Optimal. Leaf size=34 \[ 5+\frac {x}{\log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \]
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Rubi [F] time = 17.84, 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 {-8+\left (-8-16 x^2\right ) \log (x)+\left (x^2-8 x^3-2 x^4+e^x \left (1-x+2 x^2\right )\right ) \log ^2(x)+\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log \left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )}{\left (8 x^2 \log (x)+\left (-e^x x^2+4 x^3+x^4\right ) \log ^2(x)+\left (-8 \log (x)+\left (e^x-4 x-x^2\right ) \log ^2(x)\right ) \log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (\frac {8+\left (-e^x+4 x+x^2\right ) \log (x)}{x \log (x)}\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-8-8 \left (1+2 x^2\right ) \log (x)+\left (x^2-8 x^3-2 x^4+e^x \left (1-x+2 x^2\right )\right ) \log ^2(x)+\log (x) \left (8+\left (-e^x+x (4+x)\right ) \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}{\log (x) \left (8+\left (-e^x+x (4+x)\right ) \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx\\ &=\int \left (\frac {8+8 x \log (x)-4 x \log ^2(x)+2 x^2 \log ^2(x)+x^3 \log ^2(x)}{\log (x) \left (-8+e^x \log (x)-4 x \log (x)-x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}+\frac {-1+x-2 x^2+x^2 \log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right ) \log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}\right ) \, dx\\ &=\int \frac {8+8 x \log (x)-4 x \log ^2(x)+2 x^2 \log ^2(x)+x^3 \log ^2(x)}{\log (x) \left (-8+e^x \log (x)-4 x \log (x)-x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+\int \frac {-1+x-2 x^2+x^2 \log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right ) \log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx\\ &=\int \left (\frac {8}{\log (x) \left (-8+e^x \log (x)-4 x \log (x)-x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}-\frac {8 x}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}+\frac {4 x \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}-\frac {2 x^2 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}-\frac {x^3 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}\right ) \, dx+\int \frac {-1+x-2 x^2+\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx\\ &=-\left (2 \int \frac {x^2 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx\right )+4 \int \frac {x \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+8 \int \frac {1}{\log (x) \left (-8+e^x \log (x)-4 x \log (x)-x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx-8 \int \frac {x}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+\int \left (\frac {-1+x-2 x^2}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}+\frac {1}{\log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}\right ) \, dx-\int \frac {x^3 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx\\ &=-\left (2 \int \frac {x^2 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx\right )+4 \int \frac {x \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+8 \int \frac {1}{\log (x) \left (-8+e^x \log (x)-4 x \log (x)-x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx-8 \int \frac {x}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+\int \frac {-1+x-2 x^2}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx-\int \frac {x^3 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+\int \frac {1}{\log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx\\ &=-\left (2 \int \frac {x^2 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx\right )+4 \int \frac {x \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+8 \int \frac {1}{\log (x) \left (-8+e^x \log (x)-4 x \log (x)-x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx-8 \int \frac {x}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+\int \left (-\frac {1}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}+\frac {x}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}-\frac {2 x^2}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )}\right ) \, dx-\int \frac {x^3 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+\int \frac {1}{\log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx\\ &=-\left (2 \int \frac {x^2}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx\right )-2 \int \frac {x^2 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+4 \int \frac {x \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+8 \int \frac {1}{\log (x) \left (-8+e^x \log (x)-4 x \log (x)-x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx-8 \int \frac {x}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx-\int \frac {1}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+\int \frac {x}{\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx-\int \frac {x^3 \log (x)}{\left (8-e^x \log (x)+4 x \log (x)+x^2 \log (x)\right ) \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right ) \log ^2\left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx+\int \frac {1}{\log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.31, size = 32, normalized size = 0.94 \begin {gather*} \frac {x}{\log \left (x^2-\log \left (4-\frac {e^x}{x}+x+\frac {8}{x \log (x)}\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 36, normalized size = 1.06 \begin {gather*} \frac {x}{\log \left (x^{2} - \log \left (\frac {{\left (x^{2} + 4 \, x - e^{x}\right )} \log \relax (x) + 8}{x \log \relax (x)}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.85, size = 36, normalized size = 1.06 \begin {gather*} \frac {x}{\log \left (x^{2} - \log \left (x^{2} \log \relax (x) + 4 \, x \log \relax (x) - e^{x} \log \relax (x) + 8\right ) + \log \relax (x) + \log \left (\log \relax (x)\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 289, normalized size = 8.50
method | result | size |
risch | \(\frac {x}{\ln \left (\ln \relax (x )+\ln \left (\ln \relax (x )\right )-\ln \left (x^{2} \ln \relax (x )-{\mathrm e}^{x} \ln \relax (x )+4 x \ln \relax (x )+8\right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (-x^{2} \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )-4 x \ln \relax (x )-8\right )}{\ln \relax (x )}\right ) \left (\mathrm {csgn}\left (\frac {i \left (-x^{2} \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )-4 x \ln \relax (x )-8\right )}{\ln \relax (x )}\right )+\mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )\right ) \left (\mathrm {csgn}\left (\frac {i \left (-x^{2} \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )-4 x \ln \relax (x )-8\right )}{\ln \relax (x )}\right )-\mathrm {csgn}\left (i \left (-x^{2} \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )-4 x \ln \relax (x )-8\right )\right )\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (-x^{2} \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )-4 x \ln \relax (x )-8\right )}{\ln \relax (x ) x}\right ) \left (\mathrm {csgn}\left (\frac {i \left (-x^{2} \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )-4 x \ln \relax (x )-8\right )}{\ln \relax (x ) x}\right )+\mathrm {csgn}\left (\frac {i}{x}\right )\right ) \left (\mathrm {csgn}\left (\frac {i \left (-x^{2} \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )-4 x \ln \relax (x )-8\right )}{\ln \relax (x ) x}\right )-\mathrm {csgn}\left (\frac {i \left (-x^{2} \ln \relax (x )+{\mathrm e}^{x} \ln \relax (x )-4 x \ln \relax (x )-8\right )}{\ln \relax (x )}\right )\right )}{2}+x^{2}\right )}\) | \(289\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 33, normalized size = 0.97 \begin {gather*} \frac {x}{\log \left (x^{2} - \log \left ({\left (x^{2} + 4 \, x - e^{x}\right )} \log \relax (x) + 8\right ) + \log \relax (x) + \log \left (\log \relax (x)\right )\right )} \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 {\ln \left (x^2-\ln \left (\frac {\ln \relax (x)\,\left (4\,x-{\mathrm {e}}^x+x^2\right )+8}{x\,\ln \relax (x)}\right )\right )\,\left (8\,x^2\,\ln \relax (x)+{\ln \relax (x)}^2\,\left (4\,x^3-x^2\,{\mathrm {e}}^x+x^4\right )-\ln \left (\frac {\ln \relax (x)\,\left (4\,x-{\mathrm {e}}^x+x^2\right )+8}{x\,\ln \relax (x)}\right )\,\left (\left (4\,x-{\mathrm {e}}^x+x^2\right )\,{\ln \relax (x)}^2+8\,\ln \relax (x)\right )\right )-\ln \relax (x)\,\left (16\,x^2+8\right )+{\ln \relax (x)}^2\,\left ({\mathrm {e}}^x\,\left (2\,x^2-x+1\right )+x^2-8\,x^3-2\,x^4\right )-8}{{\ln \left (x^2-\ln \left (\frac {\ln \relax (x)\,\left (4\,x-{\mathrm {e}}^x+x^2\right )+8}{x\,\ln \relax (x)}\right )\right )}^2\,\left (8\,x^2\,\ln \relax (x)+{\ln \relax (x)}^2\,\left (4\,x^3-x^2\,{\mathrm {e}}^x+x^4\right )-\ln \left (\frac {\ln \relax (x)\,\left (4\,x-{\mathrm {e}}^x+x^2\right )+8}{x\,\ln \relax (x)}\right )\,\left (\left (4\,x-{\mathrm {e}}^x+x^2\right )\,{\ln \relax (x)}^2+8\,\ln \relax (x)\right )\right )} \,d x \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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