Integrand size = 117, antiderivative size = 34 \[ \int \frac {-36+12 x+\left (-8+12 x-4 x^2\right ) \log (4)+e^{2 x} \left (3+6 x-2 x^2 \log (4)\right )+e^x \left (3-3 x+3 x^2+\left (-4 x+2 x^2-x^3\right ) \log (4)\right )+\left (-36+24 x-4 x^2 \log (4)+e^x \left (12+12 x-4 x^2 \log (4)\right )\right ) \log (x)}{18-12 x \log (4)+2 x^2 \log ^2(4)} \, dx=\frac {4+\frac {1}{2} \left (3-e^x-x\right ) x \left (e^x+4 \log (x)\right )}{-3+x \log (4)} \] Output:
(1/2*x*(3-x-exp(x))*(4*ln(x)+exp(x))+4)/(2*x*ln(2)-3)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.22 (sec) , antiderivative size = 146, normalized size of antiderivative = 4.29 \[ \int \frac {-36+12 x+\left (-8+12 x-4 x^2\right ) \log (4)+e^{2 x} \left (3+6 x-2 x^2 \log (4)\right )+e^x \left (3-3 x+3 x^2+\left (-4 x+2 x^2-x^3\right ) \log (4)\right )+\left (-36+24 x-4 x^2 \log (4)+e^x \left (12+12 x-4 x^2 \log (4)\right )\right ) \log (x)}{18-12 x \log (4)+2 x^2 \log ^2(4)} \, dx=\frac {6 e^x \log ^3(4)+e^{2 x} x \log ^4(4)+e^x x^2 \log ^4(4)-4 \log ^3(4) \log (16)-e^x x \log ^3(4) \log (64)-e^x \log (4) \log (16) \log (64)-e^{\frac {3}{\log (4)}} \operatorname {ExpIntegralEi}\left (x-\frac {3}{\log (4)}\right ) (-3+x \log (4)) \left (4 \log ^3(4)-\log ^2(4) \log (256)-\log (64) \log (256)+\log (16) \log (4096)\right )+4 x \left (-3+e^x+x\right ) \log ^4(4) \log (x)}{2 \log ^4(4) (3-x \log (4))} \] Input:
Integrate[(-36 + 12*x + (-8 + 12*x - 4*x^2)*Log[4] + E^(2*x)*(3 + 6*x - 2* x^2*Log[4]) + E^x*(3 - 3*x + 3*x^2 + (-4*x + 2*x^2 - x^3)*Log[4]) + (-36 + 24*x - 4*x^2*Log[4] + E^x*(12 + 12*x - 4*x^2*Log[4]))*Log[x])/(18 - 12*x* Log[4] + 2*x^2*Log[4]^2),x]
Output:
(6*E^x*Log[4]^3 + E^(2*x)*x*Log[4]^4 + E^x*x^2*Log[4]^4 - 4*Log[4]^3*Log[1 6] - E^x*x*Log[4]^3*Log[64] - E^x*Log[4]*Log[16]*Log[64] - E^(3/Log[4])*Ex pIntegralEi[x - 3/Log[4]]*(-3 + x*Log[4])*(4*Log[4]^3 - Log[4]^2*Log[256] - Log[64]*Log[256] + Log[16]*Log[4096]) + 4*x*(-3 + E^x + x)*Log[4]^4*Log[ x])/(2*Log[4]^4*(3 - x*Log[4]))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 3.31 (sec) , antiderivative size = 699, normalized size of antiderivative = 20.56, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {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.
