Optimal. Leaf size=32 \[ \frac {5 x}{\left (\frac {1}{x}-\frac {x}{2}\right ) \log \left (-e^x+\frac {3}{x \log (2)}\right )} \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 4.52, 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 {-60 x+30 x^3+e^x \left (-20 x^3+10 x^5\right ) \log (2)+\left (-120 x+40 e^x x^2 \log (2)\right ) \log \left (\frac {3-e^x x \log (2)}{x \log (2)}\right )}{\left (-12+12 x^2-3 x^4+e^x \left (4 x-4 x^3+x^5\right ) \log (2)\right ) \log ^2\left (\frac {3-e^x x \log (2)}{x \log (2)}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {60 x-30 x^3-e^x \left (-20 x^3+10 x^5\right ) \log (2)-\left (-120 x+40 e^x x^2 \log (2)\right ) \log \left (\frac {3-e^x x \log (2)}{x \log (2)}\right )}{\left (2-x^2\right )^2 \left (3-e^x x \log (2)\right ) \log ^2\left (\frac {3-e^x x \log (2)}{x \log (2)}\right )} \, dx\\ &=\int \left (\frac {30 x (1+x)}{\left (-2+x^2\right ) \left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )}+\frac {10 x \left (-2 x+x^3+4 \log \left (-e^x+\frac {3}{x \log (2)}\right )\right )}{\left (-2+x^2\right )^2 \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )}\right ) \, dx\\ &=10 \int \frac {x \left (-2 x+x^3+4 \log \left (-e^x+\frac {3}{x \log (2)}\right )\right )}{\left (-2+x^2\right )^2 \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx+30 \int \frac {x (1+x)}{\left (-2+x^2\right ) \left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx\\ &=10 \int \left (\frac {x^2}{\left (-2+x^2\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )}+\frac {4 x}{\left (-2+x^2\right )^2 \log \left (-e^x+\frac {3}{x \log (2)}\right )}\right ) \, dx+30 \int \left (\frac {1}{\left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )}+\frac {2+x}{\left (-2+x^2\right ) \left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )}\right ) \, dx\\ &=10 \int \frac {x^2}{\left (-2+x^2\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx+30 \int \frac {1}{\left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx+30 \int \frac {2+x}{\left (-2+x^2\right ) \left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx+40 \int \frac {x}{\left (-2+x^2\right )^2 \log \left (-e^x+\frac {3}{x \log (2)}\right )} \, dx\\ &=10 \int \left (\frac {1}{\log ^2\left (-e^x+\frac {3}{x \log (2)}\right )}+\frac {2}{\left (-2+x^2\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )}\right ) \, dx+30 \int \left (\frac {2}{\left (-2+x^2\right ) \left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )}+\frac {x}{\left (-2+x^2\right ) \left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )}\right ) \, dx+30 \int \frac {1}{\left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx+40 \int \frac {x}{\left (-2+x^2\right )^2 \log \left (-e^x+\frac {3}{x \log (2)}\right )} \, dx\\ &=10 \int \frac {1}{\log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx+20 \int \frac {1}{\left (-2+x^2\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx+30 \int \frac {1}{\left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx+30 \int \frac {x}{\left (-2+x^2\right ) \left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx+40 \int \frac {x}{\left (-2+x^2\right )^2 \log \left (-e^x+\frac {3}{x \log (2)}\right )} \, dx+60 \int \frac {1}{\left (-2+x^2\right ) \left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx\\ &=10 \int \frac {1}{\log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx+20 \int \left (-\frac {1}{2 \sqrt {2} \left (\sqrt {2}-x\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )}-\frac {1}{2 \sqrt {2} \left (\sqrt {2}+x\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )}\right ) \, dx+30 \int \left (-\frac {1}{2 \left (\sqrt {2}-x\right ) \left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )}+\frac {1}{2 \left (\sqrt {2}+x\right ) \left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )}\right ) \, dx+30 \int \frac {1}{\left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx+40 \int \frac {x}{\left (-2+x^2\right )^2 \log \left (-e^x+\frac {3}{x \log (2)}\right )} \, dx+60 \int \left (-\frac {1}{2 \sqrt {2} \left (\sqrt {2}-x\right ) \left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )}-\frac {1}{2 \sqrt {2} \left (\sqrt {2}+x\right ) \left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )}\right ) \, dx\\ &=10 \int \frac {1}{\log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx-15 \int \frac {1}{\left (\sqrt {2}-x\right ) \left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx+15 \int \frac {1}{\left (\sqrt {2}+x\right ) \left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx+30 \int \frac {1}{\left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx+40 \int \frac {x}{\left (-2+x^2\right )^2 \log \left (-e^x+\frac {3}{x \log (2)}\right )} \, dx-\left (5 \sqrt {2}\right ) \int \frac {1}{\left (\sqrt {2}-x\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx-\left (5 \sqrt {2}\right ) \int \frac {1}{\left (\sqrt {2}+x\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx-\left (15 \sqrt {2}\right ) \int \frac {1}{\left (\sqrt {2}-x\right ) \left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx-\left (15 \sqrt {2}\right ) \int \frac {1}{\left (\sqrt {2}+x\right ) \left (-3+e^x x \log (2)\right ) \log ^2\left (-e^x+\frac {3}{x \log (2)}\right )} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(32)=64\).
