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 = 0.48, 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 = {1608, 28, 2326}
\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 1608
Rule 2326
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 = 0.11, 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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Maple [A]
time = 0.29, size = 31, normalized size = 1.11
method | result | size |
risch | \(-\frac {i {\mathrm e}^{\frac {3}{2 \ln \left (x \right )}}}{-i x^{2}-i \ln \left (2\right )+i \ln \left (5\right )+\pi }\) | \(31\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 19, normalized size = 0.68 \begin {gather*} \frac {e^{\left (\frac {3}{2 \, \log \left (x\right )}\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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Sympy [A]
time = 0.65, size = 20, normalized size = 0.71 \begin {gather*} \frac {e^{\frac {3}{2 \log {\left (x \right )}}}}{x^{2} - \log {\left (5 \right )} + \log {\left (2 \right )} + i \pi } \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 26, normalized size = 0.93 \begin {gather*} -\frac {i \, e^{\left (\frac {3}{2 \, \log \left (x\right )}\right )}}{\pi - i \, x^{2} + i \, \log \left (5\right ) - i \, \log \left (2\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 = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^{\frac {3}{2\,\ln \left (x\right )}}\,\left (4\,x^2\,{\ln \left (x\right )}^2+3\,x^2+\Pi \,3{}\mathrm {i}+3\,\ln \left (\frac {2}{5}\right )\right )}{{\ln \left (x\right )}^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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