Optimal. Leaf size=30 \[ 5+x+\frac {2}{3 x-\frac {e^x}{-3-e^4+e^x+\log (x)}} \]
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Rubi [F]
time = 31.77, 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 {-54 x+81 x^3+e^8 \left (-6 x+9 x^3\right )+e^{2 x} \left (-5 x-6 x^2+9 x^3\right )+e^4 \left (-36 x+54 x^3\right )+e^x \left (-2+30 x+18 x^2-54 x^3+e^4 \left (10 x+6 x^2-18 x^3\right )\right )+\left (36 x-54 x^3+e^4 \left (12 x-18 x^3\right )+e^x \left (-10 x-6 x^2+18 x^3\right )\right ) \log (x)+\left (-6 x+9 x^3\right ) \log ^2(x)}{81 x^3+54 e^4 x^3+9 e^8 x^3+e^{2 x} \left (x-6 x^2+9 x^3\right )+e^x \left (18 x^2-54 x^3+e^4 \left (6 x^2-18 x^3\right )\right )+\left (-54 x^3-18 e^4 x^3+e^x \left (-6 x^2+18 x^3\right )\right ) \log (x)+9 x^3 \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 {-54 x+81 x^3+e^8 \left (-6 x+9 x^3\right )+e^{2 x} \left (-5 x-6 x^2+9 x^3\right )+e^4 \left (-36 x+54 x^3\right )+e^x \left (-2+30 x+18 x^2-54 x^3+e^4 \left (10 x+6 x^2-18 x^3\right )\right )+\left (36 x-54 x^3+e^4 \left (12 x-18 x^3\right )+e^x \left (-10 x-6 x^2+18 x^3\right )\right ) \log (x)+\left (-6 x+9 x^3\right ) \log ^2(x)}{9 e^8 x^3+\left (81+54 e^4\right ) x^3+e^{2 x} \left (x-6 x^2+9 x^3\right )+e^x \left (18 x^2-54 x^3+e^4 \left (6 x^2-18 x^3\right )\right )+\left (-54 x^3-18 e^4 x^3+e^x \left (-6 x^2+18 x^3\right )\right ) \log (x)+9 x^3 \log ^2(x)} \, dx\\ &=\int \frac {-54 x+81 x^3+e^8 \left (-6 x+9 x^3\right )+e^{2 x} \left (-5 x-6 x^2+9 x^3\right )+e^4 \left (-36 x+54 x^3\right )+e^x \left (-2+30 x+18 x^2-54 x^3+e^4 \left (10 x+6 x^2-18 x^3\right )\right )+\left (36 x-54 x^3+e^4 \left (12 x-18 x^3\right )+e^x \left (-10 x-6 x^2+18 x^3\right )\right ) \log (x)+\left (-6 x+9 x^3\right ) \log ^2(x)}{\left (81+54 e^4+9 e^8\right ) x^3+e^{2 x} \left (x-6 x^2+9 x^3\right )+e^x \left (18 x^2-54 x^3+e^4 \left (6 x^2-18 x^3\right )\right )+\left (-54 x^3-18 e^4 x^3+e^x \left (-6 x^2+18 x^3\right )\right ) \log (x)+9 x^3 \log ^2(x)} \, dx\\ &=\int \frac {27 \left (1+\frac {1}{9} e^4 \left (6+e^4\right )\right ) x \left (-2+3 x^2\right )+e^{2 x} x \left (-5-6 x+9 x^2\right )-2 e^{4+x} x \left (-5-3 x+9 x^2\right )+e^x \left (-2+30 x+18 x^2-54 x^3\right )+2 x \left (18-27 x^2+e^4 \left (6-9 x^2\right )+e^x \left (-5-3 x+9 x^2\right )\right ) \log (x)+\left (-6 x+9 x^3\right ) \log ^2(x)}{x \left (9 \left (1+\frac {e^4}{3}\right ) x-e^x (-1+3 x)-3 x \log (x)\right )^2} \, dx\\ &=\int \left (\frac {-5-6 x+9 x^2}{(-1+3 x)^2}+\frac {2 \left (-1+18 \left (1+\frac {5 e^4}{18}\right ) x+9 \left (1+\frac {e^4}{3}\right ) x^2-5 x \log (x)-3 x^2 \log (x)\right )}{(1-3 x)^2 x \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )}+\frac {6 \left (3 \left (1+\frac {e^4}{3}\right )-\log (x)\right ) \left (-2 \left (1+\frac {e^4}{2}\right )+e^4 x-9 \left (1+\frac {e^4}{3}\right ) x^2+\log (x)-x \log (x)+3 x^2 \log (x)\right )}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2}\right ) \, dx\\ &=2 \int \frac {-1+18 \left (1+\frac {5 e^4}{18}\right ) x+9 \left (1+\frac {e^4}{3}\right ) x^2-5 x \log (x)-3 x^2 \log (x)}{(1-3 x)^2 x \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )} \, dx+6 \int \frac {\left (3 \left (1+\frac {e^4}{3}\right )-\log (x)\right ) \left (-2 \left (1+\frac {e^4}{2}\right )+e^4 x-9 \left (1+\frac {e^4}{3}\right ) x^2+\log (x)-x \log (x)+3 x^2 \log (x)\right )}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2} \, dx+\int \frac {-5-6 x+9 x^2}{(-1+3 x)^2} \, dx\\ &=2 \int \left (\frac {3 \left (-1+18 \left (1+\frac {5 e^4}{18}\right ) x+9 \left (1+\frac {e^4}{3}\right ) x^2-5 x \log (x)-3 x^2 \log (x)\right )}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )}+\frac {3 \left (-1+18 \left (1+\frac {5 e^4}{18}\right ) x+9 \left (1+\frac {e^4}{3}\right ) x^2-5 x \log (x)-3 x^2 \log (x)\right )}{(1-3 x) \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )}+\frac {-1+18 \left (1+\frac {5 e^4}{18}\right ) x+9 \left (1+\frac {e^4}{3}\right ) x^2-5 x \log (x)-3 x^2 \log (x)}{x \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )}\right ) \, dx+6 \int \left (\frac {\left (-3-e^4\right ) \left (2+e^4\right )}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2}+\frac {e^4 \left (3+e^4\right ) x}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2}-\frac {3 \left (3+e^4\right )^2 x^2}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2}+\frac {\left (2+e^4\right ) \left (1+\frac {3+e^4}{2+e^4}\right ) \log (x)}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2}-\frac {\left (2+\frac {3}{e^4}\right ) e^4 x \log (x)}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2}+\frac {6 \left (3+e^4\right ) x^2 \log (x)}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2}-\frac {\log ^2(x)}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2}+\frac {x \log ^2(x)}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2}-\frac {3 x^2 \log ^2(x)}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2}\right ) \, dx+\int \left (1-\frac {6}{(-1+3 x)^2}\right ) \, dx\\ &=-\frac {2}{1-3 x}+x+2 \int \frac {-1+18 \left (1+\frac {5 e^4}{18}\right ) x+9 \left (1+\frac {e^4}{3}\right ) x^2-5 x \log (x)-3 x^2 \log (x)}{x \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )} \, dx-6 \int \frac {\log ^2(x)}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2} \, dx+6 \int \frac {x \log ^2(x)}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2} \, dx+6 \int \frac {-1+18 \left (1+\frac {5 e^4}{18}\right ) x+9 \left (1+\frac {e^4}{3}\right ) x^2-5 x \log (x)-3 x^2 \log (x)}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )} \, dx+6 \int \frac {-1+18 \left (1+\frac {5 e^4}{18}\right ) x+9 \left (1+\frac {e^4}{3}\right ) x^2-5 x \log (x)-3 x^2 \log (x)}{(1-3 x) \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )} \, dx-18 \int \frac {x^2 \log ^2(x)}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2} \, dx+\left (36 \left (3+e^4\right )\right ) \int \frac {x^2 \log (x)}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2} \, dx+\left (6 e^4 \left (3+e^4\right )\right ) \int \frac {x}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2} \, dx-\left (6 \left (2+e^4\right ) \left (3+e^4\right )\right ) \int \frac {1}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2} \, dx-\left (18 \left (3+e^4\right )^2\right ) \int \frac {x^2}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2} \, dx-\left (6 \left (3+2 e^4\right )\right ) \int \frac {x \log (x)}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2} \, dx+\left (6 \left (5+2 e^4\right )\right ) \int \frac {\log (x)}{(1-3 x)^2 \left (e^x-3 e^x x+9 \left (1+\frac {e^4}{3}\right ) x-3 x \log (x)\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.14, size = 47, normalized size = 1.57 \begin {gather*} \frac {2}{3 x}+x+\frac {2 e^x}{3 x \left (-e^x-9 x-3 e^4 x+3 e^x x+3 x \log (x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 44, normalized size = 1.47
method | result | size |
risch | \(\frac {3 x^{2}+2}{3 x}-\frac {2 \,{\mathrm e}^{x}}{3 x \left (3 x \,{\mathrm e}^{4}-3 \,{\mathrm e}^{x} x -3 x \ln \left (x \right )+9 x +{\mathrm e}^{x}\right )}\) | \(44\) |
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. 65 vs.
