Integrand size = 132, antiderivative size = 28 \[ \int \frac {-3 x^2-6 x^3-3 x^4-2 x^3 \log (2)+\left (-3+6 x+13 x^2+6 x^3+4 x^2 \log (2)\right ) \log (4)+\left (-3-8 x-3 x^2-2 x \log (2)\right ) \log ^2(4)+e^x \left (3 x^2+2 x^3+x^4+\left (-6 x-4 x^2-2 x^3\right ) \log (4)+\left (3+2 x+x^2\right ) \log ^2(4)\right )}{x^2-2 x \log (4)+\log ^2(4)} \, dx=\left (3+x^2\right ) \left (-4+e^x-x-\log (2)+\frac {x}{x-\log (4)}\right ) \] Output:
(x^2+3)*(-4+exp(x)-x+x/(x-2*ln(2))-ln(2))
Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(28)=56\).
Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.50 \[ \int \frac {-3 x^2-6 x^3-3 x^4-2 x^3 \log (2)+\left (-3+6 x+13 x^2+6 x^3+4 x^2 \log (2)\right ) \log (4)+\left (-3-8 x-3 x^2-2 x \log (2)\right ) \log ^2(4)+e^x \left (3 x^2+2 x^3+x^4+\left (-6 x-4 x^2-2 x^3\right ) \log (4)+\left (3+2 x+x^2\right ) \log ^2(4)\right )}{x^2-2 x \log (4)+\log ^2(4)} \, dx=-x^3+e^x \left (3+x^2\right )-\frac {1}{2} x^2 (6+\log (4))+x \left (-3-2 \log ^2(4)+\log (4) (1+\log (16))\right )+\frac {-2 \log ^4(4)+\log ^3(4) (1+\log (16))+\log (64)}{x-\log (4)} \] Input:
Integrate[(-3*x^2 - 6*x^3 - 3*x^4 - 2*x^3*Log[2] + (-3 + 6*x + 13*x^2 + 6* x^3 + 4*x^2*Log[2])*Log[4] + (-3 - 8*x - 3*x^2 - 2*x*Log[2])*Log[4]^2 + E^ x*(3*x^2 + 2*x^3 + x^4 + (-6*x - 4*x^2 - 2*x^3)*Log[4] + (3 + 2*x + x^2)*L og[4]^2))/(x^2 - 2*x*Log[4] + Log[4]^2),x]
Output:
-x^3 + E^x*(3 + x^2) - (x^2*(6 + Log[4]))/2 + x*(-3 - 2*Log[4]^2 + Log[4]* (1 + Log[16])) + (-2*Log[4]^4 + Log[4]^3*(1 + Log[16]) + Log[64])/(x - Log [4])
Leaf count is larger than twice the leaf count of optimal. \(267\) vs. \(2(28)=56\).
Time = 1.53 (sec) , antiderivative size = 267, normalized size of antiderivative = 9.54, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6, 7277, 27, 7293, 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.
| \(\displaystyle \int \frac {-3 x^4-6 x^3-2 x^3 \log (2)-3 x^2+\log ^2(4) \left (-3 x^2-8 x-2 x \log (2)-3\right )+\log (4) \left (6 x^3+13 x^2+4 x^2 \log (2)+6 x-3\right )+e^x \left (x^4+2 x^3+3 x^2+\left (x^2+2 x+3\right ) \log ^2(4)+\left (-2 x^3-4 x^2-6 x\right ) \log (4)\right )}{x^2-2 x \log (4)+\log ^2(4)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-3 x^4+x^3 (-6-2 \log (2))-3 x^2+\log ^2(4) \left (-3 x^2-8 x-2 x \log (2)-3\right )+\log (4) \left (6 x^3+13 x^2+4 x^2 \log (2)+6 x-3\right )+e^x \left (x^4+2 x^3+3 x^2+\left (x^2+2 x+3\right ) \log ^2(4)+\left (-2 x^3-4 x^2-6 x\right ) \log (4)\right )}{x^2-2 x \log (4)+\log ^2(4)}dx\) |
| \(\Big \downarrow \) 7277 |
