Integrand size = 282, antiderivative size = 27 \[ \int \frac {-6 x+2 x^3+e^{10} \left (-6 x+2 x^3\right )+e^5 \left (-12 x+4 x^3\right )+e^{2-2 x} \left (-6 x+6 x^2+2 x^3-2 x^4\right )+e^{1-x} \left (-12 x+6 x^2+4 x^3-2 x^4+e^5 \left (-12 x+6 x^2+4 x^3-2 x^4\right )\right )+\left (2 x+2 x^3+4 e^5 x^3+2 e^{10} x^3+2 e^{2-2 x} x^3+e^{1-x} \left (4 x^3+4 e^5 x^3\right )\right ) \log \left (1+x^2+2 e^5 x^2+e^{10} x^2+e^{2-2 x} x^2+e^{1-x} \left (2 x^2+2 e^5 x^2\right )\right )}{1+x^2+2 e^5 x^2+e^{10} x^2+e^{2-2 x} x^2+e^{1-x} \left (2 x^2+2 e^5 x^2\right )} \, dx=\left (-3+x^2\right ) \log \left (1+\left (1+e^5+e^{1-x}\right )^2 x^2\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(94\) vs. \(2(27)=54\).
Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.48 \[ \int \frac {-6 x+2 x^3+e^{10} \left (-6 x+2 x^3\right )+e^5 \left (-12 x+4 x^3\right )+e^{2-2 x} \left (-6 x+6 x^2+2 x^3-2 x^4\right )+e^{1-x} \left (-12 x+6 x^2+4 x^3-2 x^4+e^5 \left (-12 x+6 x^2+4 x^3-2 x^4\right )\right )+\left (2 x+2 x^3+4 e^5 x^3+2 e^{10} x^3+2 e^{2-2 x} x^3+e^{1-x} \left (4 x^3+4 e^5 x^3\right )\right ) \log \left (1+x^2+2 e^5 x^2+e^{10} x^2+e^{2-2 x} x^2+e^{1-x} \left (2 x^2+2 e^5 x^2\right )\right )}{1+x^2+2 e^5 x^2+e^{10} x^2+e^{2-2 x} x^2+e^{1-x} \left (2 x^2+2 e^5 x^2\right )} \, dx=2 \left (-\frac {3}{2} \log \left (1+x^2+2 e^5 x^2+e^{10} x^2+e^{2-2 x} x^2+2 e^{1-x} x^2+2 e^{6-x} x^2\right )+\frac {1}{2} x^2 \log \left (1+e^{-2 x} \left (e+e^x+e^{5+x}\right )^2 x^2\right )\right ) \]
Integrate[(-6*x + 2*x^3 + E^10*(-6*x + 2*x^3) + E^5*(-12*x + 4*x^3) + E^(2 - 2*x)*(-6*x + 6*x^2 + 2*x^3 - 2*x^4) + E^(1 - x)*(-12*x + 6*x^2 + 4*x^3 - 2*x^4 + E^5*(-12*x + 6*x^2 + 4*x^3 - 2*x^4)) + (2*x + 2*x^3 + 4*E^5*x^3 + 2*E^10*x^3 + 2*E^(2 - 2*x)*x^3 + E^(1 - x)*(4*x^3 + 4*E^5*x^3))*Log[1 + x^2 + 2*E^5*x^2 + E^10*x^2 + E^(2 - 2*x)*x^2 + E^(1 - x)*(2*x^2 + 2*E^5*x^ 2)])/(1 + x^2 + 2*E^5*x^2 + E^10*x^2 + E^(2 - 2*x)*x^2 + E^(1 - x)*(2*x^2 + 2*E^5*x^2)),x]
2*((-3*Log[1 + x^2 + 2*E^5*x^2 + E^10*x^2 + E^(2 - 2*x)*x^2 + 2*E^(1 - x)* x^2 + 2*E^(6 - x)*x^2])/2 + (x^2*Log[1 + ((E + E^x + E^(5 + x))^2*x^2)/E^( 2*x)])/2)
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {2 x^3+e^{10} \left (2 x^3-6 x\right )+e^5 \left (4 x^3-12 x\right )+\left (2 e^{2-2 x} x^3+2 e^{10} x^3+4 e^5 x^3+2 x^3+e^{1-x} \left (4 e^5 x^3+4 x^3\right )+2 x\right ) \log \left (e^{2-2 x} x^2+e^{10} x^2+2 e^5 x^2+x^2+e^{1-x} \left (2 e^5 x^2+2 x^2\right )+1\right )+e^{2-2 x} \left (-2 x^4+2 x^3+6 x^2-6 x\right )+e^{1-x} \left (-2 x^4+4 x^3+6 x^2+e^5 \left (-2 x^4+4 x^3+6 x^2-12 x\right )-12 x\right )-6 x}{e^{2-2 x} x^2+e^{10} x^2+2 e^5 x^2+x^2+e^{1-x} \left (2 e^5 x^2+2 x^2\right )+1} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {2 x^3+e^{10} \left (2 x^3-6 x\right )+e^5 \left (4 x^3-12 x\right )+\left (2 e^{2-2 x} x^3+2 