\(\int \frac {16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{e^x+x} (e^x+x+(-5+e^x (-5+x)+x) \log (\frac {5-x}{16}))}{-5+x} \, dx\) [5327]

Optimal result

Integrand size = 79, antiderivative size = 31 \[ \int \frac {16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{e^x+x} \left (e^x+x+\left (-5+e^x (-5+x)+x\right ) \log \left (\frac {5-x}{16}\right )\right )}{-5+x} \, dx=e^{4^{-e^x-x} \left (1+\frac {1-x}{4}\right )^{e^x+x}} \]

[Out]

exp(exp((exp(x)+x)*ln(5/16-1/16*x)))

Rubi [F]

\[ \int \frac {16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{e^x+x} \left (e^x+x+\left (-5+e^x (-5+x)+x\right ) \log \left (\frac {5-x}{16}\right )\right )}{-5+x} \, dx=\int \frac {16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{e^x+x} \left (e^x+x+\left (-5+e^x (-5+x)+x\right ) \log \left (\frac {5-x}{16}\right )\right )}{-5+x} \, dx \]

[In]

Int[(16^(-E^x - x)*E^(16^(-E^x - x)*(5 - x)^(E^x + x))*(5 - x)^(E^x + x)*(E^x + x + (-5 + E^x*(-5 + x) + x)*Lo
g[(5 - x)/16]))/(-5 + x),x]

[Out]

-Defer[Int][16^(-E^x - x)*E^(16^(-E^x - x)*(5 - x)^(E^x + x) + x)*(5 - x)^(-1 + E^x + x), x] + Log[5/16 - x/16
]*Defer[Int][16^(-E^x - x)*E^(16^(-E^x - x)*(5 - x)^(E^x + x))*(5 - x)^(E^x + x), x] + Log[5/16 - x/16]*Defer[
Int][16^(-E^x - x)*E^(16^(-E^x - x)*(5 - x)^(E^x + x) + x)*(5 - x)^(E^x + x), x] - Defer[Int][16^(-E^x - x)*E^
(16^(-E^x - x)*(5 - x)^(E^x + x))*(5 - x)^(-1 + E^x + x)*x, x] - Defer[Int][Defer[Int][16^(-E^x - x)*E^(16^(-E
^x - x)*(5 - x)^(E^x + x))*(5 - x)^(E^x + x), x]/(-5 + x), x] - Defer[Int][Defer[Int][16^(-E^x - x)*E^(16^(-E^
x - x)*(5 - x)^(E^x + x) + x)*(5 - x)^(E^x + x), x]/(-5 + x), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}+x} (5-x)^{-1+e^x+x} \left (1-5 \log \left (\frac {5-x}{16}\right )+x \log \left (\frac {5-x}{16}\right )\right )-16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{-1+e^x+x} \left (x-5 \log \left (\frac {5-x}{16}\right )+x \log \left (\frac {5-x}{16}\right )\right )\right ) \, dx \\ & = -\int 16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}+x} (5-x)^{-1+e^x+x} \left (1-5 \log \left (\frac {5-x}{16}\right )+x \log \left (\frac {5-x}{16}\right )\right ) \, dx-\int 16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{-1+e^x+x} \left (x-5 \log \left (\frac {5-x}{16}\right )+x \log \left (\frac {5-x}{16}\right )\right ) \, dx \\ & = -\int 16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}+x} (5-x)^{-1+e^x+x} \left (1+(-5+x) \log \left (\frac {5-x}{16}\right )\right ) \, dx-\int 16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{-1+e^x+x} \left (x+(-5+x) \log \left (\frac {5-x}{16}\right )\right ) \, dx \\ & = -\int \left (16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{-1+e^x+x} x-16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{e^x+x} \log \left (\frac {5}{16}-\frac {x}{16}\right )\right ) \, dx-\int \left (16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}+x} (5-x)^{-1+e^x+x}-16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}+x} (5-x)^{e^x+x} \log \left (\frac {5}{16}-\frac {x}{16}\right )\right ) \, dx \\ & = -\int 16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}+x} (5-x)^{-1+e^x+x} \, dx-\int 16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{-1+e^x+x} x \, dx+\int 16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{e^x+x} \log \left (\frac {5}{16}-\frac {x}{16}\right ) \, dx+\int 16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}+x} (5-x)^{e^x+x} \log \left (\frac {5}{16}-\frac {x}{16}\right ) \, dx \\ & = \log \left (\frac {5}{16}-\frac {x}{16}\right ) \int 16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{e^x+x} \, dx+\log \left (\frac {5}{16}-\frac {x}{16}\right ) \int 16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}+x} (5-x)^{e^x+x} \, dx-\int 16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}+x} (5-x)^{-1+e^x+x} \, dx-\int 16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{-1+e^x+x} x \, dx-\int \frac {\int 16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{e^x+x} \, dx}{-5+x} \, dx-\int \frac {\int 16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}+x} (5-x)^{e^x+x} \, dx}{-5+x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 10.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{e^x+x} \left (e^x+x+\left (-5+e^x (-5+x)+x\right ) \log \left (\frac {5-x}{16}\right )\right )}{-5+x} \, dx=e^{16^{-e^x-x} (5-x)^{e^x+x}} \]

[In]

Integrate[(16^(-E^x - x)*E^(16^(-E^x - x)*(5 - x)^(E^x + x))*(5 - x)^(E^x + x)*(E^x + x + (-5 + E^x*(-5 + x) +
x)*Log[(5 - x)/16]))/(-5 + x),x]

