Optimal. Leaf size=24 \[ 1+\log \left (\frac {\frac {1}{8 \left (-11+e^{x^2}\right )^2}-x}{x}\right ) \]
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Rubi [F] time = 2.49, 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 {-11+e^{x^2} \left (1+4 x^2\right )}{11 x-10648 x^2-264 e^{2 x^2} x^2+8 e^{3 x^2} x^2+e^{x^2} \left (-x+2904 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 {-11+e^{x^2} \left (1+4 x^2\right )}{\left (11-e^{x^2}\right ) x \left (1-8 \left (-11+e^{x^2}\right )^2 x\right )} \, dx\\ &=\int \left (-\frac {44 x}{-11+e^{x^2}}+\frac {1+4 x^2-3872 x^3+352 e^{x^2} x^3}{x \left (-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x\right )}\right ) \, dx\\ &=-\left (44 \int \frac {x}{-11+e^{x^2}} \, dx\right )+\int \frac {1+4 x^2-3872 x^3+352 e^{x^2} x^3}{x \left (-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x\right )} \, dx\\ &=-\left (22 \operatorname {Subst}\left (\int \frac {1}{-11+e^x} \, dx,x,x^2\right )\right )+\int \left (\frac {1}{x \left (-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x\right )}+\frac {4 x}{-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x}-\frac {3872 x^2}{-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x}+\frac {352 e^{x^2} x^2}{-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x}\right ) \, dx\\ &=4 \int \frac {x}{-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x} \, dx-22 \operatorname {Subst}\left (\int \frac {1}{(-11+x) x} \, dx,x,e^{x^2}\right )+352 \int \frac {e^{x^2} x^2}{-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x} \, dx-3872 \int \frac {x^2}{-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x} \, dx+\int \frac {1}{x \left (-1+968 x-176 e^{x^2} x+8 e^{2 x^2} x\right )} \, dx\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{-11+x} \, dx,x,e^{x^2}\right )\right )+2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{x^2}\right )+4 \int \frac {x}{-1+8 \left (-11+e^{x^2}\right )^2 x} \, dx+352 \int \frac {e^{x^2} x^2}{-1+8 \left (-11+e^{x^2}\right )^2 x} \, dx-3872 \int \frac {x^2}{-1+8 \left (-11+e^{x^2}\right )^2 x} \, dx+\int \frac {1}{x \left (-1+8 \left (-11+e^{x^2}\right )^2 x\right )} \, dx\\ &=2 x^2-2 \log \left (11-e^{x^2}\right )+4 \int \frac {x}{-1+8 \left (-11+e^{x^2}\right )^2 x} \, dx+352 \int \frac {e^{x^2} x^2}{-1+8 \left (-11+e^{x^2}\right )^2 x} \, dx-3872 \int \frac {x^2}{-1+8 \left (-11+e^{x^2}\right )^2 x} \, dx+\int \frac {1}{x \left (-1+8 \left (-11+e^{x^2}\right )^2 x\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.31, size = 41, normalized size = 1.71 \begin {gather*} -2 \log \left (11-e^{x^2}\right )-\log (x)+\log \left (1-968 x+176 e^{x^2} x-8 e^{2 x^2} x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 36, normalized size = 1.50 \begin {gather*} \log \left (\frac {8 \, x e^{\left (2 \, x^{2}\right )} - 176 \, x e^{\left (x^{2}\right )} + 968 \, x - 1}{x}\right ) - 2 \, \log \left (e^{\left (x^{2}\right )} - 11\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 36, normalized size = 1.50 \begin {gather*} \log \left (8 \, x e^{\left (2 \, x^{2}\right )} - 176 \, x e^{\left (x^{2}\right )} + 968 \, x - 1\right ) - \log \relax (x) - 2 \, \log \left (e^{\left (x^{2}\right )} - 11\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 35, normalized size = 1.46
method | result | size |
risch | \(\ln \left ({\mathrm e}^{2 x^{2}}-22 \,{\mathrm e}^{x^{2}}+\frac {968 x -1}{8 x}\right )-2 \ln \left ({\mathrm e}^{x^{2}}-11\right )\) | \(35\) |
norman | \(-\ln \relax (x )-2 \ln \left ({\mathrm e}^{x^{2}}-11\right )+\ln \left (8 x \,{\mathrm e}^{2 x^{2}}-176 \,{\mathrm e}^{x^{2}} x +968 x -1\right )\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 37, normalized size = 1.54 \begin {gather*} \log \left (\frac {8 \, x e^{\left (2 \, x^{2}\right )} - 176 \, x e^{\left (x^{2}\right )} + 968 \, x - 1}{8 \, x}\right ) - 2 \, \log \left (e^{\left (x^{2}\right )} - 11\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.54, size = 36, normalized size = 1.50 \begin {gather*} \ln \left (968\,x-176\,x\,{\mathrm {e}}^{x^2}+8\,x\,{\mathrm {e}}^{2\,x^2}-1\right )-2\,\ln \left ({\mathrm {e}}^{x^2}-11\right )-\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 32, normalized size = 1.33 \begin {gather*} - 2 \log {\left (e^{x^{2}} - 11 \right )} + \log {\left (e^{2 x^{2}} - 22 e^{x^{2}} + \frac {968 x - 1}{8 x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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