Optimal. Leaf size=22 \[ e^{-x+x^4-\frac {4 x}{-3+\log (4+4 x)}} \]
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Rubi [F] time = 22.12, 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 {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{(1+x) (3-\log (4 (1+x)))^2} \, dx\\ &=\int \left (\frac {3 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2}+\frac {7 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x}{(1+x) (-3+\log (4 (1+x)))^2}+\frac {36 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3}{(1+x) (-3+\log (4 (1+x)))^2}+\frac {36 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^4}{(1+x) (-3+\log (4 (1+x)))^2}+\frac {2 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (1-12 x^3\right ) \log (4+4 x)}{(3-\log (4 (1+x)))^2}+\frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (-1+4 x^3\right ) \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2}\right ) \, dx\\ &=2 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (1-12 x^3\right ) \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+3 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+36 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+36 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^4}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+\int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (-1+4 x^3\right ) \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx\\ &=2 \int \left (\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \log (4+4 x)}{(3-\log (4 (1+x)))^2}-\frac {12 e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x^3 \log (4+4 x)}{(3-\log (4 (1+x)))^2}\right ) \, dx+3 \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+7 \int \left (\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(-3+\log (4 (1+x)))^2}-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(1+x) (-3+\log (4 (1+x)))^2}\right ) \, dx+36 \int \left (\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(-3+\log (4 (1+x)))^2}-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x}{(-3+\log (4 (1+x)))^2}+\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x^2}{(-3+\log (4 (1+x)))^2}-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(1+x) (-3+\log (4 (1+x)))^2}\right ) \, dx+36 \int \left (-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(-3+\log (4 (1+x)))^2}+\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x}{(-3+\log (4 (1+x)))^2}-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x^2}{(-3+\log (4 (1+x)))^2}+\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x^3}{(-3+\log (4 (1+x)))^2}+\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(1+x) (-3+\log (4 (1+x)))^2}\right ) \, dx+\int \left (-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2}+\frac {4 e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x^3 \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2}\right ) \, dx\\ &=2 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+3 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+4 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(-3+\log (4 (1+x)))^2} \, dx-7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx-24 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+36 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3}{(-3+\log (4 (1+x)))^2} \, dx-\int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx\\ &=2 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+3 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+4 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(-3+\log (4 (1+x)))^2} \, dx-7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx-24 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+36 \int \left (-\frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(-3+\log (4 (1+x)))^2}+\frac {3 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)}{(-3+\log (4 (1+x)))^2}-\frac {3 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)^2}{(-3+\log (4 (1+x)))^2}+\frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)^3}{(-3+\log (4 (1+x)))^2}\right ) \, dx-\int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx\\ &=2 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+3 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+4 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(-3+\log (4 (1+x)))^2} \, dx-7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx-24 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx-36 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(-3+\log (4 (1+x)))^2} \, dx+36 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)^3}{(-3+\log (4 (1+x)))^2} \, dx+108 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)}{(-3+\log (4 (1+x)))^2} \, dx-108 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)^2}{(-3+\log (4 (1+x)))^2} \, dx-\int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.25, size = 33, normalized size = 1.50 \begin {gather*} e^{\frac {x \left (-1-3 x^3+\left (-1+x^3\right ) \log (4 (1+x))\right )}{-3+\log (4 (1+x))}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 35, normalized size = 1.59 \begin {gather*} e^{\left (-\frac {3 \, x^{4} - {\left (x^{4} - x\right )} \log \left (4 \, x + 4\right ) + x}{\log \left (4 \, x + 4\right ) - 3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 5.31, size = 69, normalized size = 3.14 \begin {gather*} e^{\left (\frac {x^{4} \log \left (4 \, x + 4\right )}{\log \left (4 \, x + 4\right ) - 3} - \frac {3 \, x^{4}}{\log \left (4 \, x + 4\right ) - 3} - \frac {x \log \left (4 \, x + 4\right )}{\log \left (4 \, x + 4\right ) - 3} - \frac {x}{\log \left (4 \, x + 4\right ) - 3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 39, normalized size = 1.77
method | result | size |
risch | \({\mathrm e}^{\frac {x \left (\ln \left (4 x +4\right ) x^{3}-3 x^{3}-\ln \left (4 x +4\right )-1\right )}{\ln \left (4 x +4\right )-3}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.64, size = 109, normalized size = 4.95 \begin {gather*} e^{\left (\frac {2 \, x^{4} \log \relax (2)}{2 \, \log \relax (2) + \log \left (x + 1\right ) - 3} + \frac {x^{4} \log \left (x + 1\right )}{2 \, \log \relax (2) + \log \left (x + 1\right ) - 3} - \frac {3 \, x^{4}}{2 \, \log \relax (2) + \log \left (x + 1\right ) - 3} - \frac {2 \, x \log \relax (2)}{2 \, \log \relax (2) + \log \left (x + 1\right ) - 3} - \frac {x \log \left (x + 1\right )}{2 \, \log \relax (2) + \log \left (x + 1\right ) - 3} - \frac {x}{2 \, \log \relax (2) + \log \left (x + 1\right ) - 3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.99, size = 57, normalized size = 2.59 \begin {gather*} \frac {{\mathrm {e}}^{-\frac {x}{\ln \left (4\,x+4\right )-3}}\,{\mathrm {e}}^{-\frac {3\,x^4}{\ln \left (4\,x+4\right )-3}}}{{\left (4\,x+4\right )}^{\frac {x-x^4}{\ln \left (4\,x+4\right )-3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.66, size = 27, normalized size = 1.23 \begin {gather*} e^{\frac {- 3 x^{4} - x + \left (x^{4} - x\right ) \log {\left (4 x + 4 \right )}}{\log {\left (4 x + 4 \right )} - 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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