Integrand size = 68, antiderivative size = 21 \[ \int \frac {-24+8 e^x-8 x-4 x^2}{78-12 x+13 x^2-2 x^3+e^x (-13+2 x)+\left (-12+2 e^x-2 x^2\right ) \log \left (18-3 e^x+3 x^2\right )} \, dx=\log \left (\left (-\frac {13}{2}+x+\log \left (3 \left (6-e^x+x^2\right )\right )\right )^2\right ) \] Output:
ln((x-13/2+ln(-3*exp(x)+3*x^2+18))^2)
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-24+8 e^x-8 x-4 x^2}{78-12 x+13 x^2-2 x^3+e^x (-13+2 x)+\left (-12+2 e^x-2 x^2\right ) \log \left (18-3 e^x+3 x^2\right )} \, dx=2 \log \left (13-2 x-\log (9)-2 \log \left (6-e^x+x^2\right )\right ) \] Input:
Integrate[(-24 + 8*E^x - 8*x - 4*x^2)/(78 - 12*x + 13*x^2 - 2*x^3 + E^x*(- 13 + 2*x) + (-12 + 2*E^x - 2*x^2)*Log[18 - 3*E^x + 3*x^2]),x]
Output:
2*Log[13 - 2*x - Log[9] - 2*Log[6 - E^x + x^2]]
Time = 0.61 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {7292, 27, 25, 7235}
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.
\(\displaystyle \int \frac {-4 x^2-8 x+8 e^x-24}{-2 x^3+13 x^2+\left (-2 x^2+2 e^x-12\right ) \log \left (3 x^2-3 e^x+18\right )-12 x+e^x (2 x-13)+78} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {4 \left (-x^2-2 x+2 e^x-6\right )}{\left (x^2-e^x+6\right ) \left (-2 \log \left (3 \left (x^2-e^x+6\right )\right )-2 x+13\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 4 \int -\frac {x^2+2 x-2 e^x+6}{\left (x^2-e^x+6\right ) \left (-2 x-2 \log \left (3 \left (x^2-e^x+6\right )\right )+13\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -4 \int \frac {x^2+2 x-2 e^x+6}{\left (x^2-e^x+6\right ) \left (-2 x-2 \log \left (3 \left (x^2-e^x+6\right )\right )+13\right )}dx\) |
\(\Big \downarrow \) 7235 |
\(\displaystyle 2 \log \left (-2 \log \left (3 \left (x^2-e^x+6\right )\right )-2 x+13\right )\) |
Input:
Int[(-24 + 8*E^x - 8*x - 4*x^2)/(78 - 12*x + 13*x^2 - 2*x^3 + E^x*(-13 + 2 *x) + (-12 + 2*E^x - 2*x^2)*Log[18 - 3*E^x + 3*x^2]),x]
Output:
2*Log[13 - 2*x - 2*Log[3*(6 - E^x + x^2)]]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L og[RemoveContent[y, x]], x] /; !FalseQ[q]]
Time = 0.40 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90
method | result | size |
risch | \(2 \ln \left (x -\frac {13}{2}+\ln \left (-3 \,{\mathrm e}^{x}+3 x^{2}+18\right )\right )\) | \(19\) |
parallelrisch | \(2 \ln \left (x -\frac {13}{2}+\ln \left (-3 \,{\mathrm e}^{x}+3 x^{2}+18\right )\right )\) | \(19\) |
