Optimal. Leaf size=32 \[ 2 \left (e^2+e^5\right )-x-\log \left (\frac {2 \left (e^{e^{4 x^2}}+x\right )}{x}\right ) \]
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Rubi [F]
time = 0.65, 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 {-x^2+e^{e^{4 x^2}} \left (1-x-8 e^{4 x^2} x^2\right )}{e^{e^{4 x^2}} 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 {8 e^{e^{4 x^2}+4 x^2} x}{e^{e^{4 x^2}}+x}-\frac {-e^{e^{4 x^2}}+e^{e^{4 x^2}} x+x^2}{x \left (e^{e^{4 x^2}}+x\right )}\right ) \, dx\\ &=-\left (8 \int \frac {e^{e^{4 x^2}+4 x^2} x}{e^{e^{4 x^2}}+x} \, dx\right )-\int \frac {-e^{e^{4 x^2}}+e^{e^{4 x^2}} x+x^2}{x \left (e^{e^{4 x^2}}+x\right )} \, dx\\ &=-\left (8 \int \frac {e^{e^{4 x^2}+4 x^2} x}{e^{e^{4 x^2}}+x} \, dx\right )-\int \frac {e^{e^{4 x^2}} (-1+x)+x^2}{x \left (e^{e^{4 x^2}}+x\right )} \, dx\\ &=-\left (8 \int \frac {e^{e^{4 x^2}+4 x^2} x}{e^{e^{4 x^2}}+x} \, dx\right )-\int \left (\frac {-1+x}{x}+\frac {1}{e^{e^{4 x^2}}+x}\right ) \, dx\\ &=-\left (8 \int \frac {e^{e^{4 x^2}+4 x^2} x}{e^{e^{4 x^2}}+x} \, dx\right )-\int \frac {-1+x}{x} \, dx-\int \frac {1}{e^{e^{4 x^2}}+x} \, dx\\ &=-\left (8 \int \frac {e^{e^{4 x^2}+4 x^2} x}{e^{e^{4 x^2}}+x} \, dx\right )-\int \left (1-\frac {1}{x}\right ) \, dx-\int \frac {1}{e^{e^{4 x^2}}+x} \, dx\\ &=-x+\log (x)-8 \int \frac {e^{e^{4 x^2}+4 x^2} x}{e^{e^{4 x^2}}+x} \, dx-\int \frac {1}{e^{e^{4 x^2}}+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 20, normalized size = 0.62 \begin {gather*} -x+\log (x)-\log \left (e^{e^{4 x^2}}+x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 19, normalized size = 0.59
method | result | size |
norman | \(-x -\ln \left ({\mathrm e}^{{\mathrm e}^{4 x^{2}}}+x \right )+\ln \left (x \right )\) | \(19\) |
risch | \(-x -\ln \left ({\mathrm e}^{{\mathrm e}^{4 x^{2}}}+x \right )+\ln \left (x \right )\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 18, normalized size = 0.56 \begin {gather*} -x - \log \left (x + e^{\left (e^{\left (4 \, x^{2}\right )}\right )}\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 18, normalized size = 0.56 \begin {gather*} -x - \log \left (x + e^{\left (e^{\left (4 \, x^{2}\right )}\right )}\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 15, normalized size = 0.47 \begin {gather*} - x + \log {\left (x \right )} - \log {\left (x + e^{e^{4 x^{2}}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 36, normalized size = 1.12 \begin {gather*} 4 \, x^{2} - x - \log \left (x e^{\left (4 \, x^{2}\right )} + e^{\left (4 \, x^{2} + e^{\left (4 \, x^{2}\right )}\right )}\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 18, normalized size = 0.56 \begin {gather*} \ln \left (x\right )-\ln \left (x+{\mathrm {e}}^{{\mathrm {e}}^{4\,x^2}}\right )-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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