Optimal. Leaf size=30 \[ 2 \left (x-\frac {-25-e^x+x+\log (2)}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2}\right ) \]
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Rubi [F] time = 1.61, 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 {-200+e^x (-8-2 x)-144 x+8 x^2+(8+8 x) \log (2)+\left (52 x+2 e^x x\right ) \log \left (e^{2 x} x^2\right )+18 x \log ^2\left (e^{2 x} x^2\right )+2 x \log ^3\left (e^{2 x} x^2\right )}{27 x+27 x \log \left (e^{2 x} x^2\right )+9 x \log ^2\left (e^{2 x} x^2\right )+x \log ^3\left (e^{2 x} x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-200+e^x (-8-2 x)-144 x+8 x^2+(8+8 x) \log (2)+\left (52 x+2 e^x x\right ) \log \left (e^{2 x} x^2\right )+18 x \log ^2\left (e^{2 x} x^2\right )+2 x \log ^3\left (e^{2 x} x^2\right )}{x \left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx\\ &=\int \left (-\frac {144}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3}-\frac {200}{x \left (3+\log \left (e^{2 x} x^2\right )\right )^3}+\frac {8 x}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3}+\frac {8 (1+x) \log (2)}{x \left (3+\log \left (e^{2 x} x^2\right )\right )^3}+\frac {52 \log \left (e^{2 x} x^2\right )}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3}+\frac {18 \log ^2\left (e^{2 x} x^2\right )}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3}+\frac {2 \log ^3\left (e^{2 x} x^2\right )}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3}+\frac {2 e^x \left (-4-x+x \log \left (e^{2 x} x^2\right )\right )}{x \left (3+\log \left (e^{2 x} x^2\right )\right )^3}\right ) \, dx\\ &=2 \int \frac {\log ^3\left (e^{2 x} x^2\right )}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx+2 \int \frac {e^x \left (-4-x+x \log \left (e^{2 x} x^2\right )\right )}{x \left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx+8 \int \frac {x}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx+18 \int \frac {\log ^2\left (e^{2 x} x^2\right )}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx+52 \int \frac {\log \left (e^{2 x} x^2\right )}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx-144 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx-200 \int \frac {1}{x \left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx+(8 \log (2)) \int \frac {1+x}{x \left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx\\ &=-\frac {2 \log (2)}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2}+2 \int \left (-\frac {4 e^x (1+x)}{x \left (3+\log \left (e^{2 x} x^2\right )\right )^3}+\frac {e^x}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2}\right ) \, dx+2 \int \left (1-\frac {27}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3}+\frac {27}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2}-\frac {9}{3+\log \left (e^{2 x} x^2\right )}\right ) \, dx+8 \int \frac {x}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx+18 \int \left (\frac {9}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3}-\frac {6}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2}+\frac {1}{3+\log \left (e^{2 x} x^2\right )}\right ) \, dx+52 \int \left (-\frac {3}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3}+\frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2}\right ) \, dx-144 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx-200 \int \frac {1}{x \left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx\\ &=2 x-\frac {2 \log (2)}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2}+2 \int \frac {e^x}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2} \, dx+8 \int \frac {x}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx-8 \int \frac {e^x (1+x)}{x \left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx+52 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2} \, dx-54 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx+54 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2} \, dx-108 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2} \, dx-144 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx-156 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx+162 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx-200 \int \frac {1}{x \left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx\\ &=2 x-\frac {2 \log (2)}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2}+2 \int \frac {e^x}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2} \, dx+8 \int \frac {x}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx-8 \int \left (\frac {e^x}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3}+\frac {e^x}{x \left (3+\log \left (e^{2 x} x^2\right )\right )^3}\right ) \, dx+52 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2} \, dx-54 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx+54 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2} \, dx-108 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2} \, dx-144 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx-156 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx+162 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx-200 \int \frac {1}{x \left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx\\ &=2 x-\frac {2 \log (2)}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2}+2 \int \frac {e^x}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2} \, dx-8 \int \frac {e^x}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx-8 \int \frac {e^x}{x \left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx+8 \int \frac {x}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx+52 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2} \, dx-54 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx+54 