e^{10} x^3+4 e^5 x^3+2 x^3+e^{1-x} \left (4 e^5 x^3+4 x^3\right )+2 x\right ) \log \left (e^{2-2 x} x^2+e^{10} x^2+2 e^5 x^2+x^2+e^{1-x} \left (2 e^5 x^2+2 x^2\right )+1\right )+e^{2-2 x} \left (-2 x^4+2 x^3+6 x^2-6 x\right )+e^{1-x} \left (-2 x^4+4 x^3+6 x^2+e^5 \left (-2 x^4+4 x^3+6 x^2-12 x\right )-12 x\right )-6 x}{e^{2-2 x} x^2+\left (1+2 e^5\right ) x^2+e^{10} x^2+e^{1-x} \left (2 e^5 x^2+2 x^2\right )+1}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {2 x^3+e^{10} \left (2 x^3-6 x\right )+e^5 \left (4 x^3-12 x\right )+\left (2 e^{2-2 x} x^3+2 e^{10} x^3+4 e^5 x^3+2 x^3+e^{1-x} \left (4 e^5 x^3+4 x^3\right )+2 x\right ) \log \left (e^{2-2 x} x^2+e^{10} x^2+2 e^5 x^2+x^2+e^{1-x} \left (2 e^5 x^2+2 x^2\right )+1\right )+e^{2-2 x} \left (-2 x^4+2 x^3+6 x^2-6 x\right )+e^{1-x} \left (-2 x^4+4 x^3+6 x^2+e^5 \left (-2 x^4+4 x^3+6 x^2-12 x\right )-12 x\right )-6 x}{e^{2-2 x} x^2+\left (1+2 e^5+e^{10}\right ) x^2+e^{1-x} \left (2 e^5 x^2+2 x^2\right )+1}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {2 x^3+\left (1+\frac {2}{e^5}\right ) e^{10} \left (2 x^3-6 x\right )+\left (2 e^{2-2 x} x^3+2 e^{10} x^3+4 e^5 x^3+2 x^3+e^{1-x} \left (4 e^5 x^3+4 x^3\right )+2 x\right ) \log \left (e^{2-2 x} x^2+e^{10} x^2+2 e^5 x^2+x^2+e^{1-x} \left (2 e^5 x^2+2 x^2\right )+1\right )+e^{2-2 x} \left (-2 x^4+2 x^3+6 x^2-6 x\right )+e^{1-x} \left (-2 x^4+4 x^3+6 x^2+e^5 \left (-2 x^4+4 x^3+6 x^2-12 x\right )-12 x\right )-6 x}{e^{2-2 x} x^2+\left (1+2 e^5+e^{10}\right ) x^2+e^{1-x} \left (2 e^5 x^2+2 x^2\right )+1}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 x \left (\left (1+e^5 \left (2+e^5\right )\right ) x^2+\left (1+e^5 \left (2+e^5\right )\right ) x^2 \log \left (e^{-2 x} \left (e^x+e^{x+5}+e\right )^2 x^2+1\right )+\log \left (e^{-2 x} \left (e^x+e^{x+5}+e\right )^2 x^2+1\right )-3 \left (1+e^5 \left (2+e^5\right )\right )\right )}{\left (1+e^5\right )^2 x^2+1}+\frac {2 e x \left (3-x^2\right ) \left (\left (1+e^5 \left (3+3 e^5+e^{10}\right )\right ) e^x x^3+e \left (1+e^5 \left (2+e^5\right )\right ) x^3+\left (1+e^5\right ) e^x x+e x-2 \left (1+e^5\right ) e^x-e\right )}{\left (\left (1+e^5\right )^2 x^2+1\right ) \left (\left (1+e^5 \left (2+e^5\right )\right ) e^{2 x} x^2+2 \left (1+e^5\right ) e^{x+1} x^2+e^2 x^2+e^{2 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 2 x \left (\frac {\left (e^x+e^{x+5}+e\right ) \left (-e x+e^x+e^{x+5}+e\right ) \left (x^2-3\right )}{2 e^{x+1} x^2+e^{2 (x+5)} x^2+2 e^{x+6} x^2+2 e^{2 x+5} x^2+e^2 x^2+e^{2 x} \left (x^2+1\right )}+\log \left (e^{-2 x} \left (e^x+e^{x+5}+e\right )^2 x^2+1\right )\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -x \left (\frac {\left (e+e^x+e^{x+5}\right ) \left (-e x+e^x+e^{x+5}+e\right ) \left (3-x^2\right )}{2 e^{x+1} x^2+e^{2 (x+5)} x^2+2 e^{x+6} x^2+2 e^{2 x+5} x^2+e^2 x^2+e^{2 x} \left (x^2+1\right )}-\log \left (e^{-2 x} \left (e+e^x+e^{x+5}\right )^2 x^2+1\right )\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int x \left (\frac {\left (e+e^x+e^{x+5}\right ) \left (-e x+e^x+e^{x+5}+e\right ) \left (3-x^2\right )}{2 e^{x+1} x^2+e^{2 (x+5)} x^2+2 e^{x+6} x^2+2 e^{2 x+5} x^2+e^2 x^2+e^{2 x} \left (x^2+1\right )}-\log \left (e^{-2 x} \left (e+e^x+e^{x+5}\right )^2 x^2+1\right )\right )dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle -2 \int \left (\frac {e x \left (3-x^2\right ) \left (-e^x \left (1+e^5 \left (3+3 e^5+e^{10}\right )\right ) x^3-e \left (1+e^5 \left (2+e^5\right )\right ) x^3-e^x \left (1+e^5\right ) x-e x+2 e^x \left (1+e^5\right )+e\right )}{\left (\left (1+e^5\right )^2 x^2+1\right ) \left (e^{2 x} \left (1+e^5 \left (2+e^5\right )\right ) x^2+2 e^{x+1} \left (1+e^5\right ) x^2+e^2 x^2+e^{2 x}\right )}+\frac {-\left (\left (1+e^5 \left (2+e^5\right )\right ) \log \left (e^{-2 x} \left (e+e^x+e^{x+5}\right )^2 x^2+1\right ) x^3\right )-\left (1+e^5 \left (2+e^5\right )\right ) x^3-\log \left (e^{-2 x} \left (e+e^x+e^{x+5}\right )^2 x^2+1\right ) x+3 \left (1+e^5 \left (2+e^5\right )\right ) x}{\left (1+e^5\right )^2 x^2+1}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \left (-\frac {\left (e+e^x \left (1+e^5\right )\right ) x \left (-e x+e^x \left (1+e^5\right )+e\right ) \left (x^2-3\right )}{\left (1+\frac {2}{e^5}\right ) e^{2 (x+5)} x^2+2 e^{x+1} \left (1+e^5\right ) x^2+e^2 x^2+e^{2 x} \left (x^2+1\right )}-x \log \left (e^{-2 x} \left (e+e^x+e^{x+5}\right )^2 x^2+1\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {\left (e+e^x \left (1+e^5\right )\right ) x \left (-e x+e^x \left (1+e^5\right )+e\right ) \left (3-x^2\right )}{e^{2 x} \left (1+e^5 \left (2+e^5\right )\right ) x^2+2 e^{x+1} \left (1+e^5\right ) x^2+e^2 x^2+e^{2 x}}-x \log \left (e^{-2 x} \left (e+e^x+e^{x+5}\right )^2 x^2+1\right )\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -2 \int \left (\frac {\left (e+e^x \left (1+e^5\right )\right ) x \left (-e x+e^x \left (1+e^5\right )+e\right ) \left (3-x^2\right )}{e^{2 x} \left (1+e^5 \left (2+e^5\right )\right ) x^2+2 e^{x+1} \left (1+e^5\right ) x^2+e^2 x^2+e^{2 x}}-x \log \left (e^{-2 x} \left (e+e^x+e^{x+5}\right )^2 x^2+1\right )\right )dx\) |
Int[(-6*x + 2*x^3 + E^10*(-6*x + 2*x^3) + E^5*(-12*x + 4*x^3) + E^(2 - 2*x )*(-6*x + 6*x^2 + 2*x^3 - 2*x^4) + E^(1 - x)*(-12*x + 6*x^2 + 4*x^3 - 2*x^ 4 + E^5*(-12*x + 6*x^2 + 4*x^3 - 2*x^4)) + (2*x + 2*x^3 + 4*E^5*x^3 + 2*E^ 10*x^3 + 2*E^(2 - 2*x)*x^3 + E^(1 - x)*(4*x^3 + 4*E^5*x^3))*Log[1 + x^2 + 2*E^5*x^2 + E^10*x^2 + E^(2 - 2*x)*x^2 + E^(1 - x)*(2*x^2 + 2*E^5*x^2)])/( 1 + x^2 + 2*E^5*x^2 + E^10*x^2 + E^(2 - 2*x)*x^2 + E^(1 - x)*(2*x^2 + 2*E^ 5*x^2)),x]
3.18.55.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(106\) vs. \(2(25)=50\).