| \(\displaystyle 4 \int -\frac {3 x^4+2 (3+\log (2)) x^3+3 x^2-e^x \left (x^4+2 x^3+3 x^2+\left (x^2+2 x+3\right ) \log ^2(4)-2 \left (x^3+2 x^2+3 x\right ) \log (4)\right )+\left (3 x^2+2 \log (2) x+8 x+3\right ) \log ^2(4)+\left (-6 x^3-4 \log (2) x^2-13 x^2-6 x+3\right ) \log (4)}{4 (x-\log (4))^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\int \frac {3 x^4+2 (3+\log (2)) x^3+3 x^2-e^x \left (x^4+2 x^3+3 x^2+\left (x^2+2 x+3\right ) \log ^2(4)-2 \left (x^3+2 x^2+3 x\right ) \log (4)\right )+\left (3 x^2+2 \log (2) x+8 x+3\right ) \log ^2(4)+\left (-6 x^3-4 \log (2) x^2-13 x^2-6 x+3\right ) \log (4)}{(x-\log (4))^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\int \left (\frac {3 x^4}{(x-\log (4))^2}+\frac {2 (3+\log (2)) x^3}{(x-\log (4))^2}+\frac {3 x^2}{(x-\log (4))^2}-e^x \left (x^2+2 x+3\right )+\frac {\log ^2(4) \left (3 x^2+(8+\log (4)) x+3\right )}{(x-\log (4))^2}+\frac {\log (4) \left (-6 x^3-(13+\log (16)) x^2-6 x+3\right )}{(x-\log (4))^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -x^3+e^x x^2-\frac {3}{2} x^2 \log (16)+3 x^2 \log (4)-x^2 (3+\log (2))-3 x+3 e^x+\frac {3 \log ^4(4)}{x-\log (4)}-12 \log ^3(4) \log (x-\log (4))+\frac {2 (3+\log (2)) \log ^3(4)}{x-\log (4)}-12 x \log ^2(4)+2 \log (4) \left (3+9 \log ^2(4)+\log (4) (13+\log (16))\right ) \log (x-\log (4))-\log ^2(4) (8+7 \log (4)) \log (x-\log (4))-6 (3+\log (2)) \log ^2(4) \log (x-\log (4))+\frac {\log ^2(4) \left (3+4 \log ^2(4)+8 \log (4)\right )}{x-\log (4)}+\frac {3 \log ^2(4)}{x-\log (4)}+\frac {\log (4) \left (3-6 \log ^3(4)-\log ^2(4) (13+\log (16))-6 \log (4)\right )}{x-\log (4)}+13 x \log (4) \left (1+\log \left (4\ 2^{2/13}\right )\right )-2 x (3+\log (2)) \log (16)-3 \log (16) \log (x-\log (4))\) |
Input:
Int[(-3*x^2 - 6*x^3 - 3*x^4 - 2*x^3*Log[2] + (-3 + 6*x + 13*x^2 + 6*x^3 + 4*x^2*Log[2])*Log[4] + (-3 - 8*x - 3*x^2 - 2*x*Log[2])*Log[4]^2 + E^x*(3*x ^2 + 2*x^3 + x^4 + (-6*x - 4*x^2 - 2*x^3)*Log[4] + (3 + 2*x + x^2)*Log[4]^ 2))/(x^2 - 2*x*Log[4] + Log[4]^2),x]
Output:
3*E^x - 3*x + E^x*x^2 - x^3 - x^2*(3 + Log[2]) + 3*x^2*Log[4] - 12*x*Log[4 ]^2 + (3*Log[4]^2)/(x - Log[4]) + (2*(3 + Log[2])*Log[4]^3)/(x - Log[4]) + (3*Log[4]^4)/(x - Log[4]) + (Log[4]^2*(3 + 8*Log[4] + 4*Log[4]^2))/(x - L og[4]) - (3*x^2*Log[16])/2 - 2*x*(3 + Log[2])*Log[16] + (Log[4]*(3 - 6*Log [4] - 6*Log[4]^3 - Log[4]^2*(13 + Log[16])))/(x - Log[4]) + 13*x*Log[4]*(1 + Log[4*2^(2/13)]) - 6*(3 + Log[2])*Log[4]^2*Log[x - Log[4]] - 12*Log[4]^ 3*Log[x - Log[4]] - Log[4]^2*(8 + 7*Log[4])*Log[x - Log[4]] - 3*Log[16]*Lo g[x - Log[4]] + 2*Log[4]*(3 + 9*Log[4]^2 + Log[4]*(13 + Log[16]))*Log[x - Log[4]]
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_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Simp[1/(4^p*c^p) Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} , x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && !AlgebraicFu nctionQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs. \(2(27)=54\).