| \(\displaystyle \int \frac {e^{2 x} \left (-2 x^2 \log (4)+6 x+3\right )+\left (-4 x^2 \log (4)+e^x \left (-4 x^2 \log (4)+12 x+12\right )+24 x-36\right ) \log (x)+\left (-4 x^2+12 x-8\right ) \log (4)+e^x \left (3 x^2+\left (-x^3+2 x^2-4 x\right ) \log (4)-3 x+3\right )+12 x-36}{2 x^2 \log ^2(4)-12 x \log (4)+18} \, dx\) |
| \(\Big \downarrow \) 7277 |
| \(\displaystyle 8 \log ^2(4) \int -\frac {-12 x-e^{2 x} \left (-2 \log (4) x^2+6 x+3\right )-e^x \left (3 x^2-3 x-\left (x^3-2 x^2+4 x\right ) \log (4)+3\right )+4 \left (\log (4) x^2-6 x-e^x \left (-\log (4) x^2+3 x+3\right )+9\right ) \log (x)+4 \left (x^2-3 x+2\right ) \log (4)+36}{16 \log ^2(4) (3-x \log (4))^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{2} \int \frac {-12 x-e^{2 x} \left (-2 \log (4) x^2+6 x+3\right )-e^x \left (3 x^2-3 x-\left (x^3-2 x^2+4 x\right ) \log (4)+3\right )+4 \left (\log (4) x^2-6 x-e^x \left (-\log (4) x^2+3 x+3\right )+9\right ) \log (x)+4 \left (x^2-3 x+2\right ) \log (4)+36}{(3-x \log (4))^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{2} \int \left (\frac {4 \log (4) \log (x) x^2}{(x \log (4)-3)^2}-\frac {24 \log (x) x}{(x \log (4)-3)^2}-\frac {12 x}{(x \log (4)-3)^2}+\frac {e^{2 x} \left (\log (16) x^2-6 x-3\right )}{(x \log (4)-3)^2}+\frac {36 \log (x)}{(x \log (4)-3)^2}+\frac {e^x \left (\log (4) x^3+4 \log (4) \log (x) x^2-3 \left (1+\frac {\log (16)}{3}\right ) x^2-12 \log (x) x+3 \left (1+\log \left (4\ 2^{2/3}\right )\right ) x-12 \log (x)-3\right )}{(3-x \log (4))^2}+\frac {4 (x-2) (x-1) \log (4)}{(x \log (4)-3)^2}+\frac {36}{(x \log (4)-3)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {9 e^{\frac {3}{\log (4)}} (3+\log (16)) \operatorname {ExpIntegralEi}\left (-\frac {3-x \log (4)}{\log (4)}\right )}{\log ^4(4)}-\frac {27 e^{\frac {3}{\log (4)}} \operatorname {ExpIntegralEi}\left (-\frac {3-x \log (4)}{\log (4)}\right )}{\log ^4(4)}-\frac {e^{\frac {6}{\log (4)}} \log (4096) \operatorname {ExpIntegralEi}\left (-\frac {2 (3-x \log (4))}{\log (4)}\right )}{\log ^3(4)}-\frac {3 e^{\frac {3}{\log (4)}} (3+\log (256)) \operatorname {ExpIntegralEi}\left (-\frac {3-x \log (4)}{\log (4)}\right )}{\log ^3(4)}+\frac {6 e^{\frac {3}{\log (4)}} (3+\log (16)) \operatorname {ExpIntegralEi}\left (-\frac {3-x \log (4)}{\log (4)}\right )}{\log ^3(4)}-\frac {27 e^{\frac {3}{\log (4)}} \operatorname {ExpIntegralEi}\left (-\frac {3-x \log (4)}{\log (4)}\right )}{\log ^3(4)}-\frac {64^{\frac {1}{\log ^2(4)}} (3+\log (256)) \operatorname {ExpIntegralEi}\left (\frac {x \log ^2(4)-\log (64)}{\log ^2(4)}\right )}{\log ^2(4)}+\frac {6 e^{\frac {6}{\log (4)}} \operatorname {ExpIntegralEi}\left (-\frac {2 (3-x \log (4))}{\log (4)}\right )}{\log ^2(4)}+\frac {3 e^{\frac {3}{\log (4)}} \operatorname {ExpIntegralEi}\left (-\frac {3-x \log (4)}{\log (4)}\right )}{\log ^2(4)}+\frac {4 e^{\frac {3}{\log (4)}} \operatorname {ExpIntegralEi}\left (-\frac {3-x \log (4)}{\log (4)}\right )}{\log (4)}-\frac {4 x^2 \log (x)}{3-x \log (4)}-\frac {27 e^x}{\log ^3(4) (3-x \log (4))}+\frac {9 e^x (3+\log (16))}{\log ^3(4) (3-x \log (4))}-\frac {12 (2-\log (4)) \log (3-x \log (4))}{\log ^2(4)}-\frac {12 (1+\log (9)-2 \log (\log (4))) \log (x \log (4)-3)}{\log ^2(4)}+\frac {24 \left (1+\log \left (\frac {3}{\log (4)}\right )\right ) \log (x \log (4)-3)}{\log ^2(4)}+\frac {12 \log \left (x \log ^2(4)-\log (64)\right )}{\log ^2(4)}+\frac {36}{\log ^2(4) (3-x \log (4))}-\frac {3 e^x (3+\log (256))}{\log ^2(4) (3-x \log (4))}+\frac {e^x (3+\log (16))}{\log ^2(4)}-\frac {e^{2 x} \log (16)}{2 \log ^2(4)}-\frac {4 (3-\log (4)) (3-\log (16))}{\log ^2(4) (3-x \log (4))}-\frac {6 e^x}{\log ^2(4)}+\frac {24 x \log (x)}{\log (4) (3-x \log (4))}-\frac {12 x \log (x)}{3-x \log (4)}-\frac {8 x \log (x)}{\log (4)}-\frac {e^x x}{\log (4)}+\frac {12 e^x \log (x)}{\log (4) (3-x \log (4))}-\frac {4 e^x \log (x)}{\log (4)}-\frac {12 \log (3-x \log (4))}{\log (4)}+\frac {3 e^x}{\log (4) (3-x \log (4))}+\frac {3 e^{2 x}}{\log (4) (3-x \log (4))}-\frac {36}{\log (4) (3-x \log (4))}+\frac {e^x}{\log (4)}\right )\) |
Input:
Int[(-36 + 12*x + (-8 + 12*x - 4*x^2)*Log[4] + E^(2*x)*(3 + 6*x - 2*x^2*Lo g[4]) + E^x*(3 - 3*x + 3*x^2 + (-4*x + 2*x^2 - x^3)*Log[4]) + (-36 + 24*x - 4*x^2*Log[4] + E^x*(12 + 12*x - 4*x^2*Log[4]))*Log[x])/(18 - 12*x*Log[4] + 2*x^2*Log[4]^2),x]
Output:
((-27*E^(3/Log[4])*ExpIntegralEi[-((3 - x*Log[4])/Log[4])])/Log[4]^4 - (27 *E^(3/Log[4])*ExpIntegralEi[-((3 - x*Log[4])/Log[4])])/Log[4]^3 - (6*E^x)/ Log[4]^2 + (6*E^(6/Log[4])*ExpIntegralEi[(-2*(3 - x*Log[4]))/Log[4]])/Log[ 4]^2 + (3*E^(3/Log[4])*ExpIntegralEi[-((3 - x*Log[4])/Log[4])])/Log[4]^2 + E^x/Log[4] - (E^x*x)/Log[4] + (4*E^(3/Log[4])*ExpIntegralEi[-((3 - x*Log[ 4])/Log[4])])/Log[4] - (27*E^x)/(Log[4]^3*(3 - x*Log[4])) + 36/(Log[4]^2*( 3 - x*Log[4])) - 36/(Log[4]*(3 - x*Log[4])) + (3*E^x)/(Log[4]*(3 - x*Log[4 ])) + (3*E^(2*x))/(Log[4]*(3 - x*Log[4])) - (4*(3 - Log[4])*(3 - Log[16])) /(Log[4]^2*(3 - x*Log[4])) - (E^(2*x)*Log[16])/(2*Log[4]^2) + (9*E^(3/Log[ 4])*ExpIntegralEi[-((3 - x*Log[4])/Log[4])]*(3 + Log[16]))/Log[4]^4 + (6*E ^(3/Log[4])*ExpIntegralEi[-((3 - x*Log[4])/Log[4])]*(3 + Log[16]))/Log[4]^ 3 + (E^x*(3 + Log[16]))/Log[4]^2 + (9*E^x*(3 + Log[16]))/(Log[4]^3*(3 - x* Log[4])) - (3*E^(3/Log[4])*ExpIntegralEi[-((3 - x*Log[4])/Log[4])]*(3 + Lo g[256]))/Log[4]^3 - (64^Log[4]^(-2)*ExpIntegralEi[(x*Log[4]^2 - Log[64])/L og[4]^2]*(3 + Log[256]))/Log[4]^2 - (3*E^x*(3 + Log[256]))/(Log[4]^2*(3 - x*Log[4])) - (E^(6/Log[4])*ExpIntegralEi[(-2*(3 - x*Log[4]))/Log[4]]*Log[4 096])/Log[4]^3 - (4*E^x*Log[x])/Log[4] - (8*x*Log[x])/Log[4] - (12*x*Log[x ])/(3 - x*Log[4]) - (4*x^2*Log[x])/(3 - x*Log[4]) + (12*E^x*Log[x])/(Log[4 ]*(3 - x*Log[4])) + (24*x*Log[x])/(Log[4]*(3 - x*Log[4])) - (12*(2 - Log[4 ])*Log[3 - x*Log[4]])/Log[4]^2 - (12*Log[3 - x*Log[4]])/Log[4] + (24*(1...
