Optimal. Leaf size=45 \[ \frac {1}{4} (x-1)^{3/2} x (x+1)^{3/2}+\frac {1}{8} \sqrt {x-1} x \sqrt {x+1}-\frac {1}{8} \cosh ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {90, 38, 52} \[ \frac {1}{4} (x-1)^{3/2} x (x+1)^{3/2}+\frac {1}{8} \sqrt {x-1} x \sqrt {x+1}-\frac {1}{8} \cosh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 38
Rule 52
Rule 90
Rubi steps
\begin {align*} \int \sqrt {-1+x} x^2 \sqrt {1+x} \, dx &=\frac {1}{4} (-1+x)^{3/2} x (1+x)^{3/2}+\frac {1}{4} \int \sqrt {-1+x} \sqrt {1+x} \, dx\\ &=\frac {1}{8} \sqrt {-1+x} x \sqrt {1+x}+\frac {1}{4} (-1+x)^{3/2} x (1+x)^{3/2}-\frac {1}{8} \int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx\\ &=\frac {1}{8} \sqrt {-1+x} x \sqrt {1+x}+\frac {1}{4} (-1+x)^{3/2} x (1+x)^{3/2}-\frac {1}{8} \cosh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 63, normalized size = 1.40 \[ \frac {x \sqrt {x+1} \left (2 x^3-2 x^2-x+1\right )+2 \sqrt {1-x} \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{8 \sqrt {x-1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.22, size = 40, normalized size = 0.89 \[ \frac {1}{8} \, {\left (2 \, x^{3} - x\right )} \sqrt {x + 1} \sqrt {x - 1} + \frac {1}{8} \, \log \left (\sqrt {x + 1} \sqrt {x - 1} - x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.29, size = 70, normalized size = 1.56 \[ \frac {1}{24} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {x - 1} + \frac {1}{6} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {x - 1} + \frac {1}{4} \, \log \left (\sqrt {x + 1} - \sqrt {x - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 1.16 \[ -\frac {\sqrt {x -1}\, \sqrt {x +1}\, \left (-2 \sqrt {x^{2}-1}\, x^{3}+\sqrt {x^{2}-1}\, x +\ln \left (x +\sqrt {x^{2}-1}\right )\right )}{8 \sqrt {x^{2}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 37, normalized size = 0.82 \[ \frac {1}{4} \, {\left (x^{2} - 1\right )}^{\frac {3}{2}} x + \frac {1}{8} \, \sqrt {x^{2} - 1} x - \frac {1}{8} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.55, size = 362, normalized size = 8.04 \[ -\frac {\mathrm {atanh}\left (\frac {\sqrt {x-1}-\mathrm {i}}{\sqrt {x+1}-1}\right )}{2}+\frac {\frac {35\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^3}{2\,{\left (\sqrt {x+1}-1\right )}^3}+\frac {273\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^5}{2\,{\left (\sqrt {x+1}-1\right )}^5}+\frac {715\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^7}{2\,{\left (\sqrt {x+1}-1\right )}^7}+\frac {715\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^9}{2\,{\left (\sqrt {x+1}-1\right )}^9}+\frac {273\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^{11}}{2\,{\left (\sqrt {x+1}-1\right )}^{11}}+\frac {35\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^{13}}{2\,{\left (\sqrt {x+1}-1\right )}^{13}}+\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^{15}}{2\,{\left (\sqrt {x+1}-1\right )}^{15}}+\frac {\sqrt {x-1}-\mathrm {i}}{2\,\left (\sqrt {x+1}-1\right )}}{1+\frac {28\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {x+1}-1\right )}^4}-\frac {56\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {x+1}-1\right )}^6}+\frac {70\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {x+1}-1\right )}^8}-\frac {56\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {x+1}-1\right )}^{10}}+\frac {28\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {x+1}-1\right )}^{12}}-\frac {8\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {x+1}-1\right )}^{14}}+\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {x+1}-1\right )}^{16}}-\frac {8\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}} \]
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 \[ \int x^{2} \sqrt {x - 1} \sqrt {x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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