Integrand size = 175, antiderivative size = 30 \[ \int \frac {-64+16 x+8 x^2+e^{2 x^2} \left (2 x^2+4 x^4\right )+\left (32-12 x-8 x^2\right ) \log (4)+\left (-4+2 x+2 x^2\right ) \log ^2(4)+e^{x^2} \left (-8 x-8 x^2-32 x^3-8 x^4+\left (2 x+4 x^2+8 x^3+4 x^4\right ) \log (4)\right )+\left (32-16 x+(-16+12 x) \log (4)+(2-2 x) \log ^2(4)+e^{x^2} \left (8 x+16 x^3+\left (-2 x-4 x^3\right ) \log (4)\right )\right ) \log (x)}{16 x-8 x \log (4)+x \log ^2(4)} \, dx=\left (2+x-\frac {2 x+e^{x^2} x}{4-\log (4)}-\log (x)\right )^2 \]
Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {-64+16 x+8 x^2+e^{2 x^2} \left (2 x^2+4 x^4\right )+\left (32-12 x-8 x^2\right ) \log (4)+\left (-4+2 x+2 x^2\right ) \log ^2(4)+e^{x^2} \left (-8 x-8 x^2-32 x^3-8 x^4+\left (2 x+4 x^2+8 x^3+4 x^4\right ) \log (4)\right )+\left (32-16 x+(-16+12 x) \log (4)+(2-2 x) \log ^2(4)+e^{x^2} \left (8 x+16 x^3+\left (-2 x-4 x^3\right ) \log (4)\right )\right ) \log (x)}{16 x-8 x \log (4)+x \log ^2(4)} \, dx=\frac {\left (-8+x \left (-2+e^{x^2}+\log (4)\right )+\log (16)-(-4+\log (4)) \log (x)\right )^2}{(-4+\log (4))^2} \]
Integrate[(-64 + 16*x + 8*x^2 + E^(2*x^2)*(2*x^2 + 4*x^4) + (32 - 12*x - 8 *x^2)*Log[4] + (-4 + 2*x + 2*x^2)*Log[4]^2 + E^x^2*(-8*x - 8*x^2 - 32*x^3 - 8*x^4 + (2*x + 4*x^2 + 8*x^3 + 4*x^4)*Log[4]) + (32 - 16*x + (-16 + 12*x )*Log[4] + (2 - 2*x)*Log[4]^2 + E^x^2*(8*x + 16*x^3 + (-2*x - 4*x^3)*Log[4 ]))*Log[x])/(16*x - 8*x*Log[4] + x*Log[4]^2),x]
Leaf count is larger than twice the leaf count of optimal. \(98\) vs. \(2(30)=60\).
Time = 0.57 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.27, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6, 6, 27, 27, 2010, 2009}
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.
| \(\displaystyle \int \frac {8 x^2+\left (2 x^2+2 x-4\right ) \log ^2(4)+\left (-8 x^2-12 x+32\right ) \log (4)+e^{2 x^2} \left (4 x^4+2 x^2\right )+\left (e^{x^2} \left (16 x^3+\left (-4 x^3-2 x\right ) \log (4)+8 x\right )-16 x+(2-2 x) \log ^2(4)+(12 x-16) \log (4)+32\right ) \log (x)+e^{x^2} \left (-8 x^4-32 x^3-8 x^2+\left (4 x^4+8 x^3+4 x^2+2 x\right ) \log (4)-8 x\right )+16 x-64}{16 x+x \log ^2(4)-8 x \log (4)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {8 x^2+\left (2 x^2+2 x-4\right ) \log ^2(4)+\left (-8 x^2-12 x+32\right ) \log (4)+e^{2 x^2} \left (4 x^4+2 x^2\right )+\left (e^{x^2} \left (16 x^3+\left (-4 x^3-2 x\right ) \log (4)+8 x\right )-16 x+(2-2 x) \log ^2(4)+(12 x-16) \log (4)+32\right ) \log (x)+e^{x^2} \left (-8 x^4-32 x^3-8 x^2+\left (4 x^4+8 x^3+4 x^2+2 x\right ) \log (4)-8 x\right )+16 x-64}{x \log ^2(4)+x (16-8 \log (4))}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {8 x^2+\left (2 x^2+2 x-4\right ) \log ^2(4)+\left (-8 x^2-12 x+32\right ) \log (4)+e^{2 x^2} \left (4 x^4+2 x^2\right )+\left (e^{x^2} \left (16 x^3+\left (-4 x^3-2 x\right ) \log (4)+8 x\right )-16 x+(2-2 x) \log ^2(4)+(12 x-16) \log (4)+32\right ) \log (x)+e^{x^2} \left (-8 x^4-32 x^3-8 x^2+\left (4 x^4+8 x^3+4 x^2+2 x\right ) \log (4)-8 x\right )+16 x-64}{x \left (16+\log ^2(4)-8 \log (4)\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {2 \left (-4 x^2-8 x-e^{2 x^2} \left (2 x^4+x^2\right )+e^{x^2} \left (4 x^4+16 x^3+4 x^2+4 x-\left (2 x^4+4 x^3+2 x^2+x\right ) \log (4)\right )-\left (-2 \log (4) (4-3 x)-8 x+e^{x^2} \left (8 x^3+4 x-\left (2 x^3+x\right ) \log (4)\right )+(1-x) \log ^2(4)+16\right ) \log (x)+\left (-x^2-x+2\right ) \log ^2(4)-2 \left (-2 x^2-3 x+8\right ) \log (4)+32\right )}{x}dx}{(4-\log (4))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \int \frac {-4 x^2-8 x-e^{2 x^2} \left (2 x^4+x^2\right )+e^{x^2} \left (4 x^4+16 x^3+4 x^2+4 x-\left (2 x^4+4 x^3+2 x^2+x\right ) \log (4)\right )-\left (-2 \log (4) (4-3 x)-8 x+e^{x^2} \left (8 x^3+4 x-\left (2 x^3+x\right ) \log (4)\right )+(1-x) \log ^2(4)+16\right ) \log (x)+\left (-x^2-x+2\right ) \log ^2(4)-2 \left (-2 x^2-3 x+8\right ) \log (4)+32}{x}dx}{(4-\log (4))^2}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle -\frac {2 \int \left (-e^{2 x^2} x \left (2 x^2+1\right )+\frac {(-((2-\log (4)) x)-\log (4)+4) \left (2 (1-\log (2)) x-4 \left (1-\frac {\log (2)}{2}\right ) \log (x)+8 \left (1-\frac {\log (2)}{2}\right )\right )}{x}+e^{x^2} \left (4 (1-\log (2)) x^3-8 \left (1-\frac {\log (2)}{2}\right ) \log (x) x^2+16 \left (1-\frac {\log (2)}{2}\right ) x^2+4 (1-\log (2)) x-4 \left (1-\frac {\log (2)}{2}\right ) \log (x)+4 \left (1-\frac {\log (2)}{2}\right )\right )\right )dx}{(4-\log (4))^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (-\frac {1}{2} e^{2 x^2} x^2+\frac {2 e^{x^2} \left (x^3 (1-\log (2))-x^2 (2-\log (2)) \log (x)+2 x^2 (2-\log (2))\right )}{x}-2 (x (1-\log (2))-(2-\log (2)) \log (x)+4-\log (4))^2\right )}{(4-\log (4))^2}\) |
Int[(-64 + 16*x + 8*x^2 + E^(2*x^2)*(2*x^2 + 4*x^4) + (32 - 12*x - 8*x^2)* Log[4] + (-4 + 2*x + 2*x^2)*Log[4]^2 + E^x^2*(-8*x - 8*x^2 - 32*x^3 - 8*x^ 4 + (2*x + 4*x^2 + 8*x^3 + 4*x^4)*Log[4]) + (32 - 16*x + (-16 + 12*x)*Log[ 4] + (2 - 2*x)*Log[4]^2 + E^x^2*(8*x + 16*x^3 + (-2*x - 4*x^3)*Log[4]))*Lo g[x])/(16*x - 8*x*Log[4] + x*Log[4]^2),x]
(-2*(-1/2*(E^(2*x^2)*x^2) - 2*(4 + x*(1 - Log[2]) - Log[4] - (2 - Log[2])* Log[x])^2 + (2*E^x^2*(x^3*(1 - Log[2]) + 2*x^2*(2 - Log[2]) - x^2*(2 - Log [2])*Log[x]))/x))/(4 - Log[4])^2
3.12.98.3.1 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]]
Leaf count of result is larger than twice the leaf count of optimal. \(173\) vs. \(2(29)=58\).