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. \(114\) vs. \(2(31)=62\).
Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.38
\[\frac {-{\mathrm e}^{x} x^{2}+3 \,{\mathrm e}^{x} x -4 x \,{\mathrm e}^{x} \ln \left (x \right )}{4 x \ln \left (2\right )-6}+\frac {4}{2 x \ln \left (2\right )-3}-\frac {\ln \left (x \right ) x}{\ln \left (2\right )}-\frac {3 \ln \left (x \right ) x}{\ln \left (2\right ) \left (2 x \ln \left (2\right )-3\right )}+\frac {6 \ln \left (x \right ) x}{2 x \ln \left (2\right )-3}-\frac {3 \,{\mathrm e}^{2 x}}{8 \ln \left (2\right )^{2} \left (x -\frac {3}{2 \ln \left (2\right )}\right )}-\frac {{\mathrm e}^{2 x}}{4 \ln \left (2\right )}\]
Input:
int((((-8*x^2*ln(2)+12*x+12)*exp(x)-8*x^2*ln(2)+24*x-36)*ln(x)+(-4*x^2*ln( 2)+6*x+3)*exp(x)^2+(2*(-x^3+2*x^2-4*x)*ln(2)+3*x^2-3*x+3)*exp(x)+2*(-4*x^2 +12*x-8)*ln(2)+12*x-36)/(8*x^2*ln(2)^2-24*x*ln(2)+18),x)
Output:
1/2*(-exp(x)*x^2+3*exp(x)*x-4*x*exp(x)*ln(x))/(2*x*ln(2)-3)+4/(2*x*ln(2)-3 )-1/ln(2)*ln(x)*x-3/ln(2)*ln(x)*x/(2*x*ln(2)-3)+6*ln(x)*x/(2*x*ln(2)-3)-3/ 8/ln(2)^2*exp(x)^2/(x-3/2/ln(2))-1/4/ln(2)*exp(x)^2
Time = 0.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.29 \[ \int \frac {-36+12 x+\left (-8+12 x-4 x^2\right ) \log (4)+e^{2 x} \left (3+6 x-2 x^2 \log (4)\right )+e^x \left (3-3 x+3 x^2+\left (-4 x+2 x^2-x^3\right ) \log (4)\right )+\left (-36+24 x-4 x^2 \log (4)+e^x \left (12+12 x-4 x^2 \log (4)\right )\right ) \log (x)}{18-12 x \log (4)+2 x^2 \log ^2(4)} \, dx=-\frac {x e^{\left (2 \, x\right )} + {\left (x^{2} - 3 \, x\right )} e^{x} + 4 \, {\left (x^{2} + x e^{x} - 3 \, x\right )} \log \left (x\right ) - 8}{2 \, {\left (2 \, x \log \left (2\right ) - 3\right )}} \] Input:
integrate((((-8*x^2*log(2)+12*x+12)*exp(x)-8*x^2*log(2)+24*x-36)*log(x)+(- 4*x^2*log(2)+6*x+3)*exp(x)^2+(2*(-x^3+2*x^2-4*x)*log(2)+3*x^2-3*x+3)*exp(x )+2*(-4*x^2+12*x-8)*log(2)+12*x-36)/(8*x^2*log(2)^2-24*x*log(2)+18),x, alg orithm="fricas")
Output:
-1/2*(x*e^(2*x) + (x^2 - 3*x)*e^x + 4*(x^2 + x*e^x - 3*x)*log(x) - 8)/(2*x *log(2) - 3)
Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (27) = 54\).
