Optimal. Leaf size=23 \[ -\frac {2}{3} (1+x)^{3/2}+\frac {2}{5} (1+x)^{5/2} \]
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Rubi [A]
time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6263, 26, 45}
\begin {gather*} \frac {2}{5} (x+1)^{5/2}-\frac {2}{3} (x+1)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 26
Rule 45
Rule 6263
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(x)} \sqrt {1-x} x \, dx &=\int \frac {x \sqrt {1-x^2}}{\sqrt {1-x}} \, dx\\ &=\int x \sqrt {1+x} \, dx\\ &=\int \left (-\sqrt {1+x}+(1+x)^{3/2}\right ) \, dx\\ &=-\frac {2}{3} (1+x)^{3/2}+\frac {2}{5} (1+x)^{5/2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 0.70 \begin {gather*} \frac {2}{15} (1+x)^{3/2} (-2+3 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(31\) vs.
\(2(15)=30\).
time = 1.07, size = 32, normalized size = 1.39
method | result | size |
gosper | \(\frac {2 \left (1+x \right )^{2} \left (3 x -2\right ) \sqrt {1-x}}{15 \sqrt {-x^{2}+1}}\) | \(29\) |
default | \(-\frac {2 \sqrt {-x^{2}+1}\, \sqrt {1-x}\, \left (1+x \right ) \left (3 x -2\right )}{15 \left (x -1\right )}\) | \(32\) |
risch | \(-\frac {2 \sqrt {\frac {\left (-x^{2}+1\right ) \left (1-x \right )}{\left (x -1\right )^{2}}}\, \left (x -1\right ) \left (3 x^{2}+x -2\right ) \sqrt {1+x}}{15 \sqrt {-x^{2}+1}\, \sqrt {1-x}}\) | \(55\) |
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. 38 vs.
\(2 (15) = 30\).
time = 0.27, size = 38, normalized size = 1.65 \begin {gather*} \frac {2 \, {\left (3 \, x^{3} - x^{2} + 4 \, x + 8\right )}}{15 \, \sqrt {x + 1}} + \frac {2 \, {\left (x^{2} - x - 2\right )}}{3 \, \sqrt {x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (15) = 30\).
time = 0.36, size = 31, normalized size = 1.35 \begin {gather*} -\frac {2 \, {\left (3 \, x^{2} + x - 2\right )} \sqrt {-x^{2} + 1} \sqrt {-x + 1}}{15 \, {\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 \sqrt {1 - x} \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 = 0.41, size = 20, normalized size = 0.87 \begin {gather*} \frac {2}{5} \, {\left (x + 1\right )}^{\frac {5}{2}} - \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {4}{15} \, \sqrt {2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.89, size = 42, normalized size = 1.83 \begin {gather*} \frac {4\,\sqrt {1-x^2}}{15\,\sqrt {1-x}}-\frac {2\,\left (3\,x+4\right )\,\sqrt {1-x^2}\,\sqrt {1-x}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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