Optimal. Leaf size=45 \[ -\frac {1}{6} d \tanh ^{-1}\left (\frac {x}{2}\right )+\frac {1}{3} d \tanh ^{-1}(x)-\frac {1}{6} e \log \left (1-x^2\right )+\frac {1}{6} e \log \left (4-x^2\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1673, 12, 1093, 207, 1107, 616, 31} \[ -\frac {1}{6} d \tanh ^{-1}\left (\frac {x}{2}\right )+\frac {1}{3} d \tanh ^{-1}(x)-\frac {1}{6} e \log \left (1-x^2\right )+\frac {1}{6} e \log \left (4-x^2\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 207
Rule 616
Rule 1093
Rule 1107
Rule 1673
Rubi steps
\begin {align*} \int \frac {d+e x}{4-5 x^2+x^4} \, dx &=\int \frac {d}{4-5 x^2+x^4} \, dx+\int \frac {e x}{4-5 x^2+x^4} \, dx\\ &=d \int \frac {1}{4-5 x^2+x^4} \, dx+e \int \frac {x}{4-5 x^2+x^4} \, dx\\ &=\frac {1}{3} d \int \frac {1}{-4+x^2} \, dx-\frac {1}{3} d \int \frac {1}{-1+x^2} \, dx+\frac {1}{2} e \operatorname {Subst}\left (\int \frac {1}{4-5 x+x^2} \, dx,x,x^2\right )\\ &=-\frac {1}{6} d \tanh ^{-1}\left (\frac {x}{2}\right )+\frac {1}{3} d \tanh ^{-1}(x)+\frac {1}{6} e \operatorname {Subst}\left (\int \frac {1}{-4+x} \, dx,x,x^2\right )-\frac {1}{6} e \operatorname {Subst}\left (\int \frac {1}{-1+x} \, dx,x,x^2\right )\\ &=-\frac {1}{6} d \tanh ^{-1}\left (\frac {x}{2}\right )+\frac {1}{3} d \tanh ^{-1}(x)-\frac {1}{6} e \log \left (1-x^2\right )+\frac {1}{6} e \log \left (4-x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 50, normalized size = 1.11 \[ \frac {1}{12} (-2 (d+e) \log (1-x)+(d+2 e) \log (2-x)+2 (d-e) \log (x+1)-(d-2 e) \log (x+2)) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.60, size = 43, normalized size = 0.96 \[ -\frac {1}{12} \, {\left (d - 2 \, e\right )} \log \left (x + 2\right ) + \frac {1}{6} \, {\left (d - e\right )} \log \left (x + 1\right ) - \frac {1}{6} \, {\left (d + e\right )} \log \left (x - 1\right ) + \frac {1}{12} \, {\left (d + 2 \, e\right )} \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 51, normalized size = 1.13 \[ -\frac {1}{12} \, {\left (d - 2 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{6} \, {\left (d - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{6} \, {\left (d + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{12} \, {\left (d + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 1.29 \[ -\frac {d \ln \left (x +2\right )}{12}+\frac {d \ln \left (x -2\right )}{12}-\frac {d \ln \left (x -1\right )}{6}+\frac {d \ln \left (x +1\right )}{6}+\frac {e \ln \left (x +2\right )}{6}+\frac {e \ln \left (x -2\right )}{6}-\frac {e \ln \left (x -1\right )}{6}-\frac {e \ln \left (x +1\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 43, normalized size = 0.96 \[ -\frac {1}{12} \, {\left (d - 2 \, e\right )} \log \left (x + 2\right ) + \frac {1}{6} \, {\left (d - e\right )} \log \left (x + 1\right ) - \frac {1}{6} \, {\left (d + e\right )} \log \left (x - 1\right ) + \frac {1}{12} \, {\left (d + 2 \, e\right )} \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.71, size = 51, normalized size = 1.13 \[ \ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}\right )-\ln \left (x-1\right )\,\left (\frac {d}{6}+\frac {e}{6}\right )+\ln \left (x-2\right )\,\left (\frac {d}{12}+\frac {e}{6}\right )-\ln \left (x+2\right )\,\left (\frac {d}{12}-\frac {e}{6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.15, size = 515, normalized size = 11.44 \[ - \frac {\left (d - 2 e\right ) \log {\left (x + \frac {- 35 d^{4} e + \frac {51 d^{4} \left (d - 2 e\right )}{2} - 180 d^{2} e^{3} - 90 d^{2} e^{2} \left (d - 2 e\right ) + 41 d^{2} e \left (d - 2 e\right )^{2} - \frac {15 d^{2} \left (d - 2 e\right )^{3}}{2} + 320 e^{5} - 96 e^{4} \left (d - 2 e\right ) - 80 e^{3} \left (d - 2 e\right )^{2} + 24 e^{2} \left (d - 2 e\right )^{3}}{9 d^{5} - 160 d^{3} e^{2} + 256 d e^{4}} \right )}}{12} + \frac {\left (d - e\right ) \log {\left (x + \frac {- 35 d^{4} e - 51 d^{4} \left (d - e\right ) - 180 d^{2} e^{3} + 180 d^{2} e^{2} \left (d - e\right ) + 164 d^{2} e \left (d - e\right )^{2} + 60 d^{2} \left (d - e\right )^{3} + 320 e^{5} + 192 e^{4} \left (d - e\right ) - 320 e^{3} \left (d - e\right )^{2} - 192 e^{2} \left (d - e\right )^{3}}{9 d^{5} - 160 d^{3} e^{2} + 256 d e^{4}} \right )}}{6} - \frac {\left (d + e\right ) \log {\left (x + \frac {- 35 d^{4} e + 51 d^{4} \left (d + e\right ) - 180 d^{2} e^{3} - 180 d^{2} e^{2} \left (d + e\right ) + 164 d^{2} e \left (d + e\right )^{2} - 60 d^{2} \left (d + e\right )^{3} + 320 e^{5} - 192 e^{4} \left (d + e\right ) - 320 e^{3} \left (d + e\right )^{2} + 192 e^{2} \left (d + e\right )^{3}}{9 d^{5} - 160 d^{3} e^{2} + 256 d e^{4}} \right )}}{6} + \frac {\left (d + 2 e\right ) \log {\left (x + \frac {- 35 d^{4} e - \frac {51 d^{4} \left (d + 2 e\right )}{2} - 180 d^{2} e^{3} + 90 d^{2} e^{2} \left (d + 2 e\right ) + 41 d^{2} e \left (d + 2 e\right )^{2} + \frac {15 d^{2} \left (d + 2 e\right )^{3}}{2} + 320 e^{5} + 96 e^{4} \left (d + 2 e\right ) - 80 e^{3} \left (d + 2 e\right )^{2} - 24 e^{2} \left (d + 2 e\right )^{3}}{9 d^{5} - 160 d^{3} e^{2} + 256 d e^{4}} \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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