Optimal. Leaf size=33 \[ \frac {1}{2} e^{\frac {3 (-5+x)}{x+\log (x)}} \left (-e^x+x^2 (x-\log (15))\right ) \]
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Rubi [F]
time = 17.60, 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 {e^{\frac {-15+3 x}{x+\log (x)}} \left (15 x^3+12 x^4+3 x^5+e^x \left (-15-12 x-x^3\right )+\left (-15 x^2-12 x^3-2 x^4\right ) \log (15)+\left (9 x^4+e^x \left (-3 x-2 x^2\right )-7 x^3 \log (15)\right ) \log (x)+\left (-e^x x+3 x^3-2 x^2 \log (15)\right ) \log ^2(x)\right )}{2 x^3+4 x^2 \log (x)+2 x \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 {e^{\frac {-15+3 x}{x+\log (x)}} \left (15 x^3+12 x^4+3 x^5+e^x \left (-15-12 x-x^3\right )+\left (-15 x^2-12 x^3-2 x^4\right ) \log (15)+\left (9 x^4+e^x \left (-3 x-2 x^2\right )-7 x^3 \log (15)\right ) \log (x)+\left (-e^x x+3 x^3-2 x^2 \log (15)\right ) \log ^2(x)\right )}{2 x (x+\log (x))^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} \left (15 x^3+12 x^4+3 x^5+e^x \left (-15-12 x-x^3\right )+\left (-15 x^2-12 x^3-2 x^4\right ) \log (15)+\left (9 x^4+e^x \left (-3 x-2 x^2\right )-7 x^3 \log (15)\right ) \log (x)+\left (-e^x x+3 x^3-2 x^2 \log (15)\right ) \log ^2(x)\right )}{x (x+\log (x))^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {15 e^{\frac {-15+3 x}{x+\log (x)}} x^2}{(x+\log (x))^2}+\frac {12 e^{\frac {-15+3 x}{x+\log (x)}} x^3}{(x+\log (x))^2}+\frac {3 e^{\frac {-15+3 x}{x+\log (x)}} x^4}{(x+\log (x))^2}-\frac {e^{\frac {-15+3 x}{x+\log (x)}} x \left (15+12 x+2 x^2\right ) \log (15)}{(x+\log (x))^2}+\frac {9 e^{\frac {-15+3 x}{x+\log (x)}} x^3 \log (x)}{(x+\log (x))^2}-\frac {7 e^{\frac {-15+3 x}{x+\log (x)}} x^2 \log (15) \log (x)}{(x+\log (x))^2}+\frac {3 e^{\frac {-15+3 x}{x+\log (x)}} x^2 \log ^2(x)}{(x+\log (x))^2}-\frac {2 e^{\frac {-15+3 x}{x+\log (x)}} x \log (15) \log ^2(x)}{(x+\log (x))^2}-\frac {e^{x+\frac {-15+3 x}{x+\log (x)}} \left (15+12 x+x^3+3 x \log (x)+2 x^2 \log (x)+x \log ^2(x)\right )}{x (x+\log (x))^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {e^{x+\frac {-15+3 x}{x+\log (x)}} \left (15+12 x+x^3+3 x \log (x)+2 x^2 \log (x)+x \log ^2(x)\right )}{x (x+\log (x))^2} \, dx\right )+\frac {3}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^4}{(x+\log (x))^2} \, dx+\frac {3}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^2 \log ^2(x)}{(x+\log (x))^2} \, dx+\frac {9}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3 \log (x)}{(x+\log (x))^2} \, dx+6 \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{(x+\log (x))^2} \, dx+\frac {15}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^2}{(x+\log (x))^2} \, dx-\frac {1}{2} \log (15) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x \left (15+12 x+2 x^2\right )}{(x+\log (x))^2} \, dx-\log (15) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x \log ^2(x)}{(x+\log (x))^2} \, dx-\frac {1}{2} (7 \log (15)) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^2 \log (x)}{(x+\log (x))^2} \, dx\\ &=-\left (\frac {1}{2} \int \left (e^{x+\frac {-15+3 x}{x+\log (x)}}-\frac {3 e^{x+\frac {-15+3 x}{x+\log (x)}} \left (-5-4 x+x^2\right )}{x (x+\log (x))^2}+\frac {3 e^{x+\frac {-15+3 x}{x+\log (x)}}}{x+\log (x)}\right ) \, dx\right )+\frac {3}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^4}{(x+\log (x))^2} \, dx+\frac {3}{2} \int \left (e^{\frac {-15+3 x}{x+\log (x)}} x^2+\frac {e^{\frac {-15+3 x}{x+\log (x)}} x^4}{(x+\log (x))^2}-\frac {2 e^{\frac {-15+3 x}{x+\log (x)}} x^3}{x+\log (x)}\right ) \, dx+\frac {9}{2} \int \left (-\frac {e^{\frac {-15+3 x}{x+\log (x)}} x^4}{(x+\log (x))^2}+\frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{x+\log (x)}\right ) \, dx+6 \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{(x+\log (x))^2} \, dx+\frac {15}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^2}{(x+\log (x))^2} \, dx-\frac {1}{2} \log (15) \int \left (\frac {15 e^{\frac {-15+3 x}{x+\log (x)}} x}{(x+\log (x))^2}+\frac {12 e^{\frac {-15+3 x}{x+\log (x)}} x^2}{(x+\log (x))^2}+\frac {2 e^{\frac {-15+3 x}{x+\log (x)}} x^3}{(x+\log (x))^2}\right ) \, dx-\log (15) \int \left (e^{\frac {-15+3 x}{x+\log (x)}} x+\frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{(x+\log (x))^2}-\frac {2 e^{\frac {-15+3 x}{x+\log (x)}} x^2}{x+\log (x)}\right ) \, dx-\frac {1}{2} (7 \log (15)) \int \left (-\frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{(x+\log (x))^2}+\frac {e^{\frac {-15+3 x}{x+\log (x)}} x^2}{x+\log (x)}\right ) \, dx\\ &=-\left (\frac {1}{2} \int e^{x+\frac {-15+3 x}{x+\log (x)}} \, dx\right )+\frac {3}{2} \int e^{\frac {-15+3 x}{x+\log (x)}} x^2 \, dx+2 \left (\frac {3}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^4}{(x+\log (x))^2} \, dx\right )+\frac {3}{2} \int \frac {e^{x+\frac {-15+3 x}{x+\log (x)}} \left (-5-4 x+x^2\right )}{x (x+\log (x))^2} \, dx-\frac {3}{2} \int \frac {e^{x+\frac {-15+3 x}{x+\log (x)}}}{x+\log (x)} \, dx-3 \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{x+\log (x)} \, dx-\frac {9}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^4}{(x+\log (x))^2} \, dx+\frac {9}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{x+\log (x)} \, dx+6 \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{(x+\log (x))^2} \, dx+\frac {15}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^2}{(x+\log (x))^2} \, dx-\log (15) \int e^{\frac {-15+3 x}{x+\log (x)}} x \, dx-2 \left (\log (15) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{(x+\log (x))^2} \, dx\right )+(2 \log (15)) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^2}{x+\log (x)} \, dx+\frac {1}{2} (7 \log (15)) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{(x+\log (x))^2} \, dx-\frac {1}{2} (7 \log (15)) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^2}{x+\log (x)} \, dx-(6 \log (15)) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^2}{(x+\log (x))^2} \, dx-\frac {1}{2} (15 \log (15)) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x}{(x+\log (x))^2} \, dx\\ &=-\left (\frac {1}{2} \int e^{x+\frac {-15+3 x}{x+\log (x)}} \, dx\right )+\frac {3}{2} \int e^{\frac {-15+3 x}{x+\log (x)}} x^2 \, dx+2 \left (\frac {3}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^4}{(x+\log (x))^2} \, dx\right )-\frac {3}{2} \int \frac {e^{x+\frac {-15+3 x}{x+\log (x)}}}{x+\log (x)} \, dx+\frac {3}{2} \int \left (-\frac {4 e^{x+\frac {-15+3 x}{x+\log (x)}}}{(x+\log (x))^2}-\frac {5 e^{x+\frac {-15+3 x}{x+\log (x)}}}{x (x+\log (x))^2}+\frac {e^{x+\frac {-15+3 x}{x+\log (x)}} x}{(x+\log (x))^2}\right ) \, dx-3 \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{x+\log (x)} \, dx-\frac {9}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^4}{(x+\log (x))^2} \, dx+\frac {9}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{x+\log (x)} \, dx+6 \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{(x+\log (x))^2} \, dx+\frac {15}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^2}{(x+\log (x))^2} \, dx-\log (15) \int e^{\frac {-15+3 x}{x+\log (x)}} x \, dx-2 \left (\log (15) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{(x+\log (x))^2} \, dx\right )+(2 \log (15)) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^2}{x+\log (x)} \, dx+\frac {1}{2} (7 \log (15)) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{(x+\log (x))^2} \, dx-\frac {1}{2} (7 \log (15)) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^2}{x+\log (x)} \, dx-(6 \log (15)) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^2}{(x+\log (x))^2} \, dx-\frac {1}{2} (15 \log (15)) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x}{(x+\log (x))^2} \, dx\\ &=-\left (\frac {1}{2} \int e^{x+\frac {-15+3 x}{x+\log (x)}} \, dx\right )+\frac {3}{2} \int e^{\frac {-15+3 x}{x+\log (x)}} x^2 \, dx+\frac {3}{2} \int \frac {e^{x+\frac {-15+3 x}{x+\log (x)}} x}{(x+\log (x))^2} \, dx+2 \left (\frac {3}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^4}{(x+\log (x))^2} \, dx\right )-\frac {3}{2} \int \frac {e^{x+\frac {-15+3 x}{x+\log (x)}}}{x+\log (x)} \, dx-3 \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{x+\log (x)} \, dx-\frac {9}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^4}{(x+\log (x))^2} \, dx+\frac {9}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{x+\log (x)} \, dx-6 \int \frac {e^{x+\frac {-15+3 x}{x+\log (x)}}}{(x+\log (x))^2} \, dx+6 \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{(x+\log (x))^2} \, dx-\frac {15}{2} \int \frac {e^{x+\frac {-15+3 x}{x+\log (x)}}}{x (x+\log (x))^2} \, dx+\frac {15}{2} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^2}{(x+\log (x))^2} \, dx-\log (15) \int e^{\frac {-15+3 x}{x+\log (x)}} x \, dx-2 \left (\log (15) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{(x+\log (x))^2} \, dx\right )+(2 \log (15)) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^2}{x+\log (x)} \, dx+\frac {1}{2} (7 \log (15)) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^3}{(x+\log (x))^2} \, dx-\frac {1}{2} (7 \log (15)) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^2}{x+\log (x)} \, dx-(6 \log (15)) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x^2}{(x+\log (x))^2} \, dx-\frac {1}{2} (15 \log (15)) \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} x}{(x+\log (x))^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [F]
time = 0.38, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{\frac {-15+3 x}{x+\log (x)}} \left (15 x^3+12 x^4+3 x^5+e^x \left (-15-12 x-x^3\right )+\left (-15 x^2-12 x^3-2 x^4\right ) \log (15)+\left (9 x^4+e^x \left (-3 x-2 x^2\right )-7 x^3 \log (15)\right ) \log (x)+\left (-e^x x+3 x^3-2 x^2 \log (15)\right ) \log ^2(x)\right )}{2 x^3+4 x^2 \log (x)+2 x \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.37, size = 38, normalized size = 1.15
method | result | size |
risch | \(\left (\frac {x^{3}}{2}-\frac {x^{2} \ln \left (3\right )}{2}-\frac {x^{2} \ln \left (5\right )}{2}-\frac {{\mathrm e}^{x}}{2}\right ) {\mathrm e}^{\frac {3 x -15}{x +\ln \left (x \right )}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 29, normalized size = 0.88 \begin {gather*} \frac {1}{2} \, {\left (x^{3} - x^{2} \log \left (15\right ) - e^{x}\right )} e^{\left (\frac {3 \, {\left (x - 5\right )}}{x + \log \left (x\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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. 59 vs.
\(2 (29) = 58\).
time = 0.48, size = 59, normalized size = 1.79 \begin {gather*} \frac {1}{2} \, x^{3} e^{\left (\frac {3 \, {\left (x - 5\right )}}{x + \log \left (x\right )}\right )} - \frac {1}{2} \, x^{2} e^{\left (\frac {3 \, {\left (x - 5\right )}}{x + \log \left (x\right )}\right )} \log \left (15\right ) - \frac {1}{2} \, e^{\left (\frac {x^{2} + x \log \left (x\right ) + 3 \, x - 15}{x + \log \left (x\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.36, size = 56, normalized size = 1.70 \begin {gather*} \frac {x^3\,{\mathrm {e}}^{\frac {3\,x-15}{x+\ln \left (x\right )}}}{2}-\frac {{\mathrm {e}}^{\frac {3\,x-15}{x+\ln \left (x\right )}}\,{\mathrm {e}}^x}{2}-\frac {x^2\,{\mathrm {e}}^{\frac {3\,x-15}{x+\ln \left (x\right )}}\,\ln \left (15\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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