Optimal. Leaf size=28 \[ \frac {e^{\frac {3}{2 \log (x)}}}{i \pi +x^2-\log \left (\frac {5}{2}\right )} \]
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Rubi [A] time = 0.46, antiderivative size = 48, normalized size of antiderivative = 1.71, number of steps used = 3, number of rules used = 3, integrand size = 86, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {1594, 28, 2288} \begin {gather*} \frac {e^{\frac {3}{2 \log (x)}} \left (3 x^2+3 i \pi -\log \left (\frac {125}{8}\right )\right )}{3 \left (x^2+i \pi -\log \left (\frac {5}{2}\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 1594
Rule 2288
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {3}{2 \log (x)}} \left (-3 x^2-3 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-4 x^2 \log ^2(x)\right )}{x \left (2 x^4+4 x^2 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+2 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2\right ) \log ^2(x)} \, dx\\ &=2 \int \frac {e^{\frac {3}{2 \log (x)}} \left (-3 x^2-3 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-4 x^2 \log ^2(x)\right )}{x \left (2 x^2+2 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right )^2 \log ^2(x)} \, dx\\ &=\frac {e^{\frac {3}{2 \log (x)}} \left (3 i \pi +3 x^2-\log \left (\frac {125}{8}\right )\right )}{3 \left (i \pi +x^2-\log \left (\frac {5}{2}\right )\right )^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.15, size = 28, normalized size = 1.00 \begin {gather*} \frac {e^{\frac {3}{2 \log (x)}}}{i \pi +x^2-\log \left (\frac {5}{2}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 19, normalized size = 0.68 \begin {gather*} \frac {e^{\left (\frac {3}{2 \, \log \relax (x)}\right )}}{i \, \pi + x^{2} + \log \left (\frac {2}{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 26, normalized size = 0.93 \begin {gather*} -\frac {i \, e^{\left (\frac {3}{2 \, \log \relax (x)}\right )}}{\pi - i \, x^{2} + i \, \log \relax (5) - i \, \log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 28, normalized size = 1.00
method | result | size |
risch | \(-\frac {{\mathrm e}^{\frac {3}{2 \ln \relax (x )}}}{-i \pi -x^{2}+\ln \relax (5)-\ln \relax (2)}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {x^{2} e^{\left (\frac {3}{2 \, \log \relax (x)}\right )}}{x^{4} - 2 \, {\left (-i \, \pi + \log \relax (5) - \log \relax (2)\right )} x^{2} - \pi ^{2} - 2 \, \pi {\left (i \, \log \relax (5) - i \, \log \relax (2)\right )} + \log \relax (5)^{2} - 2 \, \log \relax (5) \log \relax (2) + \log \relax (2)^{2}} + \frac {i \, \pi e^{\left (\frac {3}{2 \, \log \relax (x)}\right )}}{x^{4} - 2 \, {\left (-i \, \pi + \log \relax (5) - \log \relax (2)\right )} x^{2} - \pi ^{2} - 2 \, \pi {\left (i \, \log \relax (5) - i \, \log \relax (2)\right )} + \log \relax (5)^{2} - 2 \, \log \relax (5) \log \relax (2) + \log \relax (2)^{2}} + \frac {e^{\left (\frac {3}{2 \, \log \relax (x)}\right )} \log \left (\frac {2}{5}\right )}{x^{4} - 2 \, {\left (-i \, \pi + \log \relax (5) - \log \relax (2)\right )} x^{2} - \pi ^{2} - 2 \, \pi {\left (i \, \log \relax (5) - i \, \log \relax (2)\right )} + \log \relax (5)^{2} - 2 \, \log \relax (5) \log \relax (2) + \log \relax (2)^{2}} - 2 \, \int \frac {x e^{\left (\frac {3}{2 \, \log \relax (x)}\right )}}{x^{4} - 2 \, {\left (-i \, \pi + \log \relax (5) - \log \relax (2)\right )} x^{2} - \pi ^{2} - 2 \, \pi {\left (i \, \log \relax (5) - i \, \log \relax (2)\right )} + \log \relax (5)^{2} - 2 \, \log \relax (5) \log \relax (2) + \log \relax (2)^{2}}\,{d x} \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 {{\mathrm {e}}^{\frac {3}{2\,\ln \relax (x)}}\,\left (4\,x^2\,{\ln \relax (x)}^2+3\,x^2+\Pi \,3{}\mathrm {i}+3\,\ln \left (\frac {2}{5}\right )\right )}{{\ln \relax (x)}^2\,\left (2\,x\,{\left (\ln \left (\frac {2}{5}\right )+\Pi \,1{}\mathrm {i}\right )}^2+4\,x^3\,\left (\ln \left (\frac {2}{5}\right )+\Pi \,1{}\mathrm {i}\right )+2\,x^5\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.83, size = 22, normalized size = 0.79 \begin {gather*} - \frac {e^{\frac {3}{2 \log {\relax (x )}}}}{- x^{2} - \log {\relax (2 )} + \log {\relax (5 )} - i \pi } \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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