Optimal. Leaf size=27 \[ \frac {1}{4} \left (5+\log \left (16+\frac {e^x-x^4}{5 x}\right )\right )^2 \]
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Rubi [A]
time = 0.35, antiderivative size = 28, normalized size of antiderivative = 1.04, number of steps
used = 3, number of rules used = 3, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6873, 12, 6818}
\begin {gather*} \frac {1}{4} \left (\log \left (\frac {-x^4+80 x+e^x}{5 x}\right )+5\right )^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6818
Rule 6873
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (e^x-e^x x+3 x^4\right ) \left (-5-\log \left (\frac {e^x+80 x-x^4}{5 x}\right )\right )}{2 x \left (e^x+80 x-x^4\right )} \, dx\\ &=\frac {1}{2} \int \frac {\left (e^x-e^x x+3 x^4\right ) \left (-5-\log \left (\frac {e^x+80 x-x^4}{5 x}\right )\right )}{x \left (e^x+80 x-x^4\right )} \, dx\\ &=\frac {1}{4} \left (5+\log \left (\frac {e^x+80 x-x^4}{5 x}\right )\right )^2\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 28, normalized size = 1.04 \begin {gather*} \frac {1}{4} \left (5+\log \left (\frac {e^x+80 x-x^4}{5 x}\right )\right )^2 \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 = 1.89, size = 496, normalized size = 18.37
method | result | size |
risch | \(\frac {i \ln \left (x \right ) \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x^{4}+80 x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}-x^{4}+80 x \right )}{x}\right )^{2}}{4}+\frac {i \ln \left (x \right ) \pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}-x^{4}+80 x \right )}{x}\right )^{2}}{2}+\frac {\ln \left (5\right ) \ln \left (x \right )}{2}-\frac {5 \ln \left (x \right )}{2}+\frac {\ln \left (x \right )^{2}}{4}+\frac {\ln \left (-{\mathrm e}^{x}+x^{4}-80 x \right )^{2}}{4}-\frac {i \ln \left (x \right ) \pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}-x^{4}+80 x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )}{4}-\frac {i \pi \ln \left ({\mathrm e}^{x}-x^{4}+80 x \right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x^{4}+80 x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}-x^{4}+80 x \right )}{x}\right )^{2}}{4}-\frac {i \pi \ln \left ({\mathrm e}^{x}-x^{4}+80 x \right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}-x^{4}+80 x \right )}{x}\right )^{2}}{2}-\frac {i \pi \ln \left ({\mathrm e}^{x}-x^{4}+80 x \right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x^{4}+80 x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}-x^{4}+80 x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )}{4}+\frac {i \pi \ln \left ({\mathrm e}^{x}-x^{4}+80 x \right )}{2}+\frac {i \ln \left (x \right ) \pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}-x^{4}+80 x \right )}{x}\right )^{3}}{4}-\frac {i \pi \ln \left (x \right )}{2}-\frac {i \pi \ln \left ({\mathrm e}^{x}-x^{4}+80 x \right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}-x^{4}+80 x \right )}{x}\right )^{3}}{4}+\frac {5 \ln \left ({\mathrm e}^{x}-x^{4}+80 x \right )}{2}+\frac {i \pi \ln \left ({\mathrm e}^{x}-x^{4}+80 x \right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}-x^{4}+80 x \right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )}{4}+\frac {i \ln \left (x \right ) \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-x^{4}+80 x \right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{x}-x^{4}+80 x \right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )}{4}-\frac {\ln \left (x \right ) \ln \left (-{\mathrm e}^{x}+x^{4}-80 x \right )}{2}-\frac {\ln \left (5\right ) \ln \left ({\mathrm e}^{x}-x^{4}+80 x \right )}{2}\) | \(496\) |
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. 51 vs.
\(2 (22) = 44\).
time = 0.50, size = 51, normalized size = 1.89 \begin {gather*} -\frac {1}{2} \, {\left (\log \left (5\right ) + \log \left (x\right ) - 5\right )} \log \left (-x^{4} + 80 \, x + e^{x}\right ) + \frac {1}{4} \, \log \left (-x^{4} + 80 \, x + e^{x}\right )^{2} + \frac {1}{2} \, {\left (\log \left (5\right ) - 5\right )} \log \left (x\right ) + \frac {1}{4} \, \log \left (x\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 41, normalized size = 1.52 \begin {gather*} \frac {1}{4} \, \log \left (-\frac {x^{4} - 80 \, x - e^{x}}{5 \, x}\right )^{2} + \frac {5}{2} \, \log \left (-\frac {x^{4} - 80 \, x - e^{x}}{5 \, x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs.
\(2 (20) = 40\).
time = 0.20, size = 41, normalized size = 1.52 \begin {gather*} - \frac {5 \log {\left (x \right )}}{2} + \frac {\log {\left (\frac {- \frac {x^{4}}{5} + 16 x + \frac {e^{x}}{5}}{x} \right )}^{2}}{4} + \frac {5 \log {\left (- x^{4} + 80 x + e^{x} \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 [B]
time = 5.39, size = 40, normalized size = 1.48 \begin {gather*} \frac {\ln \left (\frac {16\,x+\frac {{\mathrm {e}}^x}{5}-\frac {x^4}{5}}{x}\right )\,\left (\ln \left (\frac {16\,x+\frac {{\mathrm {e}}^x}{5}-\frac {x^4}{5}}{x}\right )+10\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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