Optimal. Leaf size=29 \[ \frac {\left (\frac {1}{4}-e^x\right ) \left (-9+x+x \left (5+(\log (4)-\log (x))^2\right )\right )}{x} \]
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Rubi [F] time = 0.88, 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 {9-2 x \log (4)+e^x \left (-36+36 x-24 x^2+8 x \log (4)-4 x^2 \log ^2(4)\right )+\left (2 x+e^x \left (-8 x+8 x^2 \log (4)\right )\right ) \log (x)-4 e^x x^2 \log ^2(x)}{4 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {9-2 x \log (4)+e^x \left (-36+36 x-24 x^2+8 x \log (4)-4 x^2 \log ^2(4)\right )+\left (2 x+e^x \left (-8 x+8 x^2 \log (4)\right )\right ) \log (x)-4 e^x x^2 \log ^2(x)}{x^2} \, dx\\ &=\frac {1}{4} \int \left (\frac {9-2 x \log (4)+2 x \log (x)}{x^2}+\frac {4 e^x \left (-9+9 x \left (1+\frac {4 \log (2)}{9}\right )-6 x^2 \left (1+\frac {\log ^2(4)}{6}\right )-2 x \log (x)+2 x^2 \log (4) \log (x)-x^2 \log ^2(x)\right )}{x^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {9-2 x \log (4)+2 x \log (x)}{x^2} \, dx+\int \frac {e^x \left (-9+9 x \left (1+\frac {4 \log (2)}{9}\right )-6 x^2 \left (1+\frac {\log ^2(4)}{6}\right )-2 x \log (x)+2 x^2 \log (4) \log (x)-x^2 \log ^2(x)\right )}{x^2} \, dx\\ &=\frac {1}{4} \int \left (\frac {9-2 x \log (4)}{x^2}+\frac {2 \log (x)}{x}\right ) \, dx+\int \left (\frac {e^x \left (-9-x^2 \left (6+\log ^2(4)\right )+x (9+\log (16))\right )}{x^2}+\frac {2 e^x (-1+x \log (4)) \log (x)}{x}-e^x \log ^2(x)\right ) \, dx\\ &=\frac {1}{4} \int \frac {9-2 x \log (4)}{x^2} \, dx+\frac {1}{2} \int \frac {\log (x)}{x} \, dx+2 \int \frac {e^x (-1+x \log (4)) \log (x)}{x} \, dx+\int \frac {e^x \left (-9-x^2 \left (6+\log ^2(4)\right )+x (9+\log (16))\right )}{x^2} \, dx-\int e^x \log ^2(x) \, dx\\ &=-2 \text {Ei}(x) \log (x)+2 e^x \log (4) \log (x)+\frac {\log ^2(x)}{4}+\frac {1}{4} \int \left (\frac {9}{x^2}-\frac {2 \log (4)}{x}\right ) \, dx-2 \int \frac {-\text {Ei}(x)+e^x \log (4)}{x} \, dx+\int \left (-\frac {9 e^x}{x^2}-6 e^x \left (1+\frac {\log ^2(4)}{6}\right )+\frac {e^x (9+\log (16))}{x}\right ) \, dx-\int e^x \log ^2(x) \, dx\\ &=-\frac {9}{4 x}-2 \text {Ei}(x) \log (x)-\frac {1}{2} \log (4) \log (x)+2 e^x \log (4) \log (x)+\frac {\log ^2(x)}{4}-2 \int \left (-\frac {\text {Ei}(x)}{x}+\frac {e^x \log (4)}{x}\right ) \, dx-9 \int \frac {e^x}{x^2} \, dx-\left (6+\log ^2(4)\right ) \int e^x \, dx+(9+\log (16)) \int \frac {e^x}{x} \, dx-\int e^x \log ^2(x) \, dx\\ &=-\frac {9}{4 x}+\frac {9 e^x}{x}-e^x \left (6+\log ^2(4)\right )+\text {Ei}(x) (9+\log (16))-2 \text {Ei}(x) \log (x)-\frac {1}{2} \log (4) \log (x)+2 e^x \log (4) \log (x)+\frac {\log ^2(x)}{4}+2 \int \frac {\text {Ei}(x)}{x} \, dx-9 \int \frac {e^x}{x} \, dx-(2 \log (4)) \int \frac {e^x}{x} \, dx-\int e^x \log ^2(x) \, dx\\ &=-\frac {9}{4 x}+\frac {9 e^x}{x}-9 \text {Ei}(x)-2 \text {Ei}(x) \log (4)-e^x \left (6+\log ^2(4)\right )+\text {Ei}(x) (9+\log (16))-2 \text {Ei}(x) \log (x)+2 (E_1(-x)+\text {Ei}(x)) \log (x)-\frac {1}{2} \log (4) \log (x)+2 e^x \log (4) \log (x)+\frac {\log ^2(x)}{4}-2 \int \frac {E_1(-x)}{x} \, dx-\int e^x \log ^2(x) \, dx\\ &=-\frac {9}{4 x}+\frac {9 e^x}{x}-9 \text {Ei}(x)+2 x \, _3F_3(1,1,1;2,2,2;x)-2 \text {Ei}(x) \log (4)-e^x \left (6+\log ^2(4)\right )+\text {Ei}(x) (9+\log (16))+\log ^2(-x)+2 \gamma \log (x)-2 \text {Ei}(x) \log (x)+2 (E_1(-x)+\text {Ei}(x)) \log (x)-\frac {1}{2} \log (4) \log (x)+2 e^x \log (4) \log (x)+\frac {\log ^2(x)}{4}-\int e^x \log ^2(x) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.42, size = 52, normalized size = 1.79 \begin {gather*} \frac {-9-4 e^x \left (-9+x \left (6+\log ^2(4)\right )\right )-x \left (\log (16)-e^x \log (65536)\right ) \log (x)+\left (x-4 e^x x\right ) \log ^2(x)}{4 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 54, normalized size = 1.86 \begin {gather*} -\frac {{\left (4 \, x e^{x} - x\right )} \log \relax (x)^{2} + 4 \, {\left (4 \, x \log \relax (2)^{2} + 6 \, x - 9\right )} e^{x} - 4 \, {\left (4 \, x e^{x} \log \relax (2) - x \log \relax (2)\right )} \log \relax (x) + 9}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 57, normalized size = 1.97 \begin {gather*} -\frac {16 \, x e^{x} \log \relax (2)^{2} - 16 \, x e^{x} \log \relax (2) \log \relax (x) + 4 \, x e^{x} \log \relax (x)^{2} + 4 \, x \log \relax (2) \log \relax (x) - x \log \relax (x)^{2} + 24 \, x e^{x} - 36 \, e^{x} + 9}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 54, normalized size = 1.86
method | result | size |
risch | \(\frac {\left (-4 \,{\mathrm e}^{x}+1\right ) \ln \relax (x )^{2}}{4}+4 \ln \relax (2) {\mathrm e}^{x} \ln \relax (x )-\frac {16 x \ln \relax (2)^{2} {\mathrm e}^{x}+4 x \ln \relax (2) \ln \relax (x )+24 \,{\mathrm e}^{x} x -36 \,{\mathrm e}^{x}+9}{4 x}\) | \(54\) |
norman | \(\frac {-\frac {9}{4}-x \ln \relax (2) \ln \relax (x )+\left (-4 \ln \relax (2)^{2}-6\right ) x \,{\mathrm e}^{x}+\frac {x \ln \relax (x )^{2}}{4}-x \,{\mathrm e}^{x} \ln \relax (x )^{2}+4 x \ln \relax (2) {\mathrm e}^{x} \ln \relax (x )+9 \,{\mathrm e}^{x}}{x}\) | \(55\) |
default | \(\frac {\left (-16 \ln \relax (2)^{2}-24\right ) x \,{\mathrm e}^{x}-4 x \,{\mathrm e}^{x} \ln \relax (x )^{2}+16 x \ln \relax (2) {\mathrm e}^{x} \ln \relax (x )+36 \,{\mathrm e}^{x}}{4 x}-\ln \relax (2) \ln \relax (x )-\frac {9}{4 x}+\frac {\ln \relax (x )^{2}}{4}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.54, size = 57, normalized size = 1.97 \begin {gather*} -4 \, e^{x} \log \relax (2)^{2} + 4 \, e^{x} \log \relax (2) \log \relax (x) - e^{x} \log \relax (x)^{2} - \log \relax (2) \log \relax (x) + \frac {1}{4} \, \log \relax (x)^{2} - \frac {9}{4 \, x} + 9 \, {\rm Ei}\relax (x) - 6 \, e^{x} - 9 \, \Gamma \left (-1, -x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.51, size = 51, normalized size = 1.76 \begin {gather*} \frac {{\ln \relax (x)}^2}{4}-4\,{\mathrm {e}}^x\,{\ln \relax (2)}^2-6\,{\mathrm {e}}^x-{\mathrm {e}}^x\,{\ln \relax (x)}^2-\ln \relax (2)\,\ln \relax (x)+\frac {9\,{\mathrm {e}}^x-\frac {9}{4}}{x}+4\,{\mathrm {e}}^x\,\ln \relax (2)\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.42, size = 53, normalized size = 1.83 \begin {gather*} \frac {\log {\relax (x )}^{2}}{4} - \log {\relax (2 )} \log {\relax (x )} + \frac {\left (- x \log {\relax (x )}^{2} + 4 x \log {\relax (2 )} \log {\relax (x )} - 6 x - 4 x \log {\relax (2 )}^{2} + 9\right ) e^{x}}{x} - \frac {9}{4 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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