Time = 3.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.96
| method | result | size |
| risch | \(x^{2} \ln \left (2 x^{2} {\mathrm e}^{-x +6}+2 x^{2} {\mathrm e}^{5}+x^{2} {\mathrm e}^{2-2 x}+x^{2} {\mathrm e}^{10}+2 x^{2} {\mathrm e}^{1-x}+x^{2}+1\right )-6 \ln \left (x \right )+6-3 \ln \left ({\mathrm e}^{2-2 x}+\left (2 \,{\mathrm e}^{5}+2\right ) {\mathrm e}^{1-x}+\frac {2 x^{2} {\mathrm e}^{5}+x^{2} {\mathrm e}^{10}+x^{2}+1}{x^{2}}\right )\) | \(107\) |
| parallelrisch | \(x^{2} \ln \left (x^{2} {\mathrm e}^{2-2 x}+\left (2 x^{2} {\mathrm e}^{5}+2 x^{2}\right ) {\mathrm e}^{1-x}+x^{2} {\mathrm e}^{10}+2 x^{2} {\mathrm e}^{5}+x^{2}+1\right )-3 \ln \left (x^{2} {\mathrm e}^{2-2 x}+\left (2 x^{2} {\mathrm e}^{5}+2 x^{2}\right ) {\mathrm e}^{1-x}+x^{2} {\mathrm e}^{10}+2 x^{2} {\mathrm e}^{5}+x^{2}+1\right )\) | \(114\) |
int(((2*x^3*exp(1-x)^2+(4*x^3*exp(5)+4*x^3)*exp(1-x)+2*x^3*exp(5)^2+4*x^3* exp(5)+2*x^3+2*x)*ln(x^2*exp(1-x)^2+(2*x^2*exp(5)+2*x^2)*exp(1-x)+x^2*exp( 5)^2+2*x^2*exp(5)+x^2+1)+(-2*x^4+2*x^3+6*x^2-6*x)*exp(1-x)^2+((-2*x^4+4*x^ 3+6*x^2-12*x)*exp(5)-2*x^4+4*x^3+6*x^2-12*x)*exp(1-x)+(2*x^3-6*x)*exp(5)^2 +(4*x^3-12*x)*exp(5)+2*x^3-6*x)/(x^2*exp(1-x)^2+(2*x^2*exp(5)+2*x^2)*exp(1 -x)+x^2*exp(5)^2+2*x^2*exp(5)+x^2+1),x,method=_RETURNVERBOSE)
x^2*ln(2*x^2*exp(-x+6)+2*x^2*exp(5)+x^2*exp(2-2*x)+x^2*exp(10)+2*x^2*exp(1 -x)+x^2+1)-6*ln(x)+6-3*ln(exp(2-2*x)+(2*exp(5)+2)*exp(1-x)+(2*x^2*exp(5)+x ^2*exp(10)+x^2+1)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (25) = 50\).
