Integrand size = 128, antiderivative size = 25 \[ \int \frac {36-e^{3 x} x+12 \log (2)+\log ^2(2)+e^{2 x} (1-12 x-2 x \log (2))+e^x \left (12-32 x+(2-12 x) \log (2)-x \log ^2(2)\right )+\left (72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))\right ) \log (x)}{36 x+e^{2 x} x+12 x \log (2)+x \log ^2(2)+e^x (12 x+2 x \log (2))} \, dx=-e^x-\frac {4}{6+e^x+\log (2)}+\log ^2(x)+\log (2 x) \] Output:
ln(2*x)-exp(x)-4/(6+ln(2)+exp(x))+ln(x)^2
\[ \int \frac {36-e^{3 x} x+12 \log (2)+\log ^2(2)+e^{2 x} (1-12 x-2 x \log (2))+e^x \left (12-32 x+(2-12 x) \log (2)-x \log ^2(2)\right )+\left (72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))\right ) \log (x)}{36 x+e^{2 x} x+12 x \log (2)+x \log ^2(2)+e^x (12 x+2 x \log (2))} \, dx=\int \frac {36-e^{3 x} x+12 \log (2)+\log ^2(2)+e^{2 x} (1-12 x-2 x \log (2))+e^x \left (12-32 x+(2-12 x) \log (2)-x \log ^2(2)\right )+\left (72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))\right ) \log (x)}{36 x+e^{2 x} x+12 x \log (2)+x \log ^2(2)+e^x (12 x+2 x \log (2))} \, dx \] Input:
Integrate[(36 - E^(3*x)*x + 12*Log[2] + Log[2]^2 + E^(2*x)*(1 - 12*x - 2*x *Log[2]) + E^x*(12 - 32*x + (2 - 12*x)*Log[2] - x*Log[2]^2) + (72 + 2*E^(2 *x) + 24*Log[2] + 2*Log[2]^2 + E^x*(24 + 4*Log[2]))*Log[x])/(36*x + E^(2*x )*x + 12*x*Log[2] + x*Log[2]^2 + E^x*(12*x + 2*x*Log[2])),x]
Output:
Integrate[(36 - E^(3*x)*x + 12*Log[2] + Log[2]^2 + E^(2*x)*(1 - 12*x - 2*x *Log[2]) + E^x*(12 - 32*x + (2 - 12*x)*Log[2] - x*Log[2]^2) + (72 + 2*E^(2 *x) + 24*Log[2] + 2*Log[2]^2 + E^x*(24 + 4*Log[2]))*Log[x])/(36*x + E^(2*x )*x + 12*x*Log[2] + x*Log[2]^2 + E^x*(12*x + 2*x*Log[2])), x]
Time = 1.42 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {6, 6, 7292, 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.
