Optimal. Leaf size=65 \[ -\frac {x}{3 \sqrt {2 x^2+1}}-\frac {2 \tanh ^{-1}\left (-\sqrt {\frac {2}{3}} x^2+\frac {\sqrt {2 x^2+1} x}{\sqrt {3}}+\sqrt {\frac {2}{3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 0.69, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {527, 12, 377, 207} \begin {gather*} -\frac {x}{3 \sqrt {2 x^2+1}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {2 x^2+1}}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 207
Rule 377
Rule 527
Rubi steps
\begin {align*} \int \frac {1+x^2}{\left (-1+x^2\right ) \left (1+2 x^2\right )^{3/2}} \, dx &=-\frac {x}{3 \sqrt {1+2 x^2}}+\frac {1}{3} \int \frac {2}{\left (-1+x^2\right ) \sqrt {1+2 x^2}} \, dx\\ &=-\frac {x}{3 \sqrt {1+2 x^2}}+\frac {2}{3} \int \frac {1}{\left (-1+x^2\right ) \sqrt {1+2 x^2}} \, dx\\ &=-\frac {x}{3 \sqrt {1+2 x^2}}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-1+3 x^2} \, dx,x,\frac {x}{\sqrt {1+2 x^2}}\right )\\ &=-\frac {x}{3 \sqrt {1+2 x^2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+2 x^2}}\right )}{3 \sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 1.31, size = 159, normalized size = 2.45 \begin {gather*} \frac {x \left (\frac {72 \left (2 x^2+1\right ) x^2 \, _2F_1\left (2,2;\frac {7}{2};\frac {3 x^2}{x^2-1}\right )}{\left (x^2-1\right )^2}+\frac {10 \left (4 x^2+3\right ) \left (\sqrt {\frac {6 x^2+3}{1-x^2}} x^2+\sqrt {\frac {x^2}{x^2-1}} \left (2 x^2+1\right ) \sin ^{-1}\left (\sqrt {3} \sqrt {\frac {x^2}{x^2-1}}\right )\right )}{\sqrt {\frac {2 x^2+1}{3-3 x^2}} x^4}+45\right )}{45 \sqrt {2 x^2+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.22, size = 65, normalized size = 1.00 \begin {gather*} -\frac {x}{3 \sqrt {2 x^2+1}}-\frac {2 \tanh ^{-1}\left (-\sqrt {\frac {2}{3}} x^2+\frac {\sqrt {2 x^2+1} x}{\sqrt {3}}+\sqrt {\frac {2}{3}}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 66, normalized size = 1.02 \begin {gather*} \frac {\sqrt {3} {\left (2 \, x^{2} + 1\right )} \log \left (\frac {2 \, \sqrt {3} \sqrt {2 \, x^{2} + 1} x - 5 \, x^{2} - 1}{x^{2} - 1}\right ) - 3 \, \sqrt {2 \, x^{2} + 1} x}{9 \, {\left (2 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 83, normalized size = 1.28 \begin {gather*} -\frac {1}{18} \, \sqrt {6} \sqrt {2} \log \left (\frac {{\left | 2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} + 1}\right )}^{2} - 4 \, \sqrt {6} - 10 \right |}}{{\left | 2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} + 1}\right )}^{2} + 4 \, \sqrt {6} - 10 \right |}}\right ) - \frac {x}{3 \, \sqrt {2 \, x^{2} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 139, normalized size = 2.14 \begin {gather*} \frac {x}{\sqrt {2 x^{2}+1}}+\frac {1}{3 \sqrt {2 \left (-1+x \right )^{2}+4 x -1}}-\frac {2 x}{3 \sqrt {2 \left (-1+x \right )^{2}+4 x -1}}-\frac {\sqrt {3}\, \arctanh \left (\frac {\left (2+4 x \right ) \sqrt {3}}{6 \sqrt {2 \left (-1+x \right )^{2}+4 x -1}}\right )}{9}-\frac {1}{3 \sqrt {2 \left (1+x \right )^{2}-4 x -1}}-\frac {2 x}{3 \sqrt {2 \left (1+x \right )^{2}-4 x -1}}+\frac {\sqrt {3}\, \arctanh \left (\frac {\left (2-4 x \right ) \sqrt {3}}{6 \sqrt {2 \left (1+x \right )^{2}-4 x -1}}\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 80, normalized size = 1.23 \begin {gather*} -\frac {1}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {2 \, \sqrt {2} x}{{\left | 2 \, x + 2 \right |}} - \frac {\sqrt {2}}{{\left | 2 \, x + 2 \right |}}\right ) - \frac {1}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {2 \, \sqrt {2} x}{{\left | 2 \, x - 2 \right |}} + \frac {\sqrt {2}}{{\left | 2 \, x - 2 \right |}}\right ) - \frac {x}{3 \, \sqrt {2 \, x^{2} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 111, normalized size = 1.71 \begin {gather*} \frac {\sqrt {3}\,\left (\ln \left (x-1\right )-\ln \left (x+\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {x^2+\frac {1}{2}}}{2}+\frac {1}{2}\right )\right )}{9}-\frac {\sqrt {3}\,\left (\ln \left (x+1\right )-\ln \left (x-\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {x^2+\frac {1}{2}}}{2}-\frac {1}{2}\right )\right )}{9}-\frac {\sqrt {2}\,\sqrt {x^2+\frac {1}{2}}}{12\,\left (x-\frac {\sqrt {2}\,1{}\mathrm {i}}{2}\right )}-\frac {\sqrt {2}\,\sqrt {x^2+\frac {1}{2}}}{12\,\left (x+\frac {\sqrt {2}\,1{}\mathrm {i}}{2}\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^{2} + 1}{\left (x - 1\right ) \left (x + 1\right ) \left (2 x^{2} + 1\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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