Optimal. Leaf size=31 \[ \log (1-x)-\frac {1}{2} \log (3-x)+\frac {3}{2} \log (x+1)-2 \log (x+3) \]
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Rubi [A] time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.32, number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1593, 1662, 12, 1107, 616, 31, 1130, 207} \begin {gather*} \frac {5}{4} \log \left (1-x^2\right )-\frac {5}{4} \log \left (9-x^2\right )-\frac {3}{2} \tanh ^{-1}\left (\frac {x}{3}\right )+\frac {1}{2} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 207
Rule 616
Rule 1107
Rule 1130
Rule 1593
Rule 1662
Rubi steps
\begin {align*} \int \frac {-20 x+4 x^2}{9-10 x^2+x^4} \, dx &=\int \frac {x (-20+4 x)}{9-10 x^2+x^4} \, dx\\ &=\int -\frac {20 x}{9-10 x^2+x^4} \, dx+\int \frac {4 x^2}{9-10 x^2+x^4} \, dx\\ &=4 \int \frac {x^2}{9-10 x^2+x^4} \, dx-20 \int \frac {x}{9-10 x^2+x^4} \, dx\\ &=-\left (\frac {1}{2} \int \frac {1}{-1+x^2} \, dx\right )+\frac {9}{2} \int \frac {1}{-9+x^2} \, dx-10 \operatorname {Subst}\left (\int \frac {1}{9-10 x+x^2} \, dx,x,x^2\right )\\ &=-\frac {3}{2} \tanh ^{-1}\left (\frac {x}{3}\right )+\frac {1}{2} \tanh ^{-1}(x)-\frac {5}{4} \operatorname {Subst}\left (\int \frac {1}{-9+x} \, dx,x,x^2\right )+\frac {5}{4} \operatorname {Subst}\left (\int \frac {1}{-1+x} \, dx,x,x^2\right )\\ &=-\frac {3}{2} \tanh ^{-1}\left (\frac {x}{3}\right )+\frac {1}{2} \tanh ^{-1}(x)+\frac {5}{4} \log \left (1-x^2\right )-\frac {5}{4} \log \left (9-x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 1.26 \begin {gather*} 4 \left (\frac {1}{4} \log (1-x)-\frac {1}{8} \log (3-x)+\frac {3}{8} \log (x+1)-\frac {1}{2} \log (x+3)\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 {-20 x+4 x^2}{9-10 x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.40, size = 23, normalized size = 0.74 \begin {gather*} -2 \, \log \left (x + 3\right ) + \frac {3}{2} \, \log \left (x + 1\right ) + \log \left (x - 1\right ) - \frac {1}{2} \, \log \left (x - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 27, normalized size = 0.87 \begin {gather*} -2 \, \log \left ({\left | x + 3 \right |}\right ) + \frac {3}{2} \, \log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x - 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x - 3 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 24, normalized size = 0.77 \begin {gather*} -\frac {\ln \left (x -3\right )}{2}+\ln \left (x -1\right )+\frac {3 \ln \left (x +1\right )}{2}-2 \ln \left (x +3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.80, size = 23, normalized size = 0.74 \begin {gather*} -2 \, \log \left (x + 3\right ) + \frac {3}{2} \, \log \left (x + 1\right ) + \log \left (x - 1\right ) - \frac {1}{2} \, \log \left (x - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 23, normalized size = 0.74 \begin {gather*} \ln \left (x-1\right )+\frac {3\,\ln \left (x+1\right )}{2}-\frac {\ln \left (x-3\right )}{2}-2\,\ln \left (x+3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 26, normalized size = 0.84 \begin {gather*} - \frac {\log {\left (x - 3 \right )}}{2} + \log {\left (x - 1 \right )} + \frac {3 \log {\left (x + 1 \right )}}{2} - 2 \log {\left (x + 3 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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