Integrand size = 47, antiderivative size = 19 \[ \int e^{-4 x+e^4 x+e^6 x} \left (1-4 x+e^4 x+e^6 x+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx=e^{\left (-4+e^4+e^6\right ) x} (x-\log (4)) \]
Leaf count is larger than twice the leaf count of optimal. \(50\) vs. \(2(19)=38\).
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.63 \[ \int e^{-4 x+e^4 x+e^6 x} \left (1-4 x+e^4 x+e^6 x+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx=\frac {e^{\left (-4+e^4+e^6\right ) x} \left (\left (-4+e^4+e^6\right ) x-e^4 \log (4)-e^6 \log (4)+\log (256)\right )}{-4+e^4+e^6} \]
(E^((-4 + E^4 + E^6)*x)*((-4 + E^4 + E^6)*x - E^4*Log[4] - E^6*Log[4] + Lo g[256]))/(-4 + E^4 + E^6)
Leaf count is larger than twice the leaf count of optimal. \(98\) vs. \(2(19)=38\).
Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 5.16, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.106, Rules used = {6, 6, 2618, 2607, 2624}
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.
\(\displaystyle \int e^{e^6 x+e^4 x-4 x} \left (e^6 x+e^4 x-4 x+1+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int e^{e^6 x+e^4 x-4 x} \left (\left (e^4-4\right ) x+e^6 x+1+\left (4-e^4-e^6\right ) \log (4)\right )dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int e^{e^6 x+e^4 x-4 x} \left (\left (-4+e^4+e^6\right ) x+1+\left (4-e^4-e^6\right ) \log (4)\right )dx\) |
\(\Big \downarrow \) 2618 |
\(\displaystyle \int e^{-\left (\left (4-e^4-e^6\right ) x\right )} \left (\left (-4+e^4+e^6\right ) x+1+\left (4-e^4-e^6\right ) \log (4)\right )dx\) |
\(\Big \downarrow \) 2607 |
\(\displaystyle -\int e^{-\left (\left (4-e^4-e^6\right ) x\right )}dx-\frac {e^{-\left (\left (4-e^4-e^6\right ) x\right )} \left (-\left (\left (4-e^4-e^6\right ) x\right )+1+\left (4-e^4-e^6\right ) \log (4)\right )}{4-e^4-e^6}\) |
\(\Big \downarrow \) 2624 |
\(\displaystyle \frac {e^{-\left (\left (4-e^4-e^6\right ) x\right )}}{4-e^4-e^6}-\frac {e^{-\left (\left (4-e^4-e^6\right ) x\right )} \left (-\left (\left (4-e^4-e^6\right ) x\right )+1+\left (4-e^4-e^6\right ) \log (4)\right )}{4-e^4-e^6}\) |
1/(E^((4 - E^4 - E^6)*x)*(4 - E^4 - E^6)) - (1 - (4 - E^4 - E^6)*x + (4 - E^4 - E^6)*Log[4])/(E^((4 - E^4 - E^6)*x)*(4 - E^4 - E^6))
3.9.65.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m _.), x_Symbol] :> Simp[(c + d*x)^m*((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Simp[d*(m/(f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x)))^ n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2* m] && !TrueQ[$UseGamma]
Int[((a_.) + (b_.)*((F_)^((g_.)*(v_)))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m _.), x_Symbol] :> Int[(c + d*x)^m*(a + b*(F^(g*ExpandToSum[v, x]))^n)^p, x] /; FreeQ[{F, a, b, c, d, g, n, p}, x] && LinearQ[v, x] && !LinearMatchQ[v , x] && IntegerQ[m]
Int[((F_)^(v_))^(n_.), x_Symbol] :> Simp[(F^v)^n/(n*Log[F]*D[v, x]), x] /; FreeQ[{F, n}, x] && LinearQ[v, x]
Time = 1.82 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89
method | result | size |
risch | \(\left (x -2 \ln \left (2\right )\right ) {\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )}\) | \(17\) |
gosper | \(-{\mathrm e}^{x \,{\mathrm e}^{4}+x \,{\mathrm e}^{6}-4 x} \left (2 \ln \left (2\right )-x \right )\) | \(26\) |
parallelrisch | \(x \,{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )}-2 \,{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} \ln \left (2\right )\) | \(30\) |
