Optimal. Leaf size=22 \[ \left (-1+e^9+x-\frac {x}{e}+\frac {5}{2+\log (x)}\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(74\) vs. \(2(22)=44\).
time = 0.73, antiderivative size = 74, normalized size of antiderivative = 3.36, number of steps
used = 15, number of rules used = 9, integrand size = 238, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6820, 12,
6874, 2339, 30, 2395, 2334, 2336, 2209} \begin {gather*} \frac {(1-e)^2 x^2}{e^2}+\frac {2 \left (1-e-e^9+e^{10}\right ) x}{e}-\frac {10 (1-e) x}{e (\log (x)+2)}-\frac {10 \left (1-e^9\right )}{\log (x)+2}+\frac {25}{(\log (x)+2)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2209
Rule 2334
Rule 2336
Rule 2339
Rule 2395
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-8 e^{10} x+8 x^2+2 e^{11} (-5+4 x)-2 e x (1+8 x)+e^2 \left (-15+2 x+8 x^2\right )+\left (-12 e^{10} x+12 x^2-3 e x (1+8 x)+e^{11} (-5+12 x)+e^2 \left (5+3 x+12 x^2\right )\right ) \log (x)+(-1+e) x \left (6 e^{10}-6 x+e (-1+6 x)\right ) \log ^2(x)+x \left (e-e^{10}+e^{11}+e^2 (-1+x)+x-2 e x\right ) \log ^3(x)\right )}{e^2 x (2+\log (x))^3} \, dx\\ &=\frac {2 \int \frac {-8 e^{10} x+8 x^2+2 e^{11} (-5+4 x)-2 e x (1+8 x)+e^2 \left (-15+2 x+8 x^2\right )+\left (-12 e^{10} x+12 x^2-3 e x (1+8 x)+e^{11} (-5+12 x)+e^2 \left (5+3 x+12 x^2\right )\right ) \log (x)+(-1+e) x \left (6 e^{10}-6 x+e (-1+6 x)\right ) \log ^2(x)+x \left (e-e^{10}+e^{11}+e^2 (-1+x)+x-2 e x\right ) \log ^3(x)}{x (2+\log (x))^3} \, dx}{e^2}\\ &=\frac {2 \int \left (e \left (1+e \left (-1-e^8+e^9\right )\right )+(1+(-2+e) e) x-\frac {25 e^2}{x (2+\log (x))^3}+\frac {5 e \left (e \left (1-e^9\right )+(1-e) x\right )}{x (2+\log (x))^2}+\frac {5 (-1+e) e}{2+\log (x)}\right ) \, dx}{e^2}\\ &=\frac {2 \left (1-e-e^9+e^{10}\right ) x}{e}+\frac {(1-e)^2 x^2}{e^2}-50 \int \frac {1}{x (2+\log (x))^3} \, dx+\frac {10 \int \frac {e \left (1-e^9\right )+(1-e) x}{x (2+\log (x))^2} \, dx}{e}-\frac {(10 (1-e)) \int \frac {1}{2+\log (x)} \, dx}{e}\\ &=\frac {2 \left (1-e-e^9+e^{10}\right ) x}{e}+\frac {(1-e)^2 x^2}{e^2}-50 \text {Subst}\left (\int \frac {1}{x^3} \, dx,x,2+\log (x)\right )+\frac {10 \int \left (\frac {1-e}{(2+\log (x))^2}+\frac {e-e^{10}}{x (2+\log (x))^2}\right ) \, dx}{e}-\frac {(10 (1-e)) \text {Subst}\left (\int \frac {e^x}{2+x} \, dx,x,\log (x)\right )}{e}\\ &=\frac {2 \left (1-e-e^9+e^{10}\right ) x}{e}+\frac {(1-e)^2 x^2}{e^2}-\frac {10 (1-e) \text {Ei}(2+\log (x))}{e^3}+\frac {25}{(2+\log (x))^2}+\frac {(10 (1-e)) \int \frac {1}{(2+\log (x))^2} \, dx}{e}+\left (10 \left (1-e^9\right )\right ) \int \frac {1}{x (2+\log (x))^2} \, dx\\ &=\frac {2 \left (1-e-e^9+e^{10}\right ) x}{e}+\frac {(1-e)^2 x^2}{e^2}-\frac {10 (1-e) \text {Ei}(2+\log (x))}{e^3}+\frac {25}{(2+\log (x))^2}-\frac {10 (1-e) x}{e (2+\log (x))}+\frac {(10 (1-e)) \int \frac {1}{2+\log (x)} \, dx}{e}+\left (10 \left (1-e^9\right )\right ) \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,2+\log (x)\right )\\ &=\frac {2 \left (1-e-e^9+e^{10}\right ) x}{e}+\frac {(1-e)^2 x^2}{e^2}-\frac {10 (1-e) \text {Ei}(2+\log (x))}{e^3}+\frac {25}{(2+\log (x))^2}-\frac {10 \left (1-e^9\right )}{2+\log (x)}-\frac {10 (1-e) x}{e (2+\log (x))}+\frac {(10 (1-e)) \text {Subst}\left (\int \frac {e^x}{2+x} \, dx,x,\log (x)\right )}{e}\\ &=\frac {2 \left (1-e-e^9+e^{10}\right ) x}{e}+\frac {(1-e)^2 x^2}{e^2}+\frac {25}{(2+\log (x))^2}-\frac {10 \left (1-e^9\right )}{2+\log (x)}-\frac {10 (1-e) x}{e (2+\log (x))}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(68\) vs. \(2(22)=44\).