Time = 2.46 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.04
| method | result | size |
| parts | \({\mathrm e}^{x} x^{2}+3 \,{\mathrm e}^{x}-x^{2} \ln \left (2\right )-x^{3}+2 x \ln \left (2\right )-3 x^{2}-3 x +\frac {2 \ln \left (2\right ) \left (4 \ln \left (2\right )^{2}+3\right )}{x -2 \ln \left (2\right )}\) | \(57\) |
| risch | \(-x^{2} \ln \left (2\right )-x^{3}+2 x \ln \left (2\right )-3 x^{2}-3 x -\frac {4 \ln \left (2\right )^{3}}{\ln \left (2\right )-\frac {x}{2}}-\frac {3 \ln \left (2\right )}{\ln \left (2\right )-\frac {x}{2}}+\left (x^{2}+3\right ) {\mathrm e}^{x}\) | \(61\) |
| norman | \(\frac {x^{4}+\left (-\ln \left (2\right )+3\right ) x^{3}+\left (-2 \ln \left (2\right )^{2}-8 \ln \left (2\right )+3\right ) x^{2}-3 \,{\mathrm e}^{x} x -{\mathrm e}^{x} x^{3}+6 \,{\mathrm e}^{x} \ln \left (2\right )+2 x^{2} \ln \left (2\right ) {\mathrm e}^{x}-12 \ln \left (2\right )^{2}-6 \ln \left (2\right )}{2 \ln \left (2\right )-x}\) | \(79\) |
| parallelrisch | \(-\frac {2 x^{2} \ln \left (2\right )^{2}+x^{3} \ln \left (2\right )-2 x^{2} \ln \left (2\right ) {\mathrm e}^{x}-x^{4}+{\mathrm e}^{x} x^{3}+8 x^{2} \ln \left (2\right )-3 x^{3}+12 \ln \left (2\right )^{2}-6 \,{\mathrm e}^{x} \ln \left (2\right )-3 x^{2}+3 \,{\mathrm e}^{x} x +6 \ln \left (2\right )}{2 \ln \left (2\right )-x}\) | \(87\) |
| default | \({\mathrm e}^{x} x^{2}+3 \,{\mathrm e}^{x}+4 \ln \left (2\right ) \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )+12 \ln \left (2\right )^{2} {\mathrm e}^{x}-128 \ln \left (2\right )^{3} \operatorname {expIntegral}_{1}\left (2 \ln \left (2\right )-x \right )+16 \ln \left (2\right )^{4} \left (-\frac {{\mathrm e}^{x}}{x -2 \ln \left (2\right )}-4 \,\operatorname {expIntegral}_{1}\left (2 \ln \left (2\right )-x \right )\right )-3 x -3 x^{2}+2 x \ln \left (2\right )+\frac {8 \ln \left (2\right )^{3}}{x -2 \ln \left (2\right )}-x^{3}-x^{2} \ln \left (2\right )+\frac {6 \ln \left (2\right )}{x -2 \ln \left (2\right )}-48 \ln \left (2\right ) \operatorname {expIntegral}_{1}\left (2 \ln \left (2\right )-x \right )+24 \ln \left (2\right )^{2} \left (-\frac {{\mathrm e}^{x}}{x -2 \ln \left (2\right )}-4 \,\operatorname {expIntegral}_{1}\left (2 \ln \left (2\right )-x \right )\right )+8 \,{\mathrm e}^{x} \ln \left (2\right )-96 \ln \left (2\right )^{2} \operatorname {expIntegral}_{1}\left (2 \ln \left (2\right )-x \right )+16 \ln \left (2\right )^{3} \left (-\frac {{\mathrm e}^{x}}{x -2 \ln \left (2\right )}-4 \,\operatorname {expIntegral}_{1}\left (2 \ln \left (2\right )-x \right )\right )-12 \ln \left (2\right ) \left (-4 \,\operatorname {expIntegral}_{1}\left (2 \ln \left (2\right )-x \right )+2 \ln \left (2\right ) \left (-\frac {{\mathrm e}^{x}}{x -2 \ln \left (2\right )}-4 \,\operatorname {expIntegral}_{1}\left (2 \ln \left (2\right )-x \right )\right )\right )-8 \ln \left (2\right ) \left ({\mathrm e}^{x}-16 \ln \left (2\right ) \operatorname {expIntegral}_{1}\left (2 \ln \left (2\right )-x \right )+4 \ln \left (2\right )^{2} \left (-\frac {{\mathrm e}^{x}}{x -2 \ln \left (2\right )}-4 \,\operatorname {expIntegral}_{1}\left (2 \ln \left (2\right )-x \right )\right )\right )-4 \ln \left (2\right ) \left ({\mathrm e}^{x} x -{\mathrm e}^{x}+4 \,{\mathrm e}^{x} \ln \left (2\right )-48 \ln \left (2\right )^{2} \operatorname {expIntegral}_{1}\left (2 \ln \left (2\right )-x \right )+8 \ln \left (2\right )^{3} \left (-\frac {{\mathrm e}^{x}}{x -2 \ln \left (2\right )}-4 \,\operatorname {expIntegral}_{1}\left (2 \ln \left (2\right )-x \right )\right )\right )+8 \ln \left (2\right )^{2} \left (-4 \,\operatorname {expIntegral}_{1}\left (2 \ln \left (2\right )-x \right )+2 \ln \left (2\right ) \left (-\frac {{\mathrm e}^{x}}{x -2 \ln \left (2\right )}-4 \,\operatorname {expIntegral}_{1}\left (2 \ln \left (2\right )-x \right )\right )\right )+4 \ln \left (2\right )^{2} \left ({\mathrm e}^{x}-16 \ln \left (2\right ) \operatorname {expIntegral}_{1}\left (2 \ln \left (2\right )-x \right )+4 \ln \left (2\right )^{2} \left (-\frac {{\mathrm e}^{x}}{x -2 \ln \left (2\right )}-4 \,\operatorname {expIntegral}_{1}\left (2 \ln \left (2\right )-x \right )\right )\right )\) | \(495\) |
Input:
int(((4*(x^2+2*x+3)*ln(2)^2+2*(-2*x^3-4*x^2-6*x)*ln(2)+x^4+2*x^3+3*x^2)*ex p(x)+4*(-2*x*ln(2)-3*x^2-8*x-3)*ln(2)^2+2*(4*x^2*ln(2)+6*x^3+13*x^2+6*x-3) *ln(2)-2*x^3*ln(2)-3*x^4-6*x^3-3*x^2)/(4*ln(2)^2-4*x*ln(2)+x^2),x,method=_ RETURNVERBOSE)
Output:
exp(x)*x^2+3*exp(x)-x^2*ln(2)-x^3+2*x*ln(2)-3*x^2-3*x+2*ln(2)*(4*ln(2)^2+3 )/(x-2*ln(2))
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (27) = 54\).
Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.86 \[ \int \frac {-3 x^2-6 x^3-3 x^4-2 x^3 \log (2)+\left (-3+6 x+13 x^2+6 x^3+4 x^2 \log (2)\right ) \log (4)+\left (-3-8 x-3 x^2-2 x \log (2)\right ) \log ^2(4)+e^x \left (3 x^2+2 x^3+x^4+\left (-6 x-4 x^2-2 x^3\right ) \log (4)+\left (3+2 x+x^2\right ) \log ^2(4)\right )}{x^2-2 x \log (4)+\log ^2(4)} \, dx=-\frac {x^{4} + 3 \, x^{3} - 2 \, {\left (x^{2} - 2 \, x\right )} \log \left (2\right )^{2} - 8 \, \log \left (2\right )^{3} + 3 \, x^{2} - {\left (x^{3} - 2 \, {\left (x^{2} + 3\right )} \log \left (2\right ) + 3 \, x\right )} e^{x} - {\left (x^{3} + 8 \, x^{2} + 6 \, x + 6\right )} \log \left (2\right )}{x - 2 \, \log \left (2\right )} \] Input:
integrate(((4*(x^2+2*x+3)*log(2)^2+2*(-2*x^3-4*x^2-6*x)*log(2)+x^4+2*x^3+3 *x^2)*exp(x)+4*(-2*x*log(2)-3*x^2-8*x-3)*log(2)^2+2*(4*x^2*log(2)+6*x^3+13 *x^2+6*x-3)*log(2)-2*x^3*log(2)-3*x^4-6*x^3-3*x^2)/(4*log(2)^2-4*x*log(2)+ x^2),x, algorithm="fricas")
Output:
-(x^4 + 3*x^3 - 2*(x^2 - 2*x)*log(2)^2 - 8*log(2)^3 + 3*x^2 - (x^3 - 2*(x^ 2 + 3)*log(2) + 3*x)*e^x - (x^3 + 8*x^2 + 6*x + 6)*log(2))/(x - 2*log(2))
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (22) = 44\).
Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {-3 x^2-6 x^3-3 x^4-2 x^3 \log (2)+\left (-3+6 x+13 x^2+6 x^3+4 x^2 \log (2)\right ) \log (4)+\left (-3-8 x-3 x^2-2 x \log (2)\right ) \log ^2(4)+e^x \left (3 x^2+2 x^3+x^4+\left (-6 x-4 x^2-2 x^3\right ) \log (4)+\left (3+2 x+x^2\right ) \log ^2(4)\right )}{x^2-2 x \log (4)+\log ^2(4)} \, dx=- x^{3} - x^{2} \left (\log {\left (2 \right )} + 3\right ) - x \left (3 - 2 \log {\left (2 \right )}\right ) + \left (x^{2} + 3\right ) e^{x} - \frac {- 6 \log {\left (2 \right )} - 8 \log {\left (2 \right )}^{3}}{x - 2 \log {\left (2 \right )}} \] Input:
integrate(((4*(x**2+2*x+3)*ln(2)**2+2*(-2*x**3-4*x**2-6*x)*ln(2)+x**4+2*x* *3+3*x**2)*exp(x)+4*(-2*x*ln(2)-3*x**2-8*x-3)*ln(2)**2+2*(4*x**2*ln(2)+6*x **3+13*x**2+6*x-3)*ln(2)-2*x**3*ln(2)-3*x**4-6*x**3-3*x**2)/(4*ln(2)**2-4* x*ln(2)+x**2),x)
Output:
-x**3 - x**2*(log(2) + 3) - x*(3 - 2*log(2)) + (x**2 + 3)*exp(x) - (-6*log (2) - 8*log(2)**3)/(x - 2*log(2))
\[ \int \frac {-3 x^2-6 x^3-3 x^4-2 x^3 \log (2)+\left (-3+6 x+13 x^2+6 x^3+4 x^2 \log (2)\right ) \log (4)+\left (-3-8 x-3 x^2-2 x \log (2)\right ) \log ^2(4)+e^x \left (3 x^2+2 x^3+x^4+\left (-6 x-4 x^2-2 x^3\right ) \log (4)+\left (3+2 x+x^2\right ) \log ^2(4)\right )}{x^2-2 x \log (4)+\log ^2(4)} \, dx=\int { -\frac {3 \, x^{4} + 2 \, x^{3} \log \left (2\right ) + 6 \, x^{3} + 4 \, {\left (3 \, x^{2} + 2 \, x \log \left (2\right ) + 8 \, x + 3\right )} \log \left (2\right )^{2} + 3 \, x^{2} - {\left (x^{4} + 2 \, x^{3} + 4 \, {\left (x^{2} + 2 \, x + 3\right )} \log \left (2\right )^{2} + 3 \, x^{2} - 4 \, {\left (x^{3} + 2 \, x^{2} + 3 \, x\right )} \log \left (2\right )\right )} e^{x} - 2 \, {\left (6 \, x^{3} + 4 \, x^{2} \log \left (2\right ) + 13 \, x^{2} + 6 \, x - 3\right )} \log \left (2\right )}{x^{2} - 4 \, x \log \left (2\right ) + 4 \, \log \left (2\right )^{2}} \,d x } \] Input:
integrate(((4*(x^2+2*x+3)*log(2)^2+2*(-2*x^3-4*x^2-6*x)*log(2)+x^4+2*x^3+3 *x^2)*exp(x)+4*(-2*x*log(2)-3*x^2-8*x-3)*log(2)^2+2*(4*x^2*log(2)+6*x^3+13 *x^2+6*x-3)*log(2)-2*x^3*log(2)-3*x^4-6*x^3-3*x^2)/(4*log(2)^2-4*x*log(2)+ x^2),x, algorithm="maxima")
Output:
8*(2*log(2)/(x - 2*log(2)) - log(x - 2*log(2)))*log(2)^3 - 96*log(2)^3*log (x - 2*log(2)) - x^3 - 6*x^2*log(2) - 4*(4*log(2)*log(x - 2*log(2)) + x - 4*log(2)^2/(x - 2*log(2)))*log(2)^2 - 36*x*log(2)^2 + 32*(2*log(2)/(x - 2* log(2)) - log(x - 2*log(2)))*log(2)^2 - 24*integrate(e^x/(x^3 - 6*x^2*log( 2) + 12*x*log(2)^2 - 8*log(2)^3), x)*log(2)^2 + 48*log(2)^4/(x - 2*log(2)) - 72*log(2)^2*log(x - 2*log(2)) - 3*x^2 + 5*(24*log(2)^2*log(x - 2*log(2) ) + x^2 + 8*x*log(2) - 16*log(2)^3/(x - 2*log(2)))*log(2) + 26*(4*log(2)*l og(x - 2*log(2)) + x - 4*log(2)^2/(x - 2*log(2)))*log(2) - 24*x*log(2) - 1 2*(2*log(2)/(x - 2*log(2)) - log(x - 2*log(2)))*log(2) - 48*exp_integral_e (2, -x + 2*log(2))*log(2)^2/(x - 2*log(2)) + 48*log(2)^3/(x - 2*log(2)) - 12*log(2)*log(x - 2*log(2)) - 3*x + (x^4 - 4*x^3*log(2) + (4*log(2)^2 + 3) *x^2 - 12*x*log(2))*e^x/(x^2 - 4*x*log(2) + 4*log(2)^2) + 24*log(2)^2/(x - 2*log(2)) + 6*log(2)/(x - 2*log(2))
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (27) = 54\).
Time = 0.12 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.43 \[ \int \frac {-3 x^2-6 x^3-3 x^4-2 x^3 \log (2)+\left (-3+6 x+13 x^2+6 x^3+4 x^2 \log (2)\right ) \log (4)+\left (-3-8 x-3 x^2-2 x \log (2)\right ) \log ^2(4)+e^x \left (3 x^2+2 x^3+x^4+\left (-6 x-4 x^2-2 x^3\right ) \log (4)+\left (3+2 x+x^2\right ) \log ^2(4)\right )}{x^2-2 x \log (4)+\log ^2(4)} \, dx=-\frac {x^{4} - x^{3} e^{x} - x^{3} \log \left (2\right ) + 2 \, x^{2} e^{x} \log \left (2\right ) - 2 \, x^{2} \log \left (2\right )^{2} + 3 \, x^{3} - 8 \, x^{2} \log \left (2\right ) + 4 \, x \log \left (2\right )^{2} - 8 \, \log \left (2\right )^{3} + 3 \, x^{2} - 3 \, x e^{x} - 6 \, x \log \left (2\right ) + 6 \, e^{x} \log \left (2\right ) - 6 \, \log \left (2\right )}{x - 2 \, \log \left (2\right )} \] Input:
integrate(((4*(x^2+2*x+3)*log(2)^2+2*(-2*x^3-4*x^2-6*x)*log(2)+x^4+2*x^3+3 *x^2)*exp(x)+4*(-2*x*log(2)-3*x^2-8*x-3)*log(2)^2+2*(4*x^2*log(2)+6*x^3+13 *x^2+6*x-3)*log(2)-2*x^3*log(2)-3*x^4-6*x^3-3*x^2)/(4*log(2)^2-4*x*log(2)+ x^2),x, algorithm="giac")
Output:
-(x^4 - x^3*e^x - x^3*log(2) + 2*x^2*e^x*log(2) - 2*x^2*log(2)^2 + 3*x^3 - 8*x^2*log(2) + 4*x*log(2)^2 - 8*log(2)^3 + 3*x^2 - 3*x*e^x - 6*x*log(2) + 6*e^x*log(2) - 6*log(2))/(x - 2*log(2))
