Integrand size = 80, antiderivative size = 25 \[ \int \frac {e^x \left (144-12 e-432 x+144 x^2\right )}{144 e^{2 x} \log (2)+e^x \left (-24 e-288 x+288 x^2\right ) \log (2)+\left (e^2+144 x^2-288 x^3+144 x^4+e \left (24 x-24 x^2\right )\right ) \log (2)} \, dx=\frac {e^x}{\left (-\frac {e}{12}+e^x-x+x^2\right ) \log (2)} \]
Time = 1.41 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {e^x \left (144-12 e-432 x+144 x^2\right )}{144 e^{2 x} \log (2)+e^x \left (-24 e-288 x+288 x^2\right ) \log (2)+\left (e^2+144 x^2-288 x^3+144 x^4+e \left (24 x-24 x^2\right )\right ) \log (2)} \, dx=-\frac {e-12 (-1+x) x}{\left (e-12 e^x-12 (-1+x) x\right ) \log (2)} \]
Integrate[(E^x*(144 - 12*E - 432*x + 144*x^2))/(144*E^(2*x)*Log[2] + E^x*( -24*E - 288*x + 288*x^2)*Log[2] + (E^2 + 144*x^2 - 288*x^3 + 144*x^4 + E*( 24*x - 24*x^2))*Log[2]),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.
\(\displaystyle \int \frac {e^x \left (144 x^2-432 x-12 e+144\right )}{e^x \left (288 x^2-288 x-24 e\right ) \log (2)+\left (144 x^4-288 x^3+144 x^2+e \left (24 x-24 x^2\right )+e^2\right ) \log (2)+144 e^{2 x} \log (2)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {12 e^x \left (12 x^2-36 x-e+12\right )}{\left (-12 (x-1) x-12 e^x+e\right )^2 \log (2)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {12 \int \frac {e^x \left (12 x^2-36 x-e+12\right )}{\left (12 (1-x) x-12 e^x+e\right )^2}dx}{\log (2)}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {12 \int \left (\frac {12 e^x x^2}{\left (12 x^2-12 x+12 e^x-e\right )^2}-\frac {36 e^x x}{\left (12 x^2-12 x+12 e^x-e\right )^2}-\frac {\left (1-\frac {12}{e}\right ) e^{x+1}}{\left (-12 x^2+12 x-12 e^x+e\right )^2}\right )dx}{\log (2)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {12 \left (-\left (\left (1-\frac {12}{e}\right ) \int \frac {e^{x+1}}{\left (-12 x^2+12 x-12 e^x+e\right )^2}dx\right )-36 \int \frac {e^x x}{\left (12 x^2-12 x+12 e^x-e\right )^2}dx+12 \int \frac {e^x x^2}{\left (12 x^2-12 x+12 e^x-e\right )^2}dx\right )}{\log (2)}\) |
Int[(E^x*(144 - 12*E - 432*x + 144*x^2))/(144*E^(2*x)*Log[2] + E^x*(-24*E - 288*x + 288*x^2)*Log[2] + (E^2 + 144*x^2 - 288*x^3 + 144*x^4 + E*(24*x - 24*x^2))*Log[2]),x]
3.6.73.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04
method | result | size |
norman | \(-\frac {12 \,{\mathrm e}^{x}}{\ln \left (2\right ) \left (-12 x^{2}+{\mathrm e}-12 \,{\mathrm e}^{x}+12 x \right )}\) | \(26\) |
risch | \(-\frac {-12 x^{2}+{\mathrm e}+12 x}{\ln \left (2\right ) \left (-12 x^{2}+{\mathrm e}-12 \,{\mathrm e}^{x}+12 x \right )}\) | \(35\) |
parallelrisch | \(-\frac {-144 x^{2}+12 \,{\mathrm e}+144 x}{12 \ln \left (2\right ) \left (-12 x^{2}+{\mathrm e}-12 \,{\mathrm e}^{x}+12 x \right )}\) | \(37\) |
int((-12*exp(1)+144*x^2-432*x+144)*exp(x)/(144*ln(2)*exp(x)^2+(-24*exp(1)+ 288*x^2-288*x)*ln(2)*exp(x)+(exp(1)^2+(-24*x^2+24*x)*exp(1)+144*x^4-288*x^ 3+144*x^2)*ln(2)),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.60 \[ \int \frac {e^x \left (144-12 e-432 x+144 x^2\right )}{144 e^{2 x} \log (2)+e^x \left (-24 e-288 x+288 x^2\right ) \log (2)+\left (e^2+144 x^2-288 x^3+144 x^4+e \left (24 x-24 x^2\right )\right ) \log (2)} \, dx=-\frac {12 \, x^{2} - 12 \, x - e}{{\left (12 \, x^{2} - 12 \, x - e\right )} \log \left (2\right ) + 12 \, e^{x} \log \left (2\right )} \]
integrate((-12*exp(1)+144*x^2-432*x+144)*exp(x)/(144*log(2)*exp(x)^2+(-24* exp(1)+288*x^2-288*x)*log(2)*exp(x)+(exp(1)^2+(-24*x^2+24*x)*exp(1)+144*x^ 4-288*x^3+144*x^2)*log(2)),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).
Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {e^x \left (144-12 e-432 x+144 x^2\right )}{144 e^{2 x} \log (2)+e^x \left (-24 e-288 x+288 x^2\right ) \log (2)+\left (e^2+144 x^2-288 x^3+144 x^4+e \left (24 x-24 x^2\right )\right ) \log (2)} \, dx=\frac {- 12 x^{2} + 12 x + e}{12 x^{2} \log {\left (2 \right )} - 12 x \log {\left (2 \right )} + 12 e^{x} \log {\left (2 \right )} - e \log {\left (2 \right )}} \]
integrate((-12*exp(1)+144*x**2-432*x+144)*exp(x)/(144*ln(2)*exp(x)**2+(-24 *exp(1)+288*x**2-288*x)*ln(2)*exp(x)+(exp(1)**2+(-24*x**2+24*x)*exp(1)+144 *x**4-288*x**3+144*x**2)*ln(2)),x)
Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {e^x \left (144-12 e-432 x+144 x^2\right )}{144 e^{2 x} \log (2)+e^x \left (-24 e-288 x+288 x^2\right ) \log (2)+\left (e^2+144 x^2-288 x^3+144 x^4+e \left (24 x-24 x^2\right )\right ) \log (2)} \, dx=-\frac {12 \, x^{2} - 12 \, x - e}{12 \, x^{2} \log \left (2\right ) - 12 \, x \log \left (2\right ) - e \log \left (2\right ) + 12 \, e^{x} \log \left (2\right )} \]
integrate((-12*exp(1)+144*x^2-432*x+144)*exp(x)/(144*log(2)*exp(x)^2+(-24* exp(1)+288*x^2-288*x)*log(2)*exp(x)+(exp(1)^2+(-24*x^2+24*x)*exp(1)+144*x^ 4-288*x^3+144*x^2)*log(2)),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {e^x \left (144-12 e-432 x+144 x^2\right )}{144 e^{2 x} \log (2)+e^x \left (-24 e-288 x+288 x^2\right ) \log (2)+\left (e^2+144 x^2-288 x^3+144 x^4+e \left (24 x-24 x^2\right )\right ) \log (2)} \, dx=-\frac {12 \, x^{2} - 12 \, x - e}{12 \, x^{2} \log \left (2\right ) - 12 \, x \log \left (2\right ) - e \log \left (2\right ) + 12 \, e^{x} \log \left (2\right )} \]
integrate((-12*exp(1)+144*x^2-432*x+144)*exp(x)/(144*log(2)*exp(x)^2+(-24* exp(1)+288*x^2-288*x)*log(2)*exp(x)+(exp(1)^2+(-24*x^2+24*x)*exp(1)+144*x^ 4-288*x^3+144*x^2)*log(2)),x, algorithm=\
Timed out. \[ \int \frac {e^x \left (144-12 e-432 x+144 x^2\right )}{144 e^{2 x} \log (2)+e^x \left (-24 e-288 x+288 x^2\right ) \log (2)+\left (e^2+144 x^2-288 x^3+144 x^4+e \left (24 x-24 x^2\right )\right ) \log (2)} \, dx=-\int \frac {{\mathrm {e}}^x\,\left (-144\,x^2+432\,x+12\,\mathrm {e}-144\right )}{144\,{\mathrm {e}}^{2\,x}\,\ln \left (2\right )+\ln \left (2\right )\,\left ({\mathrm {e}}^2+\mathrm {e}\,\left (24\,x-24\,x^2\right )+144\,x^2-288\,x^3+144\,x^4\right )-{\mathrm {e}}^x\,\ln \left (2\right )\,\left (-288\,x^2+288\,x+24\,\mathrm {e}\right )} \,d x \]
int(-(exp(x)*(432*x + 12*exp(1) - 144*x^2 - 144))/(144*exp(2*x)*log(2) + l og(2)*(exp(2) + exp(1)*(24*x - 24*x^2) + 144*x^2 - 288*x^3 + 144*x^4) - ex p(x)*log(2)*(288*x + 24*exp(1) - 288*x^2)),x)