Optimal. Leaf size=32 \[ \left (x+\frac {e^x x}{x+\frac {5 e^{-2 x}}{\log (x)}}\right ) \left (2-e^x+\log (x)\right ) \]
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Rubi [F]
time = 5.49, 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 {75+10 e^{3 x}-5 e^{4 x}+e^x (-25-25 x)+\left (25+e^{4 x} (-5-20 x)+30 e^{2 x} x+e^{3 x} \left (20+20 x-10 x^2\right )\right ) \log (x)+\left (10 e^{2 x} x+3 e^{4 x} x^2-2 e^{6 x} x^2+e^{3 x} (5+15 x)+e^{5 x} \left (x+x^2-x^3\right )\right ) \log ^2(x)+\left (e^{4 x} x^2+e^{5 x} x^2\right ) \log ^3(x)}{25+10 e^{2 x} x \log (x)+e^{4 x} x^2 \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 {75+10 e^{3 x}-5 e^{4 x}+e^x (-25-25 x)+\left (25+e^{4 x} (-5-20 x)+30 e^{2 x} x+e^{3 x} \left (20+20 x-10 x^2\right )\right ) \log (x)+\left (10 e^{2 x} x+3 e^{4 x} x^2-2 e^{6 x} x^2+e^{3 x} (5+15 x)+e^{5 x} \left (x+x^2-x^3\right )\right ) \log ^2(x)+\left (e^{4 x} x^2+e^{5 x} x^2\right ) \log ^3(x)}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx\\ &=\int \left (-2 e^{2 x}-\frac {e^x \left (-1-x+x^2-x \log (x)\right )}{x}-\frac {25 (1+\log (x)+2 x \log (x)) \left (5+2 e^x x \log (x)+e^x x \log ^2(x)\right )}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )^2}+\frac {-5-5 \log (x)+3 x^2 \log ^2(x)+x^2 \log ^3(x)}{x^2 \log ^2(x)}+\frac {5 \left (10+10 \log (x)+10 x \log (x)+2 e^x x \log (x)+2 e^x x \log ^2(x)+2 e^x x^2 \log ^2(x)+e^x x \log ^3(x)+e^x x^2 \log ^3(x)\right )}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )}\right ) \, dx\\ &=-\left (2 \int e^{2 x} \, dx\right )+5 \int \frac {10+10 \log (x)+10 x \log (x)+2 e^x x \log (x)+2 e^x x \log ^2(x)+2 e^x x^2 \log ^2(x)+e^x x \log ^3(x)+e^x x^2 \log ^3(x)}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )} \, dx-25 \int \frac {(1+\log (x)+2 x \log (x)) \left (5+2 e^x x \log (x)+e^x x \log ^2(x)\right )}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-\int \frac {e^x \left (-1-x+x^2-x \log (x)\right )}{x} \, dx+\int \frac {-5-5 \log (x)+3 x^2 \log ^2(x)+x^2 \log ^3(x)}{x^2 \log ^2(x)} \, dx\\ &=-e^{2 x}+5 \int \frac {10+2 \left (5+\left (5+e^x\right ) x\right ) \log (x)+2 e^x x (1+x) \log ^2(x)+e^x x (1+x) \log ^3(x)}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )} \, dx-25 \int \left (\frac {4 e^x}{\left (5+e^{2 x} x \log (x)\right )^2}+\frac {3 e^x}{x \left (5+e^{2 x} x \log (x)\right )^2}+\frac {5}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )^2}+\frac {5}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )^2}+\frac {10}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2}+\frac {2 e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2}+\frac {2 e^x \log (x)}{\left (5+e^{2 x} x \log (x)\right )^2}+\frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )^2}\right ) \, dx+\int \left (3-\frac {5}{x^2 \log ^2(x)}-\frac {5}{x^2 \log (x)}+\log (x)\right ) \, dx-\int \left (\frac {e^x \left (-1-x+x^2\right )}{x}-e^x \log (x)\right ) \, dx\\ &=-e^{2 x}+3 x-5 \int \frac {1}{x^2 \log ^2(x)} \, dx-5 \int \frac {1}{x^2 \log (x)} \, dx+5 \int \left (\frac {2 e^x}{5+e^{2 x} x \log (x)}+\frac {2 e^x}{x \left (5+e^{2 x} x \log (x)\right )}+\frac {10}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )}+\frac {10}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )}+\frac {10}{x \log (x) \left (5+e^{2 x} x \log (x)\right )}+\frac {2 e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )}+\frac {e^x \log (x)}{5+e^{2 x} x \log (x)}+\frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )}\right ) \, dx-25 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x \log (x)}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx-75 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-100 \int \frac {e^x}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-250 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-\int \frac {e^x \left (-1-x+x^2\right )}{x} \, dx+\int \log (x) \, dx+\int e^x \log (x) \, dx\\ &=-e^{2 x}+2 x+\frac {5}{x \log (x)}+e^x \log (x)+x \log (x)+5 \int \frac {1}{x^2 \log (x)} \, dx+5 \int \frac {e^x \log (x)}{5+e^{2 x} x \log (x)} \, dx+5 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )} \, dx-5 \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )+10 \int \frac {e^x}{5+e^{2 x} x \log (x)} \, dx+10 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )} \, dx+10 