| \(\displaystyle \int \frac {-e^{3 x} x+e^x \left (-32 x-x \log ^2(2)+(2-12 x) \log (2)+12\right )+\left (2 e^{2 x}+e^x (24+4 \log (2))+72+2 \log ^2(2)+24 \log (2)\right ) \log (x)+e^{2 x} (-12 x-2 x \log (2)+1)+36+\log ^2(2)+12 \log (2)}{e^{2 x} x+36 x+x \log ^2(2)+12 x \log (2)+e^x (12 x+2 x \log (2))} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-e^{3 x} x+e^x \left (-32 x-x \log ^2(2)+(2-12 x) \log (2)+12\right )+\left (2 e^{2 x}+e^x (24+4 \log (2))+72+2 \log ^2(2)+24 \log (2)\right ) \log (x)+e^{2 x} (-12 x-2 x \log (2)+1)+36+\log ^2(2)+12 \log (2)}{e^{2 x} x+x \log ^2(2)+x (36+12 \log (2))+e^x (12 x+2 x \log (2))}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-e^{3 x} x+e^x \left (-32 x-x \log ^2(2)+(2-12 x) \log (2)+12\right )+\left (2 e^{2 x}+e^x (24+4 \log (2))+72+2 \log ^2(2)+24 \log (2)\right ) \log (x)+e^{2 x} (-12 x-2 x \log (2)+1)+36+\log ^2(2)+12 \log (2)}{e^{2 x} x+x \left (36+\log ^2(2)+12 \log (2)\right )+e^x (12 x+2 x \log (2))}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-e^{3 x} x+e^x \left (-32 x-x \log ^2(2)+(2-12 x) \log (2)+12\right )+\left (2 e^{2 x}+e^x (24+4 \log (2))+72+2 \log ^2(2)+24 \log (2)\right ) \log (x)+e^{2 x} (-12 x-2 x \log (2)+1)+36 \left (1+\frac {1}{36} \log (2) (12+\log (2))\right )}{x \left (e^x+6 \left (1+\frac {\log (2)}{6}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-e^x+\frac {2 \log (x)+1}{x}+\frac {4}{e^x+6 \left (1+\frac {\log (2)}{6}\right )}+\frac {4 (-6-\log (2))}{\left (e^x+6 \left (1+\frac {\log (2)}{6}\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -e^x+\frac {1}{4} (2 \log (x)+1)^2-\frac {4}{e^x+6+\log (2)}\) |
Input:
Int[(36 - E^(3*x)*x + 12*Log[2] + Log[2]^2 + E^(2*x)*(1 - 12*x - 2*x*Log[2 ]) + E^x*(12 - 32*x + (2 - 12*x)*Log[2] - x*Log[2]^2) + (72 + 2*E^(2*x) + 24*Log[2] + 2*Log[2]^2 + E^x*(24 + 4*Log[2]))*Log[x])/(36*x + E^(2*x)*x + 12*x*Log[2] + x*Log[2]^2 + E^x*(12*x + 2*x*Log[2])),x]
Output:
-E^x - 4/(6 + E^x + Log[2]) + (1 + 2*Log[x])^2/4
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]
Time = 22.70 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.88
| method | result | size |
| risch | \(\ln \left (x \right )^{2}+\frac {\ln \left (2\right ) \ln \left (x \right )-{\mathrm e}^{x} \ln \left (2\right )+{\mathrm e}^{x} \ln \left (x \right )-{\mathrm e}^{2 x}+6 \ln \left (x \right )-6 \,{\mathrm e}^{x}-4}{6+\ln \left (2\right )+{\mathrm e}^{x}}\) | \(47\) |
| norman | \(\frac {\left (\ln \left (2\right )+6\right ) \ln \left (x \right )^{2}+\left (\ln \left (2\right )+6\right ) \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )^{2}+{\mathrm e}^{x} \ln \left (x \right )-{\mathrm e}^{2 x}+\ln \left (2\right )^{2}+12 \ln \left (2\right )+32}{6+\ln \left (2\right )+{\mathrm e}^{x}}\) | \(54\) |
