3.53.52 \(\int \frac {4^{-5+e^{5 e x}} (e^{-\frac {-5+x^2}{x}})^{-5+e^{5 e x}} (25+5 x^2+e^{5 e x} (-5-x^2)+5 e^{1+5 e x} x^2 \log (4 e^{-\frac {-5+x^2}{x}}))}{x^2} \, dx\)

Optimal. Leaf size=31 \[ 4^{-5+e^{5 e x}} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \]

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Rubi [F]  time = 5.94, 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 {4^{-5+e^{5 e x}} \left (e^{-\frac {-5+x^2}{x}}\right )^{-5+e^{5 e x}} \left (25+5 x^2+e^{5 e x} \left (-5-x^2\right )+5 e^{1+5 e x} x^2 \log \left (4 e^{-\frac {-5+x^2}{x}}\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4^(-5 + E^(5*E*x))*(E^(-((-5 + x^2)/x)))^(-5 + E^(5*E*x))*(25 + 5*x^2 + E^(5*E*x)*(-5 - x^2) + 5*E^(1 + 5
*E*x)*x^2*Log[4/E^((-5 + x^2)/x)]))/x^2,x]

[Out]

5*Defer[Int][4^(-5 + E^(5*E*x))*(E^(5/x - x))^(-5 + E^(5*E*x)), x] - Defer[Int][4^(-5 + E^(5*E*x))*E^(5*E*x)*(
E^(5/x - x))^(-5 + E^(5*E*x)), x] + 5*Log[4*E^(5/x - x)]*Defer[Int][4^(-5 + E^(5*E*x))*E^(1 + 5*E*x)*(E^(5/x -
x))^(-5 + E^(5*E*x)), x] + 25*Defer[Int][(4^(-5 + E^(5*E*x))*(E^(5/x - x))^(-5 + E^(5*E*x)))/x^2, x] - 5*Defe
r[Int][(4^(-5 + E^(5*E*x))*E^(5*E*x)*(E^(5/x - x))^(-5 + E^(5*E*x)))/x^2, x] + 5*Defer[Int][Defer[Int][4^(-5 +
E^(5*E*x))*E^(1 + 5*E*x)*(E^(5/x - x))^(-5 + E^(5*E*x)), x], x] + 25*Defer[Int][Defer[Int][4^(-5 + E^(5*E*x))
*E^(1 + 5*E*x)*(E^(5/x - x))^(-5 + E^(5*E*x)), x]/x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4^{-5+e^{5 e x}} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \left (25+5 x^2+e^{5 e x} \left (-5-x^2\right )+5 e^{1+5 e x} x^2 \log \left (4 e^{-\frac {-5+x^2}{x}}\right )\right )}{x^2} \, dx\\ &=\int \left (\frac {5\ 4^{-5+e^{5 e x}} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \left (5+x^2\right )}{x^2}+\frac {4^{-5+e^{5 e x}} e^{5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \left (-5-x^2+5 e x^2 \log \left (4 e^{\frac {5}{x}-x}\right )\right )}{x^2}\right ) \, dx\\ &=5 \int \frac {4^{-5+e^{5 e x}} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \left (5+x^2\right )}{x^2} \, dx+\int \frac {4^{-5+e^{5 e x}} e^{5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \left (-5-x^2+5 e x^2 \log \left (4 e^{\frac {5}{x}-x}\right )\right )}{x^2} \, dx\\ &=5 \int \left (4^{-5+e^{5 e x}} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}}+\frac {5\ 4^{-5+e^{5 e x}} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}}}{x^2}\right ) \, dx+\int \left (\frac {4^{-5+e^{5 e x}} e^{5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \left (-5-x^2\right )}{x^2}+5\ 4^{-5+e^{5 e x}} e^{1+5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \log \left (4 e^{\frac {5}{x}-x}\right )\right ) \, dx\\ &=5 \int 4^{-5+e^{5 e x}} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \, dx+5 \int 4^{-5+e^{5 e x}} e^{1+5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \log \left (4 e^{\frac {5}{x}-x}\right ) \, dx+25 \int \frac {4^{-5+e^{5 e x}} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}}}{x^2} \, dx+\int \frac {4^{-5+e^{5 e x}} e^{5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \left (-5-x^2\right )}{x^2} \, dx\\ &=5 \int 4^{-5+e^{5 e x}} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \, dx-5 \int \left (-1-\frac {5}{x^2}\right ) \int 4^{-5+e^{5 e x}} e^{1+5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \, dx \, dx+25 \int \frac {4^{-5+e^{5 e x}} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}}}{x^2} \, dx+\left (5 \log \left (4 e^{\frac {5}{x}-x}\right )\right ) \int 4^{-5+e^{5 e x}} e^{1+5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \, dx+\int \left (-4^{-5+e^{5 e x}} e^{5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}}-\frac {5\ 4^{-5+e^{5 e x}} e^{5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}}}{x^2}\right ) \, dx\\ &=5 \int 4^{-5+e^{5 e x}} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \, dx-5 \int \frac {4^{-5+e^{5 e x}} e^{5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}}}{x^2} \, dx-5 \int \left (-\int 4^{-5+e^{5 e x}} e^{1+5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \, dx-\frac {5 \int 4^{-5+e^{5 e x}} e^{1+5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \, dx}{x^2}\right ) \, dx+25 \int \frac {4^{-5+e^{5 e x}} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}}}{x^2} \, dx+\left (5 \log \left (4 e^{\frac {5}{x}-x}\right )\right ) \int 4^{-5+e^{5 e x}} e^{1+5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \, dx-\int 4^{-5+e^{5 e x}} e^{5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \, dx\\ &=5 \int 4^{-5+e^{5 e x}} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \, dx-5 \int \frac {4^{-5+e^{5 e x}} e^{5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}}}{x^2} \, dx+5 \int \left (\int 4^{-5+e^{5 e x}} e^{1+5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \, dx\right ) \, dx+25 \int \frac {4^{-5+e^{5 e x}} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}}}{x^2} \, dx+25 \int \frac {\int 4^{-5+e^{5 e x}} e^{1+5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \, dx}{x^2} \, dx+\left (5 \log \left (4 e^{\frac {5}{x}-x}\right )\right ) \int 4^{-5+e^{5 e x}} e^{1+5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \, dx-\int 4^{-5+e^{5 e x}} e^{5 e x} \left (e^{\frac {5}{x}-x}\right )^{-5+e^{5 e x}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.74, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4^{-5+e^{5 e x}} \left (e^{-\frac {-5+x^2}{x}}\right )^{-5+e^{5 e x}} \left (25+5 x^2+e^{5 e x} \left (-5-x^2\right )+5 e^{1+5 e x} x^2 \log \left (4 e^{-\frac {-5+x^2}{x}}\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(4^(-5 + E^(5*E*x))*(E^(-((-5 + x^2)/x)))^(-5 + E^(5*E*x))*(25 + 5*x^2 + E^(5*E*x)*(-5 - x^2) + 5*E^
(1 + 5*E*x)*x^2*Log[4/E^((-5 + x^2)/x)]))/x^2,x]

[Out]

Integrate[(4^(-5 + E^(5*E*x))*(E^(-((-5 + x^2)/x)))^(-5 + E^(5*E*x))*(25 + 5*x^2 + E^(5*E*x)*(-5 - x^2) + 5*E^
(1 + 5*E*x)*x^2*Log[4/E^((-5 + x^2)/x)]))/x^2, x]

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fricas [B]  time = 0.49, size = 44, normalized size = 1.42 \begin {gather*} e^{\left (-\frac {{\left (10 \, x e \log \relax (2) - 5 \, {\left (x^{2} - 5\right )} e + {\left (x^{2} - 2 \, x \log \relax (2) - 5\right )} e^{\left (5 \, x e + 1\right )}\right )} e^{\left (-1\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2*exp(1)*exp(5*x*exp(1))*log(4/exp((x^2-5)/x))+(-x^2-5)*exp(5*x*exp(1))+5*x^2+25)*exp((exp(5*x*
exp(1))-5)*log(4/exp((x^2-5)/x)))/x^2,x, algorithm="fricas")