norman | \(x \,{\mathrm e}^{x \,{\mathrm e}^{4}+x \,{\mathrm e}^{6}-4 x}-2 \ln \left (2\right ) {\mathrm e}^{x \,{\mathrm e}^{4}+x \,{\mathrm e}^{6}-4 x}\) | \(38\) |
parts | \(-\frac {2 \,{\mathrm e}^{x \,{\mathrm e}^{4}+x \,{\mathrm e}^{6}-4 x} \ln \left (2\right ) {\mathrm e}^{6}}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}+\frac {{\mathrm e}^{x \,{\mathrm e}^{4}+x \,{\mathrm e}^{6}-4 x} x \,{\mathrm e}^{6}}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}-\frac {2 \,{\mathrm e}^{x \,{\mathrm e}^{4}+x \,{\mathrm e}^{6}-4 x} {\mathrm e}^{4} \ln \left (2\right )}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}+\frac {{\mathrm e}^{x \,{\mathrm e}^{4}+x \,{\mathrm e}^{6}-4 x} x \,{\mathrm e}^{4}}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}+\frac {8 \,{\mathrm e}^{x \,{\mathrm e}^{4}+x \,{\mathrm e}^{6}-4 x} \ln \left (2\right )}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}-\frac {4 \,{\mathrm e}^{x \,{\mathrm e}^{4}+x \,{\mathrm e}^{6}-4 x} x}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}\) | \(183\) |
meijerg | \(-\frac {2 \ln \left (2\right ) {\mathrm e}^{6} \left (1-{\mathrm e}^{-x \left (-{\mathrm e}^{4}-{\mathrm e}^{6}+4\right )}\right )}{-{\mathrm e}^{4}-{\mathrm e}^{6}+4}-\frac {2 \,{\mathrm e}^{4} \ln \left (2\right ) \left (1-{\mathrm e}^{-x \left (-{\mathrm e}^{4}-{\mathrm e}^{6}+4\right )}\right )}{-{\mathrm e}^{4}-{\mathrm e}^{6}+4}+\frac {8 \ln \left (2\right ) \left (1-{\mathrm e}^{-x \left (-{\mathrm e}^{4}-{\mathrm e}^{6}+4\right )}\right )}{-{\mathrm e}^{4}-{\mathrm e}^{6}+4}+\frac {1-{\mathrm e}^{-x \left (-{\mathrm e}^{4}-{\mathrm e}^{6}+4\right )}}{-{\mathrm e}^{4}-{\mathrm e}^{6}+4}+\frac {\left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right ) \left (1-\frac {\left (2+2 x \left (-{\mathrm e}^{4}-{\mathrm e}^{6}+4\right )\right ) {\mathrm e}^{-x \left (-{\mathrm e}^{4}-{\mathrm e}^{6}+4\right )}}{2}\right )}{\left (-{\mathrm e}^{4}-{\mathrm e}^{6}+4\right )^{2}}\) | \(191\) |
derivativedivides | \(\frac {\frac {{\mathrm e}^{6} \left ({\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )-{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )}\right )}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}+\frac {{\mathrm e}^{4} \left ({\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )-{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )}\right )}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}+8 \,{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} \ln \left (2\right )-\frac {4 \left ({\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )-{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )}\right )}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}-2 \,{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} {\mathrm e}^{4} \ln \left (2\right )-2 \,{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} \ln \left (2\right ) {\mathrm e}^{6}+{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )}}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}\) | \(220\) |
