Integrand size = 258, antiderivative size = 35 \[ \int \frac {16 e^{4+x} x+e^4 \left (8 x-56 x^2+64 x^3-8 x^5\right )+e^4 \left (32 x-32 x^2-8 x^3+8 x^4\right ) \log (4)+e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+\left (16 e^{4+x}+e^4 \left (8-56 x+64 x^2-8 x^4\right )+e^4 \left (32-32 x-8 x^2+8 x^3\right ) \log (4)+e^4 \left (4 x-2 x^2\right ) \log ^2(4)\right ) \log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)} \, dx=e^4 \left (x+\log \left (4 \left (e^x+x-(2-x)^2 \left (x-\frac {\log (4)}{2}\right )^2\right )\right )\right )^2 \]
\[ \int \frac {16 e^{4+x} x+e^4 \left (8 x-56 x^2+64 x^3-8 x^5\right )+e^4 \left (32 x-32 x^2-8 x^3+8 x^4\right ) \log (4)+e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+\left (16 e^{4+x}+e^4 \left (8-56 x+64 x^2-8 x^4\right )+e^4 \left (32-32 x-8 x^2+8 x^3\right ) \log (4)+e^4 \left (4 x-2 x^2\right ) \log ^2(4)\right ) \log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)} \, dx=\int \frac {16 e^{4+x} x+e^4 \left (8 x-56 x^2+64 x^3-8 x^5\right )+e^4 \left (32 x-32 x^2-8 x^3+8 x^4\right ) \log (4)+e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+\left (16 e^{4+x}+e^4 \left (8-56 x+64 x^2-8 x^4\right )+e^4 \left (32-32 x-8 x^2+8 x^3\right ) \log (4)+e^4 \left (4 x-2 x^2\right ) \log ^2(4)\right ) \log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)} \, dx \]
Integrate[(16*E^(4 + x)*x + E^4*(8*x - 56*x^2 + 64*x^3 - 8*x^5) + E^4*(32* x - 32*x^2 - 8*x^3 + 8*x^4)*Log[4] + E^4*(4*x^2 - 2*x^3)*Log[4]^2 + (16*E^ (4 + x) + E^4*(8 - 56*x + 64*x^2 - 8*x^4) + E^4*(32 - 32*x - 8*x^2 + 8*x^3 )*Log[4] + E^4*(4*x - 2*x^2)*Log[4]^2)*Log[4*E^x + 4*x - 16*x^2 + 16*x^3 - 4*x^4 + (16*x - 16*x^2 + 4*x^3)*Log[4] + (-4 + 4*x - x^2)*Log[4]^2])/(4*E ^x + 4*x - 16*x^2 + 16*x^3 - 4*x^4 + (16*x - 16*x^2 + 4*x^3)*Log[4] + (-4 + 4*x - x^2)*Log[4]^2),x]
Integrate[(16*E^(4 + x)*x + E^4*(8*x - 56*x^2 + 64*x^3 - 8*x^5) + E^4*(32* x - 32*x^2 - 8*x^3 + 8*x^4)*Log[4] + E^4*(4*x^2 - 2*x^3)*Log[4]^2 + (16*E^ (4 + x) + E^4*(8 - 56*x + 64*x^2 - 8*x^4) + E^4*(32 - 32*x - 8*x^2 + 8*x^3 )*Log[4] + E^4*(4*x - 2*x^2)*Log[4]^2)*Log[4*E^x + 4*x - 16*x^2 + 16*x^3 - 4*x^4 + (16*x - 16*x^2 + 4*x^3)*Log[4] + (-4 + 4*x - x^2)*Log[4]^2])/(4*E ^x + 4*x - 16*x^2 + 16*x^3 - 4*x^4 + (16*x - 16*x^2 + 4*x^3)*Log[4] + (-4 + 4*x - x^2)*Log[4]^2), x]
Time = 1.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.66, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {7239, 27, 7237}
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.
