Optimal. Leaf size=32 \[ \left (-x-x^2+x \log (5)\right ) \left (x-\left (-3+\frac {2}{\log \left (e^5+x\right )}\right )^2\right ) \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.86, antiderivative size = 267, normalized size of antiderivative = 8.34, number of steps
used = 51, number of rules used = 13, integrand size = 164, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {6, 6820,
2465, 2447, 2446, 2436, 2335, 2437, 2346, 2209, 2334, 2339, 30} \begin {gather*} -16 e^5 \text {LogIntegral}\left (x+e^5\right )-4 \left (1-e^5-\log (5)\right ) \text {LogIntegral}\left (x+e^5\right )-12 \left (1-2 e^5-\log (5)\right ) \text {LogIntegral}\left (x+e^5\right )+4 \left (4-3 e^5-4 \log (5)\right ) \text {LogIntegral}\left (x+e^5\right )-x^3+x^2 (8+\log (5))+\frac {4 \left (x+e^5\right ) x}{\log ^2\left (x+e^5\right )}+\frac {4 \left (x+e^5\right ) \left (1-e^5-\log (5)\right )}{\log ^2\left (x+e^5\right )}-\frac {4 e^5 \left (1-e^5-\log (5)\right )}{\log ^2\left (x+e^5\right )}-\frac {12 \left (x+e^5\right ) x}{\log \left (x+e^5\right )}+9 x (1-\log (5))+\frac {4 e^5 \left (x+e^5\right )}{\log \left (x+e^5\right )}-\frac {4 \left (x+e^5\right ) \left (4-3 e^5-\log (625)\right )}{\log \left (x+e^5\right )}+\frac {4 \left (x+e^5\right ) \left (1-e^5-\log (5)\right )}{\log \left (x+e^5\right )}+\frac {12 e^5 \left (1-e^5-\log (5)\right )}{\log \left (x+e^5\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 30
Rule 2209
Rule 2334
Rule 2335
Rule 2339
Rule 2346
Rule 2436
Rule 2437
Rule 2446
Rule 2447
Rule 2465
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-8 x^2+x (-8+8 \log (5))+\left (16 x+20 x^2+e^5 (4+8 x)+\left (-4 e^5-16 x\right ) \log (5)\right ) \log \left (e^5+x\right )+\left (e^5 (-12-24 x)-12 x-24 x^2+\left (12 e^5+12 x\right ) \log (5)\right ) \log ^2\left (e^5+x\right )+\left (9 x+16 x^2-3 x^3+e^5 \left (9+16 x-3 x^2\right )+\left (-9 x+2 x^2+e^5 (-9+2 x)\right ) \log (5)\right ) \log ^3\left (e^5+x\right )}{\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx\\ &=\int \left (9-3 x^2-9 \log (5)+2 x (8+\log (5))-\frac {8 x (1+x-\log (5))}{\left (e^5+x\right ) \log ^3\left (e^5+x\right )}+\frac {4 \left (5 x^2+2 x \left (2+e^5-2 \log (5)\right )+e^5 (1-\log (5))\right )}{\left (e^5+x\right ) \log ^2\left (e^5+x\right )}-\frac {12 (1+2 x-\log (5))}{\log \left (e^5+x\right )}\right ) \, dx\\ &=-x^3+9 x (1-\log (5))+x^2 (8+\log (5))+4 \int \frac {5 x^2+2 x \left (2+e^5-2 \log (5)\right )+e^5 (1-\log (5))}{\left (e^5+x\right ) \log ^2\left (e^5+x\right )} \, dx-8 \int \frac {x (1+x-\log (5))}{\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx-12 \int \frac {1+2 x-\log (5)}{\log \left (e^5+x\right )} \, dx\\ &=-x^3+9 x (1-\log (5))+x^2 (8+\log (5))+4 \int \left (\frac {5 x}{\log ^2\left (e^5+x\right )}+\frac {4 \left (1-\frac {3 e^5}{4}-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {3 e^5 \left (-1+e^5+\log (5)\right )}{\left (e^5+x\right ) \log ^2\left (e^5+x\right )}\right ) \, dx-8 \int \left (\frac {x}{\log ^3\left (e^5+x\right )}+\frac {1-e^5-\log (5)}{\log ^3\left (e^5+x\right )}+\frac {e^5 \left (-1+e^5+\log (5)\right )}{\left (e^5+x\right ) \log ^3\left (e^5+x\right )}\right ) \, dx-12 \int \left (\frac {2 \left (e^5+x\right )}{\log \left (e^5+x\right )}+\frac {1-2 e^5-\log (5)}{\log \left (e^5+x\right )}\right ) \, dx\\ &=-x^3+9 x (1-\log (5))+x^2 (8+\log (5))-8 \int \frac {x}{\log ^3\left (e^5+x\right )} \, dx+20 \int \frac {x}{\log ^2\left (e^5+x\right )} \, dx-24 \int \frac {e^5+x}{\log \left (e^5+x\right )} \, dx+\left (4 \left (4-3 e^5-4 \log (5)\right )\right ) \int \frac {1}{\log ^2\left (e^5+x\right )} \, dx-\left (12 \left (1-2 e^5-\log (5)\right )\right ) \int \frac {1}{\log \left (e^5+x\right )} \, dx-\left (8 \left (1-e^5-\log (5)\right )\right ) \int \frac {1}{\log ^3\left (e^5+x\right )} \, dx+\left (8 e^5 \left (1-e^5-\log (5)\right )\right ) \int \frac {1}{\left (e^5+x\right ) \log ^3\left (e^5+x\right )} \, dx-\left (12 e^5 \left (1-e^5-\log (5)\right )\right ) \int \frac {1}{\left (e^5+x\right ) \log ^2\left (e^5+x\right )} \, dx\\ &=-x^3+9 x (1-\log (5))+x^2 (8+\log (5))+\frac {4 x \left (e^5+x\right )}{\log ^2\left (e^5+x\right )}-\frac {20 x \left (e^5+x\right )}{\log \left (e^5+x\right )}-8 \int \frac {x}{\log ^2\left (e^5+x\right )} \, dx-24 \text {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,e^5+x\right )+40 \int \frac {x}{\log \left (e^5+x\right )} \, dx-\left (4 e^5\right ) \int \frac {1}{\log ^2\left (e^5+x\right )} \, dx+\left (20 e^5\right ) \int \frac {1}{\log \left (e^5+x\right )} \, dx+\left (4 \left (4-3 e^5-4 \log (5)\right )\right ) \text {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,e^5+x\right )-\left (12 \left (1-2 e^5-\log (5)\right )\right ) \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,e^5+x\right )-\left (8 \left (1-e^5-\log (5)\right )\right ) \text {Subst}\left (\int \frac {1}{\log ^3(x)} \, dx,x,e^5+x\right )+\left (8 e^5 \left (1-e^5-\log (5)\right )\right ) \text {Subst}\left (\int \frac {1}{x \log ^3(x)} \, dx,x,e^5+x\right )-\left (12 e^5 \left (1-e^5-\log (5)\right )\right ) \text {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,e^5+x\right )\\ &=-x^3+9 x (1-\log (5))+x^2 (8+\log (5))+\frac {4 x \left (e^5+x\right )}{\log ^2\left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}-\frac {12 x \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {4 \left (e^5+x\right ) \left (4-3 e^5-4 \log (5)\right )}{\log \left (e^5+x\right )}-12 \left (1-2 e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-16 \int \frac {x}{\log \left (e^5+x\right )} \, dx-24 \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log \left (e^5+x\right )\right )+40 \int \left (-\frac {e^5}{\log \left (e^5+x\right )}+\frac {e^5+x}{\log \left (e^5+x\right )}\right ) \, dx-\left (4 e^5\right ) \text {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,e^5+x\right )-\left (8 e^5\right ) \int \frac {1}{\log \left (e^5+x\right )} \, dx+\left (20 e^5\right ) \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,e^5+x\right )+\left (4 \left (4-3 e^5-4 \log (5)\right )\right ) \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,e^5+x\right )-\left (4 \left (1-e^5-\log (5)\right )\right ) \text {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,e^5+x\right )+\left (8 e^5 \left (1-e^5-\log (5)\right )\right ) \text {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log \left (e^5+x\right )\right )-\left (12 e^5 \left (1-e^5-\log (5)\right )\right ) \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (e^5+x\right )\right )\\ &=-x^3-24 \text {Ei}\left (2 \log \left (e^5+x\right )\right )+9 x (1-\log (5))+x^2 (8+\log (5))+\frac {4 x \left (e^5+x\right )}{\log ^2\left (e^5+x\right )}-\frac {4 e^5 \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 e^5 \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {12 x \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {4 \left (e^5+x\right ) \left (4-3 e^5-4 \log (5)\right )}{\log \left (e^5+x\right )}+\frac {12 e^5 \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}+20 e^5 \text {li}\left (e^5+x\right )+4 \left (4-3 e^5-4 \log (5)\right ) \text {li}\left (e^5+x\right )-12 \left (1-2 e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-16 \int \left (-\frac {e^5}{\log \left (e^5+x\right )}+\frac {e^5+x}{\log \left (e^5+x\right )}\right ) \, dx+40 \int \frac {e^5+x}{\log \left (e^5+x\right )} \, dx-\left (4 e^5\right ) \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,e^5+x\right )-\left (8 e^5\right ) \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,e^5+x\right )-\left (40 e^5\right ) \int \frac {1}{\log \left (e^5+x\right )} \, dx-\left (4 \left (1-e^5-\log (5)\right )\right ) \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,e^5+x\right )\\ &=-x^3-24 \text {Ei}\left (2 \log \left (e^5+x\right )\right )+9 x (1-\log (5))+x^2 (8+\log (5))+\frac {4 x \left (e^5+x\right )}{\log ^2\left (e^5+x\right )}-\frac {4 e^5 \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 e^5 \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {12 