Optimal. Leaf size=43 \[ \frac {\tan ^{-1}\left (\frac {1-3 x}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {\tan ^{-1}\left (\frac {3 x+1}{\sqrt {2}}\right )}{2 \sqrt {2}} \]
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Rubi [A] time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1161, 618, 204} \begin {gather*} \frac {\tan ^{-1}\left (\frac {1-3 x}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {\tan ^{-1}\left (\frac {3 x+1}{\sqrt {2}}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 1161
Rubi steps
\begin {align*} \int \frac {1+3 x^2}{-1-2 x^2-9 x^4} \, dx &=-\left (\frac {1}{6} \int \frac {1}{\frac {1}{3}-\frac {2 x}{3}+x^2} \, dx\right )-\frac {1}{6} \int \frac {1}{\frac {1}{3}+\frac {2 x}{3}+x^2} \, dx\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-\frac {8}{9}-x^2} \, dx,x,-\frac {2}{3}+2 x\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-\frac {8}{9}-x^2} \, dx,x,\frac {2}{3}+2 x\right )\\ &=\frac {\tan ^{-1}\left (\frac {1-3 x}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {\tan ^{-1}\left (\frac {1+3 x}{\sqrt {2}}\right )}{2 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 99, normalized size = 2.30 \begin {gather*} -\frac {\left (\sqrt {2}-i\right ) \tan ^{-1}\left (\frac {3 x}{\sqrt {1-2 i \sqrt {2}}}\right )}{2 \sqrt {2 \left (1-2 i \sqrt {2}\right )}}-\frac {\left (\sqrt {2}+i\right ) \tan ^{-1}\left (\frac {3 x}{\sqrt {1+2 i \sqrt {2}}}\right )}{2 \sqrt {2 \left (1+2 i \sqrt {2}\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+3 x^2}{-1-2 x^2-9 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.92, size = 33, normalized size = 0.77 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (9 \, x^{3} + 5 \, x\right )}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {3}{4} \, \sqrt {2} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 33, normalized size = 0.77 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x + 1\right )}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x - 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 34, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {2}\, \arctan \left (\frac {\left (6 x -2\right ) \sqrt {2}}{4}\right )}{4}-\frac {\sqrt {2}\, \arctan \left (\frac {\left (6 x +2\right ) \sqrt {2}}{4}\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.49, size = 33, normalized size = 0.77 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x + 1\right )}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x - 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 29, normalized size = 0.67 \begin {gather*} -\frac {\sqrt {2}\,\left (\mathrm {atan}\left (\frac {9\,\sqrt {2}\,x^3}{4}+\frac {5\,\sqrt {2}\,x}{4}\right )+\mathrm {atan}\left (\frac {3\,\sqrt {2}\,x}{4}\right )\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 46, normalized size = 1.07 \begin {gather*} - \frac {\sqrt {2} \left (2 \operatorname {atan}{\left (\frac {3 \sqrt {2} x}{4} \right )} + 2 \operatorname {atan}{\left (\frac {9 \sqrt {2} x^{3}}{4} + \frac {5 \sqrt {2} x}{4} \right )}\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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