Time = 0.54 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.39 \[ \int \frac {-3 x^2-6 x^3-3 x^4-2 x^3 \log (2)+\left (-3+6 x+13 x^2+6 x^3+4 x^2 \log (2)\right ) \log (4)+\left (-3-8 x-3 x^2-2 x \log (2)\right ) \log ^2(4)+e^x \left (3 x^2+2 x^3+x^4+\left (-6 x-4 x^2-2 x^3\right ) \log (4)+\left (3+2 x+x^2\right ) \log ^2(4)\right )}{x^2-2 x \log (4)+\log ^2(4)} \, dx=x\,\left (26\,\ln \left (2\right )+8\,{\ln \left (2\right )}^2-4\,\ln \left (2\right )\,\left (2\,\ln \left (2\right )+6\right )-3\right )-x^2\,\left (\ln \left (2\right )+3\right )+{\mathrm {e}}^x\,\left (x^2+3\right )+\frac {6\,\ln \left (2\right )+8\,{\ln \left (2\right )}^3}{x-2\,\ln \left (2\right )}-x^3 \] Input:
int(-(2*x^3*log(2) - exp(x)*(4*log(2)^2*(2*x + x^2 + 3) - 2*log(2)*(6*x + 4*x^2 + 2*x^3) + 3*x^2 + 2*x^3 + x^4) - 2*log(2)*(6*x + 4*x^2*log(2) + 13* x^2 + 6*x^3 - 3) + 3*x^2 + 6*x^3 + 3*x^4 + 4*log(2)^2*(8*x + 2*x*log(2) + 3*x^2 + 3))/(4*log(2)^2 - 4*x*log(2) + x^2),x)
Output:
x*(26*log(2) + 8*log(2)^2 - 4*log(2)*(2*log(2) + 6) - 3) - x^2*(log(2) + 3 ) + exp(x)*(x^2 + 3) + (6*log(2) + 8*log(2)^3)/(x - 2*log(2)) - x^3
Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.11 \[ \int \frac {-3 x^2-6 x^3-3 x^4-2 x^3 \log (2)+\left (-3+6 x+13 x^2+6 x^3+4 x^2 \log (2)\right ) \log (4)+\left (-3-8 x-3 x^2-2 x \log (2)\right ) \log ^2(4)+e^x \left (3 x^2+2 x^3+x^4+\left (-6 x-4 x^2-2 x^3\right ) \log (4)+\left (3+2 x+x^2\right ) \log ^2(4)\right )}{x^2-2 x \log (4)+\log ^2(4)} \, dx=\frac {2 e^{x} \mathrm {log}\left (2\right ) x^{2}+6 e^{x} \mathrm {log}\left (2\right )-e^{x} x^{3}-3 e^{x} x -2 \mathrm {log}\left (2\right )^{2} x^{2}-\mathrm {log}\left (2\right ) x^{3}-8 \,\mathrm {log}\left (2\right ) x^{2}-6 \,\mathrm {log}\left (2\right ) x +x^{4}+3 x^{3}+3 x^{2}-3 x}{2 \,\mathrm {log}\left (2\right )-x} \] Input:
int(((4*(x^2+2*x+3)*log(2)^2+2*(-2*x^3-4*x^2-6*x)*log(2)+x^4+2*x^3+3*x^2)* exp(x)+4*(-2*x*log(2)-3*x^2-8*x-3)*log(2)^2+2*(4*x^2*log(2)+6*x^3+13*x^2+6 *x-3)*log(2)-2*x^3*log(2)-3*x^4-6*x^3-3*x^2)/(4*log(2)^2-4*x*log(2)+x^2),x )
Output:
(2*e**x*log(2)*x**2 + 6*e**x*log(2) - e**x*x**3 - 3*e**x*x - 2*log(2)**2*x **2 - log(2)*x**3 - 8*log(2)*x**2 - 6*log(2)*x + x**4 + 3*x**3 + 3*x**2 - 3*x)/(2*log(2) - x)