x \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {4 \left (e^5+x\right ) \left (4-3 e^5-4 \log (5)\right )}{\log \left (e^5+x\right )}+\frac {12 e^5 \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}+8 e^5 \text {li}\left (e^5+x\right )+4 \left (4-3 e^5-4 \log (5)\right ) \text {li}\left (e^5+x\right )-12 \left (1-2 e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-4 \left (1-e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-16 \int \frac {e^5+x}{\log \left (e^5+x\right )} \, dx+40 \text {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,e^5+x\right )+\left (16 e^5\right ) \int \frac {1}{\log \left (e^5+x\right )} \, dx-\left (40 e^5\right ) \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,e^5+x\right )\\ &=-x^3-24 \text {Ei}\left (2 \log \left (e^5+x\right )\right )+9 x (1-\log (5))+x^2 (8+\log (5))+\frac {4 x \left (e^5+x\right )}{\log ^2\left (e^5+x\right )}-\frac {4 e^5 \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 e^5 \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {12 x \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {4 \left (e^5+x\right ) \left (4-3 e^5-4 \log (5)\right )}{\log \left (e^5+x\right )}+\frac {12 e^5 \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}-32 e^5 \text {li}\left (e^5+x\right )+4 \left (4-3 e^5-4 \log (5)\right ) \text {li}\left (e^5+x\right )-12 \left (1-2 e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-4 \left (1-e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-16 \text {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,e^5+x\right )+40 \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log \left (e^5+x\right )\right )+\left (16 e^5\right ) \text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,e^5+x\right )\\ &=-x^3+16 \text {Ei}\left (2 \log \left (e^5+x\right )\right )+9 x (1-\log (5))+x^2 (8+\log (5))+\frac {4 x \left (e^5+x\right )}{\log ^2\left (e^5+x\right )}-\frac {4 e^5 \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 e^5 \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {12 x \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {4 \left (e^5+x\right ) \left (4-3 e^5-4 \log (5)\right )}{\log \left (e^5+x\right )}+\frac {12 e^5 \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}-16 e^5 \text {li}\left (e^5+x\right )+4 \left (4-3 e^5-4 \log (5)\right ) \text {li}\left (e^5+x\right )-12 \left (1-2 e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-4 \left (1-e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-16 \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log \left (e^5+x\right )\right )\\ &=-x^3+9 x (1-\log (5))+x^2 (8+\log (5))+\frac {4 x \left (e^5+x\right )}{\log ^2\left (e^5+x\right )}-\frac {4 e^5 \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log ^2\left (e^5+x\right )}+\frac {4 e^5 \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {12 x \left (e^5+x\right )}{\log \left (e^5+x\right )}-\frac {4 \left (e^5+x\right ) \left (4-3 e^5-4 \log (5)\right )}{\log \left (e^5+x\right )}+\frac {12 e^5 \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}+\frac {4 \left (e^5+x\right ) \left (1-e^5-\log (5)\right )}{\log \left (e^5+x\right )}-16 e^5 \text {li}\left (e^5+x\right )+4 \left (4-3 e^5-4 \log (5)\right ) \text {li}\left (e^5+x\right )-12 \left (1-2 e^5-\log (5)\right ) \text {li}\left (e^5+x\right )-4 \left (1-e^5-\log (5)\right ) \text {li}\left (e^5+x\right )\\ \end {aligned} \end {gather*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.21, size = 76, normalized size = 2.38 \begin {gather*} 12 \text {Ei}\left (\log \left (e^5+x\right )\right ) \left (-1+2 e^5+\log (5)\right )-\frac {x (1+x-\log (5)) \left (-4+12 \log \left (e^5+x\right )+(-9+x) \log ^2\left (e^5+x\right )\right )}{\log ^2\left (e^5+x\right )}-12 \left (-1+2 e^5+\log (5)\right ) \text {li}\left (e^5+x\right ) \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 = 0.95, size = 378, normalized size = 11.81 Too large to display
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. 68 vs.
