3.20.11 \(\int \frac {2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}} e^{2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}} (-\log (2)+e^{3-x} \log (2))}{16+e^{6-2 x}+8 x+x^2+e^{3-x} (8+2 x)+(8+2 e^{3-x}+2 x) (i \pi +\log (5-\log (5)))+(i \pi +\log (5-\log (5)))^2} \, dx\)

Optimal. Leaf size=28 \[ e^{2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}} \]

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Rubi [F]  time = 34.69, 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 {2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}} e^{2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}} \left (-\log (2)+e^{3-x} \log (2)\right )}{16+e^{6-2 x}+8 x+x^2+e^{3-x} (8+2 x)+\left (8+2 e^{3-x}+2 x\right ) (i \pi +\log (5-\log (5)))+(i \pi +\log (5-\log (5)))^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2^(4 + E^(3 - x) + I*Pi + x + Log[5 - Log[5]])^(-1)*E^2^(4 + E^(3 - x) + I*Pi + x + Log[5 - Log[5]])^(-1)
*(-Log[2] + E^(3 - x)*Log[2]))/(16 + E^(6 - 2*x) + 8*x + x^2 + E^(3 - x)*(8 + 2*x) + (8 + 2*E^(3 - x) + 2*x)*(
I*Pi + Log[5 - Log[5]]) + (I*Pi + Log[5 - Log[5]])^2),x]

[Out]

Log[2]*Defer[Int][(2^(E^(3 - x) + x + 4*(1 + (I*Pi + Log[5 - Log[5]])/4))^(-1)*E^(3 + 2^(4 + E^(3 - x) + I*Pi
+ x + Log[5 - Log[5]])^(-1) + x))/((-4 - I*Pi - x - Log[5 - Log[5]])*(I*E^3 + I*E^x*x + (4*I)*E^x*(1 + (I*Pi +
Log[5 - Log[5]])/4))^2), x] + Log[2]*Defer[Int][(2^(E^(3 - x) + x + 4*(1 + (I*Pi + Log[5 - Log[5]])/4))^(-1)*
E^(2^(4 + E^(3 - x) + I*Pi + x + Log[5 - Log[5]])^(-1) + x))/((4*I - Pi + I*x + I*Log[5 - Log[5]])*(I*E^3 + I*
E^x*x + (4*I)*E^x*(1 + (I*Pi + Log[5 - Log[5]])/4))), x] - Log[2]*Defer[Int][(2^(E^(3 - x) + x + 4*(1 + (I*Pi
+ Log[5 - Log[5]])/4))^(-1)*E^(3 + 2^(4 + E^(3 - x) + I*Pi + x + Log[5 - Log[5]])^(-1) + x))/(I*E^3 - E^x*(Pi
- I*(4 + x + Log[5 - Log[5]])))^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2^{\frac {1}{e^{3-x}+x+4 \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )}} e^{2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}+x} \left (-e^3+e^x\right ) \log (2)}{\left (i e^3-e^x (\pi -i (4+x+\log (5-\log (5))))\right )^2} \, dx\\ &=\log (2) \int \frac {2^{\frac {1}{e^{3-x}+x+4 \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )}} e^{2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}+x} \left (-e^3+e^x\right )}{\left (i e^3-e^x (\pi -i (4+x+\log (5-\log (5))))\right )^2} \, dx\\ &=\log (2) \int \left (\frac {2^{\frac {1}{e^{3-x}+x+4 \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )}} e^{2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}+x}}{(4 i-\pi +i x+i \log (5-\log (5))) \left (i e^3+i e^x x+4 i e^x \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )\right )}+\frac {2^{\frac {1}{e^{3-x}+x+4 \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )}} e^{3+2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}+x} (5 i-\pi +i x+i \log (5-\log (5)))}{(4 i-\pi +i x+i \log (5-\log (5))) \left (e^3+e^x x+4 e^x \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )\right )^2}\right ) \, dx\\ &=\log (2) \int \frac {2^{\frac {1}{e^{3-x}+x+4 \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )}} e^{2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}+x}}{(4 i-\pi +i x+i \log (5-\log (5))) \left (i e^3+i e^x x+4 i e^x \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )\right )} \, dx+\log (2) \int \frac {2^{\frac {1}{e^{3-x}+x+4 \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )}} e^{3+2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}+x} (5 i-\pi +i x+i \log (5-\log (5)))}{(4 i-\pi +i x+i \log (5-\log (5))) \left (e^3+e^x x+4 e^x \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )\right )^2} \, dx\\ &=\log (2) \int \frac {2^{\frac {1}{e^{3-x}+x+4 \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )}} e^{2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}+x}}{(4 i-\pi +i x+i \log (5-\log (5))) \left (i e^3+i e^x x+4 i e^x \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )\right )} \, dx+\log (2) \int \left (-\frac {2^{\frac {1}{e^{3-x}+x+4 \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )}} e^{3+2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}+x}}{\left (i e^3+i e^x x+4 i e^x \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )\right )^2}+\frac {2^{\frac {1}{e^{3-x}+x+4 \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )}} e^{3+2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}+x}}{(-4-i \pi -x-\log (5-\log (5))) \left (i e^3+i e^x x+4 i e^x \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )\right )^2}\right ) \, dx\\ &=-\left (\log (2) \int \frac {2^{\frac {1}{e^{3-x}+x+4 \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )}} e^{3+2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}+x}}{\left (i e^3+i e^x x+4 i e^x \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )\right )^2} \, dx\right )+\log (2) \int \frac {2^{\frac {1}{e^{3-x}+x+4 \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )}} e^{3+2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}+x}}{(-4-i \pi -x-\log (5-\log (5))) \left (i e^3+i e^x x+4 i e^x \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )\right )^2} \, dx+\log (2) \int \frac {2^{\frac {1}{e^{3-x}+x+4 \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )}} e^{2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}+x}}{(4 i-\pi +i x+i \log (5-\log (5))) \left (i e^3+i e^x x+4 i e^x \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )\right )} \, dx\\ &=\log (2) \int \frac {2^{\frac {1}{e^{3-x}+x+4 \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )}} e^{3+2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}+x}}{(-4-i \pi -x-\log (5-\log (5))) \left (i e^3+i e^x x+4 i e^x \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )\right )^2} \, dx+\log (2) \int \frac {2^{\frac {1}{e^{3-x}+x+4 \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )}} e^{2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}+x}}{(4 i-\pi +i x+i \log (5-\log (5))) \left (i e^3+i e^x x+4 i e^x \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )\right )} \, dx-\log (2) \int \frac {2^{\frac {1}{e^{3-x}+x+4 \left (1+\frac {1}{4} (i \pi +\log (5-\log (5)))\right )}} e^{3+2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}+x}}{\left (i e^3-e^x (\pi -i (4+x+\log (5-\log (5))))\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 28, normalized size = 1.00 \begin {gather*} e^{2^{\frac {1}{4+e^{3-x}+i \pi +x+\log (5-\log (5))}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2^(4 + E^(3 - x) + I*Pi + x + Log[5 - Log[5]])^(-1)*E^2^(4 + E^(3 - x) + I*Pi + x + Log[5 - Log[5]]
)^(-1)*(-Log[2] + E^(3 - x)*Log[2]))/(16 + E^(6 - 2*x) + 8*x + x^2 + E^(3 - x)*(8 + 2*x) + (8 + 2*E^(3 - x) +
2*x)*(I*Pi + Log[5 - Log[5]]) + (I*Pi + Log[5 - Log[5]])^2),x]

