Optimal. Leaf size=74 \[ \frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {-x^4-1}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-x^4-1}}\right )}{2 \sqrt {2}} \]
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Rubi [A] time = 0.06, 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 = {1318, 220, 1699, 206} \[ \frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {-x^4-1}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-x^4-1}}\right )}{2 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 220
Rule 1318
Rule 1699
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} \operatorname {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] time = 0.09, size = 60, normalized size = 0.81 \[ \frac {\sqrt [4]{-1} \sqrt {x^4+1} \left (\Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right )}{\sqrt {-x^4-1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.28, size = 0, normalized size = 0.00 \[ -\frac {1}{8} \, \sqrt {2} \log \left (\frac {\sqrt {2} x + \sqrt {-x^{4} - 1}}{x^{2} + 1}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} x - \sqrt {-x^{4} - 1}}{x^{2} + 1}\right ) + {\rm integral}\left (-\frac {\sqrt {-x^{4} - 1}}{2 \, {\left (x^{4} + 1\right )}}, x\right ) \]
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 \[ \int \frac {x^{2}}{\sqrt {-x^{4} - 1} {\left (x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 168, normalized size = 2.27 \[ \frac {\sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \EllipticF \left (\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) x , i\right )}{\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}\right ) \sqrt {-x^{4}-1}}-\frac {i \sqrt {-i}\, \sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \EllipticPi \left (\sqrt {-i}\, x , -i, \frac {\left (-1\right )^{\frac {1}{4}}}{\sqrt {-i}}\right )}{2 \sqrt {-x^{4}-1}}-\frac {\sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \EllipticPi \left (\sqrt {-i}\, x , -i, \frac {\left (-1\right )^{\frac {1}{4}}}{\sqrt {-i}}\right )}{2 \sqrt {-i}\, \sqrt {-x^{4}-1}} \]
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 \[ \int \frac {x^{2}}{\sqrt {-x^{4} - 1} {\left (x^{2} + 1\right )}}\,{d x} \]
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 \[ \int \frac {x^2}{\left (x^2+1\right )\,\sqrt {-x^4-1}} \,d x \]
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 \[ \int \frac {x^{2}}{\left (x^{2} + 1\right ) \sqrt {- x^{4} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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