Optimal. Leaf size=33 \[ 1+(1-x)^2+\log \left (1+4 e^{x/2}-e^x-x\right )+\log ^2(x) \]
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Rubi [F]
time = 0.47, 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 {3 x-4 x^2+2 x^3+e^{x/2} \left (6 x-8 x^2\right )+e^x \left (-x+2 x^2\right )+\left (-2-8 e^{x/2}+2 e^x+2 x\right ) \log (x)}{-x-4 e^{x/2} x+e^x x+x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3+e^{x/2} (6-8 x)-4 x+2 x^2+e^x (-1+2 x)}{-1-4 e^{x/2}+e^x+x}+\frac {2 \log (x)}{x}\right ) \, dx\\ &=2 \int \frac {\log (x)}{x} \, dx+\int \frac {3+e^{x/2} (6-8 x)-4 x+2 x^2+e^x (-1+2 x)}{-1-4 e^{x/2}+e^x+x} \, dx\\ &=\log ^2(x)+\int \left (-1+2 x+\frac {2+2 e^{x/2}-x}{-1-4 e^{x/2}+e^x+x}\right ) \, dx\\ &=-x+x^2+\log ^2(x)+\int \frac {2+2 e^{x/2}-x}{-1-4 e^{x/2}+e^x+x} \, dx\\ &=-x+x^2+\log ^2(x)+2 \text {Subst}\left (\int \frac {2+2 e^x-2 x}{-1-4 e^x+e^{2 x}+2 x} \, dx,x,\frac {x}{2}\right )\\ &=-x+x^2+\log ^2(x)+2 \text {Subst}\left (\int \frac {2 \left (-1-e^x+x\right )}{1+4 e^x-e^{2 x}-2 x} \, dx,x,\frac {x}{2}\right )\\ &=-x+x^2+\log ^2(x)+4 \text {Subst}\left (\int \frac {-1-e^x+x}{1+4 e^x-e^{2 x}-2 x} \, dx,x,\frac {x}{2}\right )\\ &=-x+x^2+\log ^2(x)+4 \text {Subst}\left (\int \left (\frac {1}{-1-4 e^x+e^{2 x}+2 x}+\frac {e^x}{-1-4 e^x+e^{2 x}+2 x}-\frac {x}{-1-4 e^x+e^{2 x}+2 x}\right ) \, dx,x,\frac {x}{2}\right )\\ &=-x+x^2+\log ^2(x)+4 \text {Subst}\left (\int \frac {1}{-1-4 e^x+e^{2 x}+2 x} \, dx,x,\frac {x}{2}\right )+4 \text {Subst}\left (\int \frac {e^x}{-1-4 e^x+e^{2 x}+2 x} \, dx,x,\frac {x}{2}\right )-4 \text {Subst}\left (\int \frac {x}{-1-4 e^x+e^{2 x}+2 x} \, dx,x,\frac {x}{2}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.45, size = 31, normalized size = 0.94 \begin {gather*} -2 x+x^2+\log \left (1+4 e^{x/2}-e^x-x\right )+\log ^2(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 24, normalized size = 0.73
method | result | size |
risch | \(\ln \left (x \right )^{2}+x^{2}-2 x +\ln \left ({\mathrm e}^{x}-4 \,{\mathrm e}^{\frac {x}{2}}+x -1\right )\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 23, normalized size = 0.70 \begin {gather*} x^{2} + \log \left (x\right )^{2} - 2 \, x + \log \left (x - 4 \, e^{\left (\frac {1}{2} \, x\right )} + e^{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 23, normalized size = 0.70 \begin {gather*} x^{2} + \log \left (x\right )^{2} - 2 \, x + \log \left (x - 4 \, e^{\left (\frac {1}{2} \, x\right )} + e^{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 26, normalized size = 0.79 \begin {gather*} x^{2} - 2 x + \log {\left (x \right )}^{2} + \log {\left (x - 4 e^{\frac {x}{2}} + e^{x} - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 33, normalized size = 1.00 \begin {gather*} x^{2} + 2 \, \log \left (2\right ) \log \left (\frac {1}{2} \, x\right ) + \log \left (\frac {1}{2} \, x\right )^{2} - 2 \, x + \log \left (x - 4 \, e^{\left (\frac {1}{2} \, x\right )} + e^{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.47, size = 23, normalized size = 0.70 \begin {gather*} \ln \left (x+{\mathrm {e}}^x-4\,\sqrt {{\mathrm {e}}^x}-1\right )-2\,x+{\ln \left (x\right )}^2+x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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