Optimal. Leaf size=29 \[ 5 \left (-2+e^4 \left (2+5 e^3 x+25 \left (x-\frac {x}{\log (x)}\right )^2\right )\right ) \]
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Rubi [A]
time = 0.23, antiderivative size = 39, normalized size of antiderivative = 1.34, number of steps
used = 13, number of rules used = 6, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {6873, 12,
6874, 2343, 2346, 2209} \begin {gather*} 125 e^4 x^2+\frac {125 e^4 x^2}{\log ^2(x)}-\frac {250 e^4 x^2}{\log (x)}+25 e^7 x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2209
Rule 2343
Rule 2346
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 e^4 \left (-10 x+20 x \log (x)-20 x \log ^2(x)+e^3 \log ^3(x)+10 x \log ^3(x)\right )}{\log ^3(x)} \, dx\\ &=\left (25 e^4\right ) \int \frac {-10 x+20 x \log (x)-20 x \log ^2(x)+e^3 \log ^3(x)+10 x \log ^3(x)}{\log ^3(x)} \, dx\\ &=\left (25 e^4\right ) \int \left (e^3+10 x-\frac {10 x}{\log ^3(x)}+\frac {20 x}{\log ^2(x)}-\frac {20 x}{\log (x)}\right ) \, dx\\ &=25 e^7 x+125 e^4 x^2-\left (250 e^4\right ) \int \frac {x}{\log ^3(x)} \, dx+\left (500 e^4\right ) \int \frac {x}{\log ^2(x)} \, dx-\left (500 e^4\right ) \int \frac {x}{\log (x)} \, dx\\ &=25 e^7 x+125 e^4 x^2+\frac {125 e^4 x^2}{\log ^2(x)}-\frac {500 e^4 x^2}{\log (x)}-\left (250 e^4\right ) \int \frac {x}{\log ^2(x)} \, dx-\left (500 e^4\right ) \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )+\left (1000 e^4\right ) \int \frac {x}{\log (x)} \, dx\\ &=25 e^7 x+125 e^4 x^2-500 e^4 \text {Ei}(2 \log (x))+\frac {125 e^4 x^2}{\log ^2(x)}-\frac {250 e^4 x^2}{\log (x)}-\left (500 e^4\right ) \int \frac {x}{\log (x)} \, dx+\left (1000 e^4\right ) \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=25 e^7 x+125 e^4 x^2+500 e^4 \text {Ei}(2 \log (x))+\frac {125 e^4 x^2}{\log ^2(x)}-\frac {250 e^4 x^2}{\log (x)}-\left (500 e^4\right ) \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=25 e^7 x+125 e^4 x^2+\frac {125 e^4 x^2}{\log ^2(x)}-\frac {250 e^4 x^2}{\log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 39, normalized size = 1.34 \begin {gather*} 25 e^7 x+125 e^4 x^2+\frac {125 e^4 x^2}{\log ^2(x)}-\frac {250 e^4 x^2}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.23, size = 79, normalized size = 2.72
method | result | size |
risch | \(25 \,{\mathrm e}^{4} x \left ({\mathrm e}^{3}+5 x \right )-\frac {125 x^{2} {\mathrm e}^{4} \left (2 \ln \left (x \right )-1\right )}{\ln \left (x \right )^{2}}\) | \(30\) |
norman | \(\frac {125 x^{2} {\mathrm e}^{4}-250 x^{2} {\mathrm e}^{4} \ln \left (x \right )+125 x^{2} {\mathrm e}^{4} \ln \left (x \right )^{2}+25 \,{\mathrm e}^{3} {\mathrm e}^{4} x \ln \left (x \right )^{2}}{\ln \left (x \right )^{2}}\) | \(45\) |
default | \(25 \,{\mathrm e}^{3} {\mathrm e}^{4} x +125 x^{2} {\mathrm e}^{4}+500 \,{\mathrm e}^{4} \expIntegral \left (1, -2 \ln \left (x \right )\right )+500 \,{\mathrm e}^{4} \left (-\frac {x^{2}}{\ln \left (x \right )}-2 \expIntegral \left (1, -2 \ln \left (x \right )\right )\right )-250 \,{\mathrm e}^{4} \left (-\frac {x^{2}}{2 \ln \left (x \right )^{2}}-\frac {x^{2}}{\ln \left (x \right )}-2 \expIntegral \left (1, -2 \ln \left (x \right )\right )\right )\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.34, size = 42, normalized size = 1.45 \begin {gather*} 125 \, x^{2} e^{4} + 25 \, x e^{7} - 500 \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) e^{4} + 1000 \, e^{4} \Gamma \left (-1, -2 \, \log \left (x\right )\right ) + 1000 \, e^{4} \Gamma \left (-2, -2 \, \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 41, normalized size = 1.41 \begin {gather*} -\frac {25 \, {\left (10 \, x^{2} e^{4} \log \left (x\right ) - 5 \, x^{2} e^{4} - {\left (5 \, x^{2} e^{4} + x e^{7}\right )} \log \left (x\right )^{2}\right )}}{\log \left (x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 39, normalized size = 1.34 \begin {gather*} 125 x^{2} e^{4} + 25 x e^{7} + \frac {- 250 x^{2} e^{4} \log {\left (x \right )} + 125 x^{2} e^{4}}{\log {\left (x \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 35, normalized size = 1.21 \begin {gather*} 125 \, x^{2} e^{4} + 25 \, x e^{7} - \frac {250 \, x^{2} e^{4}}{\log \left (x\right )} + \frac {125 \, x^{2} e^{4}}{\log \left (x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.30, size = 34, normalized size = 1.17 \begin {gather*} \frac {125\,x^2\,{\mathrm {e}}^4-250\,x^2\,{\mathrm {e}}^4\,\ln \left (x\right )}{{\ln \left (x\right )}^2}+25\,x\,{\mathrm {e}}^4\,\left (5\,x+{\mathrm {e}}^3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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