\(\displaystyle \int \frac {e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+e^4 \left (-8 x^5+64 x^3-56 x^2+8 x\right )+\left (e^4 \left (4 x-2 x^2\right ) \log ^2(4)+e^4 \left (-8 x^4+64 x^2-56 x+8\right )+e^4 \left (8 x^3-8 x^2-32 x+32\right ) \log (4)+16 e^{x+4}\right ) \log \left (-4 x^4+16 x^3-16 x^2+\left (-x^2+4 x-4\right ) \log ^2(4)+\left (4 x^3-16 x^2+16 x\right ) \log (4)+4 x+4 e^x\right )+e^4 \left (8 x^4-8 x^3-32 x^2+32 x\right ) \log (4)+16 e^{x+4} x}{-4 x^4+16 x^3-16 x^2+\left (-x^2+4 x-4\right ) \log ^2(4)+\left (4 x^3-16 x^2+16 x\right ) \log (4)+4 x+4 e^x} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 e^4 \left (-4 x^4+4 x^3 \log (4)-x^2 \left (-32+\log ^2(4)+\log (256)\right )+8 e^x+2 x \left (-14+\log ^2(4)-8 \log (4)\right )+4 (1+\log (256))\right ) \left (\log \left (-4 x^4+16 x^3-16 x^2+4 x+4 e^x-(x-2)^2 \log ^2(4)+4 (x-2)^2 x \log (4)\right )+x\right )}{-4 x^4+16 x^3-16 x^2+4 x+4 e^x-(x-2)^2 \log ^2(4)+4 (x-2)^2 x \log (4)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 e^4 \int \frac {\left (-4 x^4+4 \log (4) x^3+\left (32-\log ^2(4)-\log (256)\right ) x^2-2 \left (14+8 \log (4)-\log ^2(4)\right ) x+8 e^x+4 (1+\log (256))\right ) \left (x+\log \left (-4 x^4+16 x^3-16 x^2+4 (2-x)^2 \log (4) x+4 x+4 e^x-(2-x)^2 \log ^2(4)\right )\right )}{-4 x^4+16 x^3-16 x^2+4 (2-x)^2 \log (4) x+4 x+4 e^x-(2-x)^2 \log ^2(4)}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle e^4 \left (\log \left (-4 x^4+16 x^3-16 x^2+4 x+4 e^x-(2-x)^2 \log ^2(4)+4 (2-x)^2 x \log (4)\right )+x\right )^2\) |
Int[(16*E^(4 + x)*x + E^4*(8*x - 56*x^2 + 64*x^3 - 8*x^5) + E^4*(32*x - 32 *x^2 - 8*x^3 + 8*x^4)*Log[4] + E^4*(4*x^2 - 2*x^3)*Log[4]^2 + (16*E^(4 + x ) + E^4*(8 - 56*x + 64*x^2 - 8*x^4) + E^4*(32 - 32*x - 8*x^2 + 8*x^3)*Log[ 4] + E^4*(4*x - 2*x^2)*Log[4]^2)*Log[4*E^x + 4*x - 16*x^2 + 16*x^3 - 4*x^4 + (16*x - 16*x^2 + 4*x^3)*Log[4] + (-4 + 4*x - x^2)*Log[4]^2])/(4*E^x + 4 *x - 16*x^2 + 16*x^3 - 4*x^4 + (16*x - 16*x^2 + 4*x^3)*Log[4] + (-4 + 4*x - x^2)*Log[4]^2),x]
E^4*(x + Log[4*E^x + 4*x - 16*x^2 + 16*x^3 - 4*x^4 + 4*(2 - x)^2*x*Log[4] - (2 - x)^2*Log[4]^2])^2
3.9.7.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
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. \(133\) vs. \(2(33)=66\).
