Integrand size = 188, antiderivative size = 35 \[ \int \frac {176+124 x+20 x^2+e^x \left (-88-62 x-10 x^2\right )+\left (88+256 x+134 x^2+20 x^3+e^x \left (-44-216 x-129 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )+e^x \left (44 x+31 x^2+5 x^3\right ) \log \left (x^2\right ) \log \left (\frac {\left (22 x+10 x^2\right ) \log \left (x^2\right )}{20+5 x}\right )}{\left (176 x+124 x^2+20 x^3+e^x \left (-176 x-124 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )} \, dx=x+\frac {\log \left (\left (x+\frac {x+\left (-\frac {3}{5}+x\right ) x}{4+x}\right ) \log \left (x^2\right )\right )}{2-e^x} \] Output:
x+ln((x+(x*(x-3/5)+x)/(4+x))*ln(x^2))/(-exp(x)+2)
Time = 0.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {176+124 x+20 x^2+e^x \left (-88-62 x-10 x^2\right )+\left (88+256 x+134 x^2+20 x^3+e^x \left (-44-216 x-129 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )+e^x \left (44 x+31 x^2+5 x^3\right ) \log \left (x^2\right ) \log \left (\frac {\left (22 x+10 x^2\right ) \log \left (x^2\right )}{20+5 x}\right )}{\left (176 x+124 x^2+20 x^3+e^x \left (-176 x-124 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )} \, dx=x-\frac {\log \left (\frac {2 x (11+5 x) \log \left (x^2\right )}{5 (4+x)}\right )}{-2+e^x} \] Input:
Integrate[(176 + 124*x + 20*x^2 + E^x*(-88 - 62*x - 10*x^2) + (88 + 256*x + 134*x^2 + 20*x^3 + E^x*(-44 - 216*x - 129*x^2 - 20*x^3) + E^(2*x)*(44*x + 31*x^2 + 5*x^3))*Log[x^2] + E^x*(44*x + 31*x^2 + 5*x^3)*Log[x^2]*Log[((2 2*x + 10*x^2)*Log[x^2])/(20 + 5*x)])/((176*x + 124*x^2 + 20*x^3 + E^x*(-17 6*x - 124*x^2 - 20*x^3) + E^(2*x)*(44*x + 31*x^2 + 5*x^3))*Log[x^2]),x]
Output:
x - Log[(2*x*(11 + 5*x)*Log[x^2])/(5*(4 + x))]/(-2 + E^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.
| \(\displaystyle \int \frac {20 x^2+e^x \left (-10 x^2-62 x-88\right )+\left (20 x^3+134 x^2+e^x \left (-20 x^3-129 x^2-216 x-44\right )+e^{2 x} \left (5 x^3+31 x^2+44 x\right )+256 x+88\right ) \log \left (x^2\right )+e^x \left (5 x^3+31 x^2+44 x\right ) \log \left (x^2\right ) \log \left (\frac {\left (10 x^2+22 x\right ) \log \left (x^2\right )}{5 x+20}\right )+124 x+176}{\left (20 x^3+124 x^2+e^x \left (-20 x^3-124 x^2-176 x\right )+e^{2 x} \left (5 x^3+31 x^2+44 x\right )+176 x\right ) \log \left (x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\frac {4-2 e^x}{\log \left (x^2\right )}+e^x x \log \left (\frac {2 x (5 x+11) \log \left (x^2\right )}{5 (x+4)}\right )+\frac {\left (e^x-2\right ) \left (5 \left (e^x-2\right ) x^3+\left (31 e^x-67\right ) x^2+4 \left (11 e^x-32\right ) x-44\right )}{5 x^2+31 x+44}}{\left (2-e^x\right )^2 x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \log \left (\frac {2 x (5 x+11) \log \left (x^2\right )}{5 (x+4)}\right )}{\left (e^x-2\right )^2}+\frac {-10 x^2-5 x^2 \log \left (x^2\right )+31 x^2 \log \left (x^2\right ) \log \left (\frac {2 x (5 x+11) \log \left (x^2\right )}{5 (x+4)}\right )-40 x \log \left (x^2\right )+44 x \log \left (x^2\right ) \log \left (\frac {2 x (5 x+11) \log \left (x^2\right )}{5 (x+4)}\right )-44 \log \left (x^2\right )+5 x^3 \log \left (x^2\right ) \log \left (\frac {2 x (5 x+11) \log \left (x^2\right )}{5 (x+4)}\right )-62 x-88}{\left (e^x-2\right ) x (x+4) (5 x+11) \log \left (x^2\right )}+1\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {2 \log \left (\frac {2 x (5 x+11) \log \left (x^2\right )}{5 (x+4)}\right )}{\left (e^x-2\right )^2}+\frac {-10 x^2-5 x^2 \log \left (x^2\right )+31 x^2 \log \left (x^2\right ) \log \left (\frac {2 x (5 x+11) \log \left (x^2\right )}{5 (x+4)}\right )-40 x \log \left (x^2\right )+44 x \log \left (x^2\right ) \log \left (\frac {2 x (5 x+11) \log \left (x^2\right )}{5 (x+4)}\right )-44 \log \left (x^2\right )+5 x^3 \log \left (x^2\right ) \log \left (\frac {2 x (5 x+11) \log \left (x^2\right )}{5 (x+4)}\right )-62 x-88}{\left (e^x-2\right ) x (x+4) (5 x+11) \log \left (x^2\right )}+1\right )dx\) |
Input:
Int[(176 + 124*x + 20*x^2 + E^x*(-88 - 62*x - 10*x^2) + (88 + 256*x + 134* x^2 + 20*x^3 + E^x*(-44 - 216*x - 129*x^2 - 20*x^3) + E^(2*x)*(44*x + 31*x ^2 + 5*x^3))*Log[x^2] + E^x*(44*x + 31*x^2 + 5*x^3)*Log[x^2]*Log[((22*x + 10*x^2)*Log[x^2])/(20 + 5*x)])/((176*x + 124*x^2 + 20*x^3 + E^x*(-176*x - 124*x^2 - 20*x^3) + E^(2*x)*(44*x + 31*x^2 + 5*x^3))*Log[x^2]),x]
Output:
$Aborted
Time = 23.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.31
| method | result | size |
| parallelrisch | \(-\frac {-3872-2728 \,{\mathrm e}^{x} x +5456 x +1936 \,{\mathrm e}^{x}+2728 \ln \left (\frac {\left (10 x^{2}+22 x \right ) \ln \left (x^{2}\right )}{20+5 x}\right )}{2728 \left ({\mathrm e}^{x}-2\right )}\) | \(46\) |
| risch | \(\text {Expression too large to display}\) | \(1674\) |
Input:
int(((5*x^3+31*x^2+44*x)*exp(x)*ln(x^2)*ln((10*x^2+22*x)*ln(x^2)/(20+5*x)) +((5*x^3+31*x^2+44*x)*exp(x)^2+(-20*x^3-129*x^2-216*x-44)*exp(x)+20*x^3+13 4*x^2+256*x+88)*ln(x^2)+(-10*x^2-62*x-88)*exp(x)+20*x^2+124*x+176)/((5*x^3 +31*x^2+44*x)*exp(x)^2+(-20*x^3-124*x^2-176*x)*exp(x)+20*x^3+124*x^2+176*x )/ln(x^2),x,method=_RETURNVERBOSE)
Output:
-1/2728*(-3872-2728*exp(x)*x+5456*x+1936*exp(x)+2728*ln(1/5/(4+x)*(10*x^2+ 22*x)*ln(x^2)))/(exp(x)-2)
Time = 0.12 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.09 \[ \int \frac {176+124 x+20 x^2+e^x \left (-88-62 x-10 x^2\right )+\left (88+256 x+134 x^2+20 x^3+e^x \left (-44-216 x-129 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )+e^x \left (44 x+31 x^2+5 x^3\right ) \log \left (x^2\right ) \log \left (\frac {\left (22 x+10 x^2\right ) \log \left (x^2\right )}{20+5 x}\right )}{\left (176 x+124 x^2+20 x^3+e^x \left (-176 x-124 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )} \, dx=\frac {x e^{x} - 2 \, x - \log \left (\frac {2 \, {\left (5 \, x^{2} + 11 \, x\right )} \log \left (x^{2}\right )}{5 \, {\left (x + 4\right )}}\right )}{e^{x} - 2} \] Input:
integrate(((5*x^3+31*x^2+44*x)*exp(x)*log(x^2)*log((10*x^2+22*x)*log(x^2)/ (20+5*x))+((5*x^3+31*x^2+44*x)*exp(x)^2+(-20*x^3-129*x^2-216*x-44)*exp(x)+ 20*x^3+134*x^2+256*x+88)*log(x^2)+(-10*x^2-62*x-88)*exp(x)+20*x^2+124*x+17 6)/((5*x^3+31*x^2+44*x)*exp(x)^2+(-20*x^3-124*x^2-176*x)*exp(x)+20*x^3+124 *x^2+176*x)/log(x^2),x, algorithm="fricas")