time = 0.09, size = 68, normalized size = 3.09 \begin {gather*} \frac {2 \left (e \left (1-e-e^9+e^{10}\right ) x+\frac {1}{2} (-1+e)^2 x^2+\frac {25 e^2}{2 (2+\log (x))^2}+\frac {5 e \left (e^{10}+e (-1+x)-x\right )}{2+\log (x)}\right )}{e^2} \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 = 3.33, size = 879, normalized size = 39.95
method | result | size |
risch | \(\left ({\mathrm e}-1\right )^{2} x \left (2 \,{\mathrm e}^{9}+2 \,{\mathrm e}^{8}+2 \,{\mathrm e}^{7}+2 \,{\mathrm e}^{6}+2 \,{\mathrm e}^{5}+2 \,{\mathrm e}^{4}+2 \,{\mathrm e}^{3}+2 \,{\mathrm e}^{2}+2 \,{\mathrm e}+x \right ) {\mathrm e}^{-2}+\frac {5 \left (2 \,{\mathrm e}^{10} \ln \left (x \right )+4 \,{\mathrm e}^{10}+2 x \,{\mathrm e} \ln \left (x \right )+4 x \,{\mathrm e}-2 \,{\mathrm e} \ln \left (x \right )-2 x \ln \left (x \right )+{\mathrm e}-4 x \right ) {\mathrm e}^{-1}}{\left (\ln \left (x \right )+2\right )^{2}}\) | \(99\) |
norman | \(\frac {\left (10 \,{\mathrm e} \left ({\mathrm e}^{9}-1\right ) \ln \left (x \right )+\left (8 \,{\mathrm e} \,{\mathrm e}^{9}+12 \,{\mathrm e}-8 \,{\mathrm e}^{9}-12\right ) x +\left (2 \,{\mathrm e} \,{\mathrm e}^{9}-2 \,{\mathrm e}-2 \,{\mathrm e}^{9}+2\right ) x \ln \left (x \right )^{2}+\left (8 \,{\mathrm e} \,{\mathrm e}^{9}+2 \,{\mathrm e}-8 \,{\mathrm e}^{9}-2\right ) x \ln \left (x \right )+\left ({\mathrm e}^{2}-2 \,{\mathrm e}+1\right ) {\mathrm e}^{-1} x^{2} \ln \left (x \right )^{2}+4 \left ({\mathrm e}^{2}-2 \,{\mathrm e}+1\right ) {\mathrm e}^{-1} x^{2}+4 \left ({\mathrm e}^{2}-2 \,{\mathrm e}+1\right ) {\mathrm e}^{-1} x^{2} \ln \left (x \right )+5 \,{\mathrm e} \left (4 \,{\mathrm e}^{9}+1\right )\right ) {\mathrm e}^{-1}}{\left (\ln \left (x \right )+2\right )^{2}}\) | \(155\) |
default | \(\text {Expression too large to display}\) | \(879\) |
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. 189 vs.
\(2 (23) = 46\).
time = 0.32, size = 189, normalized size = 8.59 \begin {gather*} \frac {4 \, x^{2} {\left (e^{2} - 2 \, e + 1\right )} + {\left (x^{2} {\left (e^{2} - 2 \, e + 1\right )} + 2 \, x {\left (e^{11} - e^{10} - e^{2} + e\right )}\right )} \log \left (x\right )^{2} + 4 \, x {\left (2 \, e^{11} - 2 \, e^{10} + 3 \, e^{2} - 3 \, e\right )} + 2 \, {\left (2 \, x^{2} {\left (e^{2} - 2 \, e + 1\right )} + x {\left (4 \, e^{11} - 4 \, e^{10} + e^{2} - e\right )} + 5 \, e^{11} - 5 \, e^{2}\right )} \log \left (x\right ) + 10 \, e^{11} - 10 \, e^{2}}{e^{2} \log \left (x\right )^{2} + 4 \, e^{2} \log \left (x\right ) + 4 \, e^{2}} + \frac {10 \, e^{11}}{e^{2} \log \left (x\right )^{2} + 4 \, e^{2} \log \left (x\right ) + 4 \, e^{2}} + \frac {15 \, e^{2}}{e^{2} \log \left (x\right )^{2} + 4 \, e^{2} \log \left (x\right ) + 4 \, e^{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. 152 vs.