Time = 5.23 (sec) , antiderivative size = 134, normalized size of antiderivative = 3.83
method | result | size |
risch | \(x^{2} {\mathrm e}^{4}+2 \,{\mathrm e}^{4} \ln \left (4 \,{\mathrm e}^{x}+4 \left (-x^{2}+4 x -4\right ) \ln \left (2\right )^{2}+2 \left (4 x^{3}-16 x^{2}+16 x \right ) \ln \left (2\right )-4 x^{4}+16 x^{3}-16 x^{2}+4 x \right ) x +{\mathrm e}^{4} \ln \left (4 \,{\mathrm e}^{x}+4 \left (-x^{2}+4 x -4\right ) \ln \left (2\right )^{2}+2 \left (4 x^{3}-16 x^{2}+16 x \right ) \ln \left (2\right )-4 x^{4}+16 x^{3}-16 x^{2}+4 x \right )^{2}\) | \(134\) |
parallelrisch | \(x^{2} {\mathrm e}^{4}+2 \,{\mathrm e}^{4} \ln \left (4 \,{\mathrm e}^{x}+4 \left (-x^{2}+4 x -4\right ) \ln \left (2\right )^{2}+2 \left (4 x^{3}-16 x^{2}+16 x \right ) \ln \left (2\right )-4 x^{4}+16 x^{3}-16 x^{2}+4 x \right ) x +{\mathrm e}^{4} \ln \left (4 \,{\mathrm e}^{x}+4 \left (-x^{2}+4 x -4\right ) \ln \left (2\right )^{2}+2 \left (4 x^{3}-16 x^{2}+16 x \right ) \ln \left (2\right )-4 x^{4}+16 x^{3}-16 x^{2}+4 x \right )^{2}\) | \(134\) |
int(((16*exp(4)*exp(x)+4*(-2*x^2+4*x)*exp(4)*ln(2)^2+2*(8*x^3-8*x^2-32*x+3 2)*exp(4)*ln(2)+(-8*x^4+64*x^2-56*x+8)*exp(4))*ln(4*exp(x)+4*(-x^2+4*x-4)* ln(2)^2+2*(4*x^3-16*x^2+16*x)*ln(2)-4*x^4+16*x^3-16*x^2+4*x)+16*x*exp(4)*e xp(x)+4*(-2*x^3+4*x^2)*exp(4)*ln(2)^2+2*(8*x^4-8*x^3-32*x^2+32*x)*exp(4)*l n(2)+(-8*x^5+64*x^3-56*x^2+8*x)*exp(4))/(4*exp(x)+4*(-x^2+4*x-4)*ln(2)^2+2 *(4*x^3-16*x^2+16*x)*ln(2)-4*x^4+16*x^3-16*x^2+4*x),x,method=_RETURNVERBOS E)
x^2*exp(4)+2*exp(4)*ln(4*exp(x)+4*(-x^2+4*x-4)*ln(2)^2+2*(4*x^3-16*x^2+16* x)*ln(2)-4*x^4+16*x^3-16*x^2+4*x)*x+exp(4)*ln(4*exp(x)+4*(-x^2+4*x-4)*ln(2 )^2+2*(4*x^3-16*x^2+16*x)*ln(2)-4*x^4+16*x^3-16*x^2+4*x)^2
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (31) = 62\).
Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 4.20 \[ \int \frac {16 e^{4+x} x+e^4 \left (8 x-56 x^2+64 x^3-8 x^5\right )+e^4 \left (32 x-32 x^2-8 x^3+8 x^4\right ) \log (4)+e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+\left (16 e^{4+x}+e^4 \left (8-56 x+64 x^2-8 x^4\right )+e^4 \left (32-32 x-8 x^2+8 x^3\right ) \log (4)+e^4 \left (4 x-2 x^2\right ) \log ^2(4)\right ) \log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)} \, dx=x^{2} e^{4} + 2 \, x e^{4} \log \left (-4 \, {\left ({\left (x^{2} - 4 \, x + 4\right )} e^{4} \log \left (2\right )^{2} - 2 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{4} \log \left (2\right ) + {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2} - x\right )} e^{4} - e^{\left (x + 4\right )}\right )} e^{\left (-4\right )}\right ) + e^{4} \log \left (-4 \, {\left ({\left (x^{2} - 4 \, x + 4\right )} e^{4} \log \left (2\right )^{2} - 2 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{4} \log \left (2\right ) + {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2} - x\right )} e^{4} - e^{\left (x + 4\right )}\right )} e^{\left (-4\right )}\right )^{2} \]
integrate(((16*exp(4)*exp(x)+4*(-2*x^2+4*x)*exp(4)*log(2)^2+2*(8*x^3-8*x^2 -32*x+32)*exp(4)*log(2)+(-8*x^4+64*x^2-56*x+8)*exp(4))*log(4*exp(x)+4*(-x^ 2+4*x-4)*log(2)^2+2*(4*x^3-16*x^2+16*x)*log(2)-4*x^4+16*x^3-16*x^2+4*x)+16 *x*exp(4)*exp(x)+4*(-2*x^3+4*x^2)*exp(4)*log(2)^2+2*(8*x^4-8*x^3-32*x^2+32 *x)*exp(4)*log(2)+(-8*x^5+64*x^3-56*x^2+8*x)*exp(4))/(4*exp(x)+4*(-x^2+4*x -4)*log(2)^2+2*(4*x^3-16*x^2+16*x)*log(2)-4*x^4+16*x^3-16*x^2+4*x),x, algo rithm=\
x^2*e^4 + 2*x*e^4*log(-4*((x^2 - 4*x + 4)*e^4*log(2)^2 - 2*(x^3 - 4*x^2 + 4*x)*e^4*log(2) + (x^4 - 4*x^3 + 4*x^2 - x)*e^4 - e^(x + 4))*e^(-4)) + e^4 *log(-4*((x^2 - 4*x + 4)*e^4*log(2)^2 - 2*(x^3 - 4*x^2 + 4*x)*e^4*log(2) + (x^4 - 4*x^3 + 4*x^2 - x)*e^4 - e^(x + 4))*e^(-4))^2
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (29) = 58\).