time = 0.08, size = 69, normalized size = 2.16 \begin {gather*} -\frac {10 x^2 \left (-6+3 x^2+e^x x^4 \log (2)-e^x x^2 \log (4)\right )}{\left (-2+x^2\right )^2 \left (3+e^x x^2 \log (2)\right ) \log \left (-e^x+\frac {3}{x \log (2)}\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.21, size = 178, normalized size = 5.56
method | result | size |
risch | \(-\frac {20 i x^{2}}{\left (x^{2}-2\right ) \left (2 \pi \mathrm {csgn}\left (\frac {i \left (x \ln \left (2\right ) {\mathrm e}^{x}-3\right )}{x}\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (x \ln \left (2\right ) {\mathrm e}^{x}-3\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \ln \left (2\right ) {\mathrm e}^{x}-3\right )}{x}\right )^{2}+\pi \,\mathrm {csgn}\left (i \left (x \ln \left (2\right ) {\mathrm e}^{x}-3\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \ln \left (2\right ) {\mathrm e}^{x}-3\right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )-\pi \mathrm {csgn}\left (\frac {i \left (x \ln \left (2\right ) {\mathrm e}^{x}-3\right )}{x}\right )^{3}-\pi \mathrm {csgn}\left (\frac {i \left (x \ln \left (2\right ) {\mathrm e}^{x}-3\right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )-2 \pi -2 i \ln \left (\ln \left (2\right )\right )-2 i \ln \left (x \right )+2 i \ln \left (x \ln \left (2\right ) {\mathrm e}^{x}-3\right )\right )}\) | \(178\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.58, size = 45, normalized size = 1.41 \begin {gather*} \frac {10 \, x^{2}}{x^{2} \log \left (\log \left (2\right )\right ) - {\left (x^{2} - 2\right )} \log \left (-x e^{x} \log \left (2\right ) + 3\right ) + {\left (x^{2} - 2\right )} \log \left (x\right ) - 2 \, \log \left (\log \left (2\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.31, size = 32, normalized size = 1.00 \begin {gather*} -\frac {10 \, x^{2}}{{\left (x^{2} - 2\right )} \log \left (-\frac {x e^{x} \log \left (2\right ) - 3}{x \log \left (2\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.13, size = 27, normalized size = 0.84 \begin {gather*} - \frac {10 x^{2}}{\left (x^{2} - 2\right ) \log {\left (\frac {- x e^{x} \log {\left (2 \right )} + 3}{x \log {\left (2 \right )}} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.50, size = 51, normalized size = 1.59 \begin {gather*} -\frac {10 \, x^{2}}{x^{2} \log \left (-x e^{x} \log \left (2\right ) + 3\right ) - x^{2} \log \left (x \log \left (2\right )\right ) - 2 \, \log \left (-x e^{x} \log \left (2\right ) + 3\right ) + 2 \, \log \left (x \log \left (2\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.00, size = 34, normalized size = 1.06 \begin {gather*} -\frac {10\,x^2}{\left (x^2-2\right )\,\left (\ln \left (-\frac {x\,{\mathrm {e}}^x\,\ln \left (2\right )-3}{x}\right )-\ln \left (\ln \left (2\right )\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________