[Out]

e^(-(10*x*e*log(2) - 5*(x^2 - 5)*e + (x^2 - 2*x*log(2) - 5)*e^(5*x*e + 1))*e^(-1)/x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (5 \, x^{2} e^{\left (5 \, x e + 1\right )} \log \left (4 \, e^{\left (-\frac {x^{2} - 5}{x}\right )}\right ) + 5 \, x^{2} - {\left (x^{2} + 5\right )} e^{\left (5 \, x e\right )} + 25\right )} \left (4 \, e^{\left (-\frac {x^{2} - 5}{x}\right )}\right )^{e^{\left (5 \, x e\right )} - 5}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2*exp(1)*exp(5*x*exp(1))*log(4/exp((x^2-5)/x))+(-x^2-5)*exp(5*x*exp(1))+5*x^2+25)*exp((exp(5*x*
exp(1))-5)*log(4/exp((x^2-5)/x)))/x^2,x, algorithm="giac")

[Out]

integrate((5*x^2*e^(5*x*e + 1)*log(4*e^(-(x^2 - 5)/x)) + 5*x^2 - (x^2 + 5)*e^(5*x*e) + 25)*(4*e^(-(x^2 - 5)/x)
)^(e^(5*x*e) - 5)/x^2, x)

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maple [A]  time = 0.47, size = 26, normalized size = 0.84

method	result	size
default	\({\mathrm e}^{\left ({\mathrm e}^{5 x \,{\mathrm e}}-5\right ) \ln \left (4 \,{\mathrm e}^{-\frac {x^{2}-5}{x}}\right )}\)	\(26\)
risch	\({\mathrm e}^{\left ({\mathrm e}^{5 x \,{\mathrm e}}-5\right ) \left (2 \ln \relax (2)-\ln \left ({\mathrm e}^{\frac {x^{2}-5}{x}}\right )\right )}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^2*exp(1)*exp(5*x*exp(1))*ln(4/exp((x^2-5)/x))+(-x^2-5)*exp(5*x*exp(1))+5*x^2+25)*exp((exp(5*x*exp(1))
-5)*ln(4/exp((x^2-5)/x)))/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.53, size = 42, normalized size = 1.35 \begin {gather*} \frac {1}{1024} \, e^{\left (-x e^{\left (5 \, x e\right )} + 2 \, e^{\left (5 \, x e\right )} \log \relax (2) + 5 \, x + \frac {5 \, e^{\left (5 \, x e\right )}}{x} - \frac {25}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2*exp(1)*exp(5*x*exp(1))*log(4/exp((x^2-5)/x))+(-x^2-5)*exp(5*x*exp(1))+5*x^2+25)*exp((exp(5*x*
exp(1))-5)*log(4/exp((x^2-5)/x)))/x^2,x, algorithm="maxima")

[Out]

1/1024*e^(-x*e^(5*x*e) + 2*e^(5*x*e)*log(2) + 5*x + 5*e^(5*x*e)/x - 25/x)

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mupad [B]  time = 3.90, size = 44, normalized size = 1.42 \begin {gather*} \frac {2^{2\,{\mathrm {e}}^{5\,x\,\mathrm {e}}}\,{\mathrm {e}}^{5\,x}\,{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^{5\,x\,\mathrm {e}}}{x}}\,{\mathrm {e}}^{-\frac {25}{x}}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{5\,x\,\mathrm {e}}}}{1024} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(4*exp(-(x^2 - 5)/x))*(exp(5*x*exp(1)) - 5))*(5*x^2 - exp(5*x*exp(1))*(x^2 + 5) + 5*x^2*exp(1)*exp
(5*x*exp(1))*log(4*exp(-(x^2 - 5)/x)) + 25))/x^2,x)

[Out]

(2^(2*exp(5*x*exp(1)))*exp(5*x)*exp((5*exp(5*x*exp(1)))/x)*exp(-25/x)*exp(-x*exp(5*x*exp(1))))/1024

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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((5*x**2*exp(1)*exp(5*x*exp(1))*ln(4/exp((x**2-5)/x))+(-x**2-5)*exp(5*x*exp(1))+5*x**2+25)*exp((exp(5
*x*exp(1))-5)*ln(4/exp((x**2-5)/x)))/x**2,x)

[Out]

Timed out

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