Time = 0.36 (sec) , antiderivative size = 133, normalized size of antiderivative = 3.80 \[ \int \frac {16 e^{4+x} x+e^4 \left (8 x-56 x^2+64 x^3-8 x^5\right )+e^4 \left (32 x-32 x^2-8 x^3+8 x^4\right ) \log (4)+e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+\left (16 e^{4+x}+e^4 \left (8-56 x+64 x^2-8 x^4\right )+e^4 \left (32-32 x-8 x^2+8 x^3\right ) \log (4)+e^4 \left (4 x-2 x^2\right ) \log ^2(4)\right ) \log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)} \, dx=x^{2} e^{4} + 2 x e^{4} \log {\left (- 4 x^{4} + 16 x^{3} - 16 x^{2} + 4 x + \left (- 4 x^{2} + 16 x - 16\right ) \log {\left (2 \right )}^{2} + \left (8 x^{3} - 32 x^{2} + 32 x\right ) \log {\left (2 \right )} + 4 e^{x} \right )} + e^{4} \log {\left (- 4 x^{4} + 16 x^{3} - 16 x^{2} + 4 x + \left (- 4 x^{2} + 16 x - 16\right ) \log {\left (2 \right )}^{2} + \left (8 x^{3} - 32 x^{2} + 32 x\right ) \log {\left (2 \right )} + 4 e^{x} \right )}^{2} \]
integrate(((16*exp(4)*exp(x)+4*(-2*x**2+4*x)*exp(4)*ln(2)**2+2*(8*x**3-8*x **2-32*x+32)*exp(4)*ln(2)+(-8*x**4+64*x**2-56*x+8)*exp(4))*ln(4*exp(x)+4*( -x**2+4*x-4)*ln(2)**2+2*(4*x**3-16*x**2+16*x)*ln(2)-4*x**4+16*x**3-16*x**2 +4*x)+16*x*exp(4)*exp(x)+4*(-2*x**3+4*x**2)*exp(4)*ln(2)**2+2*(8*x**4-8*x* *3-32*x**2+32*x)*exp(4)*ln(2)+(-8*x**5+64*x**3-56*x**2+8*x)*exp(4))/(4*exp (x)+4*(-x**2+4*x-4)*ln(2)**2+2*(4*x**3-16*x**2+16*x)*ln(2)-4*x**4+16*x**3- 16*x**2+4*x),x)
x**2*exp(4) + 2*x*exp(4)*log(-4*x**4 + 16*x**3 - 16*x**2 + 4*x + (-4*x**2 + 16*x - 16)*log(2)**2 + (8*x**3 - 32*x**2 + 32*x)*log(2) + 4*exp(x)) + ex p(4)*log(-4*x**4 + 16*x**3 - 16*x**2 + 4*x + (-4*x**2 + 16*x - 16)*log(2)* *2 + (8*x**3 - 32*x**2 + 32*x)*log(2) + 4*exp(x))**2
Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (31) = 62\).