[Out]

E^2^(4 + E^(3 - x) + I*Pi + x + Log[5 - Log[5]])^(-1)

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fricas [A]  time = 0.72, size = 19, normalized size = 0.68 \begin {gather*} e^{\left (2^{\left (\frac {1}{x + e^{\left (-x + 3\right )} + \log \left (\log \relax (5) - 5\right ) + 4}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2)*log(2)*exp(-x+1)-log(2))*exp(log(2)/(log(log(5)-5)+exp(2)*exp(-x+1)+4+x))*exp(exp(log(2)/(lo
g(log(5)-5)+exp(2)*exp(-x+1)+4+x)))/(log(log(5)-5)^2+(2*exp(2)*exp(-x+1)+2*x+8)*log(log(5)-5)+exp(2)^2*exp(-x+
1)^2+(2*x+8)*exp(2)*exp(-x+1)+x^2+8*x+16),x, algorithm="fricas")

[Out]

e^(2^(1/(x + e^(-x + 3) + log(log(5) - 5) + 4)))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (e^{\left (-x + 3\right )} \log \relax (2) - \log \relax (2)\right )} 2^{\left (\frac {1}{x + e^{\left (-x + 3\right )} + \log \left (\log \relax (5) - 5\right ) + 4}\right )} e^{\left (2^{\left (\frac {1}{x + e^{\left (-x + 3\right )} + \log \left (\log \relax (5) - 5\right ) + 4}\right )}\right )}}{x^{2} + 2 \, {\left (x + 4\right )} e^{\left (-x + 3\right )} + 2 \, {\left (x + e^{\left (-x + 3\right )} + 4\right )} \log \left (\log \relax (5) - 5\right ) + \log \left (\log \relax (5) - 5\right )^{2} + 8 \, x + e^{\left (-2 \, x + 6\right )} + 16}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2)*log(2)*exp(-x+1)-log(2))*exp(log(2)/(log(log(5)-5)+exp(2)*exp(-x+1)+4+x))*exp(exp(log(2)/(lo
g(log(5)-5)+exp(2)*exp(-x+1)+4+x)))/(log(log(5)-5)^2+(2*exp(2)*exp(-x+1)+2*x+8)*log(log(5)-5)+exp(2)^2*exp(-x+
1)^2+(2*x+8)*exp(2)*exp(-x+1)+x^2+8*x+16),x, algorithm="giac")