\(2 (28) = 56\).
time = 0.36, size = 65, normalized size = 2.17 \begin {gather*} \frac {3 \, x^{2} {\left (e^{4} + 3\right )} - {\left (3 \, x^{2} - x + 2\right )} e^{x} - {\left (3 \, x^{2} + 2\right )} \log \left (x\right ) + 2 \, e^{4} + 6}{3 \, x {\left (e^{4} + 3\right )} - {\left (3 \, x - 1\right )} e^{x} - 3 \, x \log \left (x\right )} \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. 68 vs.
\(2 (28) = 56\).
time = 0.44, size = 68, normalized size = 2.27 \begin {gather*} \frac {9 \, x^{2} + {\left (3 \, x^{2} + 2\right )} e^{4} - {\left (3 \, x^{2} - x + 2\right )} e^{x} - {\left (3 \, x^{2} + 2\right )} \log \left (x\right ) + 6}{3 \, x e^{4} - {\left (3 \, x - 1\right )} e^{x} - 3 \, x \log \left (x\right ) + 9 \, x} \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.
\(2 (22) = 44\).
time = 0.24, size = 71, normalized size = 2.37 \begin {gather*} x + \frac {- 2 \log {\left (x \right )} + 6 + 2 e^{4}}{9 x^{2} \log {\left (x \right )} - 9 x^{2} e^{4} - 27 x^{2} - 3 x \log {\left (x \right )} + 9 x + 3 x e^{4} + \left (9 x^{2} - 6 x + 1\right ) e^{x}} + \frac {2}{3 x - 1} \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. 82 vs.
\(2 (28) = 56\).
time = 0.49, size = 82, normalized size = 2.73 \begin {gather*} \frac {3 \, x^{2} e^{4} - 3 \, x^{2} e^{x} - 3 \, x^{2} \log \left (x\right ) + 9 \, x^{2} + 3 \, x e^{4} - 2 \, x e^{x} - 3 \, x \log \left (x\right ) + 9 \, x + 2 \, e^{4} - e^{x} - 2 \, \log \left (x\right ) + 6}{3 \, x e^{4} - 3 \, x e^{x} - 3 \, x \log \left (x\right ) + 9 \, x + e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {54\,x+{\ln \left (x\right )}^2\,\left (6\,x-9\,x^3\right )+{\mathrm {e}}^8\,\left (6\,x-9\,x^3\right )+{\mathrm {e}}^4\,\left (36\,x-54\,x^3\right )+{\mathrm {e}}^{2\,x}\,\left (-9\,x^3+6\,x^2+5\,x\right )-{\mathrm {e}}^x\,\left (30\,x+{\mathrm {e}}^4\,\left (-18\,x^3+6\,x^2+10\,x\right )+18\,x^2-54\,x^3-2\right )-\ln \left (x\right )\,\left (36\,x+{\mathrm {e}}^4\,\left (12\,x-18\,x^3\right )-54\,x^3-{\mathrm {e}}^x\,\left (-18\,x^3+6\,x^2+10\,x\right )\right )-81\,x^3}{9\,x^3\,{\ln \left (x\right )}^2-\ln \left (x\right )\,\left ({\mathrm {e}}^x\,\left (6\,x^2-18\,x^3\right )+18\,x^3\,{\mathrm {e}}^4+54\,x^3\right )+{\mathrm {e}}^x\,\left ({\mathrm {e}}^4\,\left (6\,x^2-18\,x^3\right )+18\,x^2-54\,x^3\right )+54\,x^3\,{\mathrm {e}}^4+9\,x^3\,{\mathrm {e}}^8+81\,x^3+{\mathrm {e}}^{2\,x}\,\left (9\,x^3-6\,x^2+x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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