Time = 0.33 (sec) , antiderivative size = 138, normalized size of antiderivative = 3.94 \[ \int \frac {16 e^{4+x} x+e^4 \left (8 x-56 x^2+64 x^3-8 x^5\right )+e^4 \left (32 x-32 x^2-8 x^3+8 x^4\right ) \log (4)+e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+\left (16 e^{4+x}+e^4 \left (8-56 x+64 x^2-8 x^4\right )+e^4 \left (32-32 x-8 x^2+8 x^3\right ) \log (4)+e^4 \left (4 x-2 x^2\right ) \log ^2(4)\right ) \log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)} \, dx=x^{2} e^{4} + 4 \, x e^{4} \log \left (2\right ) + e^{4} \log \left (-x^{4} + 2 \, x^{3} {\left (\log \left (2\right ) + 2\right )} - {\left (\log \left (2\right )^{2} + 8 \, \log \left (2\right ) + 4\right )} x^{2} + {\left (4 \, \log \left (2\right )^{2} + 8 \, \log \left (2\right ) + 1\right )} x - 4 \, \log \left (2\right )^{2} + e^{x}\right )^{2} + 2 \, {\left (x e^{4} + 2 \, e^{4} \log \left (2\right )\right )} \log \left (-x^{4} + 2 \, x^{3} {\left (\log \left (2\right ) + 2\right )} - {\left (\log \left (2\right )^{2} + 8 \, \log \left (2\right ) + 4\right )} x^{2} + {\left (4 \, \log \left (2\right )^{2} + 8 \, \log \left (2\right ) + 1\right )} x - 4 \, \log \left (2\right )^{2} + e^{x}\right ) \]
integrate(((16*exp(4)*exp(x)+4*(-2*x^2+4*x)*exp(4)*log(2)^2+2*(8*x^3-8*x^2 -32*x+32)*exp(4)*log(2)+(-8*x^4+64*x^2-56*x+8)*exp(4))*log(4*exp(x)+4*(-x^ 2+4*x-4)*log(2)^2+2*(4*x^3-16*x^2+16*x)*log(2)-4*x^4+16*x^3-16*x^2+4*x)+16 *x*exp(4)*exp(x)+4*(-2*x^3+4*x^2)*exp(4)*log(2)^2+2*(8*x^4-8*x^3-32*x^2+32 *x)*exp(4)*log(2)+(-8*x^5+64*x^3-56*x^2+8*x)*exp(4))/(4*exp(x)+4*(-x^2+4*x -4)*log(2)^2+2*(4*x^3-16*x^2+16*x)*log(2)-4*x^4+16*x^3-16*x^2+4*x),x, algo rithm=\
x^2*e^4 + 4*x*e^4*log(2) + e^4*log(-x^4 + 2*x^3*(log(2) + 2) - (log(2)^2 + 8*log(2) + 4)*x^2 + (4*log(2)^2 + 8*log(2) + 1)*x - 4*log(2)^2 + e^x)^2 + 2*(x*e^4 + 2*e^4*log(2))*log(-x^4 + 2*x^3*(log(2) + 2) - (log(2)^2 + 8*lo g(2) + 4)*x^2 + (4*log(2)^2 + 8*log(2) + 1)*x - 4*log(2)^2 + e^x)
\[ \int \frac {16 e^{4+x} x+e^4 \left (8 x-56 x^2+64 x^3-8 x^5\right )+e^4 \left (32 x-32 x^2-8 x^3+8 x^4\right ) \log (4)+e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+\left (16 e^{4+x}+e^4 \left (8-56 x+64 x^2-8 x^4\right )+e^4 \left (32-32 x-8 x^2+8 x^3\right ) \log (4)+e^4 \left (4 x-2 x^2\right ) \log ^2(4)\right ) \log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)} \, dx=\int { \frac {2 \, {\left ({\left (x^{3} - 2 \, x^{2}\right )} e^{4} \log \left (2\right )^{2} - 2 \, {\left (x^{4} - x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{4} \log \left (2\right ) + {\left (x^{5} - 8 \, x^{3} + 7 \, x^{2} - x\right )} e^{4} - 2 \, x e^{\left (x + 4\right )} + {\left ({\left (x^{2} - 2 \, x\right )} e^{4} \log \left (2\right )^{2} - 2 \, {\left (x^{3} - x^{2} - 4 \, x + 4\right )} e^{4} \log \left (2\right ) + {\left (x^{4} - 8 \, x^{2} + 7 \, x - 1\right )} e^{4} - 2 \, e^{\left (x + 4\right )}\right )} \log \left (-4 \, x^{4} + 16 \, x^{3} - 4 \, {\left (x^{2} - 4 \, x + 4\right )} \log \left (2\right )^{2} - 16 \, x^{2} + 8 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \left (2\right ) + 4 \, x + 4 \, e^{x}\right )\right )}}{x^{4} - 4 \, x^{3} + {\left (x^{2} - 4 \, x + 4\right )} \log \left (2\right )^{2} + 4 \, x^{2} - 2 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} \log \left (2\right ) - x - e^{x}} \,d x } \]
integrate(((16*exp(4)*exp(x)+4*(-2*x^2+4*x)*exp(4)*log(2)^2+2*(8*x^3-8*x^2 -32*x+32)*exp(4)*log(2)+(-8*x^4+64*x^2-56*x+8)*exp(4))*log(4*exp(x)+4*(-x^ 2+4*x-4)*log(2)^2+2*(4*x^3-16*x^2+16*x)*log(2)-4*x^4+16*x^3-16*x^2+4*x)+16 *x*exp(4)*exp(x)+4*(-2*x^3+4*x^2)*exp(4)*log(2)^2+2*(8*x^4-8*x^3-32*x^2+32 *x)*exp(4)*log(2)+(-8*x^5+64*x^3-56*x^2+8*x)*exp(4))/(4*exp(x)+4*(-x^2+4*x -4)*log(2)^2+2*(4*x^3-16*x^2+16*x)*log(2)-4*x^4+16*x^3-16*x^2+4*x),x, algo rithm=\
integrate(2*((x^3 - 2*x^2)*e^4*log(2)^2 - 2*(x^4 - x^3 - 4*x^2 + 4*x)*e^4* log(2) + (x^5 - 8*x^3 + 7*x^2 - x)*e^4 - 2*x*e^(x + 4) + ((x^2 - 2*x)*e^4* log(2)^2 - 2*(x^3 - x^2 - 4*x + 4)*e^4*log(2) + (x^4 - 8*x^2 + 7*x - 1)*e^ 4 - 2*e^(x + 4))*log(-4*x^4 + 16*x^3 - 4*(x^2 - 4*x + 4)*log(2)^2 - 16*x^2 + 8*(x^3 - 4*x^2 + 4*x)*log(2) + 4*x + 4*e^x))/(x^4 - 4*x^3 + (x^2 - 4*x + 4)*log(2)^2 + 4*x^2 - 2*(x^3 - 4*x^2 + 4*x)*log(2) - x - e^x), x)
Time = 10.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.80 \[ \int \frac {16 e^{4+x} x+e^4 \left (8 x-56 x^2+64 x^3-8 x^5\right )+e^4 \left (32 x-32 x^2-8 x^3+8 x^4\right ) \log (4)+e^4 \left (4 x^2-2 x^3\right ) \log ^2(4)+\left (16 e^{4+x}+e^4 \left (8-56 x+64 x^2-8 x^4\right )+e^4 \left (32-32 x-8 x^2+8 x^3\right ) \log (4)+e^4 \left (4 x-2 x^2\right ) \log ^2(4)\right ) \log \left (4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)\right )}{4 e^x+4 x-16 x^2+16 x^3-4 x^4+\left (16 x-16 x^2+4 x^3\right ) \log (4)+\left (-4+4 x-x^2\right ) \log ^2(4)} \, dx={\mathrm {e}}^4\,{\left (x+\ln \left (4\,x+4\,{\mathrm {e}}^x-4\,{\ln \left (2\right )}^2\,\left (x^2-4\,x+4\right )+2\,\ln \left (2\right )\,\left (4\,x^3-16\,x^2+16\,x\right )-16\,x^2+16\,x^3-4\,x^4\right )\right )}^2 \]
int((exp(4)*(8*x - 56*x^2 + 64*x^3 - 8*x^5) - log(4*x + 4*exp(x) - 4*log(2 )^2*(x^2 - 4*x + 4) + 2*log(2)*(16*x - 16*x^2 + 4*x^3) - 16*x^2 + 16*x^3 - 4*x^4)*(exp(4)*(56*x - 64*x^2 + 8*x^4 - 8) - 16*exp(4)*exp(x) + 2*exp(4)* log(2)*(32*x + 8*x^2 - 8*x^3 - 32) - 4*exp(4)*log(2)^2*(4*x - 2*x^2)) + 2* exp(4)*log(2)*(32*x - 32*x^2 - 8*x^3 + 8*x^4) + 16*x*exp(4)*exp(x) + 4*exp (4)*log(2)^2*(4*x^2 - 2*x^3))/(4*x + 4*exp(x) - 4*log(2)^2*(x^2 - 4*x + 4) + 2*log(2)*(16*x - 16*x^2 + 4*x^3) - 16*x^2 + 16*x^3 - 4*x^4),x)