Optimal. Leaf size=29 \[ 4 \log \left (\frac {9 \left (3+e^x\right )}{4 x \left (1+\frac {1}{x}+e^{-x} x\right )}\right ) \]
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Rubi [F]
time = 1.42, 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 {-24 x+4 e^{2 x} x+12 x^2+e^x \left (-12-8 x+8 x^2\right )}{3 x^2+e^{2 x} (1+x)+e^x \left (3+3 x+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} &=\int \frac {-24 x+4 e^{2 x} x+12 x^2+e^x \left (-12-8 x+8 x^2\right )}{\left (3+e^x\right ) \left (e^x+e^x x+x^2\right )} \, dx\\ &=\int \left (-\frac {12}{3+e^x}+\frac {4 x}{1+x}+\frac {4 x \left (-2+x^2\right )}{(1+x) \left (e^x+e^x x+x^2\right )}\right ) \, dx\\ &=4 \int \frac {x}{1+x} \, dx+4 \int \frac {x \left (-2+x^2\right )}{(1+x) \left (e^x+e^x x+x^2\right )} \, dx-12 \int \frac {1}{3+e^x} \, dx\\ &=4 \int \left (1+\frac {1}{-1-x}\right ) \, dx+4 \int \left (-\frac {1}{e^x+e^x x+x^2}-\frac {x}{e^x+e^x x+x^2}+\frac {x^2}{e^x+e^x x+x^2}+\frac {1}{(1+x) \left (e^x+e^x x+x^2\right )}\right ) \, dx-12 \text {Subst}\left (\int \frac {1}{x (3+x)} \, dx,x,e^x\right )\\ &=4 x-4 \log (1+x)-4 \int \frac {1}{e^x+e^x x+x^2} \, dx-4 \int \frac {x}{e^x+e^x x+x^2} \, dx+4 \int \frac {x^2}{e^x+e^x x+x^2} \, dx+4 \int \frac {1}{(1+x) \left (e^x+e^x x+x^2\right )} \, dx-4 \text {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )+4 \text {Subst}\left (\int \frac {1}{3+x} \, dx,x,e^x\right )\\ &=4 \log \left (3+e^x\right )-4 \log (1+x)-4 \int \frac {1}{e^x+e^x x+x^2} \, dx-4 \int \frac {x}{e^x+e^x x+x^2} \, dx+4 \int \frac {x^2}{e^x+e^x x+x^2} \, dx+4 \int \frac {1}{(1+x) \left (e^x+e^x x+x^2\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 2.96, size = 25, normalized size = 0.86 \begin {gather*} 4 \left (x+\log \left (3+e^x\right )-\log \left (e^x+e^x x+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 25, normalized size = 0.86
method | result | size |
norman | \(4 x +4 \ln \left (3+{\mathrm e}^{x}\right )-4 \ln \left (x^{2}+{\mathrm e}^{x} x +{\mathrm e}^{x}\right )\) | \(25\) |
risch | \(4 x -4 \ln \left (x +1\right )+4 \ln \left (3+{\mathrm e}^{x}\right )-4 \ln \left ({\mathrm e}^{x}+\frac {x^{2}}{x +1}\right )\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 36, normalized size = 1.24 \begin {gather*} 4 \, x - 4 \, \log \left (x + 1\right ) - 4 \, \log \left (\frac {x^{2} + {\left (x + 1\right )} e^{x}}{x + 1}\right ) + 4 \, \log \left (e^{x} + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 36, normalized size = 1.24 \begin {gather*} 4 \, x - 4 \, \log \left (x + 1\right ) - 4 \, \log \left (\frac {x^{2} + {\left (x + 1\right )} e^{x}}{x + 1}\right ) + 4 \, \log \left (e^{x} + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 24, normalized size = 0.83 \begin {gather*} 4 \, x - 4 \, \log \left (x^{2} + x e^{x} + e^{x}\right ) + 4 \, \log \left (e^{x} + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.13, size = 24, normalized size = 0.83 \begin {gather*} 4\,x-4\,\ln \left ({\mathrm {e}}^x+x\,{\mathrm {e}}^x+x^2\right )+4\,\ln \left ({\mathrm {e}}^x+3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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