Time = 0.40 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {-6 x+2 x^3+e^{10} \left (-6 x+2 x^3\right )+e^5 \left (-12 x+4 x^3\right )+e^{2-2 x} \left (-6 x+6 x^2+2 x^3-2 x^4\right )+e^{1-x} \left (-12 x+6 x^2+4 x^3-2 x^4+e^5 \left (-12 x+6 x^2+4 x^3-2 x^4\right )\right )+\left (2 x+2 x^3+4 e^5 x^3+2 e^{10} x^3+2 e^{2-2 x} x^3+e^{1-x} \left (4 x^3+4 e^5 x^3\right )\right ) \log \left (1+x^2+2 e^5 x^2+e^{10} x^2+e^{2-2 x} x^2+e^{1-x} \left (2 x^2+2 e^5 x^2\right )\right )}{1+x^2+2 e^5 x^2+e^{10} x^2+e^{2-2 x} x^2+e^{1-x} \left (2 x^2+2 e^5 x^2\right )} \, dx={\left (x^{2} - 3\right )} \log \left (x^{2} e^{10} + 2 \, x^{2} e^{5} + x^{2} e^{\left (-2 \, x + 2\right )} + x^{2} + 2 \, {\left (x^{2} e^{5} + x^{2}\right )} e^{\left (-x + 1\right )} + 1\right ) \]
integrate(((2*x^3*exp(1-x)^2+(4*x^3*exp(5)+4*x^3)*exp(1-x)+2*x^3*exp(5)^2+ 4*x^3*exp(5)+2*x^3+2*x)*log(x^2*exp(1-x)^2+(2*x^2*exp(5)+2*x^2)*exp(1-x)+x ^2*exp(5)^2+2*x^2*exp(5)+x^2+1)+(-2*x^4+2*x^3+6*x^2-6*x)*exp(1-x)^2+((-2*x ^4+4*x^3+6*x^2-12*x)*exp(5)-2*x^4+4*x^3+6*x^2-12*x)*exp(1-x)+(2*x^3-6*x)*e xp(5)^2+(4*x^3-12*x)*exp(5)+2*x^3-6*x)/(x^2*exp(1-x)^2+(2*x^2*exp(5)+2*x^2 )*exp(1-x)+x^2*exp(5)^2+2*x^2*exp(5)+x^2+1),x, algorithm=\
(x^2 - 3)*log(x^2*e^10 + 2*x^2*e^5 + x^2*e^(-2*x + 2) + x^2 + 2*(x^2*e^5 + x^2)*e^(-x + 1) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (22) = 44\).
Time = 0.39 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.85 \[ \int \frac {-6 x+2 x^3+e^{10} \left (-6 x+2 x^3\right )+e^5 \left (-12 x+4 x^3\right )+e^{2-2 x} \left (-6 x+6 x^2+2 x^3-2 x^4\right )+e^{1-x} \left (-12 x+6 x^2+4 x^3-2 x^4+e^5 \left (-12 x+6 x^2+4 x^3-2 x^4\right )\right )+\left (2 x+2 x^3+4 e^5 x^3+2 e^{10} x^3+2 e^{2-2 x} x^3+e^{1-x} \left (4 x^3+4 e^5 x^3\right )\right ) \log \left (1+x^2+2 e^5 x^2+e^{10} x^2+e^{2-2 x} x^2+e^{1-x} \left (2 x^2+2 e^5 x^2\right )\right )}{1+x^2+2 e^5 x^2+e^{10} x^2+e^{2-2 x} x^2+e^{1-x} \left (2 x^2+2 e^5 x^2\right )} \, dx=x^{2} \log {\left (x^{2} e^{2 - 2 x} + x^{2} + 2 x^{2} e^{5} + x^{2} e^{10} + \left (2 x^{2} + 2 x^{2} e^{5}\right ) e^{1 - x} + 1 \right )} - 6 \log {\left (x \right )} - 3 \log {\left (\left (2 + 2 e^{5}\right ) e^{1 - x} + e^{2 - 2 x} + \frac {x^{2} + 2 x^{2} e^{5} + x^{2} e^{10} + 1}{x^{2}} \right )} \]
integrate(((2*x**3*exp(1-x)**2+(4*x**3*exp(5)+4*x**3)*exp(1-x)+2*x**3*exp( 5)**2+4*x**3*exp(5)+2*x**3+2*x)*ln(x**2*exp(1-x)**2+(2*x**2*exp(5)+2*x**2) *exp(1-x)+x**2*exp(5)**2+2*x**2*exp(5)+x**2+1)+(-2*x**4+2*x**3+6*x**2-6*x) *exp(1-x)**2+((-2*x**4+4*x**3+6*x**2-12*x)*exp(5)-2*x**4+4*x**3+6*x**2-12* x)*exp(1-x)+(2*x**3-6*x)*exp(5)**2+(4*x**3-12*x)*exp(5)+2*x**3-6*x)/(x**2* exp(1-x)**2+(2*x**2*exp(5)+2*x**2)*exp(1-x)+x**2*exp(5)**2+2*x**2*exp(5)+x **2+1),x)
x**2*log(x**2*exp(2 - 2*x) + x**2 + 2*x**2*exp(5) + x**2*exp(10) + (2*x**2 + 2*x**2*exp(5))*exp(1 - x) + 1) - 6*log(x) - 3*log((2 + 2*exp(5))*exp(1 - x) + exp(2 - 2*x) + (x**2 + 2*x**2*exp(5) + x**2*exp(10) + 1)/x**2)
Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (25) = 50\).
