Optimal. Leaf size=25 \[ 3+\frac {15}{\log \left (\left (e^x-\frac {3}{4} x \left (x+\log \left (\log \left (x^2\right )\right )\right )\right )^2\right )} \]
________________________________________________________________________________________
Rubi [A] time = 0.57, antiderivative size = 30, normalized size of antiderivative = 1.20, number of steps used = 3, number of rules used = 3, integrand size = 117, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6688, 12, 6686} \begin {gather*} \frac {15}{\log \left (\frac {1}{16} \left (-3 x^2-3 x \log \left (\log \left (x^2\right )\right )+4 e^x\right )^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 6686
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {30 \left (6-\log \left (x^2\right ) \left (4 e^x-6 x-3 \log \left (\log \left (x^2\right )\right )\right )\right )}{\log \left (x^2\right ) \left (4 e^x-3 x^2-3 x \log \left (\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {1}{16} \left (-4 e^x+3 x^2+3 x \log \left (\log \left (x^2\right )\right )\right )^2\right )} \, dx\\ &=30 \int \frac {6-\log \left (x^2\right ) \left (4 e^x-6 x-3 \log \left (\log \left (x^2\right )\right )\right )}{\log \left (x^2\right ) \left (4 e^x-3 x^2-3 x \log \left (\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {1}{16} \left (-4 e^x+3 x^2+3 x \log \left (\log \left (x^2\right )\right )\right )^2\right )} \, dx\\ &=\frac {15}{\log \left (\frac {1}{16} \left (4 e^x-3 x^2-3 x \log \left (\log \left (x^2\right )\right )\right )^2\right )}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 30, normalized size = 1.20 \begin {gather*} \frac {15}{\log \left (\frac {1}{16} \left (-4 e^x+3 x^2+3 x \log \left (\log \left (x^2\right )\right )\right )^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.54, size = 52, normalized size = 2.08 \begin {gather*} \frac {15}{\log \left (\frac {9}{16} \, x^{4} + \frac {9}{16} \, x^{2} \log \left (\log \left (x^{2}\right )\right )^{2} - \frac {3}{2} \, x^{2} e^{x} + \frac {3}{8} \, {\left (3 \, x^{3} - 4 \, x e^{x}\right )} \log \left (\log \left (x^{2}\right )\right ) + e^{\left (2 \, x\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.70, size = 63, normalized size = 2.52 \begin {gather*} -\frac {15}{4 \, \log \relax (2) - \log \left (9 \, x^{4} + 18 \, x^{3} \log \left (\log \left (x^{2}\right )\right ) + 9 \, x^{2} \log \left (\log \left (x^{2}\right )\right )^{2} - 24 \, x^{2} e^{x} - 24 \, x e^{x} \log \left (\log \left (x^{2}\right )\right ) + 16 \, e^{\left (2 \, x\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.90, size = 334, normalized size = 13.36
method | result | size |
risch | \(\frac {30 i}{\pi \mathrm {csgn}\left (i \left (-x^{2}-x \ln \left (2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )+\frac {4 \,{\mathrm e}^{x}}{3}\right )\right )^{2} \mathrm {csgn}\left (i \left (-x^{2}-x \ln \left (2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )+\frac {4 \,{\mathrm e}^{x}}{3}\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (-x^{2}-x \ln \left (2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )+\frac {4 \,{\mathrm e}^{x}}{3}\right )\right ) \mathrm {csgn}\left (i \left (-x^{2}-x \ln \left (2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )+\frac {4 \,{\mathrm e}^{x}}{3}\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (-x^{2}-x \ln \left (2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )+\frac {4 \,{\mathrm e}^{x}}{3}\right )^{2}\right )^{3}-8 i \ln \relax (2)+4 i \ln \left (x^{2}+x \ln \left (2 \ln \relax (x )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )-\frac {4 \,{\mathrm e}^{x}}{3}\right )}\) | \(334\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.56, size = 33, normalized size = 1.32 \begin {gather*} -\frac {15}{2 \, {\left (2 \, \log \relax (2) - \log \left (-3 \, x^{2} - 3 \, x \log \relax (2) - 3 \, x \log \left (\log \relax (x)\right ) + 4 \, e^{x}\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.90, size = 52, normalized size = 2.08 \begin {gather*} \frac {15}{\ln \left ({\mathrm {e}}^{2\,x}-\frac {3\,x^2\,{\mathrm {e}}^x}{2}+\frac {9\,x^2\,{\ln \left (\ln \left (x^2\right )\right )}^2}{16}-\frac {\ln \left (\ln \left (x^2\right )\right )\,\left (24\,x\,{\mathrm {e}}^x-18\,x^3\right )}{16}+\frac {9\,x^4}{16}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 12.60, size = 61, normalized size = 2.44 \begin {gather*} \frac {15}{\log {\left (\frac {9 x^{4}}{16} - \frac {3 x^{2} e^{x}}{2} + \frac {9 x^{2} \log {\left (\log {\left (x^{2} \right )} \right )}^{2}}{16} + \left (\frac {9 x^{3}}{8} - \frac {3 x e^{x}}{2}\right ) \log {\left (\log {\left (x^{2} \right )} \right )} + e^{2 x} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________