default | \(\frac {\frac {{\mathrm e}^{6} \left ({\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )-{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )}\right )}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}+\frac {{\mathrm e}^{4} \left ({\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )-{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )}\right )}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}+8 \,{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} \ln \left (2\right )-\frac {4 \left ({\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )-{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )}\right )}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}-2 \,{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} {\mathrm e}^{4} \ln \left (2\right )-2 \,{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} \ln \left (2\right ) {\mathrm e}^{6}+{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )}}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}\) | \(220\) |
int((2*(-exp(4)-exp(3)^2+4)*ln(2)+x*exp(4)+x*exp(3)^2-4*x+1)*exp(x*exp(4)+ x*exp(3)^2-4*x),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{-4 x+e^4 x+e^6 x} \left (1-4 x+e^4 x+e^6 x+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx={\left (x - 2 \, \log \left (2\right )\right )} e^{\left (x e^{6} + x e^{4} - 4 \, x\right )} \]
integrate((2*(-exp(4)-exp(3)^2+4)*log(2)+x*exp(4)+x*exp(3)^2-4*x+1)*exp(x* exp(4)+x*exp(3)^2-4*x),x, algorithm=\
Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{-4 x+e^4 x+e^6 x} \left (1-4 x+e^4 x+e^6 x+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx=\left (x - 2 \log {\left (2 \right )}\right ) e^{- 4 x + x e^{4} + x e^{6}} \]
integrate((2*(-exp(4)-exp(3)**2+4)*ln(2)+x*exp(4)+x*exp(3)**2-4*x+1)*exp(x *exp(4)+x*exp(3)**2-4*x),x)
Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (16) = 32\).
Time = 0.20 (sec) , antiderivative size = 245, normalized size of antiderivative = 12.89 \[ \int e^{-4 x+e^4 x+e^6 x} \left (1-4 x+e^4 x+e^6 x+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx=\frac {{\left (x {\left (e^{12} + e^{10} - 4 \, e^{6}\right )} - e^{6}\right )} e^{\left (x e^{6} + x e^{4} - 4 \, x\right )}}{e^{12} + 2 \, e^{10} + e^{8} - 8 \, e^{6} - 8 \, e^{4} + 16} + \frac {{\left (x {\left (e^{10} + e^{8} - 4 \, e^{4}\right )} - e^{4}\right )} e^{\left (x e^{6} + x e^{4} - 4 \, x\right )}}{e^{12} + 2 \, e^{10} + e^{8} - 8 \, e^{6} - 8 \, e^{4} + 16} - \frac {4 \, {\left (x {\left (e^{6} + e^{4} - 4\right )} - 1\right )} e^{\left (x e^{6} + x e^{4} - 4 \, x\right )}}{e^{12} + 2 \, e^{10} + e^{8} - 8 \, e^{6} - 8 \, e^{4} + 16} - \frac {2 \, e^{\left (x e^{6} + x e^{4} - 4 \, x + 6\right )} \log \left (2\right )}{e^{6} + e^{4} - 4} - \frac {2 \, e^{\left (x e^{6} + x e^{4} - 4 \, x + 4\right )} \log \left (2\right )}{e^{6} + e^{4} - 4} + \frac {8 \, e^{\left (x e^{6} + x e^{4} - 4 \, x\right )} \log \left (2\right )}{e^{6} + e^{4} - 4} + \frac {e^{\left (x e^{6} + x e^{4} - 4 \, x\right )}}{e^{6} + e^{4} - 4} \]
integrate((2*(-exp(4)-exp(3)^2+4)*log(2)+x*exp(4)+x*exp(3)^2-4*x+1)*exp(x* exp(4)+x*exp(3)^2-4*x),x, algorithm=\
(x*(e^12 + e^10 - 4*e^6) - e^6)*e^(x*e^6 + x*e^4 - 4*x)/(e^12 + 2*e^10 + e ^8 - 8*e^6 - 8*e^4 + 16) + (x*(e^10 + e^8 - 4*e^4) - e^4)*e^(x*e^6 + x*e^4 - 4*x)/(e^12 + 2*e^10 + e^8 - 8*e^6 - 8*e^4 + 16) - 4*(x*(e^6 + e^4 - 4) - 1)*e^(x*e^6 + x*e^4 - 4*x)/(e^12 + 2*e^10 + e^8 - 8*e^6 - 8*e^4 + 16) - 2*e^(x*e^6 + x*e^4 - 4*x + 6)*log(2)/(e^6 + e^4 - 4) - 2*e^(x*e^6 + x*e^4 - 4*x + 4)*log(2)/(e^6 + e^4 - 4) + 8*e^(x*e^6 + x*e^4 - 4*x)*log(2)/(e^6 + e^4 - 4) + e^(x*e^6 + x*e^4 - 4*x)/(e^6 + e^4 - 4)
Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (16) = 32\).
