Optimal. Leaf size=25 \[ \frac {1}{2} \log \left (1-x^2\right )+\frac {3}{2} \log \left (4-x^2\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1607, 1261,
646, 31} \begin {gather*} \frac {1}{2} \log \left (1-x^2\right )+\frac {3}{2} \log \left (4-x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 646
Rule 1261
Rule 1607
Rubi steps
\begin {align*} \int \frac {-7 x+4 x^3}{4-5 x^2+x^4} \, dx &=\int \frac {x \left (-7+4 x^2\right )}{4-5 x^2+x^4} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {-7+4 x}{4-5 x+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{-1+x} \, dx,x,x^2\right )+\frac {3}{2} \text {Subst}\left (\int \frac {1}{-4+x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \log \left (1-x^2\right )+\frac {3}{2} \log \left (4-x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (1-x^2\right )+\frac {3}{2} \log \left (4-x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 18, normalized size = 0.72
method | result | size |
default | \(\frac {\ln \left (x^{2}-1\right )}{2}+\frac {3 \ln \left (x^{2}-4\right )}{2}\) | \(18\) |
risch | \(\frac {\ln \left (x^{2}-1\right )}{2}+\frac {3 \ln \left (x^{2}-4\right )}{2}\) | \(18\) |
norman | \(\frac {3 \ln \left (x -2\right )}{2}+\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (1+x \right )}{2}+\frac {3 \ln \left (x +2\right )}{2}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 25, normalized size = 1.00 \begin {gather*} \frac {3}{2} \, \log \left (x + 2\right ) + \frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right ) + \frac {3}{2} \, \log \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 17, normalized size = 0.68 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} - 1\right ) + \frac {3}{2} \, \log \left (x^{2} - 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 17, normalized size = 0.68 \begin {gather*} \frac {3 \log {\left (x^{2} - 4 \right )}}{2} + \frac {\log {\left (x^{2} - 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.98, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) + \frac {3}{2} \, \log \left ({\left | x^{2} - 4 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 17, normalized size = 0.68 \begin {gather*} \frac {\ln \left (x^2-1\right )}{2}+\frac {3\,\ln \left (x^2-4\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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