Integrand size = 196, antiderivative size = 33 \[ \int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} \left (200 x-600 x^2+200 x^3 \log (3)\right )+e^x \left (360 x^2-120 x^3-180 x^4+\left (-200 x^3+60 x^4+60 x^5\right ) \log (3)\right )+\left (-60 x^2+180 x^3-60 x^4 \log (3)+e^x \left (-200 x+700 x^2-300 x^3+\left (-200 x^3+100 x^4\right ) \log (3)\right )\right ) \log \left (1-3 x+x^2 \log (3)\right )}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} \left (25-75 x+25 x^2 \log (3)\right )+e^x \left (30 x-90 x^2+30 x^3 \log (3)\right )} \, dx=\frac {4 x \left (e^x-\log (1+x (-3+x \log (3)))\right )}{\frac {3}{5}+\frac {e^x}{x}} \]
\[ \int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} \left (200 x-600 x^2+200 x^3 \log (3)\right )+e^x \left (360 x^2-120 x^3-180 x^4+\left (-200 x^3+60 x^4+60 x^5\right ) \log (3)\right )+\left (-60 x^2+180 x^3-60 x^4 \log (3)+e^x \left (-200 x+700 x^2-300 x^3+\left (-200 x^3+100 x^4\right ) \log (3)\right )\right ) \log \left (1-3 x+x^2 \log (3)\right )}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} \left (25-75 x+25 x^2 \log (3)\right )+e^x \left (30 x-90 x^2+30 x^3 \log (3)\right )} \, dx=\int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} \left (200 x-600 x^2+200 x^3 \log (3)\right )+e^x \left (360 x^2-120 x^3-180 x^4+\left (-200 x^3+60 x^4+60 x^5\right ) \log (3)\right )+\left (-60 x^2+180 x^3-60 x^4 \log (3)+e^x \left (-200 x+700 x^2-300 x^3+\left (-200 x^3+100 x^4\right ) \log (3)\right )\right ) \log \left (1-3 x+x^2 \log (3)\right )}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} \left (25-75 x+25 x^2 \log (3)\right )+e^x \left (30 x-90 x^2+30 x^3 \log (3)\right )} \, dx \]
Integrate[(180*x^3 - 120*x^4*Log[3] + E^(2*x)*(200*x - 600*x^2 + 200*x^3*L og[3]) + E^x*(360*x^2 - 120*x^3 - 180*x^4 + (-200*x^3 + 60*x^4 + 60*x^5)*L og[3]) + (-60*x^2 + 180*x^3 - 60*x^4*Log[3] + E^x*(-200*x + 700*x^2 - 300* x^3 + (-200*x^3 + 100*x^4)*Log[3]))*Log[1 - 3*x + x^2*Log[3]])/(9*x^2 - 27 *x^3 + 9*x^4*Log[3] + E^(2*x)*(25 - 75*x + 25*x^2*Log[3]) + E^x*(30*x - 90 *x^2 + 30*x^3*Log[3])),x]
Integrate[(180*x^3 - 120*x^4*Log[3] + E^(2*x)*(200*x - 600*x^2 + 200*x^3*L og[3]) + E^x*(360*x^2 - 120*x^3 - 180*x^4 + (-200*x^3 + 60*x^4 + 60*x^5)*L og[3]) + (-60*x^2 + 180*x^3 - 60*x^4*Log[3] + E^x*(-200*x + 700*x^2 - 300* x^3 + (-200*x^3 + 100*x^4)*Log[3]))*Log[1 - 3*x + x^2*Log[3]])/(9*x^2 - 27 *x^3 + 9*x^4*Log[3] + E^(2*x)*(25 - 75*x + 25*x^2*Log[3]) + E^x*(30*x - 90 *x^2 + 30*x^3*Log[3])), x]
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.
