Optimal. Leaf size=28 \[ \log ^2\left (\frac {x}{-x+\frac {1}{4} \left (5+e^x+e^{x^2}+x\right )+\log (2)}\right ) \]
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Rubi [A]
time = 2.87, antiderivative size = 23, normalized size of antiderivative = 0.82, number of steps
used = 2, number of rules used = 6, integrand size = 80, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6, 6873, 12,
6874, 14, 6818} \begin {gather*} \log ^2\left (\frac {4 x}{e^{x^2}-3 x+e^x+5+\log (16)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 14
Rule 6818
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (10+e^x (2-2 x)+e^{x^2} \left (2-4 x^2\right )+8 \log (2)\right ) \log \left (\frac {4 x}{5+e^x+e^{x^2}-3 x+4 \log (2)}\right )}{e^x x+e^{x^2} x-3 x^2+x (5+4 \log (2))} \, dx\\ &=\log ^2\left (\frac {4 x}{5+e^x+e^{x^2}-3 x+\log (16)}\right )\\ \end {aligned} \end {gather*}
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Mathematica [F]
time = 0.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (10+e^x (2-2 x)+e^{x^2} \left (2-4 x^2\right )+8 \log (2)\right ) \log \left (\frac {4 x}{5+e^x+e^{x^2}-3 x+4 \log (2)}\right )}{5 x+e^x x+e^{x^2} x-3 x^2+4 x \log (2)} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.20, size = 468, normalized size = 16.71
method | result | size |
risch | \(\ln \left (\ln \left (2\right )-\frac {3 x}{4}+\frac {5}{4}+\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{x^{2}}}{4}\right )^{2}-2 \ln \left (x \right ) \ln \left (\ln \left (2\right )-\frac {3 x}{4}+\frac {5}{4}+\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{x^{2}}}{4}\right )-i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i}{\ln \left (2\right )-\frac {3 x}{4}+\frac {5}{4}+\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{x^{2}}}{4}}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (2\right )-\frac {3 x}{4}+\frac {5}{4}+\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{x^{2}}}{4}}\right )+i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (2\right )-\frac {3 x}{4}+\frac {5}{4}+\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{x^{2}}}{4}}\right )^{2}+i \pi \ln \left (x \right ) \mathrm {csgn}\left (\frac {i}{\ln \left (2\right )-\frac {3 x}{4}+\frac {5}{4}+\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{x^{2}}}{4}}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (2\right )-\frac {3 x}{4}+\frac {5}{4}+\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{x^{2}}}{4}}\right )^{2}-i \pi \ln \left (x \right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (2\right )-\frac {3 x}{4}+\frac {5}{4}+\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{x^{2}}}{4}}\right )^{3}+i \pi \ln \left ({\mathrm e}^{x^{2}}+{\mathrm e}^{x}+4 \ln \left (2\right )-3 x +5\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i}{\ln \left (2\right )-\frac {3 x}{4}+\frac {5}{4}+\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{x^{2}}}{4}}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (2\right )-\frac {3 x}{4}+\frac {5}{4}+\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{x^{2}}}{4}}\right )-i \pi \ln \left ({\mathrm e}^{x^{2}}+{\mathrm e}^{x}+4 \ln \left (2\right )-3 x +5\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (2\right )-\frac {3 x}{4}+\frac {5}{4}+\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{x^{2}}}{4}}\right )^{2}-i \pi \ln \left ({\mathrm e}^{x^{2}}+{\mathrm e}^{x}+4 \ln \left (2\right )-3 x +5\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (2\right )-\frac {3 x}{4}+\frac {5}{4}+\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{x^{2}}}{4}}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (2\right )-\frac {3 x}{4}+\frac {5}{4}+\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{x^{2}}}{4}}\right )^{2}+i \pi \ln \left ({\mathrm e}^{x^{2}}+{\mathrm e}^{x}+4 \ln \left (2\right )-3 x +5\right ) \mathrm {csgn}\left (\frac {i x}{\ln \left (2\right )-\frac {3 x}{4}+\frac {5}{4}+\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{x^{2}}}{4}}\right )^{3}+\ln \left (x \right )^{2}\) | \(468\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs.
\(2 (27) = 54\).
time = 0.51, size = 95, normalized size = 3.39 \begin {gather*} -\log \left (x\right )^{2} + 2 \, \log \left (x\right ) \log \left (-3 \, x + e^{\left (x^{2}\right )} + e^{x} + 4 \, \log \left (2\right ) + 5\right ) - \log \left (-3 \, x + e^{\left (x^{2}\right )} + e^{x} + 4 \, \log \left (2\right ) + 5\right )^{2} + 2 \, {\left (\log \left (x\right ) - \log \left (-3 \, x + e^{\left (x^{2}\right )} + e^{x} + 4 \, \log \left (2\right ) + 5\right )\right )} \log \left (-\frac {4 \, x}{3 \, x - e^{\left (x^{2}\right )} - e^{x} - 4 \, \log \left (2\right ) - 5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 27, normalized size = 0.96 \begin {gather*} \log \left (-\frac {4 \, x}{3 \, x - e^{\left (x^{2}\right )} - e^{x} - 4 \, \log \left (2\right ) - 5}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.98, size = 24, normalized size = 0.86 \begin {gather*} \log {\left (\frac {4 x}{- 3 x + e^{x} + e^{x^{2}} + 4 \log {\left (2 \right )} + 5} \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.04 \begin {gather*} \int \frac {\ln \left (\frac {4\,x}{{\mathrm {e}}^{x^2}-3\,x+4\,\ln \left (2\right )+{\mathrm {e}}^x+5}\right )\,\left (8\,\ln \left (2\right )-{\mathrm {e}}^{x^2}\,\left (4\,x^2-2\right )-{\mathrm {e}}^x\,\left (2\,x-2\right )+10\right )}{5\,x+x\,{\mathrm {e}}^{x^2}+4\,x\,\ln \left (2\right )+x\,{\mathrm {e}}^x-3\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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