Optimal. Leaf size=31 \[ x \left (e^x-x-e^x \log \left (\frac {1+x}{5}\right )\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right ) \]
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Rubi [F] time = 22.33, 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 {-2 x-2 x^2+e^x (2+2 x)+e^x (-2-2 x) \log \left (\frac {1+x}{5}\right )+\log \left (\frac {x^2}{3}\right ) \left (-2 x-2 x^2+e^x \left (1+x+x^2\right )+e^x \left (-1-2 x-x^2\right ) \log \left (\frac {1+x}{5}\right )\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{(1+x) \log \left (\frac {x^2}{3}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 \left (x+x \log \left (\frac {x^2}{3}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )\right )}{\log \left (\frac {x^2}{3}\right )}-\frac {e^x \left (-2-2 x+2 \log \left (\frac {1+x}{5}\right )+2 x \log \left (\frac {1+x}{5}\right )-\log \left (\frac {x^2}{3}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )-x \log \left (\frac {x^2}{3}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )-x^2 \log \left (\frac {x^2}{3}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )+\log \left (\frac {x^2}{3}\right ) \log \left (\frac {1+x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )+2 x \log \left (\frac {x^2}{3}\right ) \log \left (\frac {1+x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )+x^2 \log \left (\frac {x^2}{3}\right ) \log \left (\frac {1+x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )\right )}{(1+x) \log \left (\frac {x^2}{3}\right )}\right ) \, dx\\ &=-\left (2 \int \frac {x+x \log \left (\frac {x^2}{3}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{\log \left (\frac {x^2}{3}\right )} \, dx\right )-\int \frac {e^x \left (-2-2 x+2 \log \left (\frac {1+x}{5}\right )+2 x \log \left (\frac {1+x}{5}\right )-\log \left (\frac {x^2}{3}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )-x \log \left (\frac {x^2}{3}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )-x^2 \log \left (\frac {x^2}{3}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )+\log \left (\frac {x^2}{3}\right ) \log \left (\frac {1+x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )+2 x \log \left (\frac {x^2}{3}\right ) \log \left (\frac {1+x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )+x^2 \log \left (\frac {x^2}{3}\right ) \log \left (\frac {1+x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )\right )}{(1+x) \log \left (\frac {x^2}{3}\right )} \, dx\\ &=-\left (2 \int x \left (\frac {1}{\log \left (\frac {x^2}{3}\right )}+\log \left (\log \left (\frac {x^2}{3}\right )\right )\right ) \, dx\right )-\int \frac {e^x \left (-2 (1+x)-\left (1+x+x^2\right ) \log \left (\frac {x^2}{3}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )+(1+x) \log \left (\frac {1+x}{5}\right ) \left (2+(1+x) \log \left (\frac {x^2}{3}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )\right )\right )}{(1+x) \log \left (\frac {x^2}{3}\right )} \, dx\\ &=-\left (2 \int \left (\frac {x}{\log \left (\frac {x^2}{3}\right )}+x \log \left (\log \left (\frac {x^2}{3}\right )\right )\right ) \, dx\right )-\int \left (\frac {2 e^x \left (-1+\log \left (\frac {1+x}{5}\right )\right )}{\log \left (\frac {x^2}{3}\right )}+\frac {e^x \left (-1-x-x^2+\log \left (\frac {1+x}{5}\right )+2 x \log \left (\frac {1+x}{5}\right )+x^2 \log \left (\frac {1+x}{5}\right )\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x}\right ) \, dx\\ &=-\left (2 \int \frac {x}{\log \left (\frac {x^2}{3}\right )} \, dx\right )-2 \int \frac {e^x \left (-1+\log \left (\frac {1+x}{5}\right )\right )}{\log \left (\frac {x^2}{3}\right )} \, dx-2 \int x \log \left (\log \left (\frac {x^2}{3}\right )\right ) \, dx-\int \frac {e^x \left (-1-x-x^2+\log \left (\frac {1+x}{5}\right )+2 x \log \left (\frac {1+x}{5}\right )+x^2 \log \left (\frac {1+x}{5}\right )\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x} \, dx\\ &=-x^2 \log \left (\log \left (\frac {x^2}{3}\right )\right )+2 \int \frac {x}{\log \left (\frac {x^2}{3}\right )} \, dx-2 \int \left (\frac {e^x \log \left (\frac {1}{5}+\frac {x}{5}\right )}{\log \left (\frac {x^2}{3}\right )}+\frac {e^x}{\log (3)-\log \left (x^2\right )}\right ) \, dx-\int \frac {e^x \left (-1-x-x^2+(1+x)^2 \log \left (\frac {1+x}{5}\right )\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x} \, dx-\operatorname {Subst}\left (\int \frac {1}{\log \left (\frac {x}{3}\right )} \, dx,x,x^2\right )\\ &=-x^2 \log \left (\log \left (\frac {x^2}{3}\right )\right )-3 \text {li}\left (\frac {x^2}{3}\right )-2 \int \frac {e^x \log \left (\frac {1}{5}+\frac {x}{5}\right )}{\log \left (\frac {x^2}{3}\right )} \, dx-2 \int \frac {e^x}{\log (3)-\log \left (x^2\right )} \, dx-\int \left (-\frac {e^x \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x}-\frac {e^x x \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x}-\frac {e^x x^2 \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x}+\frac {e^x \log \left (\frac {1}{5}+\frac {x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x}+\frac {2 e^x x \log \left (\frac {1}{5}+\frac {x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x}+\frac {e^x x^2 \log \left (\frac {1}{5}+\frac {x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x}\right ) \, dx+\operatorname {Subst}\left (\int \frac {1}{\log \left (\frac {x}{3}\right )} \, dx,x,x^2\right )\\ &=-x^2 \log \left (\log \left (\frac {x^2}{3}\right )\right )-2 \int \frac {e^x \log \left (\frac {1}{5}+\frac {x}{5}\right )}{\log \left (\frac {x^2}{3}\right )} \, dx-2 \int \frac {e^x}{\log (3)-\log \left (x^2\right )} \, dx-2 \int \frac {e^x x \log \left (\frac {1}{5}+\frac {x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x} \, dx+\int \frac {e^x \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x} \, dx+\int \frac {e^x x \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x} \, dx+\int \frac {e^x x^2 \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x} \, dx-\int \frac {e^x \log \left (\frac {1}{5}+\frac {x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x} \, dx-\int \frac {e^x x^2 \log \left (\frac {1}{5}+\frac {x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x} \, dx\\ &=-x^2 \log \left (\log \left (\frac {x^2}{3}\right )\right )-2 \int \frac {e^x \log \left (\frac {1}{5}+\frac {x}{5}\right )}{\log \left (\frac {x^2}{3}\right )} \, dx-2 \int \frac {e^x}{\log (3)-\log \left (x^2\right )} \, dx-2 \int \left (e^x \log \left (\frac {1}{5}+\frac {x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )+\frac {e^x \log \left (\frac {1}{5}+\frac {x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{-1-x}\right ) \, dx+\int \frac {e^x \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x} \, dx-\int \frac {e^x \log \left (\frac {1}{5}+\frac {x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x} \, dx+\int \left (e^x \log \left (\log \left (\frac {x^2}{3}\right )\right )-\frac {e^x \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x}\right ) \, dx+\int \left (-e^x \log \left (\log \left (\frac {x^2}{3}\right )\right )+e^x x \log \left (\log \left (\frac {x^2}{3}\right )\right )+\frac {e^x \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x}\right ) \, dx-\int \left (-e^x \log \left (\frac {1}{5}+\frac {x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )+e^x x \log \left (\frac {1}{5}+\frac {x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )+\frac {e^x \log \left (\frac {1}{5}+\frac {x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x}\right ) \, dx\\ &=-x^2 \log \left (\log \left (\frac {x^2}{3}\right )\right )-2 \int \frac {e^x \log \left (\frac {1}{5}+\frac {x}{5}\right )}{\log \left (\frac {x^2}{3}\right )} \, dx-2 \int \frac {e^x}{\log (3)-\log \left (x^2\right )} \, dx-2 \int e^x \log \left (\frac {1}{5}+\frac {x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right ) \, dx-2 \int \frac {e^x \log \left (\frac {1}{5}+\frac {x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{-1-x} \, dx+\int e^x x \log \left (\log \left (\frac {x^2}{3}\right )\right ) \, dx+\int \frac {e^x \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x} \, dx+\int e^x \log \left (\frac {1}{5}+\frac {x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right ) \, dx-\int e^x x \log \left (\frac {1}{5}+\frac {x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right ) \, dx-2 \int \frac {e^x \log \left (\frac {1}{5}+\frac {x}{5}\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right )}{1+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 31, normalized size = 1.00 \begin {gather*} -x \left (-e^x+x+e^x \log \left (\frac {1+x}{5}\right )\right ) \log \left (\log \left (\frac {x^2}{3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 28, normalized size = 0.90 \begin {gather*} -{\left (x e^{x} \log \left (\frac {1}{5} \, x + \frac {1}{5}\right ) + x^{2} - x e^{x}\right )} \log \left (\log \left (\frac {1}{3} \, x^{2}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.17, size = 53, normalized size = 1.71 \begin {gather*} x e^{x} \log \relax (5) \log \left (\log \left (\frac {1}{3} \, x^{2}\right )\right ) - x e^{x} \log \left (x + 1\right ) \log \left (\log \left (\frac {1}{3} \, x^{2}\right )\right ) - x^{2} \log \left (\log \left (\frac {1}{3} \, x^{2}\right )\right ) + x e^{x} \log \left (\log \left (\frac {1}{3} \, x^{2}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.90, size = 61, normalized size = 1.97
method | result | size |
risch | \(\left (-{\mathrm e}^{x} \ln \left (\frac {x}{5}+\frac {1}{5}\right ) x -x^{2}+{\mathrm e}^{x} x \right ) \ln \left (-\ln \relax (3)+2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 34, normalized size = 1.10 \begin {gather*} {\left (x {\left (\log \relax (5) + 1\right )} e^{x} - x e^{x} \log \left (x + 1\right ) - x^{2}\right )} \log \left (-\log \relax (3) + 2 \, \log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.66, size = 85, normalized size = 2.74 \begin {gather*} -\ln \left (\ln \left (\frac {x^2}{3}\right )\right )\,\left (\frac {x^4+x^3}{x\,\left (x+1\right )}+\frac {{\mathrm {e}}^x\,\left (x-x^3\right )}{x\,\left (x+1\right )}-\frac {{\mathrm {e}}^x\,\left (x^2+x\right )}{x\,\left (x+1\right )}+\frac {{\mathrm {e}}^x\,\ln \left (\frac {x}{5}+\frac {1}{5}\right )\,\left (x^3+x^2\right )}{x\,\left (x+1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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