| parallelrisch | \(\frac {\ln \left (2\right ) \ln \left (x \right )^{2}+{\mathrm e}^{x} \ln \left (x \right )^{2}+\ln \left (2\right )^{2}+\ln \left (2\right ) \ln \left (x \right )+32+6 \ln \left (x \right )^{2}+{\mathrm e}^{x} \ln \left (x \right )-{\mathrm e}^{2 x}+12 \ln \left (2\right )+6 \ln \left (x \right )}{6+\ln \left (2\right )+{\mathrm e}^{x}}\) | \(60\) |
Input:
int(((2*exp(x)^2+(4*ln(2)+24)*exp(x)+2*ln(2)^2+24*ln(2)+72)*ln(x)-x*exp(x) ^3+(-2*x*ln(2)-12*x+1)*exp(x)^2+(-x*ln(2)^2+(-12*x+2)*ln(2)-32*x+12)*exp(x )+ln(2)^2+12*ln(2)+36)/(x*exp(x)^2+(2*x*ln(2)+12*x)*exp(x)+x*ln(2)^2+12*x* ln(2)+36*x),x,method=_RETURNVERBOSE)
Output:
ln(x)^2+(ln(2)*ln(x)-exp(x)*ln(2)+exp(x)*ln(x)-exp(2*x)+6*ln(x)-6*exp(x)-4 )/(6+ln(2)+exp(x))
Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80 \[ \int \frac {36-e^{3 x} x+12 \log (2)+\log ^2(2)+e^{2 x} (1-12 x-2 x \log (2))+e^x \left (12-32 x+(2-12 x) \log (2)-x \log ^2(2)\right )+\left (72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))\right ) \log (x)}{36 x+e^{2 x} x+12 x \log (2)+x \log ^2(2)+e^x (12 x+2 x \log (2))} \, dx=\frac {{\left (e^{x} + \log \left (2\right ) + 6\right )} \log \left (x\right )^{2} - {\left (\log \left (2\right ) + 6\right )} e^{x} + {\left (e^{x} + \log \left (2\right ) + 6\right )} \log \left (x\right ) - e^{\left (2 \, x\right )} - 4}{e^{x} + \log \left (2\right ) + 6} \] Input:
integrate(((2*exp(x)^2+(4*log(2)+24)*exp(x)+2*log(2)^2+24*log(2)+72)*log(x )-x*exp(x)^3+(-2*x*log(2)-12*x+1)*exp(x)^2+(-x*log(2)^2+(-12*x+2)*log(2)-3 2*x+12)*exp(x)+log(2)^2+12*log(2)+36)/(x*exp(x)^2+(2*x*log(2)+12*x)*exp(x) +x*log(2)^2+12*x*log(2)+36*x),x, algorithm="fricas")
Output:
((e^x + log(2) + 6)*log(x)^2 - (log(2) + 6)*e^x + (e^x + log(2) + 6)*log(x ) - e^(2*x) - 4)/(e^x + log(2) + 6)
Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {36-e^{3 x} x+12 \log (2)+\log ^2(2)+e^{2 x} (1-12 x-2 x \log (2))+e^x \left (12-32 x+(2-12 x) \log (2)-x \log ^2(2)\right )+\left (72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))\right ) \log (x)}{36 x+e^{2 x} x+12 x \log (2)+x \log ^2(2)+e^x (12 x+2 x \log (2))} \, dx=- e^{x} + \log {\left (x \right )}^{2} + \log {\left (x \right )} - \frac {4}{e^{x} + \log {\left (2 \right )} + 6} \] Input:
integrate(((2*exp(x)**2+(4*ln(2)+24)*exp(x)+2*ln(2)**2+24*ln(2)+72)*ln(x)- x*exp(x)**3+(-2*x*ln(2)-12*x+1)*exp(x)**2+(-x*ln(2)**2+(-12*x+2)*ln(2)-32* x+12)*exp(x)+ln(2)**2+12*ln(2)+36)/(x*exp(x)**2+(2*x*ln(2)+12*x)*exp(x)+x* ln(2)**2+12*x*ln(2)+36*x),x)
Output:
-exp(x) + log(x)**2 + log(x) - 4/(exp(x) + log(2) + 6)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (23) = 46\).
