\(\int e^{-4 x+e^4 x+e^6 x} (1-4 x+e^4 x+e^6 x+(4-e^4-e^6) \log (4)) \, dx\) [3865]

Optimal result

Integrand size = 47, antiderivative size = 19 \[ \int e^{-4 x+e^4 x+e^6 x} \left (1-4 x+e^4 x+e^6 x+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx=e^{\left (-4+e^4+e^6\right ) x} (x-\log (4)) \]

[Out]

(x-2*ln(2))*exp(x*(exp(4)+exp(3)^2-4))

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(99\) vs. \(2(19)=38\).

Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 5.21, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {6, 2218, 2207, 2225} \[ \int e^{-4 x+e^4 x+e^6 x} \left (1-4 x+e^4 x+e^6 x+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx=\frac {e^{-\left (\left (4-e^4-e^6\right ) x\right )}}{4-e^4-e^6}-\frac {e^{-\left (\left (4-e^4-e^6\right ) x\right )} \left (-\left (\left (4-e^4-e^6\right ) x\right )+1+\log (256)-e^6 \log (4)-e^4 \log (4)\right )}{4-e^4-e^6} \]

[In]

Int[E^(-4*x + E^4*x + E^6*x)*(1 - 4*x + E^4*x + E^6*x + (4 - E^4 - E^6)*Log[4]),x]

[Out]

1/(E^((4 - E^4 - E^6)*x)*(4 - E^4 - E^6)) - (1 - (4 - E^4 - E^6)*x - E^4*Log[4] - E^6*Log[4] + Log[256])/(E^((
4 - E^4 - E^6)*x)*(4 - E^4 - E^6))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 2207

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] - Dist[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]

Rule 2218

Int[((a_.) + (b_.)*((F_)^((g_.)*(v_)))^(n_.))^(p_.)*(u_)^(m_.), x_Symbol] :> Int[NormalizePowerOfLinear[u, x]^
m*(a + b*(F^(g*ExpandToSum[v, x]))^n)^p, x] /; FreeQ[{F, a, b, g, n, p}, x] && LinearQ[v, x] && PowerOfLinearQ
[u, x] &&  !(LinearMatchQ[v, x] && PowerOfLinearMatchQ[u, x]) && IntegerQ[m]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps \begin{align*} \text {integral}& = \int e^{-4 x+e^4 x+e^6 x} \left (1+e^6 x+\left (-4+e^4\right ) x+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx \\ & = \int e^{-4 x+e^4 x+e^6 x} \left (1+\left (-4+e^4+e^6\right ) x+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx \\ & = \int e^{-\left (\left (4-e^4-e^6\right ) x\right )} \left (1-\left (4-e^4-e^6\right ) x-e^4 \log (4)-e^6 \log (4)+\log (256)\right ) \, dx \\ & = -\frac {e^{-\left (\left (4-e^4-e^6\right ) x\right )} \left (1-\left (4-e^4-e^6\right ) x-e^4 \log (4)-e^6 \log (4)+\log (256)\right )}{4-e^4-e^6}-\int e^{\left (-4+e^4+e^6\right ) x} \, dx \\ & = \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 (1-\left (4-e^4-e^6\right ) x-e^4 \log (4)-e^6 \log (4)+\log (256)\right )}{4-e^4-e^6} \\ \end{align*}

Mathematica [B] (verified)

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 \[ \int e^{-4 x+e^4 x+e^6 x} \left (1-4 x+e^4 x+e^6 x+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx=\frac {e^{\left (-4+e^4+e^6\right ) x} \left (\left (-4+e^4+e^6\right ) x-e^4 \log (4)-e^6 \log (4)+\log (256)\right )}{-4+e^4+e^6} \]

[In]

Integrate[E^(-4*x + E^4*x + E^6*x)*(1 - 4*x + E^4*x + E^6*x + (4 - E^4 - E^6)*Log[4]),x]

[Out]

(E^((-4 + E^4 + E^6)*x)*((-4 + E^4 + E^6)*x - E^4*Log[4] - E^6*Log[4] + Log[256]))/(-4 + E^4 + E^6)

Maple [A] (verified)

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	\(\frac {\frac {{\mathrm e}^{6} \left ({\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )-{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )}\right )}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}+\frac {{\mathrm e}^{4} \left ({\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )-{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )}\right )}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}+8 \,{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} \ln \left (2\right )-\frac {4 \left ({\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )-{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )}\right )}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}-2 \,{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} {\mathrm e}^{4} \ln \left (2\right )-2 \,{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )} \ln \left (2\right ) {\mathrm e}^{6}+{\mathrm e}^{x \left ({\mathrm e}^{4}+{\mathrm e}^{6}-4\right )}}{{\mathrm e}^{4}+{\mathrm e}^{6}-4}\)	\(220\)

[In]

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=_RETURNVERB
OSE)

[Out]

