Integrand size = 131, antiderivative size = 27 \[ \int \frac {-837 x+e^x \left (-3-279 x-279 x^2\right ) \log (3)+\left (-567 x+e^x \left (-189 x-189 x^2\right ) \log (3)\right ) \log (x)+\left (-96 x+e^x \left (-32 x-32 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-279 e^x x \log (3)-189 e^x x \log (3) \log (x)-32 e^x x \log (3) \log ^2(x)\right ) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )}{837 x+567 x \log (x)+96 x \log ^2(x)} \, dx=3-x-\frac {1}{3} e^x \log (3) \left (x+\log \left (32-\frac {3}{3+\log (x)}\right )\right ) \] Output:
3-x-1/3*exp(x)*ln(3)*(x+ln(32-3/(3+ln(x))))
Time = 0.46 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {-837 x+e^x \left (-3-279 x-279 x^2\right ) \log (3)+\left (-567 x+e^x \left (-189 x-189 x^2\right ) \log (3)\right ) \log (x)+\left (-96 x+e^x \left (-32 x-32 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-279 e^x x \log (3)-189 e^x x \log (3) \log (x)-32 e^x x \log (3) \log ^2(x)\right ) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )}{837 x+567 x \log (x)+96 x \log ^2(x)} \, dx=\frac {1}{3} \left (-3 x-e^x x \log (3)-e^x \log (3) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )\right ) \] Input:
Integrate[(-837*x + E^x*(-3 - 279*x - 279*x^2)*Log[3] + (-567*x + E^x*(-18 9*x - 189*x^2)*Log[3])*Log[x] + (-96*x + E^x*(-32*x - 32*x^2)*Log[3])*Log[ x]^2 + (-279*E^x*x*Log[3] - 189*E^x*x*Log[3]*Log[x] - 32*E^x*x*Log[3]*Log[ x]^2)*Log[(93 + 32*Log[x])/(3 + Log[x])])/(837*x + 567*x*Log[x] + 96*x*Log [x]^2),x]
Output:
(-3*x - E^x*x*Log[3] - E^x*Log[3]*Log[(93 + 32*Log[x])/(3 + Log[x])])/3
Leaf count is larger than twice the leaf count of optimal. \(111\) vs. \(2(27)=54\).
Time = 3.42 (sec) , antiderivative size = 111, normalized size of antiderivative = 4.11, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {7292, 27, 25, 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.
| \(\displaystyle \int \frac {\left (e^x \left (-32 x^2-32 x\right ) \log (3)-96 x\right ) \log ^2(x)+\left (e^x \left (-189 x^2-189 x\right ) \log (3)-567 x\right ) \log (x)+e^x \left (-279 x^2-279 x-3\right ) \log (3)-837 x+\left (-32 e^x x \log (3) \log ^2(x)-189 e^x x \log (3) \log (x)-279 e^x x \log (3)\right ) \log \left (\frac {32 \log (x)+93}{\log (x)+3}\right )}{837 x+96 x \log ^2(x)+567 x \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (e^x \left (-32 x^2-32 x\right ) \log (3)-96 x\right ) \log ^2(x)+\left (e^x \left (-189 x^2-189 x\right ) \log (3)-567 x\right ) \log (x)+e^x \left (-279 x^2-279 x-3\right ) \log (3)-837 x+\left (-32 e^x x \log (3) \log ^2(x)-189 e^x x \log (3) \log (x)-279 e^x x \log (3)\right ) \log \left (\frac {32 \log (x)+93}{\log (x)+3}\right )}{3 x \left (32 \log ^2(x)+189 \log (x)+279\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int -\frac {32 \left (3 x+e^x \left (x^2+x\right ) \log (3)\right ) \log ^2(x)+189 \left (3 x+e^x \left (x^2+x\right ) \log (3)\right ) \log (x)+837 x+\left (32 e^x x \log (3) \log ^2(x)+189 e^x x \log (3) \log (x)+279 e^x x \log (3)\right ) \log \left (\frac {32 \log (x)+93}{\log (x)+3}\right )+3 e^x \left (93 x^2+93 x+1\right ) \log (3)}{x \left (32 \log ^2(x)+189 \log (x)+279\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{3} \int \frac {32 \left (3 x+e^x \left (x^2+x\right ) \log (3)\right ) \log ^2(x)+189 \left (3 x+e^x \left (x^2+x\right ) \log (3)\right ) \log (x)+837 x+\left (32 e^x x \log (3) \log ^2(x)+189 e^x x \log (3) \log (x)+279 e^x x \log (3)\right ) \log \left (\frac {32 \log (x)+93}{\log (x)+3}\right )+3 e^x \left (93 x^2+93 x+1\right ) \log (3)}{x \left (32 \log ^2(x)+189 \log (x)+279\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{3} \int \left (\frac {e^x \log (3) \left (32 \log ^2(x) x^2+189 \log (x) x^2+279 x^2+32 \log ^2(x) x+189 \log (x) x+32 \log ^2(x) \log \left (\frac {32 \log (x)+93}{\log (x)+3}\right ) x+189 \log (x) \log \left (\frac {32 \log (x)+93}{\log (x)+3}\right ) x+279 \log \left (\frac {32 \log (x)+93}{\log (x)+3}\right ) x+279 x+3\right )}{x (\log (x)+3) (32 \log (x)+93)}+3\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (-\frac {e^x \log (3) \left (279 x^2+32 x^2 \log ^2(x)+189 x^2 \log (x)+32 x \log ^2(x) \log \left (\frac {32 \log (x)+93}{\log (x)+3}\right )+189 x \log (x) \log \left (\frac {32 \log (x)+93}{\log (x)+3}\right )+279 x \log \left (\frac {32 \log (x)+93}{\log (x)+3}\right )\right )}{x (\log (x)+3) (32 \log (x)+93)}-3 x\right )\) |
Input:
Int[(-837*x + E^x*(-3 - 279*x - 279*x^2)*Log[3] + (-567*x + E^x*(-189*x - 189*x^2)*Log[3])*Log[x] + (-96*x + E^x*(-32*x - 32*x^2)*Log[3])*Log[x]^2 + (-279*E^x*x*Log[3] - 189*E^x*x*Log[3]*Log[x] - 32*E^x*x*Log[3]*Log[x]^2)* Log[(93 + 32*Log[x])/(3 + Log[x])])/(837*x + 567*x*Log[x] + 96*x*Log[x]^2) ,x]
Output:
(-3*x - (E^x*Log[3]*(279*x^2 + 189*x^2*Log[x] + 32*x^2*Log[x]^2 + 279*x*Lo g[(93 + 32*Log[x])/(3 + Log[x])] + 189*x*Log[x]*Log[(93 + 32*Log[x])/(3 + Log[x])] + 32*x*Log[x]^2*Log[(93 + 32*Log[x])/(3 + Log[x])]))/(x*(3 + Log[ x])*(93 + 32*Log[x])))/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 9.94 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19
| method | result | size |
| parallelrisch | \(-\frac {x \ln \left (3\right ) {\mathrm e}^{x}}{3}-\frac {\ln \left (3\right ) \ln \left (\frac {32 \ln \left (x \right )+93}{3+\ln \left (x \right )}\right ) {\mathrm e}^{x}}{3}-x\) | \(32\) |
| risch | \(-\frac {\ln \left (3\right ) {\mathrm e}^{x} \ln \left (\frac {93}{32}+\ln \left (x \right )\right )}{3}+\frac {\ln \left (3\right ) {\mathrm e}^{x} \ln \left (3+\ln \left (x \right )\right )}{3}-x +\frac {i \pi \ln \left (3\right ) \operatorname {csgn}\left (\frac {i}{3+\ln \left (x \right )}\right ) \operatorname {csgn}\left (i \left (\frac {93}{32}+\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (\frac {i \left (\frac {93}{32}+\ln \left (x \right )\right )}{3+\ln \left (x \right )}\right ) {\mathrm e}^{x}}{6}-\frac {i \pi \ln \left (3\right ) \operatorname {csgn}\left (\frac {i}{3+\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (\frac {93}{32}+\ln \left (x \right )\right )}{3+\ln \left (x \right )}\right )^{2} {\mathrm e}^{x}}{6}-\frac {i \pi \ln \left (3\right ) \operatorname {csgn}\left (i \left (\frac {93}{32}+\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (\frac {i \left (\frac {93}{32}+\ln \left (x \right )\right )}{3+\ln \left (x \right )}\right )^{2} {\mathrm e}^{x}}{6}+\frac {i \pi \ln \left (3\right ) \operatorname {csgn}\left (\frac {i \left (\frac {93}{32}+\ln \left (x \right )\right )}{3+\ln \left (x \right )}\right )^{3} {\mathrm e}^{x}}{6}-\frac {5 \ln \left (3\right ) \ln \left (2\right ) {\mathrm e}^{x}}{3}-\frac {x \ln \left (3\right ) {\mathrm e}^{x}}{3}\) | \(172\) |
Input:
int(((-32*x*ln(3)*exp(x)*ln(x)^2-189*x*ln(3)*exp(x)*ln(x)-279*x*ln(3)*exp( x))*ln((32*ln(x)+93)/(3+ln(x)))+((-32*x^2-32*x)*ln(3)*exp(x)-96*x)*ln(x)^2 +((-189*x^2-189*x)*ln(3)*exp(x)-567*x)*ln(x)+(-279*x^2-279*x-3)*ln(3)*exp( x)-837*x)/(96*x*ln(x)^2+567*x*ln(x)+837*x),x,method=_RETURNVERBOSE)
Output:
-1/3*x*ln(3)*exp(x)-1/3*ln(3)*ln((32*ln(x)+93)/(3+ln(x)))*exp(x)-x
Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {-837 x+e^x \left (-3-279 x-279 x^2\right ) \log (3)+\left (-567 x+e^x \left (-189 x-189 x^2\right ) \log (3)\right ) \log (x)+\left (-96 x+e^x \left (-32 x-32 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-279 e^x x \log (3)-189 e^x x \log (3) \log (x)-32 e^x x \log (3) \log ^2(x)\right ) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )}{837 x+567 x \log (x)+96 x \log ^2(x)} \, dx=-\frac {1}{3} \, x e^{x} \log \left (3\right ) - \frac {1}{3} \, e^{x} \log \left (3\right ) \log \left (\frac {32 \, \log \left (x\right ) + 93}{\log \left (x\right ) + 3}\right ) - x \] Input:
integrate(((-32*x*log(3)*exp(x)*log(x)^2-189*x*log(3)*exp(x)*log(x)-279*x* log(3)*exp(x))*log((32*log(x)+93)/(3+log(x)))+((-32*x^2-32*x)*log(3)*exp(x )-96*x)*log(x)^2+((-189*x^2-189*x)*log(3)*exp(x)-567*x)*log(x)+(-279*x^2-2 79*x-3)*log(3)*exp(x)-837*x)/(96*x*log(x)^2+567*x*log(x)+837*x),x, algorit hm="fricas")
Output:
-1/3*x*e^x*log(3) - 1/3*e^x*log(3)*log((32*log(x) + 93)/(log(x) + 3)) - x
Time = 7.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {-837 x+e^x \left (-3-279 x-279 x^2\right ) \log (3)+\left (-567 x+e^x \left (-189 x-189 x^2\right ) \log (3)\right ) \log (x)+\left (-96 x+e^x \left (-32 x-32 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-279 e^x x \log (3)-189 e^x x \log (3) \log (x)-32 e^x x \log (3) \log ^2(x)\right ) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )}{837 x+567 x \log (x)+96 x \log ^2(x)} \, dx=- x + \frac {\left (- x \log {\left (3 \right )} - \log {\left (3 \right )} \log {\left (\frac {32 \log {\left (x \right )} + 93}{\log {\left (x \right )} + 3} \right )}\right ) e^{x}}{3} \] Input:
integrate(((-32*x*ln(3)*exp(x)*ln(x)**2-189*x*ln(3)*exp(x)*ln(x)-279*x*ln( 3)*exp(x))*ln((32*ln(x)+93)/(3+ln(x)))+((-32*x**2-32*x)*ln(3)*exp(x)-96*x) *ln(x)**2+((-189*x**2-189*x)*ln(3)*exp(x)-567*x)*ln(x)+(-279*x**2-279*x-3) *ln(3)*exp(x)-837*x)/(96*x*ln(x)**2+567*x*ln(x)+837*x),x)
Output:
-x + (-x*log(3) - log(3)*log((32*log(x) + 93)/(log(x) + 3)))*exp(x)/3
Time = 0.16 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {-837 x+e^x \left (-3-279 x-279 x^2\right ) \log (3)+\left (-567 x+e^x \left (-189 x-189 x^2\right ) \log (3)\right ) \log (x)+\left (-96 x+e^x \left (-32 x-32 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-279 e^x x \log (3)-189 e^x x \log (3) \log (x)-32 e^x x \log (3) \log ^2(x)\right ) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )}{837 x+567 x \log (x)+96 x \log ^2(x)} \, dx=-\frac {1}{3} \, x e^{x} \log \left (3\right ) - \frac {1}{3} \, e^{x} \log \left (3\right ) \log \left (32 \, \log \left (x\right ) + 93\right ) + \frac {1}{3} \, e^{x} \log \left (3\right ) \log \left (\log \left (x\right ) + 3\right ) - x \] Input:
integrate(((-32*x*log(3)*exp(x)*log(x)^2-189*x*log(3)*exp(x)*log(x)-279*x* log(3)*exp(x))*log((32*log(x)+93)/(3+log(x)))+((-32*x^2-32*x)*log(3)*exp(x )-96*x)*log(x)^2+((-189*x^2-189*x)*log(3)*exp(x)-567*x)*log(x)+(-279*x^2-2 79*x-3)*log(3)*exp(x)-837*x)/(96*x*log(x)^2+567*x*log(x)+837*x),x, algorit hm="maxima")