[Out]

integrate((e^(-x + 3)*log(2) - log(2))*2^(1/(x + e^(-x + 3) + log(log(5) - 5) + 4))*e^(2^(1/(x + e^(-x + 3) +
log(log(5) - 5) + 4)))/(x^2 + 2*(x + 4)*e^(-x + 3) + 2*(x + e^(-x + 3) + 4)*log(log(5) - 5) + log(log(5) - 5)^
2 + 8*x + e^(-2*x + 6) + 16), x)

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maple [A]  time = 0.12, size = 20, normalized size = 0.71

method	result	size
risch	\({\mathrm e}^{2^{\frac {1}{\ln \left (\ln \relax (5)-5\right )+{\mathrm e}^{3-x}+4+x}}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2)*ln(2)*exp(1-x)-ln(2))*exp(ln(2)/(ln(ln(5)-5)+exp(2)*exp(1-x)+4+x))*exp(exp(ln(2)/(ln(ln(5)-5)+exp(
2)*exp(1-x)+4+x)))/(ln(ln(5)-5)^2+(2*exp(2)*exp(1-x)+2*x+8)*ln(ln(5)-5)+exp(2)^2*exp(1-x)^2+(2*x+8)*exp(2)*exp
(1-x)+x^2+8*x+16),x,method=_RETURNVERBOSE)

[Out]

exp(2^(1/(ln(ln(5)-5)+exp(3-x)+4+x)))

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maxima [A]  time = 0.97, size = 22, normalized size = 0.79 \begin {gather*} e^{\left (2^{\frac {e^{x}}{{\left (x + \log \left (\log \relax (5) - 5\right ) + 4\right )} e^{x} + e^{3}}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(2)*log(2)*exp(-x+1)-log(2))*exp(log(2)/(log(log(5)-5)+exp(2)*exp(-x+1)+4+x))*exp(exp(log(2)/(lo
g(log(5)-5)+exp(2)*exp(-x+1)+4+x)))/(log(log(5)-5)^2+(2*exp(2)*exp(-x+1)+2*x+8)*log(log(5)-5)+exp(2)^2*exp(-x+
1)^2+(2*x+8)*exp(2)*exp(-x+1)+x^2+8*x+16),x, algorithm="maxima")

[Out]

e^(2^(e^x/((x + log(log(5) - 5) + 4)*e^x + e^3)))

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mupad [B]  time = 1.94, size = 20, normalized size = 0.71 \begin {gather*} {\mathrm {e}}^{2^{\frac {1}{x+\ln \left (\ln \relax (5)-5\right )+{\mathrm {e}}^{-x}\,{\mathrm {e}}^3+4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(log(2)/(x + log(log(5) - 5) + exp(2)*exp(1 - x) + 4))*exp(exp(log(2)/(x + log(log(5) - 5) + exp(2)*e
xp(1 - x) + 4)))*(log(2) - exp(2)*exp(1 - x)*log(2)))/(8*x + log(log(5) - 5)*(2*x + 2*exp(2)*exp(1 - x) + 8) +
exp(4)*exp(2 - 2*x) + log(log(5) - 5)^2 + x^2 + exp(2)*exp(1 - x)*(2*x + 8) + 16),x)

[Out]

exp(2^(1/(x + log(log(5) - 5) + exp(-x)*exp(3) + 4)))

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sympy [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((exp(2)*ln(2)*exp(-x+1)-ln(2))*exp(ln(2)/(ln(ln(5)-5)+exp(2)*exp(-x+1)+4+x))*exp(exp(ln(2)/(ln(ln(5)
-5)+exp(2)*exp(-x+1)+4+x)))/(ln(ln(5)-5)**2+(2*exp(2)*exp(-x+1)+2*x+8)*ln(ln(5)-5)+exp(2)**2*exp(-x+1)**2+(2*x
+8)*exp(2)*exp(-x+1)+x**2+8*x+16),x)

[Out]

Timed out

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