Optimal. Leaf size=33 \[ \frac {-9-\frac {1}{5 x}+\frac {5}{\log \left (e^{e^{4 x^2} x^2} x\right )}}{x} \]
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Rubi [F]
time = 1.32, 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 {-25 x+e^{4 x^2} \left (-50 x^3-200 x^5\right )-25 x \log \left (e^{e^{4 x^2} x^2} x\right )+(2+45 x) \log ^2\left (e^{e^{4 x^2} x^2} x\right )}{5 x^3 \log ^2\left (e^{e^{4 x^2} x^2} x\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {-25 x+e^{4 x^2} \left (-50 x^3-200 x^5\right )-25 x \log \left (e^{e^{4 x^2} x^2} x\right )+(2+45 x) \log ^2\left (e^{e^{4 x^2} x^2} x\right )}{x^3 \log ^2\left (e^{e^{4 x^2} x^2} x\right )} \, dx\\ &=\frac {1}{5} \int \left (-\frac {50 e^{4 x^2} \left (1+4 x^2\right )}{\log ^2\left (e^{e^{4 x^2} x^2} x\right )}+\frac {-25 x-25 x \log \left (e^{e^{4 x^2} x^2} x\right )+2 \log ^2\left (e^{e^{4 x^2} x^2} x\right )+45 x \log ^2\left (e^{e^{4 x^2} x^2} x\right )}{x^3 \log ^2\left (e^{e^{4 x^2} x^2} x\right )}\right ) \, dx\\ &=\frac {1}{5} \int \frac {-25 x-25 x \log \left (e^{e^{4 x^2} x^2} x\right )+2 \log ^2\left (e^{e^{4 x^2} x^2} x\right )+45 x \log ^2\left (e^{e^{4 x^2} x^2} x\right )}{x^3 \log ^2\left (e^{e^{4 x^2} x^2} x\right )} \, dx-10 \int \frac {e^{4 x^2} \left (1+4 x^2\right )}{\log ^2\left (e^{e^{4 x^2} x^2} x\right )} \, dx\\ &=\frac {1}{5} \int \frac {2+45 x-\frac {25 x}{\log ^2\left (e^{e^{4 x^2} x^2} x\right )}-\frac {25 x}{\log \left (e^{e^{4 x^2} x^2} x\right )}}{x^3} \, dx-10 \int \left (\frac {e^{4 x^2}}{\log ^2\left (e^{e^{4 x^2} x^2} x\right )}+\frac {4 e^{4 x^2} x^2}{\log ^2\left (e^{e^{4 x^2} x^2} x\right )}\right ) \, dx\\ &=\frac {1}{5} \int \left (\frac {2+45 x}{x^3}-\frac {25}{x^2 \log ^2\left (e^{e^{4 x^2} x^2} x\right )}-\frac {25}{x^2 \log \left (e^{e^{4 x^2} x^2} x\right )}\right ) \, dx-10 \int \frac {e^{4 x^2}}{\log ^2\left (e^{e^{4 x^2} x^2} x\right )} \, dx-40 \int \frac {e^{4 x^2} x^2}{\log ^2\left (e^{e^{4 x^2} x^2} x\right )} \, dx\\ &=\frac {1}{5} \int \frac {2+45 x}{x^3} \, dx-5 \int \frac {1}{x^2 \log ^2\left (e^{e^{4 x^2} x^2} x\right )} \, dx-5 \int \frac {1}{x^2 \log \left (e^{e^{4 x^2} x^2} x\right )} \, dx-10 \int \frac {e^{4 x^2}}{\log ^2\left (e^{e^{4 x^2} x^2} x\right )} \, dx-40 \int \frac {e^{4 x^2} x^2}{\log ^2\left (e^{e^{4 x^2} x^2} x\right )} \, dx\\ &=-\frac {(2+45 x)^2}{20 x^2}-5 \int \frac {1}{x^2 \log ^2\left (e^{e^{4 x^2} x^2} x\right )} \, dx-5 \int \frac {1}{x^2 \log \left (e^{e^{4 x^2} x^2} x\right )} \, dx-10 \int \frac {e^{4 x^2}}{\log ^2\left (e^{e^{4 x^2} x^2} x\right )} \, dx-40 \int \frac {e^{4 x^2} x^2}{\log ^2\left (e^{e^{4 x^2} x^2} x\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.22, size = 38, normalized size = 1.15 \begin {gather*} \frac {1}{5} \left (-\frac {1}{x^2}-\frac {45}{x}+\frac {25}{x \log \left (e^{e^{4 x^2} x^2} x\right )}\right ) \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 = 0.08, size = 161, normalized size = 4.88
method | result | size |
risch | \(-\frac {45 x +1}{5 x^{2}}+\frac {10 i}{x \left (\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{x^{2} {\mathrm e}^{4 x^{2}}}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x^{2} {\mathrm e}^{4 x^{2}}}\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x^{2} {\mathrm e}^{4 x^{2}}}\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{x^{2} {\mathrm e}^{4 x^{2}}}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{x^{2} {\mathrm e}^{4 x^{2}}}\right )^{2}+\pi \mathrm {csgn}\left (i x \,{\mathrm e}^{x^{2} {\mathrm e}^{4 x^{2}}}\right )^{3}+2 i \ln \left (x \right )+2 i \ln \left ({\mathrm e}^{x^{2} {\mathrm e}^{4 x^{2}}}\right )\right )}\) | \(161\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 30, normalized size = 0.91 \begin {gather*} \frac {5}{x^{3} e^{\left (4 \, x^{2}\right )} + x \log \left (x\right )} - \frac {9}{x} - \frac {1}{5 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 45, normalized size = 1.36 \begin {gather*} -\frac {{\left (45 \, x + 1\right )} \log \left (x e^{\left (x^{2} e^{\left (4 \, x^{2}\right )}\right )}\right ) - 25 \, x}{5 \, x^{2} \log \left (x e^{\left (x^{2} e^{\left (4 \, x^{2}\right )}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 29, normalized size = 0.88 \begin {gather*} \frac {5}{x \log {\left (x e^{x^{2} e^{4 x^{2}}} \right )}} + \frac {- 45 x - 1}{5 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 53, normalized size = 1.61 \begin {gather*} -\frac {45 \, x^{3} e^{\left (4 \, x^{2}\right )} + x^{2} e^{\left (4 \, x^{2}\right )} + 45 \, x \log \left (x\right ) - 25 \, x + \log \left (x\right )}{5 \, {\left (x^{4} e^{\left (4 \, x^{2}\right )} + x^{2} \log \left (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 {5\,x+5\,x\,\ln \left (x\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{4\,x^2}}\right )-\frac {{\ln \left (x\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{4\,x^2}}\right )}^2\,\left (45\,x+2\right )}{5}+\frac {{\mathrm {e}}^{4\,x^2}\,\left (200\,x^5+50\,x^3\right )}{5}}{x^3\,{\ln \left (x\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{4\,x^2}}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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