Time = 0.17 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.92 \[ \int \frac {36-e^{3 x} x+12 \log (2)+\log ^2(2)+e^{2 x} (1-12 x-2 x \log (2))+e^x \left (12-32 x+(2-12 x) \log (2)-x \log ^2(2)\right )+\left (72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))\right ) \log (x)}{36 x+e^{2 x} x+12 x \log (2)+x \log ^2(2)+e^x (12 x+2 x \log (2))} \, dx=\frac {{\left (\log \left (2\right ) + 6\right )} \log \left (x\right )^{2} + {\left (\log \left (x\right )^{2} - \log \left (2\right ) + \log \left (x\right ) - 6\right )} e^{x} + {\left (\log \left (2\right ) + 6\right )} \log \left (x\right ) - e^{\left (2 \, x\right )} - 4}{e^{x} + \log \left (2\right ) + 6} \] Input:
integrate(((2*exp(x)^2+(4*log(2)+24)*exp(x)+2*log(2)^2+24*log(2)+72)*log(x )-x*exp(x)^3+(-2*x*log(2)-12*x+1)*exp(x)^2+(-x*log(2)^2+(-12*x+2)*log(2)-3 2*x+12)*exp(x)+log(2)^2+12*log(2)+36)/(x*exp(x)^2+(2*x*log(2)+12*x)*exp(x) +x*log(2)^2+12*x*log(2)+36*x),x, algorithm="maxima")
Output:
((log(2) + 6)*log(x)^2 + (log(x)^2 - log(2) + log(x) - 6)*e^x + (log(2) + 6)*log(x) - e^(2*x) - 4)/(e^x + log(2) + 6)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (23) = 46\).
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.44 \[ \int \frac {36-e^{3 x} x+12 \log (2)+\log ^2(2)+e^{2 x} (1-12 x-2 x \log (2))+e^x \left (12-32 x+(2-12 x) \log (2)-x \log ^2(2)\right )+\left (72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))\right ) \log (x)}{36 x+e^{2 x} x+12 x \log (2)+x \log ^2(2)+e^x (12 x+2 x \log (2))} \, dx=\frac {e^{x} \log \left (x\right )^{2} + \log \left (2\right ) \log \left (x\right )^{2} - e^{x} \log \left (2\right ) + e^{x} \log \left (x\right ) + \log \left (2\right ) \log \left (x\right ) + 6 \, \log \left (x\right )^{2} - e^{\left (2 \, x\right )} - 6 \, e^{x} + 6 \, \log \left (x\right ) - 4}{e^{x} + \log \left (2\right ) + 6} \] Input:
integrate(((2*exp(x)^2+(4*log(2)+24)*exp(x)+2*log(2)^2+24*log(2)+72)*log(x )-x*exp(x)^3+(-2*x*log(2)-12*x+1)*exp(x)^2+(-x*log(2)^2+(-12*x+2)*log(2)-3 2*x+12)*exp(x)+log(2)^2+12*log(2)+36)/(x*exp(x)^2+(2*x*log(2)+12*x)*exp(x) +x*log(2)^2+12*x*log(2)+36*x),x, algorithm="giac")
Output:
(e^x*log(x)^2 + log(2)*log(x)^2 - e^x*log(2) + e^x*log(x) + log(2)*log(x) + 6*log(x)^2 - e^(2*x) - 6*e^x + 6*log(x) - 4)/(e^x + log(2) + 6)
Timed out. \[ \int \frac {36-e^{3 x} x+12 \log (2)+\log ^2(2)+e^{2 x} (1-12 x-2 x \log (2))+e^x \left (12-32 x+(2-12 x) \log (2)-x \log ^2(2)\right )+\left (72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))\right ) \log (x)}{36 x+e^{2 x} x+12 x \log (2)+x \log ^2(2)+e^x (12 x+2 x \log (2))} \, dx=\int \frac {12\,\ln \left (2\right )-x\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^{2\,x}\,\left (12\,x+2\,x\,\ln \left (2\right )-1\right )-{\mathrm {e}}^x\,\left (32\,x+\ln \left (2\right )\,\left (12\,x-2\right )+x\,{\ln \left (2\right )}^2-12\right )+\ln \left (x\right )\,\left (2\,{\mathrm {e}}^{2\,x}+24\,\ln \left (2\right )+{\mathrm {e}}^x\,\left (4\,\ln \left (2\right )+24\right )+2\,{\ln \left (2\right )}^2+72\right )+{\ln \left (2\right )}^2+36}{36\,x+x\,{\mathrm {e}}^{2\,x}+12\,x\,\ln \left (2\right )+x\,{\ln \left (2\right )}^2+{\mathrm {e}}^x\,\left (12\,x+2\,x\,\ln \left (2\right )\right )} \,d x \] Input:
int((12*log(2) - x*exp(3*x) - exp(2*x)*(12*x + 2*x*log(2) - 1) - exp(x)*(3 2*x + log(2)*(12*x - 2) + x*log(2)^2 - 12) + log(x)*(2*exp(2*x) + 24*log(2 ) + exp(x)*(4*log(2) + 24) + 2*log(2)^2 + 72) + log(2)^2 + 36)/(36*x + x*e xp(2*x) + 12*x*log(2) + x*log(2)^2 + exp(x)*(12*x + 2*x*log(2))),x)
Output:
int((12*log(2) - x*exp(3*x) - exp(2*x)*(12*x + 2*x*log(2) - 1) - exp(x)*(3 2*x + log(2)*(12*x - 2) + x*log(2)^2 - 12) + log(x)*(2*exp(2*x) + 24*log(2 ) + exp(x)*(4*log(2) + 24) + 2*log(2)^2 + 72) + log(2)^2 + 36)/(36*x + x*e xp(2*x) + 12*x*log(2) + x*log(2)^2 + exp(x)*(12*x + 2*x*log(2))), x)
Time = 0.18 (sec) , antiderivative size = 136, normalized size of antiderivative = 5.44 \[ \int \frac {36-e^{3 x} x+12 \log (2)+\log ^2(2)+e^{2 x} (1-12 x-2 x \log (2))+e^x \left (12-32 x+(2-12 x) \log (2)-x \log ^2(2)\right )+\left (72+2 e^{2 x}+24 \log (2)+2 \log ^2(2)+e^x (24+4 \log (2))\right ) \log (x)}{36 x+e^{2 x} x+12 x \log (2)+x \log ^2(2)+e^x (12 x+2 x \log (2))} \, dx=\frac {-e^{2 x} \mathrm {log}\left (2\right )-6 e^{2 x}+e^{x} \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (2\right )+6 e^{x} \mathrm {log}\left (x \right )^{2}+e^{x} \mathrm {log}\left (x \right ) \mathrm {log}\left (2\right )+6 e^{x} \mathrm {log}\left (x \right )-e^{x} \mathrm {log}\left (2\right )^{2}-12 e^{x} \mathrm {log}\left (2\right )-32 e^{x}+\mathrm {log}\left (x \right )^{2} \mathrm {log}\left (2\right )^{2}+12 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (2\right )+36 \mathrm {log}\left (x \right )^{2}+\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right )^{2}+12 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right )+36 \,\mathrm {log}\left (x \right )}{e^{x} \mathrm {log}\left (2\right )+6 e^{x}+\mathrm {log}\left (2\right )^{2}+12 \,\mathrm {log}\left (2\right )+36} \] Input:
int(((2*exp(x)^2+(4*log(2)+24)*exp(x)+2*log(2)^2+24*log(2)+72)*log(x)-x*ex p(x)^3+(-2*x*log(2)-12*x+1)*exp(x)^2+(-x*log(2)^2+(-12*x+2)*log(2)-32*x+12 )*exp(x)+log(2)^2+12*log(2)+36)/(x*exp(x)^2+(2*x*log(2)+12*x)*exp(x)+x*log (2)^2+12*x*log(2)+36*x),x)
Output:
( - e**(2*x)*log(2) - 6*e**(2*x) + e**x*log(x)**2*log(2) + 6*e**x*log(x)** 2 + e**x*log(x)*log(2) + 6*e**x*log(x) - e**x*log(2)**2 - 12*e**x*log(2) - 32*e**x + log(x)**2*log(2)**2 + 12*log(x)**2*log(2) + 36*log(x)**2 + log( x)*log(2)**2 + 12*log(x)*log(2) + 36*log(x))/(e**x*log(2) + 6*e**x + log(2 )**2 + 12*log(2) + 36)