Optimal. Leaf size=17 \[ x+\frac {x}{8+e^{(3+x)^2}+\log (x)} \]
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Rubi [F] time = 1.52, 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 {71+e^{18+12 x+2 x^2}+e^{9+6 x+x^2} \left (17-6 x-2 x^2\right )+\left (17+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(x)}{64+16 e^{9+6 x+x^2}+e^{18+12 x+2 x^2}+\left (16+2 e^{9+6 x+x^2}\right ) \log (x)+\log ^2(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 {71+e^{2 (3+x)^2}+e^{(3+x)^2} \left (17-6 x-2 x^2\right )+\left (17+2 e^{(3+x)^2}\right ) \log (x)+\log ^2(x)}{\left (8+e^{(3+x)^2}+\log (x)\right )^2} \, dx\\ &=\int \left (1-\frac {-1+6 x+2 x^2}{8+e^{(3+x)^2}+\log (x)}+\frac {-1+48 x+16 x^2+6 x \log (x)+2 x^2 \log (x)}{\left (8+e^{(3+x)^2}+\log (x)\right )^2}\right ) \, dx\\ &=x-\int \frac {-1+6 x+2 x^2}{8+e^{(3+x)^2}+\log (x)} \, dx+\int \frac {-1+48 x+16 x^2+6 x \log (x)+2 x^2 \log (x)}{\left (8+e^{(3+x)^2}+\log (x)\right )^2} \, dx\\ &=x+\int \left (-\frac {1}{\left (8+e^{(3+x)^2}+\log (x)\right )^2}+\frac {48 x}{\left (8+e^{(3+x)^2}+\log (x)\right )^2}+\frac {16 x^2}{\left (8+e^{(3+x)^2}+\log (x)\right )^2}+\frac {6 x \log (x)}{\left (8+e^{(3+x)^2}+\log (x)\right )^2}+\frac {2 x^2 \log (x)}{\left (8+e^{(3+x)^2}+\log (x)\right )^2}\right ) \, dx-\int \left (-\frac {1}{8+e^{(3+x)^2}+\log (x)}+\frac {6 x}{8+e^{(3+x)^2}+\log (x)}+\frac {2 x^2}{8+e^{(3+x)^2}+\log (x)}\right ) \, dx\\ &=x+2 \int \frac {x^2 \log (x)}{\left (8+e^{(3+x)^2}+\log (x)\right )^2} \, dx-2 \int \frac {x^2}{8+e^{(3+x)^2}+\log (x)} \, dx+6 \int \frac {x \log (x)}{\left (8+e^{(3+x)^2}+\log (x)\right )^2} \, dx-6 \int \frac {x}{8+e^{(3+x)^2}+\log (x)} \, dx+16 \int \frac {x^2}{\left (8+e^{(3+x)^2}+\log (x)\right )^2} \, dx+48 \int \frac {x}{\left (8+e^{(3+x)^2}+\log (x)\right )^2} \, dx-\int \frac {1}{\left (8+e^{(3+x)^2}+\log (x)\right )^2} \, dx+\int \frac {1}{8+e^{(3+x)^2}+\log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.48, size = 17, normalized size = 1.00 \begin {gather*} x \left (1+\frac {1}{8+e^{(3+x)^2}+\log (x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.07, size = 35, normalized size = 2.06 \begin {gather*} \frac {x e^{\left (x^{2} + 6 \, x + 9\right )} + x \log \relax (x) + 9 \, x}{e^{\left (x^{2} + 6 \, x + 9\right )} + \log \relax (x) + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 35, normalized size = 2.06 \begin {gather*} \frac {x e^{\left (x^{2} + 6 \, x + 9\right )} + x \log \relax (x) + 9 \, x}{e^{\left (x^{2} + 6 \, x + 9\right )} + \log \relax (x) + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 17, normalized size = 1.00
method | result | size |
risch | \(x +\frac {x}{\ln \relax (x )+8+{\mathrm e}^{\left (3+x \right )^{2}}}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 35, normalized size = 2.06 \begin {gather*} \frac {x e^{\left (x^{2} + 6 \, x + 9\right )} + x \log \relax (x) + 9 \, x}{e^{\left (x^{2} + 6 \, x + 9\right )} + \log \relax (x) + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.97, size = 30, normalized size = 1.76 \begin {gather*} \frac {x\,\left ({\mathrm {e}}^{x^2+6\,x+9}+\ln \relax (x)+9\right )}{{\mathrm {e}}^{x^2+6\,x+9}+\ln \relax (x)+8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 17, normalized size = 1.00 \begin {gather*} x + \frac {x}{e^{x^{2} + 6 x + 9} + \log {\relax (x )} + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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