Output:
(x*e^x - 2*x - log(2/5*(5*x^2 + 11*x)*log(x^2)/(x + 4)))/(e^x - 2)
Time = 0.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.74 \[ \int \frac {176+124 x+20 x^2+e^x \left (-88-62 x-10 x^2\right )+\left (88+256 x+134 x^2+20 x^3+e^x \left (-44-216 x-129 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )+e^x \left (44 x+31 x^2+5 x^3\right ) \log \left (x^2\right ) \log \left (\frac {\left (22 x+10 x^2\right ) \log \left (x^2\right )}{20+5 x}\right )}{\left (176 x+124 x^2+20 x^3+e^x \left (-176 x-124 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )} \, dx=x - \frac {\log {\left (\frac {\left (10 x^{2} + 22 x\right ) \log {\left (x^{2} \right )}}{5 x + 20} \right )}}{e^{x} - 2} \] Input:
integrate(((5*x**3+31*x**2+44*x)*exp(x)*ln(x**2)*ln((10*x**2+22*x)*ln(x**2 )/(20+5*x))+((5*x**3+31*x**2+44*x)*exp(x)**2+(-20*x**3-129*x**2-216*x-44)* exp(x)+20*x**3+134*x**2+256*x+88)*ln(x**2)+(-10*x**2-62*x-88)*exp(x)+20*x* *2+124*x+176)/((5*x**3+31*x**2+44*x)*exp(x)**2+(-20*x**3-124*x**2-176*x)*e xp(x)+20*x**3+124*x**2+176*x)/ln(x**2),x)
Output:
x - log((10*x**2 + 22*x)*log(x**2)/(5*x + 20))/(exp(x) - 2)
Time = 0.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.20 \[ \int \frac {176+124 x+20 x^2+e^x \left (-88-62 x-10 x^2\right )+\left (88+256 x+134 x^2+20 x^3+e^x \left (-44-216 x-129 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )+e^x \left (44 x+31 x^2+5 x^3\right ) \log \left (x^2\right ) \log \left (\frac {\left (22 x+10 x^2\right ) \log \left (x^2\right )}{20+5 x}\right )}{\left (176 x+124 x^2+20 x^3+e^x \left (-176 x-124 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )} \, dx=\frac {x e^{x} - 2 \, x + \log \left (5\right ) - 2 \, \log \left (2\right ) - \log \left (5 \, x + 11\right ) + \log \left (x + 4\right ) - \log \left (x\right ) - \log \left (\log \left (x\right )\right )}{e^{x} - 2} \] Input:
integrate(((5*x^3+31*x^2+44*x)*exp(x)*log(x^2)*log((10*x^2+22*x)*log(x^2)/ (20+5*x))+((5*x^3+31*x^2+44*x)*exp(x)^2+(-20*x^3-129*x^2-216*x-44)*exp(x)+ 20*x^3+134*x^2+256*x+88)*log(x^2)+(-10*x^2-62*x-88)*exp(x)+20*x^2+124*x+17 6)/((5*x^3+31*x^2+44*x)*exp(x)^2+(-20*x^3-124*x^2-176*x)*exp(x)+20*x^3+124 *x^2+176*x)/log(x^2),x, algorithm="maxima")
Output:
(x*e^x - 2*x + log(5) - 2*log(2) - log(5*x + 11) + log(x + 4) - log(x) - l og(log(x)))/(e^x - 2)
Time = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.20 \[ \int \frac {176+124 x+20 x^2+e^x \left (-88-62 x-10 x^2\right )+\left (88+256 x+134 x^2+20 x^3+e^x \left (-44-216 x-129 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )+e^x \left (44 x+31 x^2+5 x^3\right ) \log \left (x^2\right ) \log \left (\frac {\left (22 x+10 x^2\right ) \log \left (x^2\right )}{20+5 x}\right )}{\left (176 x+124 x^2+20 x^3+e^x \left (-176 x-124 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )} \, dx=\frac {x e^{x} - 2 \, x - \log \left (10 \, x \log \left (x^{2}\right ) + 22 \, \log \left (x^{2}\right )\right ) + \log \left (5 \, x + 20\right ) - \log \left (x\right )}{e^{x} - 2} \] Input:
integrate(((5*x^3+31*x^2+44*x)*exp(x)*log(x^2)*log((10*x^2+22*x)*log(x^2)/ (20+5*x))+((5*x^3+31*x^2+44*x)*exp(x)^2+(-20*x^3-129*x^2-216*x-44)*exp(x)+ 20*x^3+134*x^2+256*x+88)*log(x^2)+(-10*x^2-62*x-88)*exp(x)+20*x^2+124*x+17 6)/((5*x^3+31*x^2+44*x)*exp(x)^2+(-20*x^3-124*x^2-176*x)*exp(x)+20*x^3+124 *x^2+176*x)/log(x^2),x, algorithm="giac")
Output:
(x*e^x - 2*x - log(10*x*log(x^2) + 22*log(x^2)) + log(5*x + 20) - log(x))/ (e^x - 2)
Time = 4.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \frac {176+124 x+20 x^2+e^x \left (-88-62 x-10 x^2\right )+\left (88+256 x+134 x^2+20 x^3+e^x \left (-44-216 x-129 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )+e^x \left (44 x+31 x^2+5 x^3\right ) \log \left (x^2\right ) \log \left (\frac {\left (22 x+10 x^2\right ) \log \left (x^2\right )}{20+5 x}\right )}{\left (176 x+124 x^2+20 x^3+e^x \left (-176 x-124 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )} \, dx=x-\frac {\ln \left (\frac {\ln \left (x^2\right )\,\left (10\,x^2+22\,x\right )}{5\,x+20}\right )}{{\mathrm {e}}^x-2} \] Input:
int((124*x + log(x^2)*(256*x + exp(2*x)*(44*x + 31*x^2 + 5*x^3) + 134*x^2 + 20*x^3 - exp(x)*(216*x + 129*x^2 + 20*x^3 + 44) + 88) - exp(x)*(62*x + 1 0*x^2 + 88) + 20*x^2 + log((log(x^2)*(22*x + 10*x^2))/(5*x + 20))*log(x^2) *exp(x)*(44*x + 31*x^2 + 5*x^3) + 176)/(log(x^2)*(176*x + exp(2*x)*(44*x + 31*x^2 + 5*x^3) + 124*x^2 + 20*x^3 - exp(x)*(176*x + 124*x^2 + 20*x^3))), x)
Output:
x - log((log(x^2)*(22*x + 10*x^2))/(5*x + 20))/(exp(x) - 2)
Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.14 \[ \int \frac {176+124 x+20 x^2+e^x \left (-88-62 x-10 x^2\right )+\left (88+256 x+134 x^2+20 x^3+e^x \left (-44-216 x-129 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )+e^x \left (44 x+31 x^2+5 x^3\right ) \log \left (x^2\right ) \log \left (\frac {\left (22 x+10 x^2\right ) \log \left (x^2\right )}{20+5 x}\right )}{\left (176 x+124 x^2+20 x^3+e^x \left (-176 x-124 x^2-20 x^3\right )+e^{2 x} \left (44 x+31 x^2+5 x^3\right )\right ) \log \left (x^2\right )} \, dx=\frac {e^{x} \mathrm {log}\left (\mathrm {log}\left (x^{2}\right )\right )+e^{x} \mathrm {log}\left (5 x +11\right )-e^{x} \mathrm {log}\left (x +4\right )-e^{x} \mathrm {log}\left (\frac {10 \,\mathrm {log}\left (x^{2}\right ) x^{2}+22 \,\mathrm {log}\left (x^{2}\right ) x}{5 x +20}\right )+e^{x} \mathrm {log}\left (x \right )+2 e^{x} x -2 \,\mathrm {log}\left (\mathrm {log}\left (x^{2}\right )\right )-2 \,\mathrm {log}\left (5 x +11\right )+2 \,\mathrm {log}\left (x +4\right )-2 \,\mathrm {log}\left (x \right )-4 x}{2 e^{x}-4} \] Input:
int(((5*x^3+31*x^2+44*x)*exp(x)*log(x^2)*log((10*x^2+22*x)*log(x^2)/(20+5* x))+((5*x^3+31*x^2+44*x)*exp(x)^2+(-20*x^3-129*x^2-216*x-44)*exp(x)+20*x^3 +134*x^2+256*x+88)*log(x^2)+(-10*x^2-62*x-88)*exp(x)+20*x^2+124*x+176)/((5 *x^3+31*x^2+44*x)*exp(x)^2+(-20*x^3-124*x^2-176*x)*exp(x)+20*x^3+124*x^2+1 76*x)/log(x^2),x)
Output:
(e**x*log(log(x**2)) + e**x*log(5*x + 11) - e**x*log(x + 4) - e**x*log((10 *log(x**2)*x**2 + 22*log(x**2)*x)/(5*x + 20)) + e**x*log(x) + 2*e**x*x - 2 *log(log(x**2)) - 2*log(5*x + 11) + 2*log(x + 4) - 2*log(x) - 4*x)/(2*(e** x - 2))