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx-25 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x \log (x)}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx+50 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )} \, dx+50 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx+50 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx-75 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-100 \int \frac {e^x}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-250 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-\int \frac {e^x}{x} \, dx-\int \left (-e^x-\frac {e^x}{x}+e^x x\right ) \, dx\\ &=-e^{2 x}+2 x-\text {Ei}(x)-5 \text {Ei}(-\log (x))+\frac {5}{x \log (x)}+e^x \log (x)+x \log (x)+5 \int \frac {e^x \log (x)}{5+e^{2 x} x \log (x)} \, dx+5 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )} \, dx+5 \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )+10 \int \frac {e^x}{5+e^{2 x} x \log (x)} \, dx+10 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )} \, dx+10 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx-25 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x \log (x)}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx+50 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )} \, dx+50 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx+50 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx-75 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-100 \int \frac {e^x}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-250 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx+\int e^x \, dx+\int \frac {e^x}{x} \, dx-\int e^x x \, dx\\ &=e^x-e^{2 x}+2 x-e^x x+\frac {5}{x \log (x)}+e^x \log (x)+x \log (x)+5 \int \frac {e^x \log (x)}{5+e^{2 x} x \log (x)} \, dx+5 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )} \, dx+10 \int \frac {e^x}{5+e^{2 x} x \log (x)} \, dx+10 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )} \, dx+10 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx-25 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x \log (x)}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx+50 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )} \, dx+50 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx+50 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx-75 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-100 \int \frac {e^x}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-250 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx+\int e^x \, dx\\ &=2 e^x-e^{2 x}+2 x-e^x x+\frac {5}{x \log (x)}+e^x \log (x)+x \log (x)+5 \int \frac {e^x \log (x)}{5+e^{2 x} x \log (x)} \, dx+5 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )} \, dx+10 \int \frac {e^x}{5+e^{2 x} x \log (x)} \, dx+10 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )} \, dx+10 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx-25 \int \frac {e^x \log (x)}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-50 \int \frac {e^x \log (x)}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx+50 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )} \, dx+50 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx+50 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )} \, dx-75 \int \frac {e^x}{x \left (5+e^{2 x} x \log (x)\right )^2} \, dx-100 \int \frac {e^x}{\left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log ^2(x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-125 \int \frac {1}{x^2 \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx-250 \int \frac {1}{x \log (x) \left (5+e^{2 x} x \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.15, size = 40, normalized size = 1.25 \begin {gather*} -\frac {x \left (-2+e^x-\log (x)\right ) \left (5+e^{2 x} \left (e^x+x\right ) \log (x)\right )}{5+e^{2 x} x \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs.
\(2(29)=58\).
time = 0.28, size = 84, normalized size = 2.62
method | result | size |
risch | \(\left ({\mathrm e}^{x}+x \right ) \ln \left (x \right )-\frac {\left ({\mathrm e}^{2 x} x^{2}+x \,{\mathrm e}^{3 x}-2 \,{\mathrm e}^{x} x^{2}-2 x \,{\mathrm e}^{2 x}+5\right ) {\mathrm e}^{-x}}{x}+\frac {5 \left (x \,{\mathrm e}^{3 x}-2 x \,{\mathrm e}^{2 x}+5\right ) {\mathrm e}^{-x}}{x \left (x \,{\mathrm e}^{2 x} \ln \left (x \right )+5\right )}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs.