Time = 0.74 (sec) , antiderivative size = 174, normalized size of antiderivative = 5.80
| method | result | size |
| parts | \(\frac {\left (\ln \left (2\right )-1\right ) x^{2} {\mathrm e}^{x^{2}}+\left (2 \ln \left (2\right )-4\right ) x \,{\mathrm e}^{x^{2}}+\left (2-\ln \left (2\right )\right ) x \,{\mathrm e}^{x^{2}} \ln \left (x \right )}{\left (\ln \left (2\right )-2\right )^{2}}+\frac {x^{2} \ln \left (2\right )^{2}+2 x \ln \left (2\right )^{2}-2 x^{2} \ln \left (2\right )-6 x \ln \left (2\right )+x^{2}+4 x +2 \left (-2 \ln \left (2\right )^{2}+8 \ln \left (2\right )-8\right ) \ln \left (x \right )}{\ln \left (2\right )^{2}-4 \ln \left (2\right )+4}+\frac {{\mathrm e}^{2 x^{2}} x^{2}}{4 \ln \left (2\right )^{2}-16 \ln \left (2\right )+16}-\frac {2 \left (\ln \left (2\right ) \left (x \ln \left (x \right )-x \right )-\frac {\ln \left (2\right ) \ln \left (x \right )^{2}}{2}-x \ln \left (x \right )+x +\ln \left (x \right )^{2}\right )}{\ln \left (2\right )-2}\) | \(174\) |
| default | \(\frac {\left (4 \ln \left (2\right )-4\right ) x^{2} {\mathrm e}^{x^{2}}+\left (8 \ln \left (2\right )-16\right ) x \,{\mathrm e}^{x^{2}}+\left (-4 \ln \left (2\right )+8\right ) x \,{\mathrm e}^{x^{2}} \ln \left (x \right )}{4 \left (\ln \left (2\right )-2\right )^{2}}+\frac {x^{2} \ln \left (2\right )^{2}+2 x \ln \left (2\right )^{2}-2 x^{2} \ln \left (2\right )-6 x \ln \left (2\right )+x^{2}+4 x +2 \left (-2 \ln \left (2\right )^{2}+8 \ln \left (2\right )-8\right ) \ln \left (x \right )}{\ln \left (2\right )^{2}-4 \ln \left (2\right )+4}+\frac {{\mathrm e}^{2 x^{2}} x^{2}}{4 \ln \left (2\right )^{2}-16 \ln \left (2\right )+16}-\frac {2 \left (\ln \left (2\right ) \left (x \ln \left (x \right )-x \right )-\frac {\ln \left (2\right ) \ln \left (x \right )^{2}}{2}-x \ln \left (x \right )+x +\ln \left (x \right )^{2}\right )}{\ln \left (2\right )-2}\) | \(177\) |
| parallelrisch | \(\frac {-8 \ln \left (x \right ) \ln \left (2\right )^{2} x -16 \ln \left (x \right ) \ln \left (2\right )^{2}+32 x +8 x \,{\mathrm e}^{x^{2}} \ln \left (x \right )+8 x \ln \left (2\right ) {\mathrm e}^{x^{2}}+4 \,{\mathrm e}^{x^{2}} \ln \left (2\right ) x^{2}+64 \ln \left (2\right ) \ln \left (x \right )+{\mathrm e}^{2 x^{2}} x^{2}+16 x \ln \left (2\right )^{2}-16 \,{\mathrm e}^{x^{2}} x +4 x^{2} \ln \left (2\right )^{2}-48 x \ln \left (2\right )-8 x^{2} \ln \left (2\right )-16 \ln \left (2\right ) \ln \left (x \right )^{2}-16 x \ln \left (x \right )-4 x \ln \left (2\right ) {\mathrm e}^{x^{2}} \ln \left (x \right )+24 x \ln \left (2\right ) \ln \left (x \right )-64 \ln \left (x \right )+16 \ln \left (x \right )^{2}+4 x^{2}-4 x^{2} {\mathrm e}^{x^{2}}+4 \ln \left (2\right )^{2} \ln \left (x \right )^{2}}{4 \ln \left (2\right )^{2}-16 \ln \left (2\right )+16}\) | \(181\) |
