}{3 \, \mathrm {sgn}\left (x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 18, normalized size = 1.00 \begin {gather*} -\frac {{\left (\frac {x-1}{x+1}\right )}^{3/2}\,{\left (x+1\right )}^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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