\(2 (30) = 60\).
time = 0.35, size = 82, normalized size = 2.56 \begin {gather*} -\frac {x e^{\left (4 \, x\right )} \log \left (x\right ) - {\left (x \log \left (x\right )^{2} - {\left (x^{2} - 2 \, x\right )} \log \left (x\right )\right )} e^{\left (3 \, x\right )} - {\left (x^{2} \log \left (x\right )^{2} + 2 \, x^{2} \log \left (x\right )\right )} e^{\left (2 \, x\right )} + 5 \, x e^{x} - 5 \, x \log \left (x\right ) - 10 \, x}{x e^{\left (2 \, x\right )} \log \left (x\right ) + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 78 vs.
\(2 (30) = 60\).
time = 0.38, size = 78, normalized size = 2.44 \begin {gather*} \frac {{\left (x^{2} e^{\left (2 \, x\right )} + x e^{\left (3 \, x\right )}\right )} \log \left (x\right )^{2} - 5 \, x e^{x} + {\left (2 \, x^{2} e^{\left (2 \, x\right )} - x e^{\left (4 \, x\right )} - {\left (x^{2} - 2 \, x\right )} e^{\left (3 \, x\right )} + 5 \, x\right )} \log \left (x\right ) + 10 \, x}{x e^{\left (2 \, x\right )} \log \left (x\right ) + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (26) = 52\).
time = 0.22, size = 71, normalized size = 2.22 \begin {gather*} x \log {\left (x \right )} + 2 x + \frac {\left (- 5 x \log {\left (x \right )}^{2} - 10 x \log {\left (x \right )}\right ) e^{x} - 25}{x^{2} e^{2 x} \log {\left (x \right )}^{2} + 5 x \log {\left (x \right )}} + \left (- x + \log {\left (x \right )} + 2\right ) e^{x} - e^{2 x} + \frac {5}{x \log {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 204 vs.
\(2 (30) = 60\).
time = 0.49, size = 204, normalized size = 6.38 \begin {gather*} \frac {x^{4} e^{\left (5 \, x\right )} \log \left (x\right )^{3} - x^{4} e^{\left (6 \, x\right )} \log \left (x\right )^{2} + 2 \, x^{4} e^{\left (5 \, x\right )} \log \left (x\right )^{2} + x^{3} e^{\left (6 \, x\right )} \log \left (x\right )^{3} - x^{3} e^{\left (7 \, x\right )} \log \left (x\right )^{2} + 2 \, x^{3} e^{\left (6 \, x\right )} \log \left (x\right )^{2} + 10 \, x^{3} e^{\left (3 \, x\right )} \log \left (x\right )^{2} - 10 \, x^{3} e^{\left (4 \, x\right )} \log \left (x\right ) + 20 \, x^{3} e^{\left (3 \, x\right )} \log \left (x\right ) + 5 \, x^{2} e^{\left (4 \, x\right )} \log \left (x\right )^{2} - 5 \, x^{2} e^{\left (5 \, x\right )} \log \left (x\right ) + 10 \, x^{2} e^{\left (4 \, x\right )} \log \left (x\right ) + 25 \, x^{2} e^{x} \log \left (x\right ) - 25 \, x^{2} e^{\left (2 \, x\right )} + 50 \, x^{2} e^{x}}{x^{3} e^{\left (5 \, x\right )} \log \left (x\right )^{2} + 10 \, x^{2} e^{\left (3 \, x\right )} \log \left (x\right ) + 25 \, x e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.37, size = 133, normalized size = 4.16 \begin {gather*} 2\,x-{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\left (x-2\right )-\frac {5\,{\mathrm {e}}^{-x}}{x}+\ln \left (x\right )\,\left (x+{\mathrm {e}}^x\right )+\frac {5\,\left (25\,{\mathrm {e}}^x-15\,x\,{\mathrm {e}}^{3\,x}+5\,x\,{\mathrm {e}}^{4\,x}-20\,x^2\,{\mathrm {e}}^{3\,x}+10\,x^2\,{\mathrm {e}}^{4\,x}+2\,x^2\,{\mathrm {e}}^{5\,x}-x^2\,{\mathrm {e}}^{6\,x}+50\,x\,{\mathrm {e}}^x\right )}{x\,\left (x\,{\mathrm {e}}^{2\,x}\,\ln \left (x\right )+5\right )\,\left (5\,{\mathrm {e}}^{2\,x}+10\,x\,{\mathrm {e}}^{2\,x}-x\,{\mathrm {e}}^{4\,x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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