| risch | \(\ln \left (x \right )^{2}-\frac {x \left (2 \ln \left (2\right )+{\mathrm e}^{x^{2}}-2\right ) \ln \left (x \right )}{\ln \left (2\right )-2}+\frac {x^{2} {\mathrm e}^{2 x^{2}}}{4 \left (\ln \left (2\right )-2\right )^{2}}+\frac {{\mathrm e}^{x^{2}} \ln \left (2\right ) x^{2}}{\left (\ln \left (2\right )-2\right )^{2}}+\frac {\ln \left (2\right )^{2} x^{2}}{\left (\ln \left (2\right )-2\right )^{2}}+\frac {2 \,{\mathrm e}^{x^{2}} \ln \left (2\right ) x}{\left (\ln \left (2\right )-2\right )^{2}}-\frac {{\mathrm e}^{x^{2}} x^{2}}{\left (\ln \left (2\right )-2\right )^{2}}-\frac {4 \ln \left (x \right ) \ln \left (2\right )^{2}}{\left (\ln \left (2\right )-2\right )^{2}}+\frac {4 x \ln \left (2\right )^{2}}{\left (\ln \left (2\right )-2\right )^{2}}-\frac {2 \ln \left (2\right ) x^{2}}{\left (\ln \left (2\right )-2\right )^{2}}-\frac {4 \,{\mathrm e}^{x^{2}} x}{\left (\ln \left (2\right )-2\right )^{2}}+\frac {16 \ln \left (x \right ) \ln \left (2\right )}{\left (\ln \left (2\right )-2\right )^{2}}-\frac {12 x \ln \left (2\right )}{\left (\ln \left (2\right )-2\right )^{2}}+\frac {x^{2}}{\left (\ln \left (2\right )-2\right )^{2}}-\frac {16 \ln \left (x \right )}{\left (\ln \left (2\right )-2\right )^{2}}+\frac {8 x}{\left (\ln \left (2\right )-2\right )^{2}}\) | \(209\) |
int((((2*(-4*x^3-2*x)*ln(2)+16*x^3+8*x)*exp(x^2)+4*(2-2*x)*ln(2)^2+2*(12*x -16)*ln(2)-16*x+32)*ln(x)+(4*x^4+2*x^2)*exp(x^2)^2+(2*(4*x^4+8*x^3+4*x^2+2 *x)*ln(2)-8*x^4-32*x^3-8*x^2-8*x)*exp(x^2)+4*(2*x^2+2*x-4)*ln(2)^2+2*(-8*x ^2-12*x+32)*ln(2)+8*x^2+16*x-64)/(4*x*ln(2)^2-16*x*ln(2)+16*x),x,method=_R ETURNVERBOSE)
((ln(2)-1)*x^2*exp(x^2)+(2*ln(2)-4)*x*exp(x^2)+(2-ln(2))*x*exp(x^2)*ln(x)) /(ln(2)-2)^2+2/(ln(2)^2-4*ln(2)+4)*(1/2*x^2*ln(2)^2+x*ln(2)^2-x^2*ln(2)-3* x*ln(2)+1/2*x^2+2*x+(-2*ln(2)^2+8*ln(2)-8)*ln(x))+1/4/(ln(2)^2-4*ln(2)+4)* x^2*exp(x^2)^2-2/(ln(2)-2)*(ln(2)*(x*ln(x)-x)-1/2*ln(2)*ln(x)^2-x*ln(x)+x+ ln(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (30) = 60\).
Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 4.57 \[ \int \frac {-64+16 x+8 x^2+e^{2 x^2} \left (2 x^2+4 x^4\right )+\left (32-12 x-8 x^2\right ) \log (4)+\left (-4+2 x+2 x^2\right ) \log ^2(4)+e^{x^2} \left (-8 x-8 x^2-32 x^3-8 x^4+\left (2 x+4 x^2+8 x^3+4 x^4\right ) \log (4)\right )+\left (32-16 x+(-16+12 x) \log (4)+(2-2 x) \log ^2(4)+e^{x^2} \left (8 x+16 x^3+\left (-2 x-4 x^3\right ) \log (4)\right )\right ) \log (x)}{16 x-8 x \log (4)+x \log ^2(4)} \, dx=\frac {x^{2} e^{\left (2 \, x^{2}\right )} + 4 \, {\left (x^{2} + 4 \, x\right )} \log \left (2\right )^{2} + 4 \, {\left (\log \left (2\right )^{2} - 4 \, \log \left (2\right ) + 4\right )} \log \left (x\right )^{2} + 4 \, x^{2} - 4 \, {\left (x^{2} - {\left (x^{2} + 2 \, x\right )} \log \left (2\right ) + 4 \, x\right )} e^{\left (x^{2}\right )} - 8 \, {\left (x^{2} + 6 \, x\right )} \log \left (2\right ) - 4 \, {\left (2 \, {\left (x + 2\right )} \log \left (2\right )^{2} + {\left (x \log \left (2\right ) - 2 \, x\right )} e^{\left (x^{2}\right )} - 2 \, {\left (3 \, x + 8\right )} \log \left (2\right ) + 4 \, x + 16\right )} \log \left (x\right ) + 32 \, x}{4 \, {\left (\log \left (2\right )^{2} - 4 \, \log \left (2\right ) + 4\right )}} \]