Output:
-1/3*x*e^x*log(3) - 1/3*e^x*log(3)*log(32*log(x) + 93) + 1/3*e^x*log(3)*lo g(log(x) + 3) - x
Time = 0.14 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {-837 x+e^x \left (-3-279 x-279 x^2\right ) \log (3)+\left (-567 x+e^x \left (-189 x-189 x^2\right ) \log (3)\right ) \log (x)+\left (-96 x+e^x \left (-32 x-32 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-279 e^x x \log (3)-189 e^x x \log (3) \log (x)-32 e^x x \log (3) \log ^2(x)\right ) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )}{837 x+567 x \log (x)+96 x \log ^2(x)} \, dx=-\frac {1}{3} \, x e^{x} \log \left (3\right ) - \frac {1}{3} \, e^{x} \log \left (3\right ) \log \left (32 \, \log \left (x\right ) + 93\right ) + \frac {1}{3} \, e^{x} \log \left (3\right ) \log \left (\log \left (x\right ) + 3\right ) - x \] Input:
integrate(((-32*x*log(3)*exp(x)*log(x)^2-189*x*log(3)*exp(x)*log(x)-279*x* log(3)*exp(x))*log((32*log(x)+93)/(3+log(x)))+((-32*x^2-32*x)*log(3)*exp(x )-96*x)*log(x)^2+((-189*x^2-189*x)*log(3)*exp(x)-567*x)*log(x)+(-279*x^2-2 79*x-3)*log(3)*exp(x)-837*x)/(96*x*log(x)^2+567*x*log(x)+837*x),x, algorit hm="giac")
Output:
-1/3*x*e^x*log(3) - 1/3*e^x*log(3)*log(32*log(x) + 93) + 1/3*e^x*log(3)*lo g(log(x) + 3) - x
Time = 4.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {-837 x+e^x \left (-3-279 x-279 x^2\right ) \log (3)+\left (-567 x+e^x \left (-189 x-189 x^2\right ) \log (3)\right ) \log (x)+\left (-96 x+e^x \left (-32 x-32 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-279 e^x x \log (3)-189 e^x x \log (3) \log (x)-32 e^x x \log (3) \log ^2(x)\right ) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )}{837 x+567 x \log (x)+96 x \log ^2(x)} \, dx=-x-\frac {x\,{\mathrm {e}}^x\,\ln \left (3\right )}{3}-\frac {\ln \left (\frac {32\,\ln \left (x\right )+93}{\ln \left (x\right )+3}\right )\,{\mathrm {e}}^x\,\ln \left (3\right )}{3} \] Input:
int(-(837*x + log((32*log(x) + 93)/(log(x) + 3))*(279*x*exp(x)*log(3) + 18 9*x*exp(x)*log(3)*log(x) + 32*x*exp(x)*log(3)*log(x)^2) + log(x)*(567*x + exp(x)*log(3)*(189*x + 189*x^2)) + log(x)^2*(96*x + exp(x)*log(3)*(32*x + 32*x^2)) + exp(x)*log(3)*(279*x + 279*x^2 + 3))/(837*x + 96*x*log(x)^2 + 5 67*x*log(x)),x)
Output:
- x - (x*exp(x)*log(3))/3 - (log((32*log(x) + 93)/(log(x) + 3))*exp(x)*log (3))/3
Time = 0.16 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {-837 x+e^x \left (-3-279 x-279 x^2\right ) \log (3)+\left (-567 x+e^x \left (-189 x-189 x^2\right ) \log (3)\right ) \log (x)+\left (-96 x+e^x \left (-32 x-32 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-279 e^x x \log (3)-189 e^x x \log (3) \log (x)-32 e^x x \log (3) \log ^2(x)\right ) \log \left (\frac {93+32 \log (x)}{3+\log (x)}\right )}{837 x+567 x \log (x)+96 x \log ^2(x)} \, dx=-\frac {e^{x} \mathrm {log}\left (\frac {32 \,\mathrm {log}\left (x \right )+93}{\mathrm {log}\left (x \right )+3}\right ) \mathrm {log}\left (3\right )}{3}-\frac {e^{x} \mathrm {log}\left (3\right ) x}{3}-x \] Input:
int(((-32*x*log(3)*exp(x)*log(x)^2-189*x*log(3)*exp(x)*log(x)-279*x*log(3) *exp(x))*log((32*log(x)+93)/(3+log(x)))+((-32*x^2-32*x)*log(3)*exp(x)-96*x )*log(x)^2+((-189*x^2-189*x)*log(3)*exp(x)-567*x)*log(x)+(-279*x^2-279*x-3 )*log(3)*exp(x)-837*x)/(96*x*log(x)^2+567*x*log(x)+837*x),x)
Output:
( - e**x*log((32*log(x) + 93)/(log(x) + 3))*log(3) - e**x*log(3)*x - 3*x)/ 3