Time = 0.59 (sec) , antiderivative size = 153, normalized size of antiderivative = 4.50 \[ \int \frac {-36+12 x+\left (-8+12 x-4 x^2\right ) \log (4)+e^{2 x} \left (3+6 x-2 x^2 \log (4)\right )+e^x \left (3-3 x+3 x^2+\left (-4 x+2 x^2-x^3\right ) \log (4)\right )+\left (-36+24 x-4 x^2 \log (4)+e^x \left (12+12 x-4 x^2 \log (4)\right )\right ) \log (x)}{18-12 x \log (4)+2 x^2 \log ^2(4)} \, dx=\frac {\left (- 4 x^{2} \log {\left (2 \right )} + 6 x\right ) e^{2 x} + \left (- 4 x^{3} \log {\left (2 \right )} - 16 x^{2} \log {\left (2 \right )} \log {\left (x \right )} + 6 x^{2} + 12 x^{2} \log {\left (2 \right )} + 24 x \log {\left (x \right )} - 18 x\right ) e^{x}}{16 x^{2} \log {\left (2 \right )}^{2} - 48 x \log {\left (2 \right )} + 36} + \frac {3 \left (-1 + 2 \log {\left (2 \right )}\right ) \log {\left (x \right )}}{2 \log {\left (2 \right )}^{2}} + \frac {\left (- 4 x^{2} \log {\left (2 \right )}^{2} + 6 x \log {\left (2 \right )} - 9 + 18 \log {\left (2 \right )}\right ) \log {\left (x \right )}}{4 x \log {\left (2 \right )}^{3} - 6 \log {\left (2 \right )}^{2}} + \frac {4}{2 x \log {\left (2 \right )} - 3} \] Input:
integrate((((-8*x**2*ln(2)+12*x+12)*exp(x)-8*x**2*ln(2)+24*x-36)*ln(x)+(-4 *x**2*ln(2)+6*x+3)*exp(x)**2+(2*(-x**3+2*x**2-4*x)*ln(2)+3*x**2-3*x+3)*exp (x)+2*(-4*x**2+12*x-8)*ln(2)+12*x-36)/(8*x**2*ln(2)**2-24*x*ln(2)+18),x)
Output:
((-4*x**2*log(2) + 6*x)*exp(2*x) + (-4*x**3*log(2) - 16*x**2*log(2)*log(x) + 6*x**2 + 12*x**2*log(2) + 24*x*log(x) - 18*x)*exp(x))/(16*x**2*log(2)** 2 - 48*x*log(2) + 36) + 3*(-1 + 2*log(2))*log(x)/(2*log(2)**2) + (-4*x**2* log(2)**2 + 6*x*log(2) - 9 + 18*log(2))*log(x)/(4*x*log(2)**3 - 6*log(2)** 2) + 4/(2*x*log(2) - 3)
\[ \int \frac {-36+12 x+\left (-8+12 x-4 x^2\right ) \log (4)+e^{2 x} \left (3+6 x-2 x^2 \log (4)\right )+e^x \left (3-3 x+3 x^2+\left (-4 x+2 x^2-x^3\right ) \log (4)\right )+\left (-36+24 x-4 x^2 \log (4)+e^x \left (12+12 x-4 x^2 \log (4)\right )\right ) \log (x)}{18-12 x \log (4)+2 x^2 \log ^2(4)} \, dx=\int { -\frac {{\left (4 \, x^{2} \log \left (2\right ) - 6 \, x - 3\right )} e^{\left (2 \, x\right )} - {\left (3 \, x^{2} - 2 \, {\left (x^{3} - 2 \, x^{2} + 4 \, x\right )} \log \left (2\right ) - 3 \, x + 3\right )} e^{x} + 8 \, {\left (x^{2} - 3 \, x + 2\right )} \log \left (2\right ) + 4 \, {\left (2 \, x^{2} \log \left (2\right ) + {\left (2 \, x^{2} \log \left (2\right ) - 3 \, x - 3\right )} e^{x} - 6 \, x + 9\right )} \log \left (x\right ) - 12 \, x + 36}{2 \, {\left (4 \, x^{2} \log \left (2\right )^{2} - 12 \, x \log \left (2\right ) + 9\right )}} \,d x } \] Input:
