Optimal. Leaf size=25 \[ -5+e^{\left (-\frac {1}{4}-x\right ) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \]
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Rubi [F]
time = 9.31, 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 {e^{\frac {1}{4} (-1-4 x) \log ^2\left (\frac {x^2+\log ^2(x)}{x^2}\right )} \left (\left ((-1-4 x) \log (x)+(1+4 x) \log ^2(x)\right ) \log \left (\frac {x^2+\log ^2(x)}{x^2}\right )+\left (-x^3-x \log ^2(x)\right ) \log ^2\left (\frac {x^2+\log ^2(x)}{x^2}\right )\right )}{x^3+x \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (\frac {x^2+\log ^2(x)}{x^2}\right )} \left (\left ((-1-4 x) \log (x)+(1+4 x) \log ^2(x)\right ) \log \left (\frac {x^2+\log ^2(x)}{x^2}\right )+\left (-x^3-x \log ^2(x)\right ) \log ^2\left (\frac {x^2+\log ^2(x)}{x^2}\right )\right )}{x \left (x^2+\log ^2(x)\right )} \, dx\\ &=\int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \left (\left ((-1-4 x) \log (x)+(1+4 x) \log ^2(x)\right ) \log \left (\frac {x^2+\log ^2(x)}{x^2}\right )+\left (-x^3-x \log ^2(x)\right ) \log ^2\left (\frac {x^2+\log ^2(x)}{x^2}\right )\right )}{x \left (x^2+\log ^2(x)\right )} \, dx\\ &=\int \left (\frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} (1+4 x) (-1+\log (x)) \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x \left (x^2+\log ^2(x)\right )}-e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )\right ) \, dx\\ &=\int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} (1+4 x) (-1+\log (x)) \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x \left (x^2+\log ^2(x)\right )} \, dx-\int e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right ) \, dx\\ &=-\int e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right ) \, dx+\int \left (\frac {4 e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} (-1+\log (x)) \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x^2+\log ^2(x)}+\frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} (-1+\log (x)) \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x \left (x^2+\log ^2(x)\right )}\right ) \, dx\\ &=4 \int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} (-1+\log (x)) \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x^2+\log ^2(x)} \, dx+\int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} (-1+\log (x)) \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x \left (x^2+\log ^2(x)\right )} \, dx-\int e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right ) \, dx\\ &=4 \int \left (-\frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x^2+\log ^2(x)}+\frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2(x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x^2+\log ^2(x)}\right ) \, dx-\int e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right ) \, dx+\int \left (-\frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x \left (x^2+\log ^2(x)\right )}+\frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2(x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x \left (x^2+\log ^2(x)\right )}\right ) \, dx\\ &=-\left (4 \int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x^2+\log ^2(x)} \, dx\right )+4 \int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2(x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x^2+\log ^2(x)} \, dx-\int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log (x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x \left (x^2+\log ^2(x)\right )} \, dx+\int \frac {e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2(x) \log \left (1+\frac {\log ^2(x)}{x^2}\right )}{x \left (x^2+\log ^2(x)\right )} \, dx-\int e^{\frac {1}{4} (-1-4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right )} \log ^2\left (1+\frac {\log ^2(x)}{x^2}\right ) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 24, normalized size = 0.96 \begin {gather*} e^{-\frac {1}{4} (1+4 x) \log ^2\left (1+\frac {\log ^2(x)}{x^2}\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 = 0.34, size = 196, normalized size = 7.84
method | result | size |
risch | \({\mathrm e}^{-\frac {\left (4 x +1\right ) \left (-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+i \pi \mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )^{2}+x^{2}\right )}{x^{2}}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )^{2}+x^{2}\right )}{x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right )-i \pi \mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )^{2}+x^{2}\right )}{x^{2}}\right )^{2} \mathrm {csgn}\left (i \left (\ln \left (x \right )^{2}+x^{2}\right )\right )+i \pi \,\mathrm {csgn}\left (\frac {i \left (\ln \left (x \right )^{2}+x^{2}\right )}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left (\ln \left (x \right )^{2}+x^{2}\right )\right )+4 \ln \left (x \right )-2 \ln \left (\ln \left (x \right )^{2}+x^{2}\right )\right )^{2}}{16}}\) | \(196\) |
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. 68 vs.
\(2 (23) = 46\).
time = 0.55, size = 68, normalized size = 2.72 \begin {gather*} e^{\left (-x \log \left (x^{2} + \log \left (x\right )^{2}\right )^{2} + 4 \, x \log \left (x^{2} + \log \left (x\right )^{2}\right ) \log \left (x\right ) - 4 \, x \log \left (x\right )^{2} - \frac {1}{4} \, \log \left (x^{2} + \log \left (x\right )^{2}\right )^{2} + \log \left (x^{2} + \log \left (x\right )^{2}\right ) \log \left (x\right ) - \log \left (x\right )^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 23, normalized size = 0.92 \begin {gather*} e^{\left (-\frac {1}{4} \, {\left (4 \, x + 1\right )} \log \left (\frac {x^{2} + \log \left (x\right )^{2}}{x^{2}}\right )^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.74, size = 22, normalized size = 0.88 \begin {gather*} e^{\left (- x - \frac {1}{4}\right ) \log {\left (\frac {x^{2} + \log {\left (x \right )}^{2}}{x^{2}} \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.40, size = 33, normalized size = 1.32 \begin {gather*} e^{\left (-x \log \left (\frac {\log \left (x\right )^{2}}{x^{2}} + 1\right )^{2} - \frac {1}{4} \, \log \left (\frac {\log \left (x\right )^{2}}{x^{2}} + 1\right )^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.98, size = 83, normalized size = 3.32 \begin {gather*} \frac {{\mathrm {e}}^{-\frac {{\ln \left (x^2+{\ln \left (x\right )}^2\right )}^2}{4}}\,{\mathrm {e}}^{-\frac {{\ln \left (\frac {1}{x^2}\right )}^2}{4}}\,{\mathrm {e}}^{-x\,{\ln \left (x^2+{\ln \left (x\right )}^2\right )}^2}\,{\mathrm {e}}^{-x\,{\ln \left (\frac {1}{x^2}\right )}^2}\,{\mathrm {e}}^{-\frac {\ln \left (\frac {1}{x^2}\right )\,\ln \left (x^2+{\ln \left (x\right )}^2\right )}{2}}}{{\left (x^2+{\ln \left (x\right )}^2\right )}^{2\,x\,\ln \left (\frac {1}{x^2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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