Integrand size = 84, antiderivative size = 27 \[ \int \frac {e^{\frac {6 x+e^4 \left (6 x-24 x^2-x^4\right )+\left (x+e^4 \left (x-4 x^2\right )\right ) \log (5)}{6+\log (5)}} \left (6+e^4 \left (6-48 x-4 x^3\right )+\left (1+e^4 (1-8 x)\right ) \log (5)\right )}{6+\log (5)} \, dx=e^{x+e^4 \left (x-x^2 \left (4+\frac {x^2}{6+\log (5)}\right )\right )} \] Output:
exp(x+(x-(4+x^2/(ln(5)+6))*x^2)*exp(4))
Time = 4.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {e^{\frac {6 x+e^4 \left (6 x-24 x^2-x^4\right )+\left (x+e^4 \left (x-4 x^2\right )\right ) \log (5)}{6+\log (5)}} \left (6+e^4 \left (6-48 x-4 x^3\right )+\left (1+e^4 (1-8 x)\right ) \log (5)\right )}{6+\log (5)} \, dx=5^{\frac {\left (1+e^4 (1-4 x)\right ) x}{6+\log (5)}} e^{-\frac {x \left (-6+e^4 \left (-6+24 x+x^3\right )\right )}{6+\log (5)}} \] Input:
Integrate[(E^((6*x + E^4*(6*x - 24*x^2 - x^4) + (x + E^4*(x - 4*x^2))*Log[ 5])/(6 + Log[5]))*(6 + E^4*(6 - 48*x - 4*x^3) + (1 + E^4*(1 - 8*x))*Log[5] ))/(6 + Log[5]),x]
Output:
5^(((1 + E^4*(1 - 4*x))*x)/(6 + Log[5]))/E^((x*(-6 + E^4*(-6 + 24*x + x^3) ))/(6 + Log[5]))
Time = 0.51 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {27, 2726}
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.
\(\displaystyle \int \frac {\left (e^4 \left (-4 x^3-48 x+6\right )+\left (e^4 (1-8 x)+1\right ) \log (5)+6\right ) \exp \left (\frac {\left (e^4 \left (x-4 x^2\right )+x\right ) \log (5)+e^4 \left (-x^4-24 x^2+6 x\right )+6 x}{6+\log (5)}\right )}{6+\log (5)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int 5^{\frac {x+e^4 \left (x-4 x^2\right )}{6+\log (5)}} \exp \left (\frac {6 x+e^4 \left (-x^4-24 x^2+6 x\right )}{6+\log (5)}\right ) \left (\log (5) \left (e^4 (1-8 x)+1\right )+2 e^4 \left (-2 x^3-24 x+3\right )+6\right )dx}{6+\log (5)}\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle 5^{\frac {e^4 \left (x-4 x^2\right )+x}{6+\log (5)}} \exp \left (\frac {e^4 \left (-x^4-24 x^2+6 x\right )+6 x}{6+\log (5)}\right )\) |
Input:
Int[(E^((6*x + E^4*(6*x - 24*x^2 - x^4) + (x + E^4*(x - 4*x^2))*Log[5])/(6 + Log[5]))*(6 + E^4*(6 - 48*x - 4*x^3) + (1 + E^4*(1 - 8*x))*Log[5]))/(6 + Log[5]),x]
Output:
5^((x + E^4*(x - 4*x^2))/(6 + Log[5]))*E^((6*x + E^4*(6*x - 24*x^2 - x^4)) /(6 + Log[5]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Time = 0.56 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67
method | result | size |
