\(\int (-16+e^{2 x} (32-2 \log (\frac {6561}{625}))+\log (\frac {6561}{625})) \, dx\) [5904]

Optimal result

Integrand size = 20, antiderivative size = 18 \[ \int \left (-16+e^{2 x} \left (32-2 \log \left (\frac {6561}{625}\right )\right )+\log \left (\frac {6561}{625}\right )\right ) \, dx=\left (e^{2 x}-x\right ) \left (16-\log \left (\frac {6561}{625}\right )\right ) \]

[Out]

(exp(x)^2-x)*(16+ln(625/6561))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2225} \[ \int \left (-16+e^{2 x} \left (32-2 \log \left (\frac {6561}{625}\right )\right )+\log \left (\frac {6561}{625}\right )\right ) \, dx=e^{2 x} \left (16-\log \left (\frac {6561}{625}\right )\right )-x \left (16-\log \left (\frac {6561}{625}\right )\right ) \]

[In]

Int[-16 + E^(2*x)*(32 - 2*Log[6561/625]) + Log[6561/625],x]

[Out]

E^(2*x)*(16 - Log[6561/625]) - x*(16 - Log[6561/625])

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}& = -x \left (16-\log \left (\frac {6561}{625}\right )\right )+\left (2 \left (16-\log \left (\frac {6561}{625}\right )\right )\right ) \int e^{2 x} \, dx \\ & = e^{2 x} \left (16-\log \left (\frac {6561}{625}\right )\right )-x \left (16-\log \left (\frac {6561}{625}\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \left (-16+e^{2 x} \left (32-2 \log \left (\frac {6561}{625}\right )\right )+\log \left (\frac {6561}{625}\right )\right ) \, dx=-\left (\left (e^{2 x}-x\right ) \left (-16+\log \left (\frac {6561}{625}\right )\right )\right ) \]

[In]

Integrate[-16 + E^(2*x)*(32 - 2*Log[6561/625]) + Log[6561/625],x]

[Out]

-((E^(2*x) - x)*(-16 + Log[6561/625]))

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(\left (16+\ln \left (\frac {625}{6561}\right )\right ) \left ({\mathrm e}^{2 x}-\ln \left ({\mathrm e}^{x}\right )\right )\)	\(16\)
default	\(-16 x +\frac {\left (2 \ln \left (\frac {625}{6561}\right )+32\right ) {\mathrm e}^{2 x}}{2}-\ln \left (\frac {625}{6561}\right ) x\)	\(22\)
parallelrisch	\(\frac {\left (2 \ln \left (\frac {625}{6561}\right )+32\right ) {\mathrm e}^{2 x}}{2}+\left (-\ln \left (\frac {625}{6561}\right )-16\right ) x\)	\(22\)
parts	\(-16 x +\frac {\left (2 \ln \left (\frac {625}{6561}\right )+32\right ) {\mathrm e}^{2 x}}{2}-\ln \left (\frac {625}{6561}\right ) x\)	\(22\)
norman	\(\left (-4 \ln \left (5\right )+8 \ln \left (3\right )-16\right ) x +\left (4 \ln \left (5\right )-8 \ln \left (3\right )+16\right ) {\mathrm e}^{2 x}\)	\(29\)
risch	\(-8 \,{\mathrm e}^{2 x} \ln \left (3\right )+8 x \ln \left (3\right )+4 \,{\mathrm e}^{2 x} \ln \left (5\right )-4 x \ln \left (5\right )+16 \,{\mathrm e}^{2 x}-16 x\)	\(37\)

[In]

int((2*ln(625/6561)+32)*exp(x)^2-ln(625/6561)-16,x,method=_RETURNVERBOSE)

[Out]

(16+ln(625/6561))*(exp(x)^2-ln(exp(x)))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \left (-16+e^{2 x} \left (32-2 \log \left (\frac {6561}{625}\right )\right )+\log \left (\frac {6561}{625}\right )\right ) \, dx={\left (\log \left (\frac {625}{6561}\right ) + 16\right )} e^{\left (2 \, x\right )} - x \log \left (\frac {625}{6561}\right ) - 16 \, x \]

[In]

integrate((2*log(625/6561)+32)*exp(x)^2-log(625/6561)-16,x, algorithm="fricas")

[Out]

(log(625/6561) + 16)*e^(2*x) - x*log(625/6561) - 16*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).

Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.61 \[ \int \left (-16+e^{2 x} \left (32-2 \log \left (\frac {6561}{625}\right )\right )+\log \left (\frac {6561}{625}\right )\right ) \, dx=x \left (-16 - 4 \log {\left (5 \right )} + 8 \log {\left (3 \right )}\right ) + \left (- 8 \log {\left (3 \right )} + 4 \log {\left (5 \right )} + 16\right ) e^{2 x} \]

[In]

integrate((2*ln(625/6561)+32)*exp(x)**2-ln(625/6561)-16,x)

[Out]

x*(-16 - 4*log(5) + 8*log(3)) + (-8*log(3) + 4*log(5) + 16)*exp(2*x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \left (-16+e^{2 x} \left (32-2 \log \left (\frac {6561}{625}\right )\right )+\log \left (\frac {6561}{625}\right )\right ) \, dx={\left (\log \left (\frac {625}{6561}\right ) + 16\right )} e^{\left (2 \, x\right )} - x \log \left (\frac {625}{6561}\right ) - 16 \, x \]

[In]

integrate((2*log(625/6561)+32)*exp(x)^2-log(625/6561)-16,x, algorithm="maxima")

[Out]

(log(625/6561) + 16)*e^(2*x) - x*log(625/6561) - 16*x

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \left (-16+e^{2 x} \left (32-2 \log \left (\frac {6561}{625}\right )\right )+\log \left (\frac {6561}{625}\right )\right ) \, dx={\left (\log \left (\frac {625}{6561}\right ) + 16\right )} e^{\left (2 \, x\right )} - x \log \left (\frac {625}{6561}\right ) - 16 \, x \]

[In]

integrate((2*log(625/6561)+32)*exp(x)^2-log(625/6561)-16,x, algorithm="giac")

[Out]

(log(625/6561) + 16)*e^(2*x) - x*log(625/6561) - 16*x

Mupad [B] (verification not implemented)

Time = 11.92 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \left (-16+e^{2 x} \left (32-2 \log \left (\frac {6561}{625}\right )\right )+\log \left (\frac {6561}{625}\right )\right ) \, dx={\mathrm {e}}^{2\,x}\,\left (\ln \left (\frac {625}{6561}\right )+16\right )+x\,\left (\ln \left (\frac {6561}{625}\right )-16\right ) \]

[In]

int(exp(2*x)*(2*log(625/6561) + 32) - log(625/6561) - 16,x)

[Out]

exp(2*x)*(log(625/6561) + 16) + x*(log(6561/625) - 16)