Time = 0.30 (sec) , antiderivative size = 127, normalized size of antiderivative = 4.70 \[ \int \frac {-6 x+2 x^3+e^{10} \left (-6 x+2 x^3\right )+e^5 \left (-12 x+4 x^3\right )+e^{2-2 x} \left (-6 x+6 x^2+2 x^3-2 x^4\right )+e^{1-x} \left (-12 x+6 x^2+4 x^3-2 x^4+e^5 \left (-12 x+6 x^2+4 x^3-2 x^4\right )\right )+\left (2 x+2 x^3+4 e^5 x^3+2 e^{10} x^3+2 e^{2-2 x} x^3+e^{1-x} \left (4 x^3+4 e^5 x^3\right )\right ) \log \left (1+x^2+2 e^5 x^2+e^{10} x^2+e^{2-2 x} x^2+e^{1-x} \left (2 x^2+2 e^5 x^2\right )\right )}{1+x^2+2 e^5 x^2+e^{10} x^2+e^{2-2 x} x^2+e^{1-x} \left (2 x^2+2 e^5 x^2\right )} \, dx=-2 \, x^{3} + x^{2} \log \left (2 \, x^{2} {\left (e^{6} + e\right )} e^{x} + x^{2} e^{2} + {\left (x^{2} {\left (e^{10} + 2 \, e^{5} + 1\right )} + 1\right )} e^{\left (2 \, x\right )}\right ) + 6 \, x - 3 \, \log \left (x^{2} {\left (e^{10} + 2 \, e^{5} + 1\right )} + 1\right ) - 3 \, \log \left (\frac {2 \, x^{2} {\left (e^{6} + e\right )} e^{x} + x^{2} e^{2} + {\left (x^{2} {\left (e^{10} + 2 \, e^{5} + 1\right )} + 1\right )} e^{\left (2 \, x\right )}}{x^{2} {\left (e^{10} + 2 \, e^{5} + 1\right )} + 1}\right ) \]
integrate(((2*x^3*exp(1-x)^2+(4*x^3*exp(5)+4*x^3)*exp(1-x)+2*x^3*exp(5)^2+ 4*x^3*exp(5)+2*x^3+2*x)*log(x^2*exp(1-x)^2+(2*x^2*exp(5)+2*x^2)*exp(1-x)+x ^2*exp(5)^2+2*x^2*exp(5)+x^2+1)+(-2*x^4+2*x^3+6*x^2-6*x)*exp(1-x)^2+((-2*x ^4+4*x^3+6*x^2-12*x)*exp(5)-2*x^4+4*x^3+6*x^2-12*x)*exp(1-x)+(2*x^3-6*x)*e xp(5)^2+(4*x^3-12*x)*exp(5)+2*x^3-6*x)/(x^2*exp(1-x)^2+(2*x^2*exp(5)+2*x^2 )*exp(1-x)+x^2*exp(5)^2+2*x^2*exp(5)+x^2+1),x, algorithm=\
-2*x^3 + x^2*log(2*x^2*(e^6 + e)*e^x + x^2*e^2 + (x^2*(e^10 + 2*e^5 + 1) + 1)*e^(2*x)) + 6*x - 3*log(x^2*(e^10 + 2*e^5 + 1) + 1) - 3*log((2*x^2*(e^6 + e)*e^x + x^2*e^2 + (x^2*(e^10 + 2*e^5 + 1) + 1)*e^(2*x))/(x^2*(e^10 + 2 *e^5 + 1) + 1))
Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (25) = 50\).