[Out]

E^(16^(-E^x - x)*(5 - x)^(E^x + x))

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.39

method	result	size
risch	\({\mathrm e}^{\left (\frac {5}{16}-\frac {x}{16}\right )^{{\mathrm e}^{x}+x}}\)	\(12\)
parallelrisch	\({\mathrm e}^{{\mathrm e}^{\left ({\mathrm e}^{x}+x \right ) \ln \left (\frac {5}{16}-\frac {x}{16}\right )}}\)	\(14\)

[In]

int((((-5+x)*exp(x)+x-5)*ln(5/16-1/16*x)+exp(x)+x)*exp((exp(x)+x)*ln(5/16-1/16*x))*exp(exp((exp(x)+x)*ln(5/16-
1/16*x)))/(-5+x),x,method=_RETURNVERBOSE)

[Out]

exp((5/16-1/16*x)^(exp(x)+x))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.35 \[ \int \frac {16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{e^x+x} \left (e^x+x+\left (-5+e^x (-5+x)+x\right ) \log \left (\frac {5-x}{16}\right )\right )}{-5+x} \, dx=e^{\left ({\left (-\frac {1}{16} \, x + \frac {5}{16}\right )}^{x + e^{x}}\right )} \]

[In]

integrate((((-5+x)*exp(x)+x-5)*log(5/16-1/16*x)+exp(x)+x)*exp((exp(x)+x)*log(5/16-1/16*x))*exp(exp((exp(x)+x)*
log(5/16-1/16*x)))/(-5+x),x, algorithm="fricas")

[Out]

e^((-1/16*x + 5/16)^(x + e^x))

Sympy [A] (verification not implemented)

Time = 13.47 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.48 \[ \int \frac {16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{e^x+x} \left (e^x+x+\left (-5+e^x (-5+x)+x\right ) \log \left (\frac {5-x}{16}\right )\right )}{-5+x} \, dx=e^{e^{\left (x + e^{x}\right ) \log {\left (\frac {5}{16} - \frac {x}{16} \right )}}} \]

[In]

integrate((((-5+x)*exp(x)+x-5)*ln(5/16-1/16*x)+exp(x)+x)*exp((exp(x)+x)*ln(5/16-1/16*x))*exp(exp((exp(x)+x)*ln
(5/16-1/16*x)))/(-5+x),x)

[Out]

exp(exp((x + exp(x))*log(5/16 - x/16)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (11) = 22\).

Time = 0.44 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{e^x+x} \left (e^x+x+\left (-5+e^x (-5+x)+x\right ) \log \left (\frac {5-x}{16}\right )\right )}{-5+x} \, dx=e^{\left (e^{\left (-4 \, x \log \left (2\right ) - 4 \, e^{x} \log \left (2\right ) + x \log \left (-x + 5\right ) + e^{x} \log \left (-x + 5\right )\right )}\right )} \]

[In]

integrate((((-5+x)*exp(x)+x-5)*log(5/16-1/16*x)+exp(x)+x)*exp((exp(x)+x)*log(5/16-1/16*x))*exp(exp((exp(x)+x)*
log(5/16-1/16*x)))/(-5+x),x, algorithm="maxima")

[Out]

e^(e^(-4*x*log(2) - 4*e^x*log(2) + x*log(-x + 5) + e^x*log(-x + 5)))

Giac [F]

\[ \int \frac {16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{e^x+x} \left (e^x+x+\left (-5+e^x (-5+x)+x\right ) \log \left (\frac {5-x}{16}\right )\right )}{-5+x} \, dx=\int { \frac {{\left ({\left ({\left (x - 5\right )} e^{x} + x - 5\right )} \log \left (-\frac {1}{16} \, x + \frac {5}{16}\right ) + x + e^{x}\right )} {\left (-\frac {1}{16} \, x + \frac {5}{16}\right )}^{x + e^{x}} e^{\left ({\left (-\frac {1}{16} \, x + \frac {5}{16}\right )}^{x + e^{x}}\right )}}{x - 5} \,d x } \]

[In]

integrate((((-5+x)*exp(x)+x-5)*log(5/16-1/16*x)+exp(x)+x)*exp((exp(x)+x)*log(5/16-1/16*x))*exp(exp((exp(x)+x)*
log(5/16-1/16*x)))/(-5+x),x, algorithm="giac")

[Out]

integrate((((x - 5)*e^x + x - 5)*log(-1/16*x + 5/16) + x + e^x)*(-1/16*x + 5/16)^(x + e^x)*e^((-1/16*x + 5/16)
^(x + e^x))/(x - 5), x)

Mupad [B] (verification not implemented)

Time = 11.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.35 \[ \int \frac {16^{-e^x-x} e^{16^{-e^x-x} (5-x)^{e^x+x}} (5-x)^{e^x+x} \left (e^x+x+\left (-5+e^x (-5+x)+x\right ) \log \left (\frac {5-x}{16}\right )\right )}{-5+x} \, dx={\mathrm {e}}^{{\left (\frac {5}{16}-\frac {x}{16}\right )}^{x+{\mathrm {e}}^x}} \]

[In]

int((exp(log(5/16 - x/16)*(x + exp(x)))*exp(exp(log(5/16 - x/16)*(x + exp(x))))*(x + exp(x) + log(5/16 - x/16)
*(x + exp(x)*(x - 5) - 5)))/(x - 5),x)

[Out]

exp((5/16 - x/16)^(x + exp(x)))