\(2 (23) = 46\).
time = 0.42, size = 152, normalized size = 6.91 \begin {gather*} \frac {{\left (x^{2} + 2 \, x e^{11} - 2 \, x e^{10} + {\left (x^{2} - 2 \, x\right )} e^{2} - 2 \, {\left (x^{2} - x\right )} e\right )} \log \left (x\right )^{2} + 4 \, x^{2} + 4 \, {\left (2 \, x + 5\right )} e^{11} - 8 \, x e^{10} + {\left (4 \, x^{2} + 12 \, x + 5\right )} e^{2} - 4 \, {\left (2 \, x^{2} + 3 \, x\right )} e + 2 \, {\left (2 \, x^{2} + {\left (4 \, x + 5\right )} e^{11} - 4 \, x e^{10} + {\left (2 \, x^{2} + x - 5\right )} e^{2} - {\left (4 \, x^{2} + x\right )} e\right )} \log \left (x\right )}{e^{2} \log \left (x\right )^{2} + 4 \, e^{2} \log \left (x\right ) + 4 \, e^{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. 104 vs.
\(2 (19) = 38\).
time = 0.10, size = 104, normalized size = 4.73 \begin {gather*} \frac {x^{2} \left (- 2 e + 1 + e^{2}\right )}{e^{2}} + \frac {x \left (- 2 e^{9} - 2 e + 2 + 2 e^{10}\right )}{e} + \frac {- 20 x + 20 e x + \left (- 10 x + 10 e x - 10 e + 10 e^{10}\right ) \log {\left (x \right )} + 5 e + 20 e^{10}}{e \log {\left (x \right )}^{2} + 4 e \log {\left (x \right )} + 4 e} \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. 205 vs.
\(2 (23) = 46\).
time = 0.40, size = 205, normalized size = 9.32 \begin {gather*} \frac {x^{2} e^{3} \log \left (x\right )^{2} - 2 \, x^{2} e^{2} \log \left (x\right )^{2} + x^{2} e \log \left (x\right )^{2} + 4 \, x^{2} e^{3} \log \left (x\right ) - 8 \, x^{2} e^{2} \log \left (x\right ) + 4 \, x^{2} e \log \left (x\right ) + 2 \, x e^{12} \log \left (x\right )^{2} - 2 \, x e^{11} \log \left (x\right )^{2} - 2 \, x e^{3} \log \left (x\right )^{2} + 2 \, x e^{2} \log \left (x\right )^{2} + 4 \, x^{2} e^{3} - 8 \, x^{2} e^{2} + 4 \, x^{2} e + 8 \, x e^{12} \log \left (x\right ) - 8 \, x e^{11} \log \left (x\right ) + 2 \, x e^{3} \log \left (x\right ) - 2 \, x e^{2} \log \left (x\right ) + 8 \, x e^{12} - 8 \, x e^{11} + 12 \, x e^{3} - 12 \, x e^{2} + 10 \, e^{12} \log \left (x\right ) - 10 \, e^{3} \log \left (x\right ) + 20 \, e^{12} + 5 \, e^{3}}{e^{3} \log \left (x\right )^{2} + 4 \, e^{3} \log \left (x\right ) + 4 \, e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.91, size = 170, normalized size = 7.73 \begin {gather*} \frac {{\left (\mathrm {e}-1\right )}^2\,x^4\,{\ln \left (x\right )}^2+4\,{\left (\mathrm {e}-1\right )}^2\,x^4\,\ln \left (x\right )+4\,{\left (\mathrm {e}-1\right )}^2\,x^4+\left (2\,\mathrm {e}-2\,{\mathrm {e}}^2-2\,{\mathrm {e}}^{10}+2\,{\mathrm {e}}^{11}\right )\,x^3\,{\ln \left (x\right )}^2+2\,\mathrm {e}\,\left (\mathrm {e}-4\,{\mathrm {e}}^9+4\,{\mathrm {e}}^{10}-1\right )\,x^3\,\ln \left (x\right )+4\,\mathrm {e}\,\left (3\,\mathrm {e}-2\,{\mathrm {e}}^9+2\,{\mathrm {e}}^{10}-3\right )\,x^3+\left (-\frac {5\,{\mathrm {e}}^2}{4}-5\,{\mathrm {e}}^{11}\right )\,x^2\,{\ln \left (x\right )}^2+\left (-15\,{\mathrm {e}}^2-10\,{\mathrm {e}}^{11}\right )\,x^2\,\ln \left (x\right )}{{\mathrm {e}}^2\,x^2\,{\ln \left (x\right )}^2+4\,{\mathrm {e}}^2\,x^2\,\ln \left (x\right )+4\,{\mathrm {e}}^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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