integrate((((-8*x^2*log(2)+12*x+12)*exp(x)-8*x^2*log(2)+24*x-36)*log(x)+(- 4*x^2*log(2)+6*x+3)*exp(x)^2+(2*(-x^3+2*x^2-4*x)*log(2)+3*x^2-3*x+3)*exp(x )+2*(-4*x^2+12*x-8)*log(2)+12*x-36)/(8*x^2*log(2)^2-24*x*log(2)+18),x, alg orithm="maxima")
Output:
1/2*(9/(2*x*log(2)^4 - 3*log(2)^3) - 2*x/log(2)^2 - 6*log(2*x*log(2) - 3)/ log(2)^3)*log(2) - 3*(3/(2*x*log(2)^3 - 3*log(2)^2) - log(2*x*log(2) - 3)/ log(2)^2)*log(2) - 1/2*(4*x*e^x*log(2)^2*log(x) - 4*x^2*log(2)^2 + x*e^(2* x)*log(2)^2 + 6*x*log(2) + (4*x^2*log(2)^2 - 6*x*log(2) - 18*log(2) + 9)*l og(x))/(2*x*log(2)^3 - 3*log(2)^2) - 3/4*e^(3/2/log(2))*exp_integral_e(2, -1/2*(2*x*log(2) - 3)/log(2))/((2*x*log(2) - 3)*log(2)) + 4*log(2)/(2*x*lo g(2)^2 - 3*log(2)) - 3/2*(2*log(2) - 1)*log(2*x*log(2) - 3)/log(2)^2 + 3/2 *(2*log(2) - 1)*log(x)/log(2)^2 - 9/2/(2*x*log(2)^3 - 3*log(2)^2) + 9/(2*x *log(2)^2 - 3*log(2)) + 3/2*log(2*x*log(2) - 3)/log(2)^2 - 1/2*integrate(( 2*x^3*log(2) - x^2*(4*log(2) + 3) + 3*x + 12)*e^x/(4*x^2*log(2)^2 - 12*x*l og(2) + 9), x)
Time = 0.13 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {-36+12 x+\left (-8+12 x-4 x^2\right ) \log (4)+e^{2 x} \left (3+6 x-2 x^2 \log (4)\right )+e^x \left (3-3 x+3 x^2+\left (-4 x+2 x^2-x^3\right ) \log (4)\right )+\left (-36+24 x-4 x^2 \log (4)+e^x \left (12+12 x-4 x^2 \log (4)\right )\right ) \log (x)}{18-12 x \log (4)+2 x^2 \log ^2(4)} \, dx=-\frac {x^{2} e^{x} + 4 \, x^{2} \log \left (x\right ) + 4 \, x e^{x} \log \left (x\right ) + x e^{\left (2 \, x\right )} - 3 \, x e^{x} - 12 \, x \log \left (x\right ) - 8}{2 \, {\left (2 \, x \log \left (2\right ) - 3\right )}} \] Input:
integrate((((-8*x^2*log(2)+12*x+12)*exp(x)-8*x^2*log(2)+24*x-36)*log(x)+(- 4*x^2*log(2)+6*x+3)*exp(x)^2+(2*(-x^3+2*x^2-4*x)*log(2)+3*x^2-3*x+3)*exp(x )+2*(-4*x^2+12*x-8)*log(2)+12*x-36)/(8*x^2*log(2)^2-24*x*log(2)+18),x, alg orithm="giac")
Output:
-1/2*(x^2*e^x + 4*x^2*log(x) + 4*x*e^x*log(x) + x*e^(2*x) - 3*x*e^x - 12*x *log(x) - 8)/(2*x*log(2) - 3)
