Optimal. Leaf size=27 \[ 5+x+x \left (e^x+\frac {-2+e^{2 x}}{3+x \log ^2(x)}\right ) \]
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Rubi [F]
time = 0.67, 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+e^{2 x} (3+6 x)+e^x (9+9 x)+\left (4 x-2 e^{2 x} x\right ) \log (x)+\left (6 x+2 e^{2 x} x^2+e^x \left (6 x+6 x^2\right )\right ) \log ^2(x)+\left (x^2+e^x \left (x^2+x^3\right )\right ) \log ^4(x)}{9+6 x \log ^2(x)+x^2 \log ^4(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 {3+e^{2 x} (3+6 x)+e^x (9+9 x)+\left (4 x-2 e^{2 x} x\right ) \log (x)+\left (6 x+2 e^{2 x} x^2+e^x \left (6 x+6 x^2\right )\right ) \log ^2(x)+\left (x^2+e^x \left (x^2+x^3\right )\right ) \log ^4(x)}{\left (3+x \log ^2(x)\right )^2} \, dx\\ &=\int \left (e^x (1+x)+\frac {3}{\left (3+x \log ^2(x)\right )^2}+\frac {4 x \log (x)}{\left (3+x \log ^2(x)\right )^2}+\frac {6 x \log ^2(x)}{\left (3+x \log ^2(x)\right )^2}+\frac {x^2 \log ^4(x)}{\left (3+x \log ^2(x)\right )^2}+\frac {e^{2 x} \left (3+6 x-2 x \log (x)+2 x^2 \log ^2(x)\right )}{\left (3+x \log ^2(x)\right )^2}\right ) \, dx\\ &=3 \int \frac {1}{\left (3+x \log ^2(x)\right )^2} \, dx+4 \int \frac {x \log (x)}{\left (3+x \log ^2(x)\right )^2} \, dx+6 \int \frac {x \log ^2(x)}{\left (3+x \log ^2(x)\right )^2} \, dx+\int e^x (1+x) \, dx+\int \frac {x^2 \log ^4(x)}{\left (3+x \log ^2(x)\right )^2} \, dx+\int \frac {e^{2 x} \left (3+6 x-2 x \log (x)+2 x^2 \log ^2(x)\right )}{\left (3+x \log ^2(x)\right )^2} \, dx\\ &=e^x (1+x)+\frac {e^{2 x} \left (3 x+x^2 \log ^2(x)\right )}{\left (3+x \log ^2(x)\right )^2}+3 \int \frac {1}{\left (3+x \log ^2(x)\right )^2} \, dx+4 \int \frac {x \log (x)}{\left (3+x \log ^2(x)\right )^2} \, dx+6 \int \left (-\frac {3}{\left (3+x \log ^2(x)\right )^2}+\frac {1}{3+x \log ^2(x)}\right ) \, dx-\int e^x \, dx+\int \left (1+\frac {9}{\left (3+x \log ^2(x)\right )^2}-\frac {6}{3+x \log ^2(x)}\right ) \, dx\\ &=-e^x+x+e^x (1+x)+\frac {e^{2 x} \left (3 x+x^2 \log ^2(x)\right )}{\left (3+x \log ^2(x)\right )^2}+3 \int \frac {1}{\left (3+x \log ^2(x)\right )^2} \, dx+4 \int \frac {x \log (x)}{\left (3+x \log ^2(x)\right )^2} \, dx+9 \int \frac {1}{\left (3+x \log ^2(x)\right )^2} \, dx-18 \int \frac {1}{\left (3+x \log ^2(x)\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 26, normalized size = 0.96 \begin {gather*} x+e^x x+\frac {\left (-2+e^{2 x}\right ) x}{3+x \log ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.20, size = 25, normalized size = 0.93
method | result | size |
risch | \({\mathrm e}^{x} x +x +\frac {x \left ({\mathrm e}^{2 x}-2\right )}{x \ln \left (x \right )^{2}+3}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 42, normalized size = 1.56 \begin {gather*} \frac {x^{2} \log \left (x\right )^{2} + x e^{\left (2 \, x\right )} + {\left (x^{2} \log \left (x\right )^{2} + 3 \, x\right )} e^{x} + x}{x \log \left (x\right )^{2} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 39, normalized size = 1.44 \begin {gather*} \frac {{\left (x^{2} e^{x} + x^{2}\right )} \log \left (x\right )^{2} + x e^{\left (2 \, x\right )} + 3 \, x e^{x} + x}{x \log \left (x\right )^{2} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.31, size = 42, normalized size = 1.56 \begin {gather*} x - \frac {2 x}{x \log {\left (x \right )}^{2} + 3} + \frac {x e^{2 x} + \left (x^{2} \log {\left (x \right )}^{2} + 3 x\right ) e^{x}}{x \log {\left (x \right )}^{2} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 42, normalized size = 1.56 \begin {gather*} \frac {x^{2} e^{x} \log \left (x\right )^{2} + x^{2} \log \left (x\right )^{2} + x e^{\left (2 \, x\right )} + 3 \, x e^{x} + x}{x \log \left (x\right )^{2} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.45, size = 36, normalized size = 1.33 \begin {gather*} \frac {x\,\left ({\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^x+x\,{\ln \left (x\right )}^2+x\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2+1\right )}{x\,{\ln \left (x\right )}^2+3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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