Optimal. Leaf size=48 \[ \frac {\tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {442} \begin {gather*} \frac {\tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 442
Rubi steps
\begin {align*} \int \frac {x^2}{\left (-2+x^2\right ) \left (-1+x^2\right )^{3/4}} \, dx &=\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 48, normalized size = 1.00 \begin {gather*} -\frac {x^3 \left (1-x^2\right )^{3/4} F_1\left (\frac {3}{2};\frac {3}{4},1;\frac {5}{2};x^2,\frac {x^2}{2}\right )}{6 \left (x^2-1\right )^{3/4}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 2.02, size = 48, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 91, normalized size = 1.90 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (-\frac {x^{4} - 2 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{2} - 1} x^{2} - 4 \, \sqrt {2} {\left (x^{2} - 1\right )}^{\frac {3}{4}} x + 4 \, x^{2} - 4}{x^{4} - 4 \, x^{2} + 4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{{\left (x^{2} - 1\right )}^{\frac {3}{4}} {\left (x^{2} - 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.46, size = 121, normalized size = 2.52
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\left (x^{2}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+2\right )-x \sqrt {x^{2}-1}-\RootOf \left (\textit {\_Z}^{2}+2\right ) \left (x^{2}-1\right )^{\frac {1}{4}}+x}{x^{2}-2}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\left (x^{2}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}-2\right )-x \sqrt {x^{2}-1}+\RootOf \left (\textit {\_Z}^{2}-2\right ) \left (x^{2}-1\right )^{\frac {1}{4}}-x}{x^{2}-2}\right )}{2}\) | \(121\) |
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*} \int \frac {x^{2}}{{\left (x^{2} - 1\right )}^{\frac {3}{4}} {\left (x^{2} - 2\right )}}\,{d x} \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 = -0.02 \begin {gather*} \int \frac {x^2}{{\left (x^2-1\right )}^{3/4}\,\left (x^2-2\right )} \,d x \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^{2}}{\left (\left (x - 1\right ) \left (x + 1\right )\right )^{\frac {3}{4}} \left (x^{2} - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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