Timed out. \[ \int \frac {-36+12 x+\left (-8+12 x-4 x^2\right ) \log (4)+e^{2 x} \left (3+6 x-2 x^2 \log (4)\right )+e^x \left (3-3 x+3 x^2+\left (-4 x+2 x^2-x^3\right ) \log (4)\right )+\left (-36+24 x-4 x^2 \log (4)+e^x \left (12+12 x-4 x^2 \log (4)\right )\right ) \log (x)}{18-12 x \log (4)+2 x^2 \log ^2(4)} \, dx=\int \frac {12\,x+\ln \left (x\right )\,\left (24\,x+{\mathrm {e}}^x\,\left (-8\,\ln \left (2\right )\,x^2+12\,x+12\right )-8\,x^2\,\ln \left (2\right )-36\right )-2\,\ln \left (2\right )\,\left (4\,x^2-12\,x+8\right )+{\mathrm {e}}^{2\,x}\,\left (-4\,\ln \left (2\right )\,x^2+6\,x+3\right )-{\mathrm {e}}^x\,\left (3\,x+2\,\ln \left (2\right )\,\left (x^3-2\,x^2+4\,x\right )-3\,x^2-3\right )-36}{8\,{\ln \left (2\right )}^2\,x^2-24\,\ln \left (2\right )\,x+18} \,d x \] Input:
int((12*x + log(x)*(24*x + exp(x)*(12*x - 8*x^2*log(2) + 12) - 8*x^2*log(2 ) - 36) - 2*log(2)*(4*x^2 - 12*x + 8) + exp(2*x)*(6*x - 4*x^2*log(2) + 3) - exp(x)*(3*x + 2*log(2)*(4*x - 2*x^2 + x^3) - 3*x^2 - 3) - 36)/(8*x^2*log (2)^2 - 24*x*log(2) + 18),x)
Output:
int((12*x + log(x)*(24*x + exp(x)*(12*x - 8*x^2*log(2) + 12) - 8*x^2*log(2 ) - 36) - 2*log(2)*(4*x^2 - 12*x + 8) + exp(2*x)*(6*x - 4*x^2*log(2) + 3) - exp(x)*(3*x + 2*log(2)*(4*x - 2*x^2 + x^3) - 3*x^2 - 3) - 36)/(8*x^2*log (2)^2 - 24*x*log(2) + 18), x)
Time = 0.19 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.47 \[ \int \frac {-36+12 x+\left (-8+12 x-4 x^2\right ) \log (4)+e^{2 x} \left (3+6 x-2 x^2 \log (4)\right )+e^x \left (3-3 x+3 x^2+\left (-4 x+2 x^2-x^3\right ) \log (4)\right )+\left (-36+24 x-4 x^2 \log (4)+e^x \left (12+12 x-4 x^2 \log (4)\right )\right ) \log (x)}{18-12 x \log (4)+2 x^2 \log ^2(4)} \, dx=\frac {x \left (-3 e^{2 x}-12 e^{x} \mathrm {log}\left (x \right )-3 e^{x} x +9 e^{x}-12 \,\mathrm {log}\left (x \right ) x +36 \,\mathrm {log}\left (x \right )+16 \,\mathrm {log}\left (2\right )\right )}{12 \,\mathrm {log}\left (2\right ) x -18} \] Input:
int((((-8*x^2*log(2)+12*x+12)*exp(x)-8*x^2*log(2)+24*x-36)*log(x)+(-4*x^2* log(2)+6*x+3)*exp(x)^2+(2*(-x^3+2*x^2-4*x)*log(2)+3*x^2-3*x+3)*exp(x)+2*(- 4*x^2+12*x-8)*log(2)+12*x-36)/(8*x^2*log(2)^2-24*x*log(2)+18),x)
Output:
(x*( - 3*e**(2*x) - 12*e**x*log(x) - 3*e**x*x + 9*e**x - 12*log(x)*x + 36* log(x) + 16*log(2)))/(6*(2*log(2)*x - 3))