integrate((((2*(-4*x^3-2*x)*log(2)+16*x^3+8*x)*exp(x^2)+4*(2-2*x)*log(2)^2 +2*(12*x-16)*log(2)-16*x+32)*log(x)+(4*x^4+2*x^2)*exp(x^2)^2+(2*(4*x^4+8*x ^3+4*x^2+2*x)*log(2)-8*x^4-32*x^3-8*x^2-8*x)*exp(x^2)+4*(2*x^2+2*x-4)*log( 2)^2+2*(-8*x^2-12*x+32)*log(2)+8*x^2+16*x-64)/(4*x*log(2)^2-16*x*log(2)+16 *x),x, algorithm=\
1/4*(x^2*e^(2*x^2) + 4*(x^2 + 4*x)*log(2)^2 + 4*(log(2)^2 - 4*log(2) + 4)* log(x)^2 + 4*x^2 - 4*(x^2 - (x^2 + 2*x)*log(2) + 4*x)*e^(x^2) - 8*(x^2 + 6 *x)*log(2) - 4*(2*(x + 2)*log(2)^2 + (x*log(2) - 2*x)*e^(x^2) - 2*(3*x + 8 )*log(2) + 4*x + 16)*log(x) + 32*x)/(log(2)^2 - 4*log(2) + 4)
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (24) = 48\).
Time = 0.47 (sec) , antiderivative size = 240, normalized size of antiderivative = 8.00 \[ \int \frac {-64+16 x+8 x^2+e^{2 x^2} \left (2 x^2+4 x^4\right )+\left (32-12 x-8 x^2\right ) \log (4)+\left (-4+2 x+2 x^2\right ) \log ^2(4)+e^{x^2} \left (-8 x-8 x^2-32 x^3-8 x^4+\left (2 x+4 x^2+8 x^3+4 x^4\right ) \log (4)\right )+\left (32-16 x+(-16+12 x) \log (4)+(2-2 x) \log ^2(4)+e^{x^2} \left (8 x+16 x^3+\left (-2 x-4 x^3\right ) \log (4)\right )\right ) \log (x)}{16 x-8 x \log (4)+x \log ^2(4)} \, dx=\frac {\left (- 2 x \log {\left (2 \right )} + 2 x\right ) \log {\left (x \right )}}{-2 + \log {\left (2 \right )}} + \frac {\left (- 4 x^{2} \log {\left (2 \right )} + x^{2} \log {\left (2 \right )}^{2} + 4 x^{2}\right ) e^{2 x^{2}} + \left (- 16 x^{2} - 20 x^{2} \log {\left (2 \right )}^{2} + 4 x^{2} \log {\left (2 \right )}^{3} + 32 x^{2} \log {\left (2 \right )} - 48 x \log {\left (2 \right )} \log {\left (x \right )} - 4 x \log {\left (2 \right )}^{3} \log {\left (x \right )} + 24 x \log {\left (2 \right )}^{2} \log {\left (x \right )} + 32 x \log {\left (x \right )} - 64 x - 48 x \log {\left (2 \right )}^{2} + 8 x \log {\left (2 \right )}^{3} + 96 x \log {\left (2 \right )}\right ) e^{x^{2}}}{- 128 \log {\left (2 \right )} - 32 \log {\left (2 \right )}^{3} + 4 \log {\left (2 \right )}^{4} + 96 \log {\left (2 \right )}^{2} + 64} + \frac {x^{2} \left (- 2 \log {\left (2 \right )} + \log {\left (2 \right )}^{2} + 1\right ) + x \left (- 12 \log {\left (2 \right )} + 4 \log {\left (2 \right )}^{2} + 8\right ) - 4 \left (-2 + \log {\left (2 \right )}\right )^{2} \log {\left (x \right )}}{- 4 \log {\left (2 \right )} + \log {\left (2 \right )}^{2} + 4} + \log {\left (x \right )}^{2} \]
integrate((((2*(-4*x**3-2*x)*ln(2)+16*x**3+8*x)*exp(x**2)+4*(2-2*x)*ln(2)* *2+2*(12*x-16)*ln(2)-16*x+32)*ln(x)+(4*x**4+2*x**2)*exp(x**2)**2+(2*(4*x** 4+8*x**3+4*x**2+2*x)*ln(2)-8*x**4-32*x**3-8*x**2-8*x)*exp(x**2)+4*(2*x**2+ 2*x-4)*ln(2)**2+2*(-8*x**2-12*x+32)*ln(2)+8*x**2+16*x-64)/(4*x*ln(2)**2-16 *x*ln(2)+16*x),x)