norman | \(2 \ln \left (2 \ln \left (-3 \,{\mathrm e}^{x}+3 x^{2}+18\right )+2 x -13\right )\) | \(23\) |
Input:
int((8*exp(x)-4*x^2-8*x-24)/((2*exp(x)-2*x^2-12)*ln(-3*exp(x)+3*x^2+18)+(2 *x-13)*exp(x)-2*x^3+13*x^2-12*x+78),x,method=_RETURNVERBOSE)
Output:
2*ln(x-13/2+ln(-3*exp(x)+3*x^2+18))
Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {-24+8 e^x-8 x-4 x^2}{78-12 x+13 x^2-2 x^3+e^x (-13+2 x)+\left (-12+2 e^x-2 x^2\right ) \log \left (18-3 e^x+3 x^2\right )} \, dx=2 \, \log \left (2 \, x + 2 \, \log \left (3 \, x^{2} - 3 \, e^{x} + 18\right ) - 13\right ) \] Input:
integrate((8*exp(x)-4*x^2-8*x-24)/((2*exp(x)-2*x^2-12)*log(-3*exp(x)+3*x^2 +18)+(2*x-13)*exp(x)-2*x^3+13*x^2-12*x+78),x, algorithm="fricas")
Output:
2*log(2*x + 2*log(3*x^2 - 3*e^x + 18) - 13)
Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {-24+8 e^x-8 x-4 x^2}{78-12 x+13 x^2-2 x^3+e^x (-13+2 x)+\left (-12+2 e^x-2 x^2\right ) \log \left (18-3 e^x+3 x^2\right )} \, dx=2 \log {\left (x + \log {\left (3 x^{2} - 3 e^{x} + 18 \right )} - \frac {13}{2} \right )} \] Input:
integrate((8*exp(x)-4*x**2-8*x-24)/((2*exp(x)-2*x**2-12)*ln(-3*exp(x)+3*x* *2+18)+(2*x-13)*exp(x)-2*x**3+13*x**2-12*x+78),x)
Output:
2*log(x + log(3*x**2 - 3*exp(x) + 18) - 13/2)
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-24+8 e^x-8 x-4 x^2}{78-12 x+13 x^2-2 x^3+e^x (-13+2 x)+\left (-12+2 e^x-2 x^2\right ) \log \left (18-3 e^x+3 x^2\right )} \, dx=2 \, \log \left (i \, \pi + x + \log \left (3\right ) + \log \left (-x^{2} + e^{x} - 6\right ) - \frac {13}{2}\right ) \] Input:
integrate((8*exp(x)-4*x^2-8*x-24)/((2*exp(x)-2*x^2-12)*log(-3*exp(x)+3*x^2 +18)+(2*x-13)*exp(x)-2*x^3+13*x^2-12*x+78),x, algorithm="maxima")
Output:
2*log(I*pi + x + log(3) + log(-x^2 + e^x - 6) - 13/2)
Time = 0.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {-24+8 e^x-8 x-4 x^2}{78-12 x+13 x^2-2 x^3+e^x (-13+2 x)+\left (-12+2 e^x-2 x^2\right ) \log \left (18-3 e^x+3 x^2\right )} \, dx=2 \, \log \left (2 \, x + 2 \, \log \left (3 \, x^{2} - 3 \, e^{x} + 18\right ) - 13\right ) \] Input:
integrate((8*exp(x)-4*x^2-8*x-24)/((2*exp(x)-2*x^2-12)*log(-3*exp(x)+3*x^2 +18)+(2*x-13)*exp(x)-2*x^3+13*x^2-12*x+78),x, algorithm="giac")
Output:
2*log(2*x + 2*log(3*x^2 - 3*e^x + 18) - 13)