| \(\displaystyle \int \frac {-120 x^4 \log (3)+180 x^3+e^{2 x} \left (200 x^3 \log (3)-600 x^2+200 x\right )+\left (-60 x^4 \log (3)+180 x^3-60 x^2+e^x \left (-300 x^3+700 x^2+\left (100 x^4-200 x^3\right ) \log (3)-200 x\right )\right ) \log \left (x^2 \log (3)-3 x+1\right )+e^x \left (-180 x^4-120 x^3+360 x^2+\left (60 x^5+60 x^4-200 x^3\right ) \log (3)\right )}{9 x^4 \log (3)-27 x^3+9 x^2+e^{2 x} \left (25 x^2 \log (3)-75 x+25\right )+e^x \left (30 x^3 \log (3)-90 x^2+30 x\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-120 x^4 \log (3)+180 x^3+e^{2 x} \left (200 x^3 \log (3)-600 x^2+200 x\right )+\left (-60 x^4 \log (3)+180 x^3-60 x^2+e^x \left (-300 x^3+700 x^2+\left (100 x^4-200 x^3\right ) \log (3)-200 x\right )\right ) \log \left (x^2 \log (3)-3 x+1\right )+e^x \left (-180 x^4-120 x^3+360 x^2+\left (60 x^5+60 x^4-200 x^3\right ) \log (3)\right )}{\left (3 x+5 e^x\right )^2 \left (x^2 \log (3)-3 x+1\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (-\frac {12 (x-1) x^2 \left (5 \log \left (x^2 \log (3)-3 x+1\right )+3 x\right )}{\left (3 x+5 e^x\right )^2}+\frac {4 x \left (x^4 \log (27)-9 x^3 (1+\log (3))-15 x^2 \left (1+\frac {2 \log (3)}{3}\right ) \log \left (x^2 \log (3)-3 x+1\right )+30 x^2 \left (1-\frac {\log (3)}{3}\right )+35 x \log \left (x^2 \log (3)-3 x+1\right )-10 \log \left (x^2 \log (3)-3 x+1\right )+5 x^3 \log (3) \log \left (x^2 \log (3)-3 x+1\right )+6 x\right )}{\left (3 x+5 e^x\right ) \left (x^2 \log (3)-3 x+1\right )}+8 x\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (-\frac {12 (x-1) x^2 \left (5 \log \left (x^2 \log (3)-3 x+1\right )+3 x\right )}{\left (3 x+5 e^x\right )^2}+\frac {4 x \left (x^4 \log (27)-9 x^3 (1+\log (3))-15 x^2 \left (1+\frac {2 \log (3)}{3}\right ) \log \left (x^2 \log (3)-3 x+1\right )+30 x^2 \left (1-\frac {\log (3)}{3}\right )+35 x \log \left (x^2 \log (3)-3 x+1\right )-10 \log \left (x^2 \log (3)-3 x+1\right )+5 x^3 \log (3) \log \left (x^2 \log (3)-3 x+1\right )+6 x\right )}{\left (3 x+5 e^x\right ) \left (x^2 \log (3)-3 x+1\right )}+8 x\right )dx\) |
Int[(180*x^3 - 120*x^4*Log[3] + E^(2*x)*(200*x - 600*x^2 + 200*x^3*Log[3]) + E^x*(360*x^2 - 120*x^3 - 180*x^4 + (-200*x^3 + 60*x^4 + 60*x^5)*Log[3]) + (-60*x^2 + 180*x^3 - 60*x^4*Log[3] + E^x*(-200*x + 700*x^2 - 300*x^3 + (-200*x^3 + 100*x^4)*Log[3]))*Log[1 - 3*x + x^2*Log[3]])/(9*x^2 - 27*x^3 + 9*x^4*Log[3] + E^(2*x)*(25 - 75*x + 25*x^2*Log[3]) + E^x*(30*x - 90*x^2 + 30*x^3*Log[3])),x]
3.14.87.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 1.31 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15
| method | result | size |
| parallelrisch | \(\frac {100 \,{\mathrm e}^{x} x^{2}-100 \ln \left (x^{2} \ln \left (3\right )-3 x +1\right ) x^{2}}{15 x +25 \,{\mathrm e}^{x}}\) | \(38\) |
