Integrand size = 404, antiderivative size = 29 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\log \left (1+e^x\right )+\frac {5}{\left (-x+2 e^4 \log \left (3+e^{2 x}+x\right )\right )^2} \]
[Out]
\[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {20 e^4+20 e^{4+x}+40 e^{4+3 x}-10 e^{2 x} \left (1-4 e^4\right )-10 (3+x)+e^{3 x} \left (-10+x^3\right )+e^x \left (-30-10 x+3 x^3+x^4\right )-6 e^{4+x} x^2 \left (3+e^{2 x}+x\right ) \log \left (3+e^{2 x}+x\right )+12 e^{8+x} x \left (3+e^{2 x}+x\right ) \log ^2\left (3+e^{2 x}+x\right )-8 e^{12+x} \left (3+e^{2 x}+x\right ) \log ^3\left (3+e^{2 x}+x\right )}{\left (1+e^x\right ) \left (3+e^{2 x}+x\right ) \left (x-2 e^4 \log \left (3+e^{2 x}+x\right )\right )^3} \, dx \\ & = \int \left (-\frac {1}{1+e^x}+\frac {20 e^4 (5+2 x)}{\left (3+e^{2 x}+x\right ) \left (-x+2 e^4 \log \left (3+e^{2 x}+x\right )\right )^3}+\frac {-10 \left (1-4 e^4\right )+x^3-6 e^4 x^2 \log \left (3+e^{2 x}+x\right )+12 e^8 x \log ^2\left (3+e^{2 x}+x\right )-8 e^{12} \log ^3\left (3+e^{2 x}+x\right )}{\left (x-2 e^4 \log \left (3+e^{2 x}+x\right )\right )^3}\right ) \, dx \\ & = \left (20 e^4\right ) \int \frac {5+2 x}{\left (3+e^{2 x}+x\right ) \left (-x+2 e^4 \log \left (3+e^{2 x}+x\right )\right )^3} \, dx-\int \frac {1}{1+e^x} \, dx+\int \frac {-10 \left (1-4 e^4\right )+x^3-6 e^4 x^2 \log \left (3+e^{2 x}+x\right )+12 e^8 x \log ^2\left (3+e^{2 x}+x\right )-8 e^{12} \log ^3\left (3+e^{2 x}+x\right )}{\left (x-2 e^4 \log \left (3+e^{2 x}+x\right )\right )^3} \, dx \\ & = \left (20 e^4\right ) \int \left (-\frac {5}{\left (3+e^{2 x}+x\right ) \left (x-2 e^4 \log \left (3+e^{2 x}+x\right )\right )^3}-\frac {2 x}{\left (3+e^{2 x}+x\right ) \left (x-2 e^4 \log \left (3+e^{2 x}+x\right )\right )^3}\right ) \, dx+\int \left (1+\frac {10 \left (-1+4 e^4\right )}{\left (x-2 e^4 \log \left (3+e^{2 x}+x\right )\right )^3}\right ) \, dx-\text {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,e^x\right ) \\ & = x-\left (40 e^4\right ) \int \frac {x}{\left (3+e^{2 x}+x\right ) \left (x-2 e^4 \log \left (3+e^{2 x}+x\right )\right )^3} \, dx-\left (100 e^4\right ) \int \frac {1}{\left (3+e^{2 x}+x\right ) \left (x-2 e^4 \log \left (3+e^{2 x}+x\right )\right )^3} \, dx-\left (10 \left (1-4 e^4\right )\right ) \int \frac {1}{\left (x-2 e^4 \log \left (3+e^{2 x}+x\right )\right )^3} \, dx-\text {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )+\text {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^x\right ) \\ & = \log \left (1+e^x\right )-\left (40 e^4\right ) \int \frac {x}{\left (3+e^{2 x}+x\right ) \left (x-2 e^4 \log \left (3+e^{2 x}+x\right )\right )^3} \, dx-\left (100 e^4\right ) \int \frac {1}{\left (3+e^{2 x}+x\right ) \left (x-2 e^4 \log \left (3+e^{2 x}+x\right )\right )^3} \, dx-\left (10 \left (1-4 e^4\right )\right ) \int \frac {1}{\left (x-2 e^4 \log \left (3+e^{2 x}+x\right )\right )^3} \, dx \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\log \left (1+e^x\right )+\frac {5}{\left (-x+2 e^4 \log \left (3+e^{2 x}+x\right )\right )^2} \]
[In]
[Out]
Time = 6.47 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93
method | result | size |
risch | \(\frac {5}{{\left (2 \ln \left ({\mathrm e}^{2 x}+3+x \right ) {\mathrm e}^{4}-x \right )}^{2}}+\ln \left ({\mathrm e}^{x}+1\right )\) | \(27\) |
parallelrisch | \(\frac {4 \,{\mathrm e}^{8} \ln \left ({\mathrm e}^{x}+1\right ) \ln \left ({\mathrm e}^{2 x}+3+x \right )^{2}-4 \,{\mathrm e}^{4} \ln \left ({\mathrm e}^{x}+1\right ) \ln \left ({\mathrm e}^{2 x}+3+x \right ) x +\ln \left ({\mathrm e}^{x}+1\right ) x^{2}+5}{4 \,{\mathrm e}^{8} \ln \left ({\mathrm e}^{2 x}+3+x \right )^{2}-4 \,{\mathrm e}^{4} \ln \left ({\mathrm e}^{2 x}+3+x \right ) x +x^{2}}\) | \(87\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 130, normalized size of antiderivative = 4.48 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\frac {4 \, x e^{4} \log \left ({\left ({\left (x + 3\right )} e^{24} + e^{\left (2 \, x + 24\right )}\right )} e^{\left (-24\right )}\right ) \log \left (e^{12} + e^{\left (x + 12\right )}\right ) - 4 \, e^{8} \log \left ({\left ({\left (x + 3\right )} e^{24} + e^{\left (2 \, x + 24\right )}\right )} e^{\left (-24\right )}\right )^{2} \log \left (e^{12} + e^{\left (x + 12\right )}\right ) - x^{2} \log \left (e^{12} + e^{\left (x + 12\right )}\right ) - 5}{4 \, x e^{4} \log \left ({\left ({\left (x + 3\right )} e^{24} + e^{\left (2 \, x + 24\right )}\right )} e^{\left (-24\right )}\right ) - 4 \, e^{8} \log \left ({\left ({\left (x + 3\right )} e^{24} + e^{\left (2 \, x + 24\right )}\right )} e^{\left (-24\right )}\right )^{2} - x^{2}} \]
[In]
[Out]
Time = 0.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\log {\left (e^{x} + 1 \right )} + \frac {5}{x^{2} - 4 x e^{4} \log {\left (x + e^{2 x} + 3 \right )} + 4 e^{8} \log {\left (x + e^{2 x} + 3 \right )}^{2}} \]
[In]
[Out]
none
Time = 0.90 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=-\frac {5}{4 \, x e^{4} \log \left (x + e^{\left (2 \, x\right )} + 3\right ) - 4 \, e^{8} \log \left (x + e^{\left (2 \, x\right )} + 3\right )^{2} - x^{2}} + \log \left (e^{x} + 1\right ) \]
[In]
[Out]
Timed out. \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\text {Timed out} \]
[In]
[Out]
Time = 8.91 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\ln \left ({\mathrm {e}}^x+1\right )+\frac {5}{{\left (x-2\,{\mathrm {e}}^4\,\ln \left (x+{\mathrm {e}}^{2\,x}+3\right )\right )}^2} \]
[In]
[Out]