gosper | \({\mathrm e}^{-\frac {x \left (x^{3} {\mathrm e}^{4}+4 \,{\mathrm e}^{4} \ln \left (5\right ) x -{\mathrm e}^{4} \ln \left (5\right )+24 x \,{\mathrm e}^{4}-\ln \left (5\right )-6 \,{\mathrm e}^{4}-6\right )}{\ln \left (5\right )+6}}\) | \(45\) |
derivativedivides | \({\mathrm e}^{\frac {\left (\left (-4 x^{2}+x \right ) {\mathrm e}^{4}+x \right ) \ln \left (5\right )+\left (-x^{4}-24 x^{2}+6 x \right ) {\mathrm e}^{4}+6 x}{\ln \left (5\right )+6}}\) | \(45\) |
default | \({\mathrm e}^{\frac {\left (\left (-4 x^{2}+x \right ) {\mathrm e}^{4}+x \right ) \ln \left (5\right )+\left (-x^{4}-24 x^{2}+6 x \right ) {\mathrm e}^{4}+6 x}{\ln \left (5\right )+6}}\) | \(45\) |
norman | \({\mathrm e}^{\frac {\left (\left (-4 x^{2}+x \right ) {\mathrm e}^{4}+x \right ) \ln \left (5\right )+\left (-x^{4}-24 x^{2}+6 x \right ) {\mathrm e}^{4}+6 x}{\ln \left (5\right )+6}}\) | \(45\) |
risch | \({\mathrm e}^{-\frac {x \left (x^{3} {\mathrm e}^{4}+4 \,{\mathrm e}^{4} \ln \left (5\right ) x -{\mathrm e}^{4} \ln \left (5\right )+24 x \,{\mathrm e}^{4}-\ln \left (5\right )-6 \,{\mathrm e}^{4}-6\right )}{\ln \left (5\right )+6}}\) | \(45\) |
parallelrisch | \(\frac {\ln \left (5\right ) {\mathrm e}^{\frac {\left (\left (-4 x^{2}+x \right ) {\mathrm e}^{4}+x \right ) \ln \left (5\right )+\left (-x^{4}-24 x^{2}+6 x \right ) {\mathrm e}^{4}+6 x}{\ln \left (5\right )+6}}+6 \,{\mathrm e}^{\frac {\left (\left (-4 x^{2}+x \right ) {\mathrm e}^{4}+x \right ) \ln \left (5\right )+\left (-x^{4}-24 x^{2}+6 x \right ) {\mathrm e}^{4}+6 x}{\ln \left (5\right )+6}}}{\ln \left (5\right )+6}\) | \(102\) |
Input:
int((((-8*x+1)*exp(4)+1)*ln(5)+(-4*x^3-48*x+6)*exp(4)+6)*exp((((-4*x^2+x)* exp(4)+x)*ln(5)+(-x^4-24*x^2+6*x)*exp(4)+6*x)/(ln(5)+6))/(ln(5)+6),x,metho d=_RETURNVERBOSE)
Output:
exp(-x*(x^3*exp(4)+4*exp(4)*ln(5)*x-exp(4)*ln(5)+24*x*exp(4)-ln(5)-6*exp(4 )-6)/(ln(5)+6))
Time = 0.08 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {e^{\frac {6 x+e^4 \left (6 x-24 x^2-x^4\right )+\left (x+e^4 \left (x-4 x^2\right )\right ) \log (5)}{6+\log (5)}} \left (6+e^4 \left (6-48 x-4 x^3\right )+\left (1+e^4 (1-8 x)\right ) \log (5)\right )}{6+\log (5)} \, dx=e^{\left (-\frac {{\left (x^{4} + 24 \, x^{2} - 6 \, x\right )} e^{4} + {\left ({\left (4 \, x^{2} - x\right )} e^{4} - x\right )} \log \left (5\right ) - 6 \, x}{\log \left (5\right ) + 6}\right )} \] Input:
integrate((((-8*x+1)*exp(4)+1)*log(5)+(-4*x^3-48*x+6)*exp(4)+6)*exp((((-4* x^2+x)*exp(4)+x)*log(5)+(-x^4-24*x^2+6*x)*exp(4)+6*x)/(log(5)+6))/(log(5)+ 6),x, algorithm="fricas")