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2} \, dx-108 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2} \, dx-144 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx-156 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx+162 \int \frac {1}{\left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx-200 \int \frac {1}{x \left (3+\log \left (e^{2 x} x^2\right )\right )^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 36, normalized size = 1.20 \begin {gather*} 2 \left (x+\frac {100+4 e^x-4 x-\log (16)}{4 \left (3+\log \left (e^{2 x} x^2\right )\right )^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.12, size = 64, normalized size = 2.13 \begin {gather*} \frac {2 \, {\left (x \log \left (x^{2} e^{\left (2 \, x\right )}\right )^{2} + 6 \, x \log \left (x^{2} e^{\left (2 \, x\right )}\right ) + 8 \, x + e^{x} - \log \relax (2) + 25\right )}}{\log \left (x^{2} e^{\left (2 \, x\right )}\right )^{2} + 6 \, \log \left (x^{2} e^{\left (2 \, x\right )}\right ) + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.52, size = 78, normalized size = 2.60 \begin {gather*} \frac {2 \, {\left (4 \, x^{3} + 4 \, x^{2} \log \left (x^{2}\right ) + x \log \left (x^{2}\right )^{2} + 12 \, x^{2} + 6 \, x \log \left (x^{2}\right ) + 8 \, x + e^{x} - \log \relax (2) + 25\right )}}{4 \, x^{2} + 4 \, x \log \left (x^{2}\right ) + \log \left (x^{2}\right )^{2} + 12 \, x + 6 \, \log \left (x^{2}\right ) + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.12, size = 215, normalized size = 7.17
method | result | size |
risch | \(2 x +\frac {8 \ln \relax (2)-8 \,{\mathrm e}^{x}-200+8 x}{\left (\pi \mathrm {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{3}-\pi \mathrm {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{2} \mathrm {csgn}\left (i x^{2}\right )-\pi \mathrm {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )+\pi \,\mathrm {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )-2 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{2}+\pi \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}+4 i \ln \relax (x )+4 i \ln \left ({\mathrm e}^{x}\right )+6 i\right )^{2}}\) | \(215\) |
default | \(\frac {2 \left (-4 \ln \left ({\mathrm e}^{2 x} x^{2}\right )+8 \ln \relax (x )+8 x -12\right ) \ln \relax (x )^{2}+2 \left (-3 \left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )-2 \ln \left ({\mathrm e}^{x}\right )-2 \ln \relax (x )\right )^{2}-12 \left (\ln \left ({\mathrm e}^{x}\right )-x \right ) \left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )-2 \ln \left ({\mathrm e}^{x}\right )-2 \ln \relax (x )\right )-12 \left (\ln \left ({\mathrm e}^{x}\right )-x \right )^{2}-18 \ln \left ({\mathrm e}^{2 x} x^{2}\right )+36 \ln \relax (x )+36 x -28\right ) x +2 \left (-4 \left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )-2 \ln \left ({\mathrm e}^{x}\right )-2 \ln \relax (x )\right )^{2}-16 \left (\ln \left ({\mathrm e}^{x}\right )-x \right ) \left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )-2 \ln \left ({\mathrm e}^{x}\right )-2 \ln \relax (x )\right )-16 \left (\ln \left ({\mathrm e}^{x}\right )-x \right )^{2}-24 \ln \left ({\mathrm e}^{2 x} x^{2}\right )+48 \ln \relax (x )+48 x -36\right ) \ln \relax (x )+8 x \ln \relax (x )^{2}+16 x^{2} \ln \relax (x )+2 \left (-4 \ln \left ({\mathrm e}^{2 x} x^{2}\right )+8 \ln \relax (x )+8 x -12\right ) x \ln \relax (x )+8 x^{3}-4-2 \left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )-2 \ln \left ({\mathrm e}^{x}\right )-2 \ln \relax (x )\right )^{3}-12 \left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )-2 \ln \left ({\mathrm e}^{x}\right )-2 \ln \relax (x )\right )^{2} \left (\ln \left ({\mathrm e}^{x}\right )-x \right )-24 \left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )-2 \ln \left ({\mathrm e}^{x}\right )-2 \ln \relax (x )\right ) \left (\ln \left ({\mathrm e}^{x}\right )-x \right )^{2}-16 \left (\ln \left ({\mathrm e}^{x}\right )-x \right )^{3}-18 \left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )-2 \ln \left ({\mathrm e}^{x}\right )-2 \ln \relax (x )\right )^{2}-72 \left (\ln \left ({\mathrm e}^{x}\right )-x \right ) \left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )-2 \ln \left ({\mathrm e}^{x}\right )-2 \ln \relax (x )\right )-72 \left (\ln \left ({\mathrm e}^{x}\right )-x \right )^{2}-54 \ln \left ({\mathrm e}^{2 x} x^{2}\right )+108 \ln \relax (x )+108 x}{\left (3+\ln \left ({\mathrm e}^{2 x} x^{2}\right )\right )^{2}}-\frac {2 \ln \relax (2)}{\left (3+\ln \left ({\mathrm e}^{2 x} x^{2}\right )\right )^{2}}+\frac {2 \,{\mathrm e}^{x}}{\left (3+\ln \left ({\mathrm e}^{2 x} x^{2}\right )\right )^{2}}\) | \(464\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 70, normalized size = 2.33 \begin {gather*} \frac {2 \, {\left (4 \, x^{3} + 4 \, x \log \relax (x)^{2} + 12 \, x^{2} + 4 \, {\left (2 \, x^{2} + 3 \, x\right )} \log \relax (x) + 8 \, x + e^{x} - \log \relax (2) + 25\right )}}{4 \, x^{2} + 4 \, {\left (2 \, x + 3\right )} \log \relax (x) + 4 \, \log \relax (x)^{2} + 12 \, x + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.50, size = 30, normalized size = 1.00 \begin {gather*} 2\,x-\frac {2\,x+\ln \relax (4)-2\,{\mathrm {e}}^x-50}{{\left (\ln \left (x^2\,{\mathrm {e}}^{2\,x}\right )+3\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 42, normalized size = 1.40 \begin {gather*} 2 x + \frac {- 2 x + 2 e^{x} - 2 \log {\relax (2 )} + 50}{\log {\left (x^{2} e^{2 x} \right )}^{2} + 6 \log {\left (x^{2} e^{2 x} \right )} + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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