(-2*x*log(2) + 2*x)*log(x)/(-2 + log(2)) + ((-4*x**2*log(2) + x**2*log(2)* *2 + 4*x**2)*exp(2*x**2) + (-16*x**2 - 20*x**2*log(2)**2 + 4*x**2*log(2)** 3 + 32*x**2*log(2) - 48*x*log(2)*log(x) - 4*x*log(2)**3*log(x) + 24*x*log( 2)**2*log(x) + 32*x*log(x) - 64*x - 48*x*log(2)**2 + 8*x*log(2)**3 + 96*x* log(2))*exp(x**2))/(-128*log(2) - 32*log(2)**3 + 4*log(2)**4 + 96*log(2)** 2 + 64) + (x**2*(-2*log(2) + log(2)**2 + 1) + x*(-12*log(2) + 4*log(2)**2 + 8) - 4*(-2 + log(2))**2*log(x))/(-4*log(2) + log(2)**2 + 4) + log(x)**2
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.34 (sec) , antiderivative size = 728, normalized size of antiderivative = 24.27 \[ \int \frac {-64+16 x+8 x^2+e^{2 x^2} \left (2 x^2+4 x^4\right )+\left (32-12 x-8 x^2\right ) \log (4)+\left (-4+2 x+2 x^2\right ) \log ^2(4)+e^{x^2} \left (-8 x-8 x^2-32 x^3-8 x^4+\left (2 x+4 x^2+8 x^3+4 x^4\right ) \log (4)\right )+\left (32-16 x+(-16+12 x) \log (4)+(2-2 x) \log ^2(4)+e^{x^2} \left (8 x+16 x^3+\left (-2 x-4 x^3\right ) \log (4)\right )\right ) \log (x)}{16 x-8 x \log (4)+x \log ^2(4)} \, dx=\text {Too large to display} \]
integrate((((2*(-4*x^3-2*x)*log(2)+16*x^3+8*x)*exp(x^2)+4*(2-2*x)*log(2)^2 +2*(12*x-16)*log(2)-16*x+32)*log(x)+(4*x^4+2*x^2)*exp(x^2)^2+(2*(4*x^4+8*x ^3+4*x^2+2*x)*log(2)-8*x^4-32*x^3-8*x^2-8*x)*exp(x^2)+4*(2*x^2+2*x-4)*log( 2)^2+2*(-8*x^2-12*x+32)*log(2)+8*x^2+16*x-64)/(4*x*log(2)^2-16*x*log(2)+16 *x),x, algorithm=\
x^2*log(2)^2/(log(2)^2 - 4*log(2) + 4) - (2*log(x*log(2)^2 - 4*x*log(2) + 4*x)*log(x)/(log(2)^2 - 4*log(2) + 4) - log(x)^2/(log(2)^2 - 4*log(2) + 4) )*log(2)^2 - 2*x*log(2)^2*log(x)/(log(2)^2 - 4*log(2) + 4) + 2*log(2)^2*lo g(x*log(2)^2 - 4*x*log(2) + 4*x)*log(x)/(log(2)^2 - 4*log(2) + 4) - 2*x^2* log(2)/(log(2)^2 - 4*log(2) + 4) + (2*x*e^(x^2)/(log(2)^2 - 4*log(2) + 4) + I*sqrt(pi)*erf(I*x)/(log(2)^2 - 4*log(2) + 4))*log(2) + 4*(2*log(x*log(2 )^2 - 4*x*log(2) + 4*x)*log(x)/(log(2)^2 - 4*log(2) + 4) - log(x)^2/(log(2 )^2 - 4*log(2) + 4))*log(2) + (x^2 - 1)*e^(x^2)*log(2)/(log(2)^2 - 4*log(2 ) + 4) + 4*x*log(2)^2/(log(2)^2 - 4*log(2) + 4) - 4*log(2)^2*log(x*log(2)^ 2 - 4*x*log(2) + 4*x)/(log(2)^2 - 4*log(2) + 4) - x*e^(x^2)*log(x)/(log(2) - 2) + 6*x*log(2)*log(x)/(log(2)^2 - 4*log(2) + 4) - 8*log(2)*log(x*log(2 )^2 - 4*x*log(2) + 4*x)*log(x)/(log(2)^2 - 4*log(2) + 4) - 1/2*I*sqrt(pi)* erf(I*x)*log(2)/(log(2)^2 - 4*log(2) + 4) + x^2/(log(2)^2 - 4*log(2) + 4) + 1/8*(2*x^2 - 1)*e^(2*x^2)/(log(2)^2 - 4*log(2) + 4) - (x^2 - 1)*e^(x^2)/ (log(2)^2 - 4*log(2) + 4) - 4*x*e^(x^2)/(log(2)^2 - 4*log(2) + 4) - 12*x*l og(2)/(log(2)^2 - 4*log(2) + 4) + e^(x^2)*log(2)/(log(2)^2 - 4*log(2) + 4) + 16*log(2)*log(x*log(2)^2 - 4*x*log(2) + 4*x)/(log(2)^2 - 4*log(2) + 4) - 4*x*log(x)/(log(2)^2 - 4*log(2) + 4) + 4*log(x)^2/(log(2)^2 - 4*log(2) + 4) - I*sqrt(pi)*erf(I*x)/(log(2)^2 - 4*log(2) + 4) - 1/2*I*sqrt(pi)*erf(I *x)/(log(2) - 2) + 8*x/(log(2)^2 - 4*log(2) + 4) + 1/8*e^(2*x^2)/(log(2...
