Integrand size = 171, antiderivative size = 31 \[ \int \frac {e^{5+e^{2 x}} \left (6+e^{16+x} \left (-9+9 x-18 e^{2 x} x\right )+e^{16+x} \left (3-3 x+6 e^{2 x} x\right ) \log \left (x^2\right )\right )+e^{5+e^{2 x}} \left (9+18 e^{2 x} x+\left (-3-6 e^{2 x} x\right ) \log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{-3 e^{32+2 x}+e^{32+2 x} \log \left (x^2\right )+\left (6 e^{16+x}-2 e^{16+x} \log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )+\left (-3+\log \left (x^2\right )\right ) \log ^2\left (3-\log \left (x^2\right )\right )} \, dx=\frac {3 e^{5+e^{2 x}} x}{e^{16+x}-\log \left (3-\log \left (x^2\right )\right )} \] Output:
3*exp(exp(x)^2+5)/(exp(x+16)-ln(3-ln(x^2)))*x
Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {e^{5+e^{2 x}} \left (6+e^{16+x} \left (-9+9 x-18 e^{2 x} x\right )+e^{16+x} \left (3-3 x+6 e^{2 x} x\right ) \log \left (x^2\right )\right )+e^{5+e^{2 x}} \left (9+18 e^{2 x} x+\left (-3-6 e^{2 x} x\right ) \log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{-3 e^{32+2 x}+e^{32+2 x} \log \left (x^2\right )+\left (6 e^{16+x}-2 e^{16+x} \log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )+\left (-3+\log \left (x^2\right )\right ) \log ^2\left (3-\log \left (x^2\right )\right )} \, dx=-\frac {3 e^{5+e^{2 x}} x}{-e^{16+x}+\log \left (3-\log \left (x^2\right )\right )} \] Input:
Integrate[(E^(5 + E^(2*x))*(6 + E^(16 + x)*(-9 + 9*x - 18*E^(2*x)*x) + E^( 16 + x)*(3 - 3*x + 6*E^(2*x)*x)*Log[x^2]) + E^(5 + E^(2*x))*(9 + 18*E^(2*x )*x + (-3 - 6*E^(2*x)*x)*Log[x^2])*Log[3 - Log[x^2]])/(-3*E^(32 + 2*x) + E ^(32 + 2*x)*Log[x^2] + (6*E^(16 + x) - 2*E^(16 + x)*Log[x^2])*Log[3 - Log[ x^2]] + (-3 + Log[x^2])*Log[3 - Log[x^2]]^2),x]
Output:
(-3*E^(5 + E^(2*x))*x)/(-E^(16 + x) + Log[3 - Log[x^2]])
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.
| \(\displaystyle \int \frac {e^{e^{2 x}+5} \left (e^{x+16} \left (6 e^{2 x} x-3 x+3\right ) \log \left (x^2\right )+e^{x+16} \left (-18 e^{2 x} x+9 x-9\right )+6\right )+e^{e^{2 x}+5} \left (\left (-6 e^{2 x} x-3\right ) \log \left (x^2\right )+18 e^{2 x} x+9\right ) \log \left (3-\log \left (x^2\right )\right )}{\left (\log \left (x^2\right )-3\right ) \log ^2\left (3-\log \left (x^2\right )\right )+\left (6 e^{x+16}-2 e^{x+16} \log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )+e^{2 x+32} \log \left (x^2\right )-3 e^{2 x+32}} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-e^{e^{2 x}+5} \left (e^{x+16} \left (6 e^{2 x} x-3 x+3\right ) \log \left (x^2\right )+e^{x+16} \left (-18 e^{2 x} x+9 x-9\right )+6\right )-e^{e^{2 x}+5} \left (\left (-6 e^{2 x} x-3\right ) \log \left (x^2\right )+18 e^{2 x} x+9\right ) \log \left (3-\log \left (x^2\right )\right )}{\left (3-\log \left (x^2\right )\right ) \left (e^{x+16}-\log \left (3-\log \left (x^2\right )\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 e^{e^{2 x}} \left (-2 x \log ^2\left (3-\log \left (x^2\right )\right )+e^{32} x-e^{32}\right )}{e^{x+43}-e^{27} \log \left (3-\log \left (x^2\right )\right )}+6 e^{e^{2 x}-27} x \log \left (3-\log \left (x^2\right )\right )-\frac {3 e^{e^{2 x}+5} \left (-3 x \log \left (3-\log \left (x^2\right )\right )+x \log \left (x^2\right ) \log \left (3-\log \left (x^2\right )\right )-2\right )}{\left (\log \left (x^2\right )-3\right ) \left (e^{x+16}-\log \left (3-\log \left (x^2\right )\right )\right )^2}+6 e^{x+e^{2 x}-11} x\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -6 \int \frac {e^{e^{2 x}} x \log ^2\left (3-\log \left (x^2\right )\right )}{e^{27} \log \left (3-\log \left (x^2\right )\right )-e^{x+43}}dx+6 \int \frac {e^{5+e^{2 x}}}{\left (\log \left (x^2\right )-3\right ) \left (e^{x+16}-\log \left (3-\log \left (x^2\right )\right )\right )^2}dx+3 \int \frac {e^{5+e^{2 x}}}{e^{x+16}-\log \left (3-\log \left (x^2\right )\right )}dx-3 \int \frac {e^{5+e^{2 x}} x}{e^{x+16}-\log \left (3-\log \left (x^2\right )\right )}dx+6 \int e^{-27+e^{2 x}} x \log \left (3-\log \left (x^2\right )\right )dx+9 \int \frac {e^{5+e^{2 x}} x \log \left (3-\log \left (x^2\right )\right )}{\left (\log \left (x^2\right )-3\right ) \left (e^{x+16}-\log \left (3-\log \left (x^2\right )\right )\right )^2}dx-3 \int \frac {e^{5+e^{2 x}} x \log \left (x^2\right ) \log \left (3-\log \left (x^2\right )\right )}{\left (\log \left (x^2\right )-3\right ) \left (e^{x+16}-\log \left (3-\log \left (x^2\right )\right )\right )^2}dx+6 \int e^{x+e^{2 x}-11} xdx\) |
Input:
Int[(E^(5 + E^(2*x))*(6 + E^(16 + x)*(-9 + 9*x - 18*E^(2*x)*x) + E^(16 + x )*(3 - 3*x + 6*E^(2*x)*x)*Log[x^2]) + E^(5 + E^(2*x))*(9 + 18*E^(2*x)*x + (-3 - 6*E^(2*x)*x)*Log[x^2])*Log[3 - Log[x^2]])/(-3*E^(32 + 2*x) + E^(32 + 2*x)*Log[x^2] + (6*E^(16 + x) - 2*E^(16 + x)*Log[x^2])*Log[3 - Log[x^2]] + (-3 + Log[x^2])*Log[3 - Log[x^2]]^2),x]
Output:
$Aborted
Time = 223.81 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94
| method | result | size |
| parallelrisch | \(\frac {3 \,{\mathrm e}^{{\mathrm e}^{2 x}+5} x}{{\mathrm e}^{x +16}-\ln \left (3-\ln \left (x^{2}\right )\right )}\) | \(29\) |
| risch | \(\frac {3 \,{\mathrm e}^{{\mathrm e}^{2 x}+5} x}{{\mathrm e}^{x +16}-\ln \left (3-2 \ln \left (x \right )+\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )}\) | \(55\) |
Input:
int((((-6*x*exp(x)^2-3)*ln(x^2)+18*x*exp(x)^2+9)*exp(exp(x)^2+5)*ln(3-ln(x ^2))+((6*x*exp(x)^2-3*x+3)*exp(x+16)*ln(x^2)+(-18*x*exp(x)^2+9*x-9)*exp(x+ 16)+6)*exp(exp(x)^2+5))/((ln(x^2)-3)*ln(3-ln(x^2))^2+(-2*exp(x+16)*ln(x^2) +6*exp(x+16))*ln(3-ln(x^2))+exp(x+16)^2*ln(x^2)-3*exp(x+16)^2),x,method=_R ETURNVERBOSE)
Output:
3*exp(exp(x)^2+5)/(exp(x+16)-ln(3-ln(x^2)))*x