| risch | \(-\frac {20 x^{2} \ln \left (x^{2} \ln \left (3\right )-3 x +1\right )}{3 x +5 \,{\mathrm e}^{x}}+\frac {20 x^{2} {\mathrm e}^{x}}{3 x +5 \,{\mathrm e}^{x}}\) | \(46\) |
int(((((100*x^4-200*x^3)*ln(3)-300*x^3+700*x^2-200*x)*exp(x)-60*x^4*ln(3)+ 180*x^3-60*x^2)*ln(x^2*ln(3)-3*x+1)+(200*x^3*ln(3)-600*x^2+200*x)*exp(x)^2 +((60*x^5+60*x^4-200*x^3)*ln(3)-180*x^4-120*x^3+360*x^2)*exp(x)-120*x^4*ln (3)+180*x^3)/((25*x^2*ln(3)-75*x+25)*exp(x)^2+(30*x^3*ln(3)-90*x^2+30*x)*e xp(x)+9*x^4*ln(3)-27*x^3+9*x^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} \left (200 x-600 x^2+200 x^3 \log (3)\right )+e^x \left (360 x^2-120 x^3-180 x^4+\left (-200 x^3+60 x^4+60 x^5\right ) \log (3)\right )+\left (-60 x^2+180 x^3-60 x^4 \log (3)+e^x \left (-200 x+700 x^2-300 x^3+\left (-200 x^3+100 x^4\right ) \log (3)\right )\right ) \log \left (1-3 x+x^2 \log (3)\right )}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} \left (25-75 x+25 x^2 \log (3)\right )+e^x \left (30 x-90 x^2+30 x^3 \log (3)\right )} \, dx=\frac {20 \, {\left (x^{2} e^{x} - x^{2} \log \left (x^{2} \log \left (3\right ) - 3 \, x + 1\right )\right )}}{3 \, x + 5 \, e^{x}} \]
integrate(((((100*x^4-200*x^3)*log(3)-300*x^3+700*x^2-200*x)*exp(x)-60*x^4 *log(3)+180*x^3-60*x^2)*log(x^2*log(3)-3*x+1)+(200*x^3*log(3)-600*x^2+200* x)*exp(x)^2+((60*x^5+60*x^4-200*x^3)*log(3)-180*x^4-120*x^3+360*x^2)*exp(x )-120*x^4*log(3)+180*x^3)/((25*x^2*log(3)-75*x+25)*exp(x)^2+(30*x^3*log(3) -90*x^2+30*x)*exp(x)+9*x^4*log(3)-27*x^3+9*x^2),x, algorithm=\
Time = 0.15 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} \left (200 x-600 x^2+200 x^3 \log (3)\right )+e^x \left (360 x^2-120 x^3-180 x^4+\left (-200 x^3+60 x^4+60 x^5\right ) \log (3)\right )+\left (-60 x^2+180 x^3-60 x^4 \log (3)+e^x \left (-200 x+700 x^2-300 x^3+\left (-200 x^3+100 x^4\right ) \log (3)\right )\right ) \log \left (1-3 x+x^2 \log (3)\right )}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} \left (25-75 x+25 x^2 \log (3)\right )+e^x \left (30 x-90 x^2+30 x^3 \log (3)\right )} \, dx=4 x^{2} + \frac {- 12 x^{3} - 20 x^{2} \log {\left (x^{2} \log {\left (3 \right )} - 3 x + 1 \right )}}{3 x + 5 e^{x}} \]
integrate(((((100*x**4-200*x**3)*ln(3)-300*x**3+700*x**2-200*x)*exp(x)-60* x**4*ln(3)+180*x**3-60*x**2)*ln(x**2*ln(3)-3*x+1)+(200*x**3*ln(3)-600*x**2 +200*x)*exp(x)**2+((60*x**5+60*x**4-200*x**3)*ln(3)-180*x**4-120*x**3+360* x**2)*exp(x)-120*x**4*ln(3)+180*x**3)/((25*x**2*ln(3)-75*x+25)*exp(x)**2+( 30*x**3*ln(3)-90*x**2+30*x)*exp(x)+9*x**4*ln(3)-27*x**3+9*x**2),x)