(x-2*ln(2))*exp(x*(exp(4)+exp(6)-4))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{-4 x+e^4 x+e^6 x} \left (1-4 x+e^4 x+e^6 x+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx={\left (x - 2 \, \log \left (2\right )\right )} e^{\left (x e^{6} + x e^{4} - 4 \, x\right )} \]

[In]

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=
"fricas")

[Out]

(x - 2*log(2))*e^(x*e^6 + x*e^4 - 4*x)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{-4 x+e^4 x+e^6 x} \left (1-4 x+e^4 x+e^6 x+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx=\left (x - 2 \log {\left (2 \right )}\right ) e^{- 4 x + x e^{4} + x e^{6}} \]

[In]

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)

[Out]

(x - 2*log(2))*exp(-4*x + x*exp(4) + x*exp(6))

Maxima [B] (verification not implemented)

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 \[ \int e^{-4 x+e^4 x+e^6 x} \left (1-4 x+e^4 x+e^6 x+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx=\frac {{\left (x {\left (e^{12} + e^{10} - 4 \, e^{6}\right )} - e^{6}\right )} e^{\left (x e^{6} + x e^{4} - 4 \, x\right )}}{e^{12} + 2 \, e^{10} + e^{8} - 8 \, e^{6} - 8 \, e^{4} + 16} + \frac {{\left (x {\left (e^{10} + e^{8} - 4 \, e^{4}\right )} - e^{4}\right )} e^{\left (x e^{6} + x e^{4} - 4 \, x\right )}}{e^{12} + 2 \, e^{10} + e^{8} - 8 \, e^{6} - 8 \, e^{4} + 16} - \frac {4 \, {\left (x {\left (e^{6} + e^{4} - 4\right )} - 1\right )} e^{\left (x e^{6} + x e^{4} - 4 \, x\right )}}{e^{12} + 2 \, e^{10} + e^{8} - 8 \, e^{6} - 8 \, e^{4} + 16} - \frac {2 \, e^{\left (x e^{6} + x e^{4} - 4 \, x + 6\right )} \log \left (2\right )}{e^{6} + e^{4} - 4} - \frac {2 \, e^{\left (x e^{6} + x e^{4} - 4 \, x + 4\right )} \log \left (2\right )}{e^{6} + e^{4} - 4} + \frac {8 \, e^{\left (x e^{6} + x e^{4} - 4 \, x\right )} \log \left (2\right )}{e^{6} + e^{4} - 4} + \frac {e^{\left (x e^{6} + x e^{4} - 4 \, x\right )}}{e^{6} + e^{4} - 4} \]

[In]

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=
"maxima")

[Out]

(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)

Giac [B] (verification not implemented)

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 \[ \int e^{-4 x+e^4 x+e^6 x} \left (1-4 x+e^4 x+e^6 x+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx=\frac {{\left (x e^{6} + x e^{4} - 2 \, e^{6} \log \left (2\right ) - 2 \, e^{4} \log \left (2\right ) - 4 \, x + 8 \, \log \left (2\right ) - 1\right )} e^{\left (x e^{6} + x e^{4} - 4 \, x + 6\right )}}{e^{12} + 2 \, e^{10} + e^{8} - 8 \, e^{6} - 8 \, e^{4} + 16} + \frac {{\left (x e^{6} + x e^{4} - 2 \, e^{6} \log \left (2\right ) - 2 \, e^{4} \log \left (2\right ) - 4 \, x + 8 \, \log \left (2\right ) - 1\right )} e^{\left (x e^{6} + x e^{4} - 4 \, x + 4\right )}}{e^{12} + 2 \, e^{10} + e^{8} - 8 \, e^{6} - 8 \, e^{4} + 16} - \frac {{\left (4 \, x e^{6} + 4 \, x e^{4} - 8 \, e^{6} \log \left (2\right ) - 8 \, e^{4} \log \left (2\right ) - 16 \, x - e^{6} - e^{4} + 32 \, \log \left (2\right )\right )} e^{\left (x e^{6} + x e^{4} - 4 \, x\right )}}{e^{12} + 2 \, e^{10} + e^{8} - 8 \, e^{6} - 8 \, e^{4} + 16} \]

[In]

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=
"giac")

[Out]

(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)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int e^{-4 x+e^4 x+e^6 x} \left (1-4 x+e^4 x+e^6 x+\left (4-e^4-e^6\right ) \log (4)\right ) \, dx={\mathrm {e}}^{-4\,x}\,{\mathrm {e}}^{x\,{\mathrm {e}}^4}\,{\mathrm {e}}^{x\,{\mathrm {e}}^6}\,\left (x-\ln \left (4\right )\right ) \]

[In]

int(exp(x*exp(4) - 4*x + x*exp(6))*(x*exp(4) - 4*x + x*exp(6) - 2*log(2)*(exp(4) + exp(6) - 4) + 1),x)

[Out]

exp(-4*x)*exp(x*exp(4))*exp(x*exp(6))*(x - log(4))