Time = 0.25 (sec) , antiderivative size = 202, normalized size of antiderivative = 10.63 \[ \int e^{-4 x+e^4 x+e^6 x} \left (1-4 x+e^4 x+e^6 x+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx=\frac {{\left (x e^{6} + x e^{4} - 2 \, e^{6} \log \left (2\right ) - 2 \, e^{4} \log \left (2\right ) - 4 \, x + 8 \, \log \left (2\right ) - 1\right )} e^{\left (x e^{6} + x e^{4} - 4 \, x + 6\right )}}{e^{12} + 2 \, e^{10} + e^{8} - 8 \, e^{6} - 8 \, e^{4} + 16} + \frac {{\left (x e^{6} + x e^{4} - 2 \, e^{6} \log \left (2\right ) - 2 \, e^{4} \log \left (2\right ) - 4 \, x + 8 \, \log \left (2\right ) - 1\right )} e^{\left (x e^{6} + x e^{4} - 4 \, x + 4\right )}}{e^{12} + 2 \, e^{10} + e^{8} - 8 \, e^{6} - 8 \, e^{4} + 16} - \frac {{\left (4 \, x e^{6} + 4 \, x e^{4} - 8 \, e^{6} \log \left (2\right ) - 8 \, e^{4} \log \left (2\right ) - 16 \, x - e^{6} - e^{4} + 32 \, \log \left (2\right )\right )} e^{\left (x e^{6} + x e^{4} - 4 \, x\right )}}{e^{12} + 2 \, e^{10} + e^{8} - 8 \, e^{6} - 8 \, e^{4} + 16} \]
integrate((2*(-exp(4)-exp(3)^2+4)*log(2)+x*exp(4)+x*exp(3)^2-4*x+1)*exp(x* exp(4)+x*exp(3)^2-4*x),x, algorithm=\
(x*e^6 + x*e^4 - 2*e^6*log(2) - 2*e^4*log(2) - 4*x + 8*log(2) - 1)*e^(x*e^ 6 + x*e^4 - 4*x + 6)/(e^12 + 2*e^10 + e^8 - 8*e^6 - 8*e^4 + 16) + (x*e^6 + x*e^4 - 2*e^6*log(2) - 2*e^4*log(2) - 4*x + 8*log(2) - 1)*e^(x*e^6 + x*e^ 4 - 4*x + 4)/(e^12 + 2*e^10 + e^8 - 8*e^6 - 8*e^4 + 16) - (4*x*e^6 + 4*x*e ^4 - 8*e^6*log(2) - 8*e^4*log(2) - 16*x - e^6 - e^4 + 32*log(2))*e^(x*e^6 + x*e^4 - 4*x)/(e^12 + 2*e^10 + e^8 - 8*e^6 - 8*e^4 + 16)
Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int e^{-4 x+e^4 x+e^6 x} \left (1-4 x+e^4 x+e^6 x+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx={\mathrm {e}}^{-4\,x}\,{\mathrm {e}}^{x\,{\mathrm {e}}^4}\,{\mathrm {e}}^{x\,{\mathrm {e}}^6}\,\left (x-\ln \left (4\right )\right ) \]