Time = 0.37 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} \left (200 x-600 x^2+200 x^3 \log (3)\right )+e^x \left (360 x^2-120 x^3-180 x^4+\left (-200 x^3+60 x^4+60 x^5\right ) \log (3)\right )+\left (-60 x^2+180 x^3-60 x^4 \log (3)+e^x \left (-200 x+700 x^2-300 x^3+\left (-200 x^3+100 x^4\right ) \log (3)\right )\right ) \log \left (1-3 x+x^2 \log (3)\right )}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} \left (25-75 x+25 x^2 \log (3)\right )+e^x \left (30 x-90 x^2+30 x^3 \log (3)\right )} \, dx=\frac {20 \, {\left (x^{2} e^{x} - x^{2} \log \left (x^{2} \log \left (3\right ) - 3 \, x + 1\right )\right )}}{3 \, x + 5 \, e^{x}} \]
integrate(((((100*x^4-200*x^3)*log(3)-300*x^3+700*x^2-200*x)*exp(x)-60*x^4 *log(3)+180*x^3-60*x^2)*log(x^2*log(3)-3*x+1)+(200*x^3*log(3)-600*x^2+200* x)*exp(x)^2+((60*x^5+60*x^4-200*x^3)*log(3)-180*x^4-120*x^3+360*x^2)*exp(x )-120*x^4*log(3)+180*x^3)/((25*x^2*log(3)-75*x+25)*exp(x)^2+(30*x^3*log(3) -90*x^2+30*x)*exp(x)+9*x^4*log(3)-27*x^3+9*x^2),x, algorithm=\
Time = 0.33 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} \left (200 x-600 x^2+200 x^3 \log (3)\right )+e^x \left (360 x^2-120 x^3-180 x^4+\left (-200 x^3+60 x^4+60 x^5\right ) \log (3)\right )+\left (-60 x^2+180 x^3-60 x^4 \log (3)+e^x \left (-200 x+700 x^2-300 x^3+\left (-200 x^3+100 x^4\right ) \log (3)\right )\right ) \log \left (1-3 x+x^2 \log (3)\right )}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} \left (25-75 x+25 x^2 \log (3)\right )+e^x \left (30 x-90 x^2+30 x^3 \log (3)\right )} \, dx=\frac {20 \, {\left (x^{2} e^{x} - x^{2} \log \left (x^{2} \log \left (3\right ) - 3 \, x + 1\right )\right )}}{3 \, x + 5 \, e^{x}} \]
integrate(((((100*x^4-200*x^3)*log(3)-300*x^3+700*x^2-200*x)*exp(x)-60*x^4 *log(3)+180*x^3-60*x^2)*log(x^2*log(3)-3*x+1)+(200*x^3*log(3)-600*x^2+200* x)*exp(x)^2+((60*x^5+60*x^4-200*x^3)*log(3)-180*x^4-120*x^3+360*x^2)*exp(x )-120*x^4*log(3)+180*x^3)/((25*x^2*log(3)-75*x+25)*exp(x)^2+(30*x^3*log(3) -90*x^2+30*x)*exp(x)+9*x^4*log(3)-27*x^3+9*x^2),x, algorithm=\
Time = 14.64 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {180 x^3-120 x^4 \log (3)+e^{2 x} \left (200 x-600 x^2+200 x^3 \log (3)\right )+e^x \left (360 x^2-120 x^3-180 x^4+\left (-200 x^3+60 x^4+60 x^5\right ) \log (3)\right )+\left (-60 x^2+180 x^3-60 x^4 \log (3)+e^x \left (-200 x+700 x^2-300 x^3+\left (-200 x^3+100 x^4\right ) \log (3)\right )\right ) \log \left (1-3 x+x^2 \log (3)\right )}{9 x^2-27 x^3+9 x^4 \log (3)+e^{2 x} \left (25-75 x+25 x^2 \log (3)\right )+e^x \left (30 x-90 x^2+30 x^3 \log (3)\right )} \, dx=-\frac {20\,x^2\,\left (\ln \left (\ln \left (3\right )\,x^2-3\,x+1\right )-{\mathrm {e}}^x\right )}{3\,x+5\,{\mathrm {e}}^x} \]
int((exp(x)*(log(3)*(60*x^4 - 200*x^3 + 60*x^5) + 360*x^2 - 120*x^3 - 180* x^4) - log(x^2*log(3) - 3*x + 1)*(exp(x)*(200*x + log(3)*(200*x^3 - 100*x^ 4) - 700*x^2 + 300*x^3) + 60*x^4*log(3) + 60*x^2 - 180*x^3) - 120*x^4*log( 3) + exp(2*x)*(200*x + 200*x^3*log(3) - 600*x^2) + 180*x^3)/(exp(x)*(30*x + 30*x^3*log(3) - 90*x^2) + exp(2*x)*(25*x^2*log(3) - 75*x + 25) + 9*x^4*l og(3) + 9*x^2 - 27*x^3),x)