Optimal. Leaf size=34 \[ -\frac {4}{3} (1+x)^{3/2}+\frac {6}{5} (1+x)^{5/2}-\frac {2}{7} (1+x)^{7/2} \]
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Rubi [A]
time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.38, number of steps
used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {6263, 809, 641,
45} \begin {gather*} -\frac {2}{7} \sqrt {1-x} \left (1-x^2\right )^{3/2}+\frac {2}{35} (x+1)^{5/2}-\frac {4}{21} (x+1)^{3/2} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 45
Rule 641
Rule 809
Rule 6263
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(x)} (1-x)^{3/2} x \, dx &=\int \sqrt {1-x} x \sqrt {1-x^2} \, dx\\ &=-\frac {2}{7} \sqrt {1-x} \left (1-x^2\right )^{3/2}-\frac {1}{7} \int \sqrt {1-x} \sqrt {1-x^2} \, dx\\ &=-\frac {2}{7} \sqrt {1-x} \left (1-x^2\right )^{3/2}-\frac {1}{7} \int (1-x) \sqrt {1+x} \, dx\\ &=-\frac {2}{7} \sqrt {1-x} \left (1-x^2\right )^{3/2}-\frac {1}{7} \int \left (2 \sqrt {1+x}-(1+x)^{3/2}\right ) \, dx\\ &=-\frac {4}{21} (1+x)^{3/2}+\frac {2}{35} (1+x)^{5/2}-\frac {2}{7} \sqrt {1-x} \left (1-x^2\right )^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 21, normalized size = 0.62 \begin {gather*} -\frac {2}{105} (1+x)^{3/2} \left (22-33 x+15 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.13, size = 37, normalized size = 1.09
method | result | size |
gosper | \(-\frac {2 \left (1+x \right )^{2} \left (15 x^{2}-33 x +22\right ) \sqrt {1-x}}{105 \sqrt {-x^{2}+1}}\) | \(34\) |
default | \(\frac {2 \sqrt {-x^{2}+1}\, \sqrt {1-x}\, \left (1+x \right ) \left (15 x^{2}-33 x +22\right )}{105 \left (x -1\right )}\) | \(37\) |
risch | \(\frac {2 \sqrt {\frac {\left (-x^{2}+1\right ) \left (1-x \right )}{\left (x -1\right )^{2}}}\, \left (x -1\right ) \left (15 x^{3}-18 x^{2}-11 x +22\right ) \sqrt {1+x}}{105 \sqrt {-x^{2}+1}\, \sqrt {1-x}}\) | \(62\) |
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. 48 vs.
\(2 (22) = 44\).
time = 0.27, size = 48, normalized size = 1.41 \begin {gather*} -\frac {2 \, {\left (15 \, x^{4} - 24 \, x^{3} + 13 \, x^{2} - 52 \, x - 104\right )}}{105 \, \sqrt {x + 1}} - \frac {2 \, {\left (x^{3} - 2 \, x^{2} + 3 \, x + 6\right )}}{5 \, \sqrt {x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 38, normalized size = 1.12 \begin {gather*} \frac {2 \, {\left (15 \, x^{3} - 18 \, x^{2} - 11 \, x + 22\right )} \sqrt {-x^{2} + 1} \sqrt {-x + 1}}{105 \, {\left (x - 1\right )}} \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 \frac {x \left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 27, normalized size = 0.79 \begin {gather*} -\frac {2}{7} \, {\left (x + 1\right )}^{\frac {7}{2}} + \frac {6}{5} \, {\left (x + 1\right )}^{\frac {5}{2}} - \frac {4}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} + \frac {16}{105} \, \sqrt {2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.92, size = 33, normalized size = 0.97 \begin {gather*} \frac {2\,\sqrt {1-x^2}\,\left (-15\,x^3+18\,x^2+11\,x-22\right )}{105\,\sqrt {1-x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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