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {e^{5+e^{2 x}} \left (6+e^{16+x} \left (-9+9 x-18 e^{2 x} x\right )+e^{16+x} \left (3-3 x+6 e^{2 x} x\right ) \log \left (x^2\right )\right )+e^{5+e^{2 x}} \left (9+18 e^{2 x} x+\left (-3-6 e^{2 x} x\right ) \log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{-3 e^{32+2 x}+e^{32+2 x} \log \left (x^2\right )+\left (6 e^{16+x}-2 e^{16+x} \log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )+\left (-3+\log \left (x^2\right )\right ) \log ^2\left (3-\log \left (x^2\right )\right )} \, dx=\frac {3 \, x e^{\left ({\left (5 \, e^{32} + e^{\left (2 \, x + 32\right )}\right )} e^{\left (-32\right )}\right )}}{e^{\left (x + 16\right )} - \log \left (-\log \left (x^{2}\right ) + 3\right )} \] Input:
integrate((((-6*x*exp(x)^2-3)*log(x^2)+18*x*exp(x)^2+9)*exp(exp(x)^2+5)*lo g(3-log(x^2))+((6*x*exp(x)^2-3*x+3)*exp(x+16)*log(x^2)+(-18*x*exp(x)^2+9*x -9)*exp(x+16)+6)*exp(exp(x)^2+5))/((log(x^2)-3)*log(3-log(x^2))^2+(-2*exp( x+16)*log(x^2)+6*exp(x+16))*log(3-log(x^2))+exp(x+16)^2*log(x^2)-3*exp(x+1 6)^2),x, algorithm="fricas")
Output:
3*x*e^((5*e^32 + e^(2*x + 32))*e^(-32))/(e^(x + 16) - log(-log(x^2) + 3))
Timed out. \[ \int \frac {e^{5+e^{2 x}} \left (6+e^{16+x} \left (-9+9 x-18 e^{2 x} x\right )+e^{16+x} \left (3-3 x+6 e^{2 x} x\right ) \log \left (x^2\right )\right )+e^{5+e^{2 x}} \left (9+18 e^{2 x} x+\left (-3-6 e^{2 x} x\right ) \log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{-3 e^{32+2 x}+e^{32+2 x} \log \left (x^2\right )+\left (6 e^{16+x}-2 e^{16+x} \log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )+\left (-3+\log \left (x^2\right )\right ) \log ^2\left (3-\log \left (x^2\right )\right )} \, dx=\text {Timed out} \] Input:
integrate((((-6*x*exp(x)**2-3)*ln(x**2)+18*x*exp(x)**2+9)*exp(exp(x)**2+5) *ln(3-ln(x**2))+((6*x*exp(x)**2-3*x+3)*exp(x+16)*ln(x**2)+(-18*x*exp(x)**2 +9*x-9)*exp(x+16)+6)*exp(exp(x)**2+5))/((ln(x**2)-3)*ln(3-ln(x**2))**2+(-2 *exp(x+16)*ln(x**2)+6*exp(x+16))*ln(3-ln(x**2))+exp(x+16)**2*ln(x**2)-3*ex p(x+16)**2),x)
Output:
Timed out
Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {e^{5+e^{2 x}} \left (6+e^{16+x} \left (-9+9 x-18 e^{2 x} x\right )+e^{16+x} \left (3-3 x+6 e^{2 x} x\right ) \log \left (x^2\right )\right )+e^{5+e^{2 x}} \left (9+18 e^{2 x} x+\left (-3-6 e^{2 x} x\right ) \log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{-3 e^{32+2 x}+e^{32+2 x} \log \left (x^2\right )+\left (6 e^{16+x}-2 e^{16+x} \log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )+\left (-3+\log \left (x^2\right )\right ) \log ^2\left (3-\log \left (x^2\right )\right )} \, dx=\frac {3 \, x e^{\left (e^{\left (2 \, x\right )} + 5\right )}}{e^{\left (x + 16\right )} - \log \left (-2 \, \log \left (x\right ) + 3\right )} \] Input:
integrate((((-6*x*exp(x)^2-3)*log(x^2)+18*x*exp(x)^2+9)*exp(exp(x)^2+5)*lo g(3-log(x^2))+((6*x*exp(x)^2-3*x+3)*exp(x+16)*log(x^2)+(-18*x*exp(x)^2+9*x -9)*exp(x+16)+6)*exp(exp(x)^2+5))/((log(x^2)-3)*log(3-log(x^2))^2+(-2*exp( x+16)*log(x^2)+6*exp(x+16))*log(3-log(x^2))+exp(x+16)^2*log(x^2)-3*exp(x+1 6)^2),x, algorithm="maxima")
Output:
3*x*e^(e^(2*x) + 5)/(e^(x + 16) - log(-2*log(x) + 3))
Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (28) = 56\).
Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 4.10 \[ \int \frac {e^{5+e^{2 x}} \left (6+e^{16+x} \left (-9+9 x-18 e^{2 x} x\right )+e^{16+x} \left (3-3 x+6 e^{2 x} x\right ) \log \left (x^2\right )\right )+e^{5+e^{2 x}} \left (9+18 e^{2 x} x+\left (-3-6 e^{2 x} x\right ) \log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{-3 e^{32+2 x}+e^{32+2 x} \log \left (x^2\right )+\left (6 e^{16+x}-2 e^{16+x} \log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )+\left (-3+\log \left (x^2\right )\right ) \log ^2\left (3-\log \left (x^2\right )\right )} \, dx=-\frac {3 \, {\left (2 \, x^{2} e^{\left (x + e^{\left (2 \, x\right )} + 21\right )} \log \left (x \mathrm {sgn}\left (x\right )\right ) - 3 \, x^{2} e^{\left (x + e^{\left (2 \, x\right )} + 21\right )} - 2 \, x e^{\left (e^{\left (2 \, x\right )} + 5\right )}\right )}}{2 \, x e^{\left (x + 16\right )} \log \left (x \mathrm {sgn}\left (x\right )\right ) \log \left (-2 \, \log \left (x \mathrm {sgn}\left (x\right )\right ) + 3\right ) - 2 \, x e^{\left (2 \, x + 32\right )} \log \left (x \mathrm {sgn}\left (x\right )\right ) - 3 \, x e^{\left (x + 16\right )} \log \left (-2 \, \log \left (x \mathrm {sgn}\left (x\right )\right ) + 3\right ) + 3 \, x e^{\left (2 \, x + 32\right )} + 2 \, e^{\left (x + 16\right )} - 2 \, \log \left (-2 \, \log \left (x \mathrm {sgn}\left (x\right )\right ) + 3\right )} \] Input:
integrate((((-6*x*exp(x)^2-3)*log(x^2)+18*x*exp(x)^2+9)*exp(exp(x)^2+5)*lo g(3-log(x^2))+((6*x*exp(x)^2-3*x+3)*exp(x+16)*log(x^2)+(-18*x*exp(x)^2+9*x -9)*exp(x+16)+6)*exp(exp(x)^2+5))/((log(x^2)-3)*log(3-log(x^2))^2+(-2*exp( x+16)*log(x^2)+6*exp(x+16))*log(3-log(x^2))+exp(x+16)^2*log(x^2)-3*exp(x+1 6)^2),x, algorithm="giac")
Output:
-3*(2*x^2*e^(x + e^(2*x) + 21)*log(x*sgn(x)) - 3*x^2*e^(x + e^(2*x) + 21) - 2*x*e^(e^(2*x) + 5))/(2*x*e^(x + 16)*log(x*sgn(x))*log(-2*log(x*sgn(x)) + 3) - 2*x*e^(2*x + 32)*log(x*sgn(x)) - 3*x*e^(x + 16)*log(-2*log(x*sgn(x) ) + 3) + 3*x*e^(2*x + 32) + 2*e^(x + 16) - 2*log(-2*log(x*sgn(x)) + 3))
Timed out. \[ \int \frac {e^{5+e^{2 x}} \left (6+e^{16+x} \left (-9+9 x-18 e^{2 x} x\right )+e^{16+x} \left (3-3 x+6 e^{2 x} x\right ) \log \left (x^2\right )\right )+e^{5+e^{2 x}} \left (9+18 e^{2 x} x+\left (-3-6 e^{2 x} x\right ) \log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{-3 e^{32+2 x}+e^{32+2 x} \log \left (x^2\right )+\left (6 e^{16+x}-2 e^{16+x} \log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )+\left (-3+\log \left (x^2\right )\right ) \log ^2\left (3-\log \left (x^2\right )\right )} \, dx=\int \frac {{\mathrm {e}}^{{\mathrm {e}}^{2\,x}+5}\,\left (\ln \left (x^2\right )\,{\mathrm {e}}^{x+16}\,\left (6\,x\,{\mathrm {e}}^{2\,x}-3\,x+3\right )-{\mathrm {e}}^{x+16}\,\left (18\,x\,{\mathrm {e}}^{2\,x}-9\,x+9\right )+6\right )+{\mathrm {e}}^{{\mathrm {e}}^{2\,x}+5}\,\ln \left (3-\ln \left (x^2\right )\right )\,\left (18\,x\,{\mathrm {e}}^{2\,x}-\ln \left (x^2\right )\,\left (6\,x\,{\mathrm {e}}^{2\,x}+3\right )+9\right )}{\left (\ln \left (x^2\right )-3\right )\,{\ln \left (3-\ln \left (x^2\right )\right )}^2+\left (6\,{\mathrm {e}}^{x+16}-2\,\ln \left (x^2\right )\,{\mathrm {e}}^{x+16}\right )\,\ln \left (3-\ln \left (x^2\right )\right )-3\,{\mathrm {e}}^{2\,x+32}+\ln \left (x^2\right )\,{\mathrm {e}}^{2\,x+32}} \,d x \] Input:
int((exp(exp(2*x) + 5)*(log(x^2)*exp(x + 16)*(6*x*exp(2*x) - 3*x + 3) - ex p(x + 16)*(18*x*exp(2*x) - 9*x + 9) + 6) + exp(exp(2*x) + 5)*log(3 - log(x ^2))*(18*x*exp(2*x) - log(x^2)*(6*x*exp(2*x) + 3) + 9))/(log(x^2)*exp(2*x + 32) - 3*exp(2*x + 32) + log(3 - log(x^2))*(6*exp(x + 16) - 2*log(x^2)*ex p(x + 16)) + log(3 - log(x^2))^2*(log(x^2) - 3)),x)
Output:
int((exp(exp(2*x) + 5)*(log(x^2)*exp(x + 16)*(6*x*exp(2*x) - 3*x + 3) - ex p(x + 16)*(18*x*exp(2*x) - 9*x + 9) + 6) + exp(exp(2*x) + 5)*log(3 - log(x ^2))*(18*x*exp(2*x) - log(x^2)*(6*x*exp(2*x) + 3) + 9))/(log(x^2)*exp(2*x + 32) - 3*exp(2*x + 32) + log(3 - log(x^2))*(6*exp(x + 16) - 2*log(x^2)*ex p(x + 16)) + log(3 - log(x^2))^2*(log(x^2) - 3)), x)
Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {e^{5+e^{2 x}} \left (6+e^{16+x} \left (-9+9 x-18 e^{2 x} x\right )+e^{16+x} \left (3-3 x+6 e^{2 x} x\right ) \log \left (x^2\right )\right )+e^{5+e^{2 x}} \left (9+18 e^{2 x} x+\left (-3-6 e^{2 x} x\right ) \log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )}{-3 e^{32+2 x}+e^{32+2 x} \log \left (x^2\right )+\left (6 e^{16+x}-2 e^{16+x} \log \left (x^2\right )\right ) \log \left (3-\log \left (x^2\right )\right )+\left (-3+\log \left (x^2\right )\right ) \log ^2\left (3-\log \left (x^2\right )\right )} \, dx=\frac {3 e^{e^{2 x}} e^{5} x}{e^{x} e^{16}-\mathrm {log}\left (-\mathrm {log}\left (x^{2}\right )+3\right )} \] Input:
int((((-6*x*exp(x)^2-3)*log(x^2)+18*x*exp(x)^2+9)*exp(exp(x)^2+5)*log(3-lo g(x^2))+((6*x*exp(x)^2-3*x+3)*exp(x+16)*log(x^2)+(-18*x*exp(x)^2+9*x-9)*ex p(x+16)+6)*exp(exp(x)^2+5))/((log(x^2)-3)*log(3-log(x^2))^2+(-2*exp(x+16)* log(x^2)+6*exp(x+16))*log(3-log(x^2))+exp(x+16)^2*log(x^2)-3*exp(x+16)^2), x)
Output:
(3*e**(e**(2*x))*e**5*x)/(e**x*e**16 - log( - log(x**2) + 3))