Optimal. Leaf size=40 \[ -\frac {\sqrt {\left (x^2+1\right )^2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{x^2+1}\right )}{\sqrt {2} \left (x^2+1\right )} \]
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Rubi [C] time = 0.66, antiderivative size = 74, normalized size of antiderivative = 1.85, number of steps used = 12, number of rules used = 7, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {6725, 1147, 8, 1148, 388, 206, 203} \begin {gather*} \frac {\sqrt {x^4+2 x^2+1} \tanh ^{-1}\left ((-1)^{3/4} x\right )}{\sqrt {2} \left (x^2+1\right )}+\frac {i \sqrt {x^4+2 x^2+1} \tan ^{-1}\left ((-1)^{3/4} x\right )}{\sqrt {2} \left (x^2+1\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 203
Rule 206
Rule 388
Rule 1147
Rule 1148
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt {1+2 x^2+x^4}}{\left (1+x^2\right ) \left (1+x^4\right )} \, dx &=\int \left (\frac {\sqrt {1+2 x^2+x^4}}{-1-x^2}+\frac {x^2 \sqrt {1+2 x^2+x^4}}{1+x^4}\right ) \, dx\\ &=\int \frac {\sqrt {1+2 x^2+x^4}}{-1-x^2} \, dx+\int \frac {x^2 \sqrt {1+2 x^2+x^4}}{1+x^4} \, dx\\ &=\frac {\sqrt {1+2 x^2+x^4} \int 1 \, dx}{-1-x^2}+\int \left (-\frac {\sqrt {1+2 x^2+x^4}}{2 \left (i-x^2\right )}+\frac {\sqrt {1+2 x^2+x^4}}{2 \left (i+x^2\right )}\right ) \, dx\\ &=-\frac {x \sqrt {1+2 x^2+x^4}}{1+x^2}-\frac {1}{2} \int \frac {\sqrt {1+2 x^2+x^4}}{i-x^2} \, dx+\frac {1}{2} \int \frac {\sqrt {1+2 x^2+x^4}}{i+x^2} \, dx\\ &=-\frac {x \sqrt {1+2 x^2+x^4}}{1+x^2}-\frac {\sqrt {1+2 x^2+x^4} \int \frac {1+x^2}{i-x^2} \, dx}{2 \left (1+x^2\right )}+\frac {\sqrt {1+2 x^2+x^4} \int \frac {1+x^2}{i+x^2} \, dx}{2 \left (1+x^2\right )}\\ &=\frac {\left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {1+2 x^2+x^4}\right ) \int \frac {1}{i+x^2} \, dx}{1+x^2}-\frac {\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {1+2 x^2+x^4}\right ) \int \frac {1}{i-x^2} \, dx}{1+x^2}\\ &=\frac {i \sqrt {1+2 x^2+x^4} \tan ^{-1}\left ((-1)^{3/4} x\right )}{\sqrt {2} \left (1+x^2\right )}+\frac {\sqrt {1+2 x^2+x^4} \tanh ^{-1}\left ((-1)^{3/4} x\right )}{\sqrt {2} \left (1+x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 58, normalized size = 1.45 \begin {gather*} \frac {\sqrt {\left (x^2+1\right )^2} \left (\log \left (-x^2+\sqrt {2} x-1\right )-\log \left (x^2+\sqrt {2} x+1\right )\right )}{2 \sqrt {2} \left (x^2+1\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 29, normalized size = 0.72 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+2 x^2+1}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 34, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{4} + 4 \, x^{2} - 2 \, \sqrt {2} {\left (x^{3} + x\right )} + 1}{x^{4} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 34, normalized size = 0.85 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 79, normalized size = 1.98 \begin {gather*} -\frac {\sqrt {\left (x^{2}+1\right )^{2}}\, \sqrt {2}\, \left (\ln \left (-\frac {x^{2}+\sqrt {2}\, x +1}{\sqrt {2}\, x -x^{2}-1}\right )-\ln \left (-\frac {\sqrt {2}\, x -x^{2}-1}{x^{2}+\sqrt {2}\, x +1}\right )\right )}{8 \left (x^{2}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 34, normalized size = 0.85 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 18, normalized size = 0.45 \begin {gather*} -\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,x}{x^2+1}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 49, normalized size = 1.22 \begin {gather*} \frac {\sqrt {2} \log {\left (- \sqrt {2} x + \sqrt {\left (x^{2} + 1\right )^{2}} \right )}}{4} - \frac {\sqrt {2} \log {\left (\sqrt {2} x + \sqrt {\left (x^{2} + 1\right )^{2}} \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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