Optimal. Leaf size=27 \[ x \left (-3+\frac {3}{\log \left (\frac {x}{\left (-4+e^x\right ) \left (-\frac {1}{x}+x\right )}\right )}\right ) \]
________________________________________________________________________________________
Rubi [F] time = 1.26, 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 {-24+e^x \left (6-3 x+3 x^3\right )+\left (12-12 x^2+e^x \left (-3+3 x^2\right )\right ) \log \left (\frac {x^2}{4-4 x^2+e^x \left (-1+x^2\right )}\right )+\left (-12+12 x^2+e^x \left (3-3 x^2\right )\right ) \log ^2\left (\frac {x^2}{4-4 x^2+e^x \left (-1+x^2\right )}\right )}{\left (4-4 x^2+e^x \left (-1+x^2\right )\right ) \log ^2\left (\frac {x^2}{4-4 x^2+e^x \left (-1+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 3 \left (-1+\frac {-8+e^x \left (2-x+x^3\right )}{\left (-4+e^x\right ) \left (-1+x^2\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}+\frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}\right ) \, dx\\ &=3 \int \left (-1+\frac {-8+e^x \left (2-x+x^3\right )}{\left (-4+e^x\right ) \left (-1+x^2\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}+\frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}\right ) \, dx\\ &=-3 x+3 \int \frac {-8+e^x \left (2-x+x^3\right )}{\left (-4+e^x\right ) \left (-1+x^2\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+3 \int \frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx\\ &=-3 x+3 \int \left (\frac {4 x}{\left (-4+e^x\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}+\frac {2-x+x^3}{\left (-1+x^2\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}\right ) \, dx+3 \int \frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx\\ &=-3 x+3 \int \frac {2-x+x^3}{\left (-1+x^2\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+3 \int \frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+12 \int \frac {x}{\left (-4+e^x\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx\\ &=-3 x+3 \int \left (\frac {x}{\log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}+\frac {2}{\left (-1+x^2\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}\right ) \, dx+3 \int \frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+12 \int \frac {x}{\left (-4+e^x\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx\\ &=-3 x+3 \int \frac {x}{\log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+3 \int \frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+6 \int \frac {1}{\left (-1+x^2\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+12 \int \frac {x}{\left (-4+e^x\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx\\ &=-3 x+3 \int \frac {x}{\log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+3 \int \frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+6 \int \left (\frac {1}{2 (-1+x) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}-\frac {1}{2 (1+x) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}\right ) \, dx+12 \int \frac {x}{\left (-4+e^x\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx\\ &=-3 x+3 \int \frac {1}{(-1+x) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+3 \int \frac {x}{\log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx-3 \int \frac {1}{(1+x) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+3 \int \frac {1}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx+12 \int \frac {x}{\left (-4+e^x\right ) \log ^2\left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.44, size = 29, normalized size = 1.07 \begin {gather*} 3 \left (-x+\frac {x}{\log \left (\frac {x^2}{\left (-4+e^x\right ) \left (-1+x^2\right )}\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.61, size = 58, normalized size = 2.15 \begin {gather*} -\frac {3 \, {\left (x \log \left (-\frac {x^{2}}{4 \, x^{2} - {\left (x^{2} - 1\right )} e^{x} - 4}\right ) - x\right )}}{\log \left (-\frac {x^{2}}{4 \, x^{2} - {\left (x^{2} - 1\right )} e^{x} - 4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.29, size = 58, normalized size = 2.15 \begin {gather*} -\frac {3 \, {\left (x \log \left (\frac {x^{2}}{x^{2} e^{x} - 4 \, x^{2} - e^{x} + 4}\right ) - x\right )}}{\log \left (\frac {x^{2}}{x^{2} e^{x} - 4 \, x^{2} - e^{x} + 4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.27, size = 348, normalized size = 12.89
method | result | size |
risch | \(-3 x +\frac {6 i x}{\pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{x}-4}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}-1}\right ) \mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}-4\right ) \left (x^{2}-1\right )}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{x}-4}\right ) \mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}-4\right ) \left (x^{2}-1\right )}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{x^{2}-1}\right ) \mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}-4\right ) \left (x^{2}-1\right )}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}-4\right ) \left (x^{2}-1\right )}\right )^{3}-\pi \,\mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}-4\right ) \left (x^{2}-1\right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\left (x^{2}-1\right ) \left ({\mathrm e}^{x}-4\right )}\right )^{2}+\pi \,\mathrm {csgn}\left (\frac {i}{\left ({\mathrm e}^{x}-4\right ) \left (x^{2}-1\right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\left (x^{2}-1\right ) \left ({\mathrm e}^{x}-4\right )}\right ) \mathrm {csgn}\left (i x^{2}\right )+\pi \mathrm {csgn}\left (\frac {i x^{2}}{\left (x^{2}-1\right ) \left ({\mathrm e}^{x}-4\right )}\right )^{3}-\pi \mathrm {csgn}\left (\frac {i x^{2}}{\left (x^{2}-1\right ) \left ({\mathrm e}^{x}-4\right )}\right )^{2} \mathrm {csgn}\left (i x^{2}\right )+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+4 i \ln \relax (x )-2 i \ln \left (x^{2}-1\right )-2 i \ln \left ({\mathrm e}^{x}-4\right )}\) | \(348\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.45, size = 48, normalized size = 1.78 \begin {gather*} -\frac {3 \, {\left (x \log \left (x + 1\right ) + x \log \left (x - 1\right ) - 2 \, x \log \relax (x) + x \log \left (e^{x} - 4\right ) + x\right )}}{\log \left (x + 1\right ) + \log \left (x - 1\right ) - 2 \, \log \relax (x) + \log \left (e^{x} - 4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\left ({\mathrm {e}}^x\,\left (3\,x^2-3\right )-12\,x^2+12\right )\,{\ln \left (\frac {x^2}{{\mathrm {e}}^x\,\left (x^2-1\right )-4\,x^2+4}\right )}^2+\left (12\,x^2-{\mathrm {e}}^x\,\left (3\,x^2-3\right )-12\right )\,\ln \left (\frac {x^2}{{\mathrm {e}}^x\,\left (x^2-1\right )-4\,x^2+4}\right )-{\mathrm {e}}^x\,\left (3\,x^3-3\,x+6\right )+24}{{\ln \left (\frac {x^2}{{\mathrm {e}}^x\,\left (x^2-1\right )-4\,x^2+4}\right )}^2\,\left ({\mathrm {e}}^x\,\left (x^2-1\right )-4\,x^2+4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.30, size = 26, normalized size = 0.96 \begin {gather*} - 3 x + \frac {3 x}{\log {\left (\frac {x^{2}}{- 4 x^{2} + \left (x^{2} - 1\right ) e^{x} + 4} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________