Time = 2.18 (sec) , antiderivative size = 434, normalized size of antiderivative = 16.07 \[ \int \frac {-6 x+2 x^3+e^{10} \left (-6 x+2 x^3\right )+e^5 \left (-12 x+4 x^3\right )+e^{2-2 x} \left (-6 x+6 x^2+2 x^3-2 x^4\right )+e^{1-x} \left (-12 x+6 x^2+4 x^3-2 x^4+e^5 \left (-12 x+6 x^2+4 x^3-2 x^4\right )\right )+\left (2 x+2 x^3+4 e^5 x^3+2 e^{10} x^3+2 e^{2-2 x} x^3+e^{1-x} \left (4 x^3+4 e^5 x^3\right )\right ) \log \left (1+x^2+2 e^5 x^2+e^{10} x^2+e^{2-2 x} x^2+e^{1-x} \left (2 x^2+2 e^5 x^2\right )\right )}{1+x^2+2 e^5 x^2+e^{10} x^2+e^{2-2 x} x^2+e^{1-x} \left (2 x^2+2 e^5 x^2\right )} \, dx={\left (x - 1\right )}^{2} \log \left ({\left (x - 1\right )}^{2} e^{10} + 2 \, {\left (x - 1\right )}^{2} e^{5} + 2 \, {\left (x - 1\right )}^{2} e^{\left (-x + 6\right )} + 2 \, {\left (x - 1\right )}^{2} e^{\left (-x + 1\right )} + {\left (x - 1\right )}^{2} e^{\left (-2 \, x + 2\right )} + {\left (x - 1\right )}^{2} + 2 \, {\left (x - 1\right )} e^{10} + 4 \, {\left (x - 1\right )} e^{5} + 4 \, {\left (x - 1\right )} e^{\left (-x + 6\right )} + 4 \, {\left (x - 1\right )} e^{\left (-x + 1\right )} + 2 \, {\left (x - 1\right )} e^{\left (-2 \, x + 2\right )} + 2 \, x + e^{10} + 2 \, e^{5} + 2 \, e^{\left (-x + 6\right )} + 2 \, e^{\left (-x + 1\right )} + e^{\left (-2 \, x + 2\right )}\right ) + 2 \, {\left (x - 1\right )} \log \left ({\left (x - 1\right )}^{2} e^{10} + 2 \, {\left (x - 1\right )}^{2} e^{5} + 2 \, {\left (x - 1\right )}^{2} e^{\left (-x + 6\right )} + 2 \, {\left (x - 1\right )}^{2} e^{\left (-x + 1\right )} + {\left (x - 1\right )}^{2} e^{\left (-2 \, x + 2\right )} + {\left (x - 1\right )}^{2} + 2 \, {\left (x - 1\right )} e^{10} + 4 \, {\left (x - 1\right )} e^{5} + 4 \, {\left (x - 1\right )} e^{\left (-x + 6\right )} + 4 \, {\left (x - 1\right )} e^{\left (-x + 1\right )} + 2 \, {\left (x - 1\right )} e^{\left (-2 \, x + 2\right )} + 2 \, x + e^{10} + 2 \, e^{5} + 2 \, e^{\left (-x + 6\right )} + 2 \, e^{\left (-x + 1\right )} + e^{\left (-2 \, x + 2\right )}\right ) - 2 \, \log \left ({\left (x - 1\right )}^{2} e^{10} + 2 \, {\left (x - 1\right )}^{2} e^{5} + 2 \, {\left (x - 1\right )}^{2} e^{\left (-x + 6\right )} + 2 \, {\left (x - 1\right )}^{2} e^{\left (-x + 1\right )} + {\left (x - 1\right )}^{2} e^{\left (-2 \, x + 2\right )} + {\left (x - 1\right )}^{2} + 2 \, {\left (x - 1\right )} e^{10} + 4 \, {\left (x - 1\right )} e^{5} + 4 \, {\left (x - 1\right )} e^{\left (-x + 6\right )} + 4 \, {\left (x - 1\right )} e^{\left (-x + 1\right )} + 2 \, {\left (x - 1\right )} e^{\left (-2 \, x + 2\right )} + 2 \, x + e^{10} + 2 \, e^{5} + 2 \, e^{\left (-x + 6\right )} + 2 \, e^{\left (-x + 1\right )} + e^{\left (-2 \, x + 2\right )}\right ) \]
