Optimal. Leaf size=18 \[ x^4 \left (2+\log \left (-1+e x^2-\log (9)\right )\right ) \]
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Rubi [A] time = 0.25, antiderivative size = 22, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 7, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {6, 6725, 446, 77, 2454, 2395, 43} \begin {gather*} 2 x^4+x^4 \log \left (e x^2-1-\log (9)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 43
Rule 77
Rule 446
Rule 2395
Rule 2454
Rule 6725
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-10 e x^5+x^3 (8+8 \log (9))+\left (4 x^3-4 e x^5+4 x^3 \log (9)\right ) \log \left (-1+e x^2-\log (9)\right )}{1-e x^2+\log (9)} \, dx\\ &=\int \left (\frac {2 x^3 \left (-4+5 e x^2-4 \log (9)\right )}{-1+e x^2-\log (9)}+4 x^3 \log \left (-1+e x^2-\log (9)\right )\right ) \, dx\\ &=2 \int \frac {x^3 \left (-4+5 e x^2-4 \log (9)\right )}{-1+e x^2-\log (9)} \, dx+4 \int x^3 \log \left (-1+e x^2-\log (9)\right ) \, dx\\ &=2 \operatorname {Subst}\left (\int x \log (-1+e x-\log (9)) \, dx,x,x^2\right )+\operatorname {Subst}\left (\int \frac {x (-4+5 e x-4 \log (9))}{-1+e x-\log (9)} \, dx,x,x^2\right )\\ &=x^4 \log \left (-1+e x^2-\log (9)\right )-e \operatorname {Subst}\left (\int \frac {x^2}{-1+e x-\log (9)} \, dx,x,x^2\right )+\operatorname {Subst}\left (\int \left (5 x+\frac {1+\log (9)}{e}+\frac {(1+\log (9))^2}{e (-1+e x-\log (9))}\right ) \, dx,x,x^2\right )\\ &=\frac {5 x^4}{2}+\frac {x^2 (1+\log (9))}{e}+x^4 \log \left (-1+e x^2-\log (9)\right )+\frac {(1+\log (9))^2 \log \left (1-e x^2+\log (9)\right )}{e^2}-e \operatorname {Subst}\left (\int \left (\frac {x}{e}+\frac {1+\log (9)}{e^2}+\frac {(1+\log (9))^2}{e^2 (-1+e x-\log (9))}\right ) \, dx,x,x^2\right )\\ &=2 x^4+x^4 \log \left (-1+e x^2-\log (9)\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 25, normalized size = 1.39 \begin {gather*} 2 \left (x^4+\frac {1}{2} x^4 \log \left (-1+e x^2-\log (9)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 23, normalized size = 1.28 \begin {gather*} x^{4} \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right ) + 2 \, x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 181, normalized size = 10.06 \begin {gather*} {\left (4 \, x^{2} e + {\left (x^{2} e - 2 \, \log \relax (3) - 1\right )}^{2} \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right ) + 4 \, {\left (x^{2} e - 2 \, \log \relax (3) - 1\right )} \log \relax (3) \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right ) + 4 \, \log \relax (3)^{2} \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right ) + 2 \, {\left (x^{2} e - 2 \, \log \relax (3) - 1\right )}^{2} + 8 \, {\left (x^{2} e - 2 \, \log \relax (3) - 1\right )} \log \relax (3) + 2 \, {\left (x^{2} e - 2 \, \log \relax (3) - 1\right )} \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right ) + 4 \, \log \relax (3) \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right ) - 8 \, \log \relax (3) + \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right ) - 4\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 24, normalized size = 1.33
method | result | size |
norman | \(x^{4} \ln \left (-2 \ln \relax (3)+x^{2} {\mathrm e}-1\right )+2 x^{4}\) | \(24\) |
risch | \(x^{4} \ln \left (-2 \ln \relax (3)+x^{2} {\mathrm e}-1\right )+2 x^{4}\) | \(24\) |
