\(\int \frac {36 e^{1+\log ^2(\frac {1}{6} (24+6 e^x-x^2))} (e^{e^{2 x}} (-12 e^x+4 x+e^{2 x} (48+12 e^x-2 x^2))+e^{e^{2 x}} (12 e^x-4 x) \log (\frac {1}{6} (24+6 e^x-x^2)))}{(24+6 e^x-x^2)^3} \, dx\) [7833]

Optimal result

Integrand size = 109, antiderivative size = 25 \[ \int \frac {36 e^{1+\log ^2\left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )} \left (e^{e^{2 x}} \left (-12 e^x+4 x+e^{2 x} \left (48+12 e^x-2 x^2\right )\right )+e^{e^{2 x}} \left (12 e^x-4 x\right ) \log \left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )\right )}{\left (24+6 e^x-x^2\right )^3} \, dx=e^{e^{2 x}+\left (-1+\log \left (4+e^x-\frac {x^2}{6}\right )\right )^2} \]

[Out]

exp((ln(exp(x)-1/6*x^2+4)-1)^2)*exp(exp(2*x))

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {12, 2326} \[ \int \frac {36 e^{1+\log ^2\left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )} \left (e^{e^{2 x}} \left (-12 e^x+4 x+e^{2 x} \left (48+12 e^x-2 x^2\right )\right )+e^{e^{2 x}} \left (12 e^x-4 x\right ) \log \left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )\right )}{\left (24+6 e^x-x^2\right )^3} \, dx=\frac {36 e^{\log ^2\left (\frac {1}{6} \left (-x^2+6 e^x+24\right )\right )+e^{2 x}+1}}{\left (-x^2+6 e^x+24\right )^2} \]

[In]

Int[(36*E^(1 + Log[(24 + 6*E^x - x^2)/6]^2)*(E^E^(2*x)*(-12*E^x + 4*x + E^(2*x)*(48 + 12*E^x - 2*x^2)) + E^E^(
2*x)*(12*E^x - 4*x)*Log[(24 + 6*E^x - x^2)/6]))/(24 + 6*E^x - x^2)^3,x]

[Out]

(36*E^(1 + E^(2*x) + Log[(24 + 6*E^x - x^2)/6]^2))/(24 + 6*E^x - x^2)^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2326

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]

Rubi steps \begin{align*} \text {integral}& = 36 \int \frac {e^{1+\log ^2\left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )} \left (e^{e^{2 x}} \left (-12 e^x+4 x+e^{2 x} \left (48+12 e^x-2 x^2\right )\right )+e^{e^{2 x}} \left (12 e^x-4 x\right ) \log \left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )\right )}{\left (24+6 e^x-x^2\right )^3} \, dx \\ & = \frac {36 e^{1+e^{2 x}+\log ^2\left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )}}{\left (24+6 e^x-x^2\right )^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.60 \[ \int \frac {36 e^{1+\log ^2\left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )} \left (e^{e^{2 x}} \left (-12 e^x+4 x+e^{2 x} \left (48+12 e^x-2 x^2\right )\right )+e^{e^{2 x}} \left (12 e^x-4 x\right ) \log \left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )\right )}{\left (24+6 e^x-x^2\right )^3} \, dx=\frac {36 e^{1+e^{2 x}+\log ^2\left (4+e^x-\frac {x^2}{6}\right )}}{\left (24+6 e^x-x^2\right )^2} \]

[In]

Integrate[(36*E^(1 + Log[(24 + 6*E^x - x^2)/6]^2)*(E^E^(2*x)*(-12*E^x + 4*x + E^(2*x)*(48 + 12*E^x - 2*x^2)) +
E^E^(2*x)*(12*E^x - 4*x)*Log[(24 + 6*E^x - x^2)/6]))/(24 + 6*E^x - x^2)^3,x]

[Out]

(36*E^(1 + E^(2*x) + Log[4 + E^x - x^2/6]^2))/(24 + 6*E^x - x^2)^2

Maple [A] (verified)

Time = 18.60 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28

method	result	size
risch	\(\frac {{\mathrm e}^{{\mathrm e}^{2 x}+\ln \left ({\mathrm e}^{x}-\frac {x^{2}}{6}+4\right )^{2}+1}}{\left ({\mathrm e}^{x}-\frac {x^{2}}{6}+4\right )^{2}}\)	\(32\)
parallelrisch	\({\mathrm e}^{{\mathrm e}^{2 x}} {\mathrm e}^{\ln \left ({\mathrm e}^{x}-\frac {x^{2}}{6}+4\right )^{2}-2 \ln \left ({\mathrm e}^{x}-\frac {x^{2}}{6}+4\right )+1}\)	\(34\)