Time = 0.33 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {-24+8 e^x-8 x-4 x^2}{78-12 x+13 x^2-2 x^3+e^x (-13+2 x)+\left (-12+2 e^x-2 x^2\right ) \log \left (18-3 e^x+3 x^2\right )} \, dx=2\,\ln \left (x+\ln \left (3\,x^2-3\,{\mathrm {e}}^x+18\right )-\frac {13}{2}\right ) \] Input:
int((8*x - 8*exp(x) + 4*x^2 + 24)/(12*x + log(3*x^2 - 3*exp(x) + 18)*(2*x^ 2 - 2*exp(x) + 12) - exp(x)*(2*x - 13) - 13*x^2 + 2*x^3 - 78),x)
Output:
2*log(x + log(3*x^2 - 3*exp(x) + 18) - 13/2)
\[ \int \frac {-24+8 e^x-8 x-4 x^2}{78-12 x+13 x^2-2 x^3+e^x (-13+2 x)+\left (-12+2 e^x-2 x^2\right ) \log \left (18-3 e^x+3 x^2\right )} \, dx=8 \left (\int \frac {e^{x}}{2 e^{x} \mathrm {log}\left (-3 e^{x}+3 x^{2}+18\right )+2 e^{x} x -13 e^{x}-2 \,\mathrm {log}\left (-3 e^{x}+3 x^{2}+18\right ) x^{2}-12 \,\mathrm {log}\left (-3 e^{x}+3 x^{2}+18\right )-2 x^{3}+13 x^{2}-12 x +78}d x \right )-4 \left (\int \frac {x^{2}}{2 e^{x} \mathrm {log}\left (-3 e^{x}+3 x^{2}+18\right )+2 e^{x} x -13 e^{x}-2 \,\mathrm {log}\left (-3 e^{x}+3 x^{2}+18\right ) x^{2}-12 \,\mathrm {log}\left (-3 e^{x}+3 x^{2}+18\right )-2 x^{3}+13 x^{2}-12 x +78}d x \right )-8 \left (\int \frac {x}{2 e^{x} \mathrm {log}\left (-3 e^{x}+3 x^{2}+18\right )+2 e^{x} x -13 e^{x}-2 \,\mathrm {log}\left (-3 e^{x}+3 x^{2}+18\right ) x^{2}-12 \,\mathrm {log}\left (-3 e^{x}+3 x^{2}+18\right )-2 x^{3}+13 x^{2}-12 x +78}d x \right )-24 \left (\int \frac {1}{2 e^{x} \mathrm {log}\left (-3 e^{x}+3 x^{2}+18\right )+2 e^{x} x -13 e^{x}-2 \,\mathrm {log}\left (-3 e^{x}+3 x^{2}+18\right ) x^{2}-12 \,\mathrm {log}\left (-3 e^{x}+3 x^{2}+18\right )-2 x^{3}+13 x^{2}-12 x +78}d x \right ) \] Input:
int((8*exp(x)-4*x^2-8*x-24)/((2*exp(x)-2*x^2-12)*log(-3*exp(x)+3*x^2+18)+( 2*x-13)*exp(x)-2*x^3+13*x^2-12*x+78),x)
Output:
4*(2*int(e**x/(2*e**x*log( - 3*e**x + 3*x**2 + 18) + 2*e**x*x - 13*e**x - 2*log( - 3*e**x + 3*x**2 + 18)*x**2 - 12*log( - 3*e**x + 3*x**2 + 18) - 2* x**3 + 13*x**2 - 12*x + 78),x) - int(x**2/(2*e**x*log( - 3*e**x + 3*x**2 + 18) + 2*e**x*x - 13*e**x - 2*log( - 3*e**x + 3*x**2 + 18)*x**2 - 12*log( - 3*e**x + 3*x**2 + 18) - 2*x**3 + 13*x**2 - 12*x + 78),x) - 2*int(x/(2*e* *x*log( - 3*e**x + 3*x**2 + 18) + 2*e**x*x - 13*e**x - 2*log( - 3*e**x + 3 *x**2 + 18)*x**2 - 12*log( - 3*e**x + 3*x**2 + 18) - 2*x**3 + 13*x**2 - 12 *x + 78),x) - 6*int(1/(2*e**x*log( - 3*e**x + 3*x**2 + 18) + 2*e**x*x - 13 *e**x - 2*log( - 3*e**x + 3*x**2 + 18)*x**2 - 12*log( - 3*e**x + 3*x**2 + 18) - 2*x**3 + 13*x**2 - 12*x + 78),x))