\(2 (28) = 56\).
time = 0.54, size = 68, normalized size = 2.12 \begin {gather*} -\frac {{\left (x^{3} - x^{2} {\left (\log \left (5\right ) + 8\right )} + 9 \, x {\left (\log \left (5\right ) - 1\right )}\right )} \log \left (x + e^{5}\right )^{2} - 4 \, x^{2} + 4 \, x {\left (\log \left (5\right ) - 1\right )} + 12 \, {\left (x^{2} - x {\left (\log \left (5\right ) - 1\right )}\right )} \log \left (x + e^{5}\right )}{\log \left (x + e^{5}\right )^{2}} \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. 71 vs.
\(2 (28) = 56\).
time = 0.40, size = 71, normalized size = 2.22 \begin {gather*} -\frac {{\left (x^{3} - 8 \, x^{2} - {\left (x^{2} - 9 \, x\right )} \log \left (5\right ) - 9 \, x\right )} \log \left (x + e^{5}\right )^{2} - 4 \, x^{2} + 4 \, x \log \left (5\right ) + 12 \, {\left (x^{2} - x \log \left (5\right ) + x\right )} \log \left (x + e^{5}\right ) - 4 \, x}{\log \left (x + e^{5}\right )^{2}} \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. 65 vs.
\(2 (22) = 44\).
time = 0.08, size = 65, normalized size = 2.03 \begin {gather*} - x^{3} + x^{2} \left (\log {\left (5 \right )} + 8\right ) + x \left (9 - 9 \log {\left (5 \right )}\right ) + \frac {4 x^{2} - 4 x \log {\left (5 \right )} + 4 x + \left (- 12 x^{2} - 12 x + 12 x \log {\left (5 \right )}\right ) \log {\left (x + e^{5} \right )}}{\log {\left (x + e^{5} \right )}^{2}} \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. 110 vs.
\(2 (28) = 56\).
time = 0.43, size = 110, normalized size = 3.44 \begin {gather*} -\frac {x^{3} \log \left (x + e^{5}\right )^{2} - x^{2} \log \left (5\right ) \log \left (x + e^{5}\right )^{2} - 8 \, x^{2} \log \left (x + e^{5}\right )^{2} + 9 \, x \log \left (5\right ) \log \left (x + e^{5}\right )^{2} + 12 \, x^{2} \log \left (x + e^{5}\right ) - 12 \, x \log \left (5\right ) \log \left (x + e^{5}\right ) - 9 \, x \log \left (x + e^{5}\right )^{2} - 4 \, x^{2} + 4 \, x \log \left (5\right ) + 12 \, x \log \left (x + e^{5}\right ) - 4 \, x}{\log \left (x + e^{5}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.66, size = 93, normalized size = 2.91 \begin {gather*} 9\,x-\frac {12\,x^2}{\ln \left (x+{\mathrm {e}}^5\right )}+\frac {4\,x^2}{{\ln \left (x+{\mathrm {e}}^5\right )}^2}-9\,x\,\ln \left (5\right )+x^2\,\ln \left (5\right )+8\,x^2-x^3-\frac {12\,x}{\ln \left (x+{\mathrm {e}}^5\right )}+\frac {4\,x}{{\ln \left (x+{\mathrm {e}}^5\right )}^2}+\frac {12\,x\,\ln \left (5\right )}{\ln \left (x+{\mathrm {e}}^5\right )}-\frac {4\,x\,\ln \left (5\right )}{{\ln \left (x+{\mathrm {e}}^5\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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