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (30) = 60\).
Time = 0.29 (sec) , antiderivative size = 180, normalized size of antiderivative = 6.00 \[ \int \frac {-64+16 x+8 x^2+e^{2 x^2} \left (2 x^2+4 x^4\right )+\left (32-12 x-8 x^2\right ) \log (4)+\left (-4+2 x+2 x^2\right ) \log ^2(4)+e^{x^2} \left (-8 x-8 x^2-32 x^3-8 x^4+\left (2 x+4 x^2+8 x^3+4 x^4\right ) \log (4)\right )+\left (32-16 x+(-16+12 x) \log (4)+(2-2 x) \log ^2(4)+e^{x^2} \left (8 x+16 x^3+\left (-2 x-4 x^3\right ) \log (4)\right )\right ) \log (x)}{16 x-8 x \log (4)+x \log ^2(4)} \, dx=\frac {4 \, x^{2} e^{\left (x^{2}\right )} \log \left (2\right ) + 4 \, x^{2} \log \left (2\right )^{2} - 4 \, x e^{\left (x^{2}\right )} \log \left (2\right ) \log \left (x\right ) - 8 \, x \log \left (2\right )^{2} \log \left (x\right ) + 4 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + x^{2} e^{\left (2 \, x^{2}\right )} - 4 \, x^{2} e^{\left (x^{2}\right )} - 8 \, x^{2} \log \left (2\right ) + 8 \, x e^{\left (x^{2}\right )} \log \left (2\right ) + 16 \, x \log \left (2\right )^{2} + 8 \, x e^{\left (x^{2}\right )} \log \left (x\right ) + 24 \, x \log \left (2\right ) \log \left (x\right ) - 16 \, \log \left (2\right )^{2} \log \left (x\right ) - 16 \, \log \left (2\right ) \log \left (x\right )^{2} + 4 \, x^{2} - 16 \, x e^{\left (x^{2}\right )} - 48 \, x \log \left (2\right ) - 16 \, x \log \left (x\right ) + 64 \, \log \left (2\right ) \log \left (x\right ) + 16 \, \log \left (x\right )^{2} + 32 \, x - 64 \, \log \left (x\right )}{4 \, {\left (\log \left (2\right )^{2} - 4 \, \log \left (2\right ) + 4\right )}} \]
integrate((((2*(-4*x^3-2*x)*log(2)+16*x^3+8*x)*exp(x^2)+4*(2-2*x)*log(2)^2 +2*(12*x-16)*log(2)-16*x+32)*log(x)+(4*x^4+2*x^2)*exp(x^2)^2+(2*(4*x^4+8*x ^3+4*x^2+2*x)*log(2)-8*x^4-32*x^3-8*x^2-8*x)*exp(x^2)+4*(2*x^2+2*x-4)*log( 2)^2+2*(-8*x^2-12*x+32)*log(2)+8*x^2+16*x-64)/(4*x*log(2)^2-16*x*log(2)+16 *x),x, algorithm=\
1/4*(4*x^2*e^(x^2)*log(2) + 4*x^2*log(2)^2 - 4*x*e^(x^2)*log(2)*log(x) - 8 *x*log(2)^2*log(x) + 4*log(2)^2*log(x)^2 + x^2*e^(2*x^2) - 4*x^2*e^(x^2) - 8*x^2*log(2) + 8*x*e^(x^2)*log(2) + 16*x*log(2)^2 + 8*x*e^(x^2)*log(x) + 24*x*log(2)*log(x) - 16*log(2)^2*log(x) - 16*log(2)*log(x)^2 + 4*x^2 - 16* x*e^(x^2) - 48*x*log(2) - 16*x*log(x) + 64*log(2)*log(x) + 16*log(x)^2 + 3 2*x - 64*log(x))/(log(2)^2 - 4*log(2) + 4)
