Optimal. Leaf size=31 \[ -x+\frac {-3+\frac {x}{\log \left (-x^2\right )}}{\frac {e^3}{4 x}+x} \]
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Rubi [F] time = 0.51, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-8 e^3 x-32 x^3+8 e^3 x \log \left (-x^2\right )+\left (-e^6+48 x^2-16 x^4+e^3 \left (-12-8 x^2\right )\right ) \log ^2\left (-x^2\right )}{\left (e^6+8 e^3 x^2+16 x^4\right ) \log ^2\left (-x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=16 \int \frac {-8 e^3 x-32 x^3+8 e^3 x \log \left (-x^2\right )+\left (-e^6+48 x^2-16 x^4+e^3 \left (-12-8 x^2\right )\right ) \log ^2\left (-x^2\right )}{\left (4 e^3+16 x^2\right )^2 \log ^2\left (-x^2\right )} \, dx\\ &=16 \int \left (\frac {-e^3 \left (12+e^3\right )+8 \left (6-e^3\right ) x^2-16 x^4}{16 \left (e^3+4 x^2\right )^2}-\frac {x}{2 \left (e^3+4 x^2\right ) \log ^2\left (-x^2\right )}+\frac {e^3 x}{2 \left (e^3+4 x^2\right )^2 \log \left (-x^2\right )}\right ) \, dx\\ &=-\left (8 \int \frac {x}{\left (e^3+4 x^2\right ) \log ^2\left (-x^2\right )} \, dx\right )+\left (8 e^3\right ) \int \frac {x}{\left (e^3+4 x^2\right )^2 \log \left (-x^2\right )} \, dx+\int \frac {-e^3 \left (12+e^3\right )+8 \left (6-e^3\right ) x^2-16 x^4}{\left (e^3+4 x^2\right )^2} \, dx\\ &=-\frac {12 x}{e^3+4 x^2}-4 \operatorname {Subst}\left (\int \frac {1}{\left (e^3+4 x\right ) \log ^2(-x)} \, dx,x,x^2\right )-\frac {\int \frac {2 e^6+8 e^3 x^2}{e^3+4 x^2} \, dx}{2 e^3}+\left (4 e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (e^3+4 x\right )^2 \log (-x)} \, dx,x,x^2\right )\\ &=-\frac {12 x}{e^3+4 x^2}-4 \operatorname {Subst}\left (\int \frac {1}{\left (e^3+4 x\right ) \log ^2(-x)} \, dx,x,x^2\right )+\left (4 e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (e^3+4 x\right )^2 \log (-x)} \, dx,x,x^2\right )-\int 1 \, dx\\ &=-x-\frac {12 x}{e^3+4 x^2}-4 \operatorname {Subst}\left (\int \frac {1}{\left (e^3+4 x\right ) \log ^2(-x)} \, dx,x,x^2\right )+\left (4 e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (e^3+4 x\right )^2 \log (-x)} \, dx,x,x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 39, normalized size = 1.26 \begin {gather*} x \left (-1-\frac {12}{e^3+4 x^2}+\frac {4 x}{\left (e^3+4 x^2\right ) \log \left (-x^2\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 46, normalized size = 1.48 \begin {gather*} \frac {4 \, x^{2} - {\left (4 \, x^{3} + x e^{3} + 12 \, x\right )} \log \left (-x^{2}\right )}{{\left (4 \, x^{2} + e^{3}\right )} \log \left (-x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.30, size = 61, normalized size = 1.97 \begin {gather*} -\frac {4 \, x^{3} \log \left (-x^{2}\right ) + x e^{3} \log \left (-x^{2}\right ) - 4 \, x^{2} + 24 \, x \log \left (-x^{2}\right )}{4 \, x^{2} \log \left (-x^{2}\right ) + e^{3} \log \left (-x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 47, normalized size = 1.52
method | result | size |
risch | \(-\frac {x \left (4 x^{2}+{\mathrm e}^{3}+12\right )}{{\mathrm e}^{3}+4 x^{2}}+\frac {4 x^{2}}{\left ({\mathrm e}^{3}+4 x^{2}\right ) \ln \left (-x^{2}\right )}\) | \(47\) |
norman | \(\frac {\left (-{\mathrm e}^{3}-12\right ) x \ln \left (-x^{2}\right )+4 x^{2}-4 x^{3} \ln \left (-x^{2}\right )}{\left ({\mathrm e}^{3}+4 x^{2}\right ) \ln \left (-x^{2}\right )}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.55, size = 67, normalized size = 2.16 \begin {gather*} -\frac {4 i \, \pi x^{3} + {\left (12 i \, \pi + i \, \pi e^{3}\right )} x - 4 \, x^{2} + 2 \, {\left (4 \, x^{3} + x {\left (e^{3} + 12\right )}\right )} \log \relax (x)}{4 i \, \pi x^{2} + i \, \pi e^{3} + 2 \, {\left (4 \, x^{2} + e^{3}\right )} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 53, normalized size = 1.71 \begin {gather*} -\frac {x\,\left (12\,\ln \left (-x^2\right )-4\,x+{\mathrm {e}}^3\,\ln \left (-x^2\right )+4\,x^2\,\ln \left (-x^2\right )\right )}{\ln \left (-x^2\right )\,\left (4\,x^2+{\mathrm {e}}^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 32, normalized size = 1.03 \begin {gather*} \frac {4 x^{2}}{\left (4 x^{2} + e^{3}\right ) \log {\left (- x^{2} \right )}} - x - \frac {12 x}{4 x^{2} + e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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