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

Optimal. Leaf size=31 \[ e^{4^{-e^x-x} \left (1+\frac {1-x}{4}\right )^{e^x+x}} \]

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Rubi [F]  time = 15.32, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 9.52, size = 25, normalized size = 0.81 \begin {gather*} e^{16^{-e^x-x} (5-x)^{e^x+x}} \end {gather*}

Antiderivative was successfully verified.

[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))

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fricas [A]  time = 0.67, size = 11, normalized size = 0.35 \begin {gather*} e^{\left ({\left (-\frac {1}{16} \, x + \frac {5}{16}\right )}^{x + e^{x}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-5)*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)*l
og(5/16-1/16*x)))/(x-5),x, algorithm="fricas")

[Out]

e^((-1/16*x + 5/16)^(x + e^x))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-5)*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)*l
og(5/16-1/16*x)))/(x-5),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.20, size = 12, normalized size = 0.39

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x-5)*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)))/(x-5),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.66, size = 31, normalized size = 1.00 \begin {gather*} e^{\left (e^{\left (-4 \, x \log \relax (2) - 4 \, e^{x} \log \relax (2) + x \log \left (-x + 5\right ) + e^{x} \log \left (-x + 5\right )\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-5)*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)*l
og(5/16-1/16*x)))/(x-5),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.78, size = 11, normalized size = 0.35 \begin {gather*} {\mathrm {e}}^{{\left (\frac {5}{16}-\frac {x}{16}\right )}^{x+{\mathrm {e}}^x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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sympy [A]  time = 11.12, size = 15, normalized size = 0.48 \begin {gather*} e^{e^{\left (x + e^{x}\right ) \log {\left (\frac {5}{16} - \frac {x}{16} \right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-5)*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)))/(x-5),x)

[Out]

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

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