[In]

int(((12*exp(x)-4*x)*exp(exp(2*x))*ln(exp(x)-1/6*x^2+4)+((12*exp(x)-2*x^2+48)*exp(2*x)-12*exp(x)+4*x)*exp(exp(
2*x)))*exp(ln(exp(x)-1/6*x^2+4)^2-2*ln(exp(x)-1/6*x^2+4)+1)/(6*exp(x)-x^2+24),x,method=_RETURNVERBOSE)

[Out]

1/(exp(x)-1/6*x^2+4)^2*exp(exp(2*x)+ln(exp(x)-1/6*x^2+4)^2+1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {36 e^{1+\log ^2\left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )} \left (e^{e^{2 x}} \left (-12 e^x+4 x+e^{2 x} \left (48+12 e^x-2 x^2\right )\right )+e^{e^{2 x}} \left (12 e^x-4 x\right ) \log \left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )\right )}{\left (24+6 e^x-x^2\right )^3} \, dx=e^{\left (\log \left (-\frac {1}{6} \, x^{2} + e^{x} + 4\right )^{2} + e^{\left (2 \, x\right )} - 2 \, \log \left (-\frac {1}{6} \, x^{2} + e^{x} + 4\right ) + 1\right )} \]

[In]

integrate(((12*exp(x)-4*x)*exp(exp(2*x))*log(exp(x)-1/6*x^2+4)+((12*exp(x)-2*x^2+48)*exp(2*x)-12*exp(x)+4*x)*e
xp(exp(2*x)))*exp(log(exp(x)-1/6*x^2+4)^2-2*log(exp(x)-1/6*x^2+4)+1)/(6*exp(x)-x^2+24),x, algorithm="fricas")

[Out]

e^(log(-1/6*x^2 + e^x + 4)^2 + e^(2*x) - 2*log(-1/6*x^2 + e^x + 4) + 1)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (22) = 44\).

Time = 1.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16 \[ \int \frac {36 e^{1+\log ^2\left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )} \left (e^{e^{2 x}} \left (-12 e^x+4 x+e^{2 x} \left (48+12 e^x-2 x^2\right )\right )+e^{e^{2 x}} \left (12 e^x-4 x\right ) \log \left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )\right )}{\left (24+6 e^x-x^2\right )^3} \, dx=\frac {36 e^{\log {\left (- \frac {x^{2}}{6} + e^{x} + 4 \right )}^{2} + 1} e^{e^{2 x}}}{x^{4} - 12 x^{2} e^{x} - 48 x^{2} + 36 e^{2 x} + 288 e^{x} + 576} \]

[In]

integrate(((12*exp(x)-4*x)*exp(exp(2*x))*ln(exp(x)-1/6*x**2+4)+((12*exp(x)-2*x**2+48)*exp(2*x)-12*exp(x)+4*x)*
exp(exp(2*x)))*exp(ln(exp(x)-1/6*x**2+4)**2-2*ln(exp(x)-1/6*x**2+4)+1)/(6*exp(x)-x**2+24),x)

[Out]

36*exp(log(-x**2/6 + exp(x) + 4)**2 + 1)*exp(exp(2*x))/(x**4 - 12*x**2*exp(x) - 48*x**2 + 36*exp(2*x) + 288*ex
p(x) + 576)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (20) = 40\).

Time = 0.43 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.92 \[ \int \frac {36 e^{1+\log ^2\left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )} \left (e^{e^{2 x}} \left (-12 e^x+4 x+e^{2 x} \left (48+12 e^x-2 x^2\right )\right )+e^{e^{2 x}} \left (12 e^x-4 x\right ) \log \left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )\right )}{\left (24+6 e^x-x^2\right )^3} \, dx=\frac {9 \cdot 2^{2 \, \log \left (3\right ) + 2} e^{\left (\log \left (3\right )^{2} + \log \left (2\right )^{2} - 2 \, \log \left (3\right ) \log \left (-x^{2} + 6 \, e^{x} + 24\right ) - 2 \, \log \left (2\right ) \log \left (-x^{2} + 6 \, e^{x} + 24\right ) + \log \left (-x^{2} + 6 \, e^{x} + 24\right )^{2} + e^{\left (2 \, x\right )} + 1\right )}}{x^{4} - 48 \, x^{2} - 12 \, {\left (x^{2} - 24\right )} e^{x} + 36 \, e^{\left (2 \, x\right )} + 576} \]

