Optimal. Leaf size=23 \[ \left (\frac {x}{\log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )}\right )^{\frac {1}{-1+e^2}} \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 2.74, 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 {\left (\frac {x}{\log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )}\right )^{\frac {1}{-1+e^2}} \left (-1+\log \left (\frac {x}{2}\right ) \log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )\right )}{\left (-x+e^2 x\right ) \log \left (\frac {x}{2}\right ) \log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (\frac {x}{\log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )}\right )^{\frac {1}{-1+e^2}} \left (-1+\log \left (\frac {x}{2}\right ) \log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )\right )}{\left (-1+e^2\right ) x \log \left (\frac {x}{2}\right ) \log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )} \, dx\\ &=\frac {\int \frac {\left (\frac {x}{\log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )}\right )^{\frac {1}{-1+e^2}} \left (-1+\log \left (\frac {x}{2}\right ) \log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )\right )}{x \log \left (\frac {x}{2}\right ) \log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )} \, dx}{-1+e^2}\\ &=\frac {\left (x^{-\frac {1}{-1+e^2}} \left (\frac {x}{\log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )}\right )^{\frac {1}{-1+e^2}} \sqrt [-1+e^2]{\log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )}\right ) \int \frac {x^{-1+\frac {1}{-1+e^2}} \log ^{-1-\frac {1}{-1+e^2}}\left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right ) \left (-1+\log \left (\frac {x}{2}\right ) \log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )\right )}{\log \left (\frac {x}{2}\right )} \, dx}{-1+e^2}\\ &=\frac {\left (x^{-\frac {1}{-1+e^2}} \left (\frac {x}{\log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )}\right )^{\frac {1}{-1+e^2}} \sqrt [-1+e^2]{\log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )}\right ) \int \left (x^{-1+\frac {1}{-1+e^2}} \sqrt [1-e^2]{\log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )}+\frac {x^{-1+\frac {1}{-1+e^2}} \log ^{\frac {e^2}{1-e^2}}\left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )}{\log (2)-\log (x)}\right ) \, dx}{-1+e^2}\\ &=\frac {\left (x^{-\frac {1}{-1+e^2}} \left (\frac {x}{\log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )}\right )^{\frac {1}{-1+e^2}} \sqrt [-1+e^2]{\log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )}\right ) \int x^{-1+\frac {1}{-1+e^2}} \sqrt [1-e^2]{\log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )} \, dx}{-1+e^2}+\frac {\left (x^{-\frac {1}{-1+e^2}} \left (\frac {x}{\log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )}\right )^{\frac {1}{-1+e^2}} \sqrt [-1+e^2]{\log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )}\right ) \int \frac {x^{-1+\frac {1}{-1+e^2}} \log ^{\frac {e^2}{1-e^2}}\left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )}{\log (2)-\log (x)} \, dx}{-1+e^2}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.16, size = 23, normalized size = 1.00 \begin {gather*} \left (\frac {x}{\log \left (\frac {3}{4} \log \left (\frac {x}{2}\right )\right )}\right )^{\frac {1}{-1+e^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.97, size = 0, normalized size = 0.00 \[\int \frac {\left (\ln \left (\frac {x}{2}\right ) \ln \left (\frac {3 \ln \left (\frac {x}{2}\right )}{4}\right )-1\right ) {\mathrm e}^{\frac {\ln \left (\frac {x}{\ln \left (\frac {3 \ln \left (\frac {x}{2}\right )}{4}\right )}\right )}{{\mathrm e}^{2}-1}}}{\left ({\mathrm e}^{2} x -x \right ) \ln \left (\frac {x}{2}\right ) \ln \left (\frac {3 \ln \left (\frac {x}{2}\right )}{4}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.41, size = 18, normalized size = 0.78 \begin {gather*} \left (\frac {x}{\log \left (\frac {3}{4} \, \log \left (\frac {1}{2} \, x\right )\right )}\right )^{\left (\frac {1}{e^{2} - 1}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 143.21, size = 17, normalized size = 0.74 \begin {gather*} \left (\frac {x}{\log {\left (\frac {3 \log {\left (\frac {x}{2} \right )}}{4} \right )}}\right )^{\frac {1}{-1 + e^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 18, normalized size = 0.78 \begin {gather*} \left (\frac {x}{\log \left (\frac {3}{4} \, \log \left (\frac {1}{2} \, x\right )\right )}\right )^{\left (\frac {1}{e^{2} - 1}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.95, size = 18, normalized size = 0.78 \begin {gather*} {\left (\frac {x}{\ln \left (\frac {3\,\ln \left (\frac {x}{2}\right )}{4}\right )}\right )}^{\frac {1}{{\mathrm {e}}^2-1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________