Output:
e^(-((x^4 + 24*x^2 - 6*x)*e^4 + ((4*x^2 - x)*e^4 - x)*log(5) - 6*x)/(log(5 ) + 6))
Time = 0.14 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\frac {6 x+e^4 \left (6 x-24 x^2-x^4\right )+\left (x+e^4 \left (x-4 x^2\right )\right ) \log (5)}{6+\log (5)}} \left (6+e^4 \left (6-48 x-4 x^3\right )+\left (1+e^4 (1-8 x)\right ) \log (5)\right )}{6+\log (5)} \, dx=e^{\frac {6 x + \left (x + \left (- 4 x^{2} + x\right ) e^{4}\right ) \log {\left (5 \right )} + \left (- x^{4} - 24 x^{2} + 6 x\right ) e^{4}}{\log {\left (5 \right )} + 6}} \] Input:
integrate((((-8*x+1)*exp(4)+1)*ln(5)+(-4*x**3-48*x+6)*exp(4)+6)*exp((((-4* x**2+x)*exp(4)+x)*ln(5)+(-x**4-24*x**2+6*x)*exp(4)+6*x)/(ln(5)+6))/(ln(5)+ 6),x)
Output:
exp((6*x + (x + (-4*x**2 + x)*exp(4))*log(5) + (-x**4 - 24*x**2 + 6*x)*exp (4))/(log(5) + 6))
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (27) = 54\).
Time = 0.37 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.15 \[ \int \frac {e^{\frac {6 x+e^4 \left (6 x-24 x^2-x^4\right )+\left (x+e^4 \left (x-4 x^2\right )\right ) \log (5)}{6+\log (5)}} \left (6+e^4 \left (6-48 x-4 x^3\right )+\left (1+e^4 (1-8 x)\right ) \log (5)\right )}{6+\log (5)} \, dx=e^{\left (-\frac {x^{4} e^{4}}{\log \left (5\right ) + 6} - \frac {4 \, x^{2} e^{4} \log \left (5\right )}{\log \left (5\right ) + 6} - \frac {24 \, x^{2} e^{4}}{\log \left (5\right ) + 6} + \frac {x e^{4} \log \left (5\right )}{\log \left (5\right ) + 6} + \frac {6 \, x e^{4}}{\log \left (5\right ) + 6} + \frac {x \log \left (5\right )}{\log \left (5\right ) + 6} + \frac {6 \, x}{\log \left (5\right ) + 6}\right )} \] Input:
integrate((((-8*x+1)*exp(4)+1)*log(5)+(-4*x^3-48*x+6)*exp(4)+6)*exp((((-4* x^2+x)*exp(4)+x)*log(5)+(-x^4-24*x^2+6*x)*exp(4)+6*x)/(log(5)+6))/(log(5)+ 6),x, algorithm="maxima")
Output:
e^(-x^4*e^4/(log(5) + 6) - 4*x^2*e^4*log(5)/(log(5) + 6) - 24*x^2*e^4/(log (5) + 6) + x*e^4*log(5)/(log(5) + 6) + 6*x*e^4/(log(5) + 6) + x*log(5)/(lo g(5) + 6) + 6*x/(log(5) + 6))
Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 4.33 \[ \int \frac {e^{\frac {6 x+e^4 \left (6 x-24 x^2-x^4\right )+\left (x+e^4 \left (x-4 x^2\right )\right ) \log (5)}{6+\log (5)}} \left (6+e^4 \left (6-48 x-4 x^3\right )+\left (1+e^4 (1-8 x)\right ) \log (5)\right )}{6+\log (5)} \, dx=\frac {e^{\left (-\frac {x^{4} e^{4} + 4 \, x^{2} e^{4} \log \left (5\right ) + 24 \, x^{2} e^{4} - x e^{4} \log \left (5\right ) - 6 \, x e^{4} - x \log \left (5\right ) - 6 \, x}{\log \left (5\right ) + 6}\right )} \log \left (5\right ) + 6 \, e^{\left (-\frac {x^{4} e^{4} + 4 \, x^{2} e^{4} \log \left (5\right ) + 24 \, x^{2} e^{4} - x e^{4} \log \left (5\right ) - 6 \, x e^{4} - x \log \left (5\right ) - 6 \, x}{\log \left (5\right ) + 6}\right )}}{\log \left (5\right ) + 6} \] Input:
integrate((((-8*x+1)*exp(4)+1)*log(5)+(-4*x^3-48*x+6)*exp(4)+6)*exp((((-4* x^2+x)*exp(4)+x)*log(5)+(-x^4-24*x^2+6*x)*exp(4)+6*x)/(log(5)+6))/(log(5)+ 6),x, algorithm="giac")
Output:
(e^(-(x^4*e^4 + 4*x^2*e^4*log(5) + 24*x^2*e^4 - x*e^4*log(5) - 6*x*e^4 - x *log(5) - 6*x)/(log(5) + 6))*log(5) + 6*e^(-(x^4*e^4 + 4*x^2*e^4*log(5) + 24*x^2*e^4 - x*e^4*log(5) - 6*x*e^4 - x*log(5) - 6*x)/(log(5) + 6)))/(log( 5) + 6)
Time = 0.43 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.70 \[ \int \frac {e^{\frac {6 x+e^4 \left (6 x-24 x^2-x^4\right )+\left (x+e^4 \left (x-4 x^2\right )\right ) \log (5)}{6+\log (5)}} \left (6+e^4 \left (6-48 x-4 x^3\right )+\left (1+e^4 (1-8 x)\right ) \log (5)\right )}{6+\log (5)} \, dx=5^{\frac {x+x\,{\mathrm {e}}^4-4\,x^2\,{\mathrm {e}}^4}{\ln \left (5\right )+6}}\,{\mathrm {e}}^{-\frac {x^4\,{\mathrm {e}}^4}{\ln \left (5\right )+6}}\,{\mathrm {e}}^{-\frac {24\,x^2\,{\mathrm {e}}^4}{\ln \left (5\right )+6}}\,{\mathrm {e}}^{\frac {6\,x}{\ln \left (5\right )+6}}\,{\mathrm {e}}^{\frac {6\,x\,{\mathrm {e}}^4}{\ln \left (5\right )+6}} \] Input:
int(-(exp((6*x - exp(4)*(24*x^2 - 6*x + x^4) + log(5)*(x + exp(4)*(x - 4*x ^2)))/(log(5) + 6))*(log(5)*(exp(4)*(8*x - 1) - 1) + exp(4)*(48*x + 4*x^3 - 6) - 6))/(log(5) + 6),x)
Output:
5^((x + x*exp(4) - 4*x^2*exp(4))/(log(5) + 6))*exp(-(x^4*exp(4))/(log(5) + 6))*exp(-(24*x^2*exp(4))/(log(5) + 6))*exp((6*x)/(log(5) + 6))*exp((6*x*e xp(4))/(log(5) + 6))
Time = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {e^{\frac {6 x+e^4 \left (6 x-24 x^2-x^4\right )+\left (x+e^4 \left (x-4 x^2\right )\right ) \log (5)}{6+\log (5)}} \left (6+e^4 \left (6-48 x-4 x^3\right )+\left (1+e^4 (1-8 x)\right ) \log (5)\right )}{6+\log (5)} \, dx=\frac {e^{e^{4} x +x}}{e^{\frac {4 \,\mathrm {log}\left (5\right ) e^{4} x^{2}+e^{4} x^{4}+24 e^{4} x^{2}}{\mathrm {log}\left (5\right )+6}}} \] Input:
int((((-8*x+1)*exp(4)+1)*log(5)+(-4*x^3-48*x+6)*exp(4)+6)*exp((((-4*x^2+x) *exp(4)+x)*log(5)+(-x^4-24*x^2+6*x)*exp(4)+6*x)/(log(5)+6))/(log(5)+6),x)
Output:
e**(e**4*x + x)/e**((4*log(5)*e**4*x**2 + e**4*x**4 + 24*e**4*x**2)/(log(5 ) + 6))