default | \(x^{4} \ln \left (-2 \ln \relax (3)+x^{2} {\mathrm e}-1\right )+2 x^{4}-4 \,{\mathrm e}^{-2} \ln \left (-2 \ln \relax (3)+x^{2} {\mathrm e}-1\right ) \ln \relax (3)^{2}-4 \,{\mathrm e}^{-2} \ln \left (-2 \ln \relax (3)+x^{2} {\mathrm e}-1\right ) \ln \relax (3)-{\mathrm e}^{-2} \ln \left (-2 \ln \relax (3)+x^{2} {\mathrm e}-1\right )+4 \left ({\mathrm e}^{-1}\right )^{2} \ln \left (-2 \ln \relax (3)+x^{2} {\mathrm e}-1\right ) \ln \relax (3)^{2}+4 \left ({\mathrm e}^{-1}\right )^{2} \ln \left (-2 \ln \relax (3)+x^{2} {\mathrm e}-1\right ) \ln \relax (3)+\left ({\mathrm e}^{-1}\right )^{2} \ln \left (-2 \ln \relax (3)+x^{2} {\mathrm e}-1\right )\) | \(149\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 467, normalized size = 25.94 \begin {gather*} -4 \, x^{2} e^{\left (-1\right )} + 2 \, {\left (2 \, x^{2} e + {\left (2 \, \log \relax (3) + 1\right )} \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right )^{2} + 2 \, {\left (2 \, \log \relax (3) + 1\right )} \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right )\right )} e^{\left (-2\right )} \log \relax (3) + {\left (2 \, {\left (4 \, \log \relax (3)^{2} + 4 \, \log \relax (3) + 1\right )} e^{\left (-3\right )} \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right ) + {\left (x^{4} e + 2 \, x^{2} {\left (2 \, \log \relax (3) + 1\right )}\right )} e^{\left (-2\right )}\right )} e \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right ) - 4 \, {\left (2 \, \log \relax (3) + 1\right )} e^{\left (-2\right )} \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right ) - 4 \, {\left (x^{2} e^{\left (-1\right )} + {\left (2 \, \log \relax (3) + 1\right )} e^{\left (-2\right )} \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right )\right )} \log \relax (3) \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right ) + \frac {5}{2} \, {\left (2 \, {\left (4 \, \log \relax (3)^{2} + 4 \, \log \relax (3) + 1\right )} e^{\left (-3\right )} \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right ) + {\left (x^{4} e + 2 \, x^{2} {\left (2 \, \log \relax (3) + 1\right )}\right )} e^{\left (-2\right )}\right )} e - \frac {1}{2} \, {\left (x^{4} e^{2} + 6 \, x^{2} {\left (2 \, \log \relax (3) + 1\right )} e + 2 \, {\left (4 \, \log \relax (3)^{2} + 4 \, \log \relax (3) + 1\right )} \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right )^{2} + 6 \, {\left (4 \, \log \relax (3)^{2} + 4 \, \log \relax (3) + 1\right )} \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right )\right )} e^{\left (-2\right )} + {\left (2 \, x^{2} e + {\left (2 \, \log \relax (3) + 1\right )} \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right )^{2} + 2 \, {\left (2 \, \log \relax (3) + 1\right )} \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right )\right )} e^{\left (-2\right )} - 8 \, {\left (x^{2} e^{\left (-1\right )} + {\left (2 \, \log \relax (3) + 1\right )} e^{\left (-2\right )} \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right )\right )} \log \relax (3) - 2 \, {\left (x^{2} e^{\left (-1\right )} + {\left (2 \, \log \relax (3) + 1\right )} e^{\left (-2\right )} \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right )\right )} \log \left (x^{2} e - 2 \, \log \relax (3) - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 20.72, size = 156, normalized size = 8.67 \begin {gather*} x^4\,\ln \left (\mathrm {e}\,x^2-2\,\ln \relax (3)-1\right )-4\,x^2\,{\mathrm {e}}^{-1}+2\,x^4-8\,x^2\,{\mathrm {e}}^{-1}\,\ln \relax (3)-\ln \left (\mathrm {e}\,x^2-\ln \relax (9)-1\right )\,{\mathrm {e}}^{-2}\,\left (4\,\ln \relax (9)+4\right )+\ln \left (\mathrm {e}\,x^2-\ln \relax (9)-1\right )\,{\mathrm {e}}^{-2}\,\left (10\,\ln \relax (9)+5\,{\ln \relax (9)}^2+5\right )-\ln \left (\mathrm {e}\,x^2-\ln \relax (9)-1\right )\,{\mathrm {e}}^{-2}\,\left (8\,\ln \relax (3)+8\,\ln \relax (3)\,\ln \relax (9)\right )+4\,x^2\,{\mathrm {e}}^{-1}\,\left (\ln \relax (9)+1\right )-\ln \left (\mathrm {e}\,x^2-\ln \relax (9)-1\right )\,{\mathrm {e}}^{-2}\,\left (2\,\ln \relax (9)+{\ln \relax (9)}^2+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 22, normalized size = 1.22 \begin {gather*} x^{4} \log {\left (e x^{2} - 2 \log {\relax (3 )} - 1 \right )} + 2 x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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