Optimal. Leaf size=30 \[ \frac {5 x}{\log \left (\frac {e^{9-e^x-e^x x-x^3}}{x^2}\right )} \]
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Rubi [F]
time = 1.05, 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 {10+15 x^3+e^x \left (10 x+5 x^2\right )+5 \log \left (\frac {e^{9+e^x (-1-x)-x^3}}{x^2}\right )}{\log ^2\left (\frac {e^{9+e^x (-1-x)-x^3}}{x^2}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \left (2+2 e^x x+e^x x^2+3 x^3+\log \left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )\right )}{\log ^2\left (\frac {e^{9+e^x (-1-x)-x^3}}{x^2}\right )} \, dx\\ &=5 \int \frac {2+2 e^x x+e^x x^2+3 x^3+\log \left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )}{\log ^2\left (\frac {e^{9+e^x (-1-x)-x^3}}{x^2}\right )} \, dx\\ &=5 \int \left (\frac {e^x x (2+x)}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )}+\frac {2+3 x^3+\log \left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )}\right ) \, dx\\ &=5 \int \frac {e^x x (2+x)}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )} \, dx+5 \int \frac {2+3 x^3+\log \left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )} \, dx\\ &=5 \int \left (\frac {2 e^x x}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )}+\frac {e^x x^2}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )}\right ) \, dx+5 \int \left (\frac {2+3 x^3}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )}+\frac {1}{\log \left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )}\right ) \, dx\\ &=5 \int \frac {e^x x^2}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )} \, dx+5 \int \frac {2+3 x^3}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )} \, dx+5 \int \frac {1}{\log \left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )} \, dx+10 \int \frac {e^x x}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )} \, dx\\ &=5 \int \left (\frac {2}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )}+\frac {3 x^3}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )}\right ) \, dx+5 \int \frac {e^x x^2}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )} \, dx+5 \int \frac {1}{\log \left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )} \, dx+10 \int \frac {e^x x}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )} \, dx\\ &=5 \int \frac {e^x x^2}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )} \, dx+5 \int \frac {1}{\log \left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )} \, dx+10 \int \frac {1}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )} \, dx+10 \int \frac {e^x x}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )} \, dx+15 \int \frac {x^3}{\log ^2\left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 27, normalized size = 0.90 \begin {gather*} \frac {5 x}{\log \left (\frac {e^{9-x^3-e^x (1+x)}}{x^2}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.32, size = 246, normalized size = 8.20
method | result | size |
risch | \(-\frac {10 i x}{\pi \,\mathrm {csgn}\left (i {\mathrm e}^{9-{\mathrm e}^{x} x -x^{3}-{\mathrm e}^{x}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{9-{\mathrm e}^{x} x -x^{3}-{\mathrm e}^{x}}}{x^{2}}\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{9-{\mathrm e}^{x} x -x^{3}-{\mathrm e}^{x}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{9-{\mathrm e}^{x} x -x^{3}-{\mathrm e}^{x}}}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}}\right )-\pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{9-{\mathrm e}^{x} x -x^{3}-{\mathrm e}^{x}}}{x^{2}}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{9-{\mathrm e}^{x} x -x^{3}-{\mathrm e}^{x}}}{x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (x \right )-2 i \ln \left ({\mathrm e}^{9-{\mathrm e}^{x} x -x^{3}-{\mathrm e}^{x}}\right )}\) | \(246\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 20, normalized size = 0.67 \begin {gather*} -\frac {5 \, x}{x^{3} + {\left (x + 1\right )} e^{x} + 2 \, \log \left (x\right ) - 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 25, normalized size = 0.83 \begin {gather*} \frac {5 \, x}{\log \left (\frac {e^{\left (-x^{3} - {\left (x + 1\right )} e^{x} + 9\right )}}{x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 22, normalized size = 0.73 \begin {gather*} \frac {5 x}{\log {\left (\frac {e^{- x^{3} + \left (- x - 1\right ) e^{x} + 9}}{x^{2}} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 20, normalized size = 0.67 \begin {gather*} -\frac {5 \, x}{x^{3} + x e^{x} + e^{x} + \log \left (x^{2}\right ) - 9} \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 {5\,\ln \left (\frac {{\mathrm {e}}^{9-x^3-{\mathrm {e}}^x\,\left (x+1\right )}}{x^2}\right )+{\mathrm {e}}^x\,\left (5\,x^2+10\,x\right )+15\,x^3+10}{{\ln \left (\frac {{\mathrm {e}}^{9-x^3-{\mathrm {e}}^x\,\left (x+1\right )}}{x^2}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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