Optimal. Leaf size=74 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1-x^4}}\right )}{2 \sqrt {2}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {-1-x^4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1332, 226,
1713, 212} \begin {gather*} \frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \text {ArcTan}(x)\left |\frac {1}{2}\right .\right )}{4 \sqrt {-x^4-1}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-x^4-1}}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 226
Rule 1332
Rule 1713
Rubi steps
\begin {align*} \int \frac {x^2}{\left (1+x^2\right ) \sqrt {-1-x^4}} \, dx &=\frac {1}{2} \int \frac {1}{\sqrt {-1-x^4}} \, dx-\frac {1}{2} \int \frac {1-x^2}{\left (1+x^2\right ) \sqrt {-1-x^4}} \, dx\\ &=\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {-1-x^4}}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {-1-x^4}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1-x^4}}\right )}{2 \sqrt {2}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {-1-x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.08, size = 60, normalized size = 0.81 \begin {gather*} \frac {\sqrt [4]{-1} \sqrt {1+x^4} \left (-F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+\Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right )}{\sqrt {-1-x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.19, size = 168, normalized size = 2.27
method | result | size |
default | \(\frac {\sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \EllipticF \left (\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) x , i\right )}{\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) \sqrt {-x^{4}-1}}-\frac {i \sqrt {-i}\, \sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \EllipticPi \left (\sqrt {-i}\, x , -i, \frac {\left (-1\right )^{\frac {1}{4}}}{\sqrt {-i}}\right )}{2 \sqrt {-x^{4}-1}}-\frac {\sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \EllipticPi \left (\sqrt {-i}\, x , -i, \frac {\left (-1\right )^{\frac {1}{4}}}{\sqrt {-i}}\right )}{2 \sqrt {-i}\, \sqrt {-x^{4}-1}}\) | \(168\) |
elliptic | \(\frac {\sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \EllipticF \left (\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) x , i\right )}{\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) \sqrt {-x^{4}-1}}-\frac {i \sqrt {-i}\, \sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \EllipticPi \left (\sqrt {-i}\, x , -i, \frac {\left (-1\right )^{\frac {1}{4}}}{\sqrt {-i}}\right )}{2 \sqrt {-x^{4}-1}}-\frac {\sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \EllipticPi \left (\sqrt {-i}\, x , -i, \frac {\left (-1\right )^{\frac {1}{4}}}{\sqrt {-i}}\right )}{2 \sqrt {-i}\, \sqrt {-x^{4}-1}}\) | \(168\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 0.10, size = 74, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \, \sqrt {i} {\rm ellipticF}\left (\sqrt {i} x, -1\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\frac {\sqrt {2} x + \sqrt {-x^{4} - 1}}{x^{2} + 1}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} x - \sqrt {-x^{4} - 1}}{x^{2} + 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^{2}}{\left (x^{2} + 1\right ) \sqrt {- x^{4} - 1}}\, dx \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*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int \frac {x^2}{\left (x^2+1\right )\,\sqrt {-x^4-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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