[In]

integrate(((12*exp(x)-4*x)*exp(exp(2*x))*log(exp(x)-1/6*x^2+4)+((12*exp(x)-2*x^2+48)*exp(2*x)-12*exp(x)+4*x)*e
xp(exp(2*x)))*exp(log(exp(x)-1/6*x^2+4)^2-2*log(exp(x)-1/6*x^2+4)+1)/(6*exp(x)-x^2+24),x, algorithm="maxima")

[Out]

9*2^(2*log(3) + 2)*e^(log(3)^2 + log(2)^2 - 2*log(3)*log(-x^2 + 6*e^x + 24) - 2*log(2)*log(-x^2 + 6*e^x + 24)
+ log(-x^2 + 6*e^x + 24)^2 + e^(2*x) + 1)/(x^4 - 48*x^2 - 12*(x^2 - 24)*e^x + 36*e^(2*x) + 576)

Giac [F]

\[ \int \frac {36 e^{1+\log ^2\left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )} \left (e^{e^{2 x}} \left (-12 e^x+4 x+e^{2 x} \left (48+12 e^x-2 x^2\right )\right )+e^{e^{2 x}} \left (12 e^x-4 x\right ) \log \left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )\right )}{\left (24+6 e^x-x^2\right )^3} \, dx=\int { \frac {2 \, {\left (2 \, {\left (x - 3 \, e^{x}\right )} e^{\left (e^{\left (2 \, x\right )}\right )} \log \left (-\frac {1}{6} \, x^{2} + e^{x} + 4\right ) + {\left ({\left (x^{2} - 6 \, e^{x} - 24\right )} e^{\left (2 \, x\right )} - 2 \, x + 6 \, e^{x}\right )} e^{\left (e^{\left (2 \, x\right )}\right )}\right )} e^{\left (\log \left (-\frac {1}{6} \, x^{2} + e^{x} + 4\right )^{2} - 2 \, \log \left (-\frac {1}{6} \, x^{2} + e^{x} + 4\right ) + 1\right )}}{x^{2} - 6 \, e^{x} - 24} \,d x } \]

[In]

integrate(((12*exp(x)-4*x)*exp(exp(2*x))*log(exp(x)-1/6*x^2+4)+((12*exp(x)-2*x^2+48)*exp(2*x)-12*exp(x)+4*x)*e
xp(exp(2*x)))*exp(log(exp(x)-1/6*x^2+4)^2-2*log(exp(x)-1/6*x^2+4)+1)/(6*exp(x)-x^2+24),x, algorithm="giac")

[Out]

integrate(2*(2*(x - 3*e^x)*e^(e^(2*x))*log(-1/6*x^2 + e^x + 4) + ((x^2 - 6*e^x - 24)*e^(2*x) - 2*x + 6*e^x)*e^
(e^(2*x)))*e^(log(-1/6*x^2 + e^x + 4)^2 - 2*log(-1/6*x^2 + e^x + 4) + 1)/(x^2 - 6*e^x - 24), x)

Mupad [B] (verification not implemented)

Time = 14.14 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.00 \[ \int \frac {36 e^{1+\log ^2\left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )} \left (e^{e^{2 x}} \left (-12 e^x+4 x+e^{2 x} \left (48+12 e^x-2 x^2\right )\right )+e^{e^{2 x}} \left (12 e^x-4 x\right ) \log \left (\frac {1}{6} \left (24+6 e^x-x^2\right )\right )\right )}{\left (24+6 e^x-x^2\right )^3} \, dx=\frac {\mathrm {e}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{{\ln \left ({\mathrm {e}}^x-\frac {x^2}{6}+4\right )}^2}}{{\mathrm {e}}^{2\,x}+8\,{\mathrm {e}}^x-\frac {x^2\,{\mathrm {e}}^x}{3}-\frac {4\,x^2}{3}+\frac {x^4}{36}+16} \]

[In]

int((exp(log(exp(x) - x^2/6 + 4)^2 - 2*log(exp(x) - x^2/6 + 4) + 1)*(exp(exp(2*x))*(4*x - 12*exp(x) + exp(2*x)
*(12*exp(x) - 2*x^2 + 48)) - exp(exp(2*x))*log(exp(x) - x^2/6 + 4)*(4*x - 12*exp(x))))/(6*exp(x) - x^2 + 24),x
)

[Out]

(exp(1)*exp(exp(2*x))*exp(log(exp(x) - x^2/6 + 4)^2))/(exp(2*x) + 8*exp(x) - (x^2*exp(x))/3 - (4*x^2)/3 + x^4/
36 + 16)