integrate(((2*x^3*exp(1-x)^2+(4*x^3*exp(5)+4*x^3)*exp(1-x)+2*x^3*exp(5)^2+ 4*x^3*exp(5)+2*x^3+2*x)*log(x^2*exp(1-x)^2+(2*x^2*exp(5)+2*x^2)*exp(1-x)+x ^2*exp(5)^2+2*x^2*exp(5)+x^2+1)+(-2*x^4+2*x^3+6*x^2-6*x)*exp(1-x)^2+((-2*x ^4+4*x^3+6*x^2-12*x)*exp(5)-2*x^4+4*x^3+6*x^2-12*x)*exp(1-x)+(2*x^3-6*x)*e xp(5)^2+(4*x^3-12*x)*exp(5)+2*x^3-6*x)/(x^2*exp(1-x)^2+(2*x^2*exp(5)+2*x^2 )*exp(1-x)+x^2*exp(5)^2+2*x^2*exp(5)+x^2+1),x, algorithm=\
(x - 1)^2*log((x - 1)^2*e^10 + 2*(x - 1)^2*e^5 + 2*(x - 1)^2*e^(-x + 6) + 2*(x - 1)^2*e^(-x + 1) + (x - 1)^2*e^(-2*x + 2) + (x - 1)^2 + 2*(x - 1)*e^ 10 + 4*(x - 1)*e^5 + 4*(x - 1)*e^(-x + 6) + 4*(x - 1)*e^(-x + 1) + 2*(x - 1)*e^(-2*x + 2) + 2*x + e^10 + 2*e^5 + 2*e^(-x + 6) + 2*e^(-x + 1) + e^(-2 *x + 2)) + 2*(x - 1)*log((x - 1)^2*e^10 + 2*(x - 1)^2*e^5 + 2*(x - 1)^2*e^ (-x + 6) + 2*(x - 1)^2*e^(-x + 1) + (x - 1)^2*e^(-2*x + 2) + (x - 1)^2 + 2 *(x - 1)*e^10 + 4*(x - 1)*e^5 + 4*(x - 1)*e^(-x + 6) + 4*(x - 1)*e^(-x + 1 ) + 2*(x - 1)*e^(-2*x + 2) + 2*x + e^10 + 2*e^5 + 2*e^(-x + 6) + 2*e^(-x + 1) + e^(-2*x + 2)) - 2*log((x - 1)^2*e^10 + 2*(x - 1)^2*e^5 + 2*(x - 1)^2 *e^(-x + 6) + 2*(x - 1)^2*e^(-x + 1) + (x - 1)^2*e^(-2*x + 2) + (x - 1)^2 + 2*(x - 1)*e^10 + 4*(x - 1)*e^5 + 4*(x - 1)*e^(-x + 6) + 4*(x - 1)*e^(-x + 1) + 2*(x - 1)*e^(-2*x + 2) + 2*x + e^10 + 2*e^5 + 2*e^(-x + 6) + 2*e^(- x + 1) + e^(-2*x + 2))
Time = 0.81 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.41 \[ \int \frac {-6 x+2 x^3+e^{10} \left (-6 x+2 x^3\right )+e^5 \left (-12 x+4 x^3\right )+e^{2-2 x} \left (-6 x+6 x^2+2 x^3-2 x^4\right )+e^{1-x} \left (-12 x+6 x^2+4 x^3-2 x^4+e^5 \left (-12 x+6 x^2+4 x^3-2 x^4\right )\right )+\left (2 x+2 x^3+4 e^5 x^3+2 e^{10} x^3+2 e^{2-2 x} x^3+e^{1-x} \left (4 x^3+4 e^5 x^3\right )\right ) \log \left (1+x^2+2 e^5 x^2+e^{10} x^2+e^{2-2 x} x^2+e^{1-x} \left (2 x^2+2 e^5 x^2\right )\right )}{1+x^2+2 e^5 x^2+e^{10} x^2+e^{2-2 x} x^2+e^{1-x} \left (2 x^2+2 e^5 x^2\right )} \, dx=x^2\,\ln \left (2\,x^2\,{\mathrm {e}}^5+x^2\,{\mathrm {e}}^{10}+x^2+2\,x^2\,{\mathrm {e}}^{-x}\,\mathrm {e}+x^2\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^2+2\,x^2\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^6+1\right )-3\,\ln \left (2\,x^2\,{\mathrm {e}}^5+x^2\,{\mathrm {e}}^{10}+x^2+2\,x^2\,{\mathrm {e}}^{-x}\,\mathrm {e}+x^2\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^2+2\,x^2\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^6+1\right )-6\,\ln \left (x\right )-3\,\ln \left (\frac {1}{x^2}\right ) \]
int(-(6*x + exp(10)*(6*x - 2*x^3) + exp(5)*(12*x - 4*x^3) + exp(1 - x)*(12 *x + exp(5)*(12*x - 6*x^2 - 4*x^3 + 2*x^4) - 6*x^2 - 4*x^3 + 2*x^4) + exp( 2 - 2*x)*(6*x - 6*x^2 - 2*x^3 + 2*x^4) - 2*x^3 - log(exp(1 - x)*(2*x^2*exp (5) + 2*x^2) + 2*x^2*exp(5) + x^2*exp(10) + x^2*exp(2 - 2*x) + x^2 + 1)*(2 *x + exp(1 - x)*(4*x^3*exp(5) + 4*x^3) + 4*x^3*exp(5) + 2*x^3*exp(10) + 2* x^3*exp(2 - 2*x) + 2*x^3))/(exp(1 - x)*(2*x^2*exp(5) + 2*x^2) + 2*x^2*exp( 5) + x^2*exp(10) + x^2*exp(2 - 2*x) + x^2 + 1),x)