Time = 13.33 (sec) , antiderivative size = 216, normalized size of antiderivative = 7.20 \[ \int \frac {-64+16 x+8 x^2+e^{2 x^2} \left (2 x^2+4 x^4\right )+\left (32-12 x-8 x^2\right ) \log (4)+\left (-4+2 x+2 x^2\right ) \log ^2(4)+e^{x^2} \left (-8 x-8 x^2-32 x^3-8 x^4+\left (2 x+4 x^2+8 x^3+4 x^4\right ) \log (4)\right )+\left (32-16 x+(-16+12 x) \log (4)+(2-2 x) \log ^2(4)+e^{x^2} \left (8 x+16 x^3+\left (-2 x-4 x^3\right ) \log (4)\right )\right ) \log (x)}{16 x-8 x \log (4)+x \log ^2(4)} \, dx=\frac {4\,x}{{\ln \left (2\right )}^2-\ln \left (16\right )+4}+\frac {x^2\,{\mathrm {e}}^{x^2}\,\left (\ln \left (2\right )-1\right )+x\,{\mathrm {e}}^{x^2}\,\left (\ln \left (4\right )-4\right )}{{\ln \left (2\right )}^2-\ln \left (16\right )+4}+\frac {x^2}{{\ln \left (2\right )}^2-\ln \left (16\right )+4}-\frac {16\,\ln \left (x\right )}{{\ln \left (2\right )}^2-\ln \left (16\right )+4}+\frac {x^3\,\left (\ln \left (4\right )-2\right )+x^2\,{\ln \left (x\right )}^2\,\left (\ln \left (2\right )-2\right )-x^3\,{\mathrm {e}}^{x^2}\,\ln \left (x\right )-x^3\,\ln \left (x\right )\,\left (\ln \left (4\right )-2\right )}{x^2\,\left (\ln \left (2\right )-2\right )}-\frac {2\,\ln \left (2\right )\,\left (3\,x-8\,\ln \left (x\right )+x^2\right )}{{\ln \left (2\right )}^2-\ln \left (16\right )+4}+\frac {x^2\,{\mathrm {e}}^{2\,x^2}}{2\,\left (2\,{\ln \left (2\right )}^2-\ln \left (256\right )+8\right )}+\frac {{\ln \left (2\right )}^2\,\left (2\,x-4\,\ln \left (x\right )+x^2\right )}{{\ln \left (2\right )}^2-\ln \left (16\right )+4} \]
int((16*x - exp(x^2)*(8*x - 2*log(2)*(2*x + 4*x^2 + 8*x^3 + 4*x^4) + 8*x^2 + 32*x^3 + 8*x^4) + log(x)*(2*log(2)*(12*x - 16) - 16*x + exp(x^2)*(8*x - 2*log(2)*(2*x + 4*x^3) + 16*x^3) - 4*log(2)^2*(2*x - 2) + 32) - 2*log(2)* (12*x + 8*x^2 - 32) + 4*log(2)^2*(2*x + 2*x^2 - 4) + exp(2*x^2)*(2*x^2 + 4 *x^4) + 8*x^2 - 64)/(16*x - 16*x*log(2) + 4*x*log(2)^2),x)
(4*x)/(log(2)^2 - log(16) + 4) + (x^2*exp(x^2)*(log(2) - 1) + x*exp(x^2)*( log(4) - 4))/(log(2)^2 - log(16) + 4) + x^2/(log(2)^2 - log(16) + 4) - (16 *log(x))/(log(2)^2 - log(16) + 4) + (x^3*(log(4) - 2) + x^2*log(x)^2*(log( 2) - 2) - x^3*exp(x^2)*log(x) - x^3*log(x)*(log(4) - 2))/(x^2*(log(2) - 2) ) - (2*log(2)*(3*x - 8*log(x) + x^2))/(log(2)^2 - log(16) + 4) + (x^2*exp( 2*x^2))/(2*(2*log(2)^2 - log(256) + 8)) + (log(2)^2*(2*x - 4*log(x) + x^2) )/(log(2)^2 - log(16) + 4)