\(\int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} ((-3600+240 \sqrt {e}-4 e) e^{2 e^x}-32 e^x x)}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx\) [8313]

Optimal result

Integrand size = 80, antiderivative size = 25 \[ \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx=\frac {4 e^{\frac {4 e^{-2 e^x}}{\left (-30+\sqrt {e}\right )^2}}}{x} \]

[Out]

4*exp(4/exp(exp(x))^2/(exp(1/4)^2-30)^2)/x

Rubi [F]

\[ \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx=\int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx \]

[In]

Int[(E^(-2*E^x + 4/(E^(2*E^x)*(900 - 60*Sqrt[E] + E)))*((-3600 + 240*Sqrt[E] - 4*E)*E^(2*E^x) - 32*E^x*x))/(90
0*x^2 - 60*Sqrt[E]*x^2 + E*x^2),x]

[Out]

-4*Defer[Int][E^(4/((-30 + Sqrt[E])^2*E^(2*E^x)))/x^2, x] - (32*Defer[Int][E^(-2*E^x + 4/(E^(2*E^x)*(900 - 60*
Sqrt[E] + E)) + x)/x, x])/(30 - Sqrt[E])^2

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{\left (900-60 \sqrt {e}\right ) x^2+e x^2} \, dx \\ & = \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{\left (900-60 \sqrt {e}+e\right ) x^2} \, dx \\ & = \frac {\int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{x^2} \, dx}{900-60 \sqrt {e}+e} \\ & = \frac {\int \left (-\frac {4 \left (-30+\sqrt {e}\right )^2 e^{\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}}}{x^2}-\frac {32 e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}+x}}{x}\right ) \, dx}{900-60 \sqrt {e}+e} \\ & = -\left (4 \int \frac {e^{\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}}}{x^2} \, dx\right )-\frac {32 \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}+x}}{x} \, dx}{\left (30-\sqrt {e}\right )^2} \\ & = -\left (4 \int \frac {e^{\frac {4 e^{-2 e^x}}{\left (-30+\sqrt {e}\right )^2}}}{x^2} \, dx\right )-\frac {32 \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}+x}}{x} \, dx}{\left (30-\sqrt {e}\right )^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.49 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx=\frac {4 e^{\frac {4 e^{-2 e^x}}{\left (-30+\sqrt {e}\right )^2}}}{x} \]

[In]

Integrate[(E^(-2*E^x + 4/(E^(2*E^x)*(900 - 60*Sqrt[E] + E)))*((-3600 + 240*Sqrt[E] - 4*E)*E^(2*E^x) - 32*E^x*x
))/(900*x^2 - 60*Sqrt[E]*x^2 + E*x^2),x]

[Out]

(4*E^(4/((-30 + Sqrt[E])^2*E^(2*E^x))))/x

Maple [A] (verified)

Time = 2.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96

method	result	size
risch	\(\frac {4 \,{\mathrm e}^{\frac {4 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}}}{{\mathrm e}-60 \,{\mathrm e}^{\frac {1}{2}}+900}}}{x}\)	\(24\)
parallelrisch	\(\frac {4 \,{\mathrm e} \,{\mathrm e}^{\frac {4 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}}}{{\mathrm e}-60 \,{\mathrm e}^{\frac {1}{2}}+900}}-240 \,{\mathrm e}^{\frac {1}{2}} {\mathrm e}^{\frac {4 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}}}{{\mathrm e}-60 \,{\mathrm e}^{\frac {1}{2}}+900}}+3600 \,{\mathrm e}^{\frac {4 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}}}{{\mathrm e}-60 \,{\mathrm e}^{\frac {1}{2}}+900}}}{\left ({\mathrm e}-60 \,{\mathrm e}^{\frac {1}{2}}+900\right ) x}\)	\(100\)

[In]

int(((-4*exp(1/4)^4+240*exp(1/4)^2-3600)*exp(exp(x))^2-32*exp(x)*x)*exp(4/(exp(1/4)^4-60*exp(1/4)^2+900)/exp(e
xp(x))^2)/(x^2*exp(1/4)^4-60*x^2*exp(1/4)^2+900*x^2)/exp(exp(x))^2,x,method=_RETURNVERBOSE)

[Out]

4/x*exp(4/(exp(1)-60*exp(1/2)+900)*exp(-2*exp(x)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (19) = 38\).

Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx=\frac {4 \, e^{\left (-\frac {2 \, {\left ({\left (e - 60 \, e^{\frac {1}{2}} + 900\right )} e^{\left (x + 2 \, e^{x}\right )} - 2\right )} e^{\left (-2 \, e^{x}\right )}}{e - 60 \, e^{\frac {1}{2}} + 900} + 2 \, e^{x}\right )}}{x} \]

[In]

integrate(((-4*exp(1/4)^4+240*exp(1/4)^2-3600)*exp(exp(x))^2-32*exp(x)*x)*exp(4/(exp(1/4)^4-60*exp(1/4)^2+900)
/exp(exp(x))^2)/(x^2*exp(1/4)^4-60*x^2*exp(1/4)^2+900*x^2)/exp(exp(x))^2,x, algorithm="fricas")

[Out]

4*e^(-2*((e - 60*e^(1/2) + 900)*e^(x + 2*e^x) - 2)*e^(-2*e^x)/(e - 60*e^(1/2) + 900) + 2*e^x)/x

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx=\frac {4 e^{\frac {4 e^{- 2 e^{x}}}{- 60 e^{\frac {1}{2}} + e + 900}}}{x} \]

[In]

integrate(((-4*exp(1/4)**4+240*exp(1/4)**2-3600)*exp(exp(x))**2-32*exp(x)*x)*exp(4/(exp(1/4)**4-60*exp(1/4)**2
+900)/exp(exp(x))**2)/(x**2*exp(1/4)**4-60*x**2*exp(1/4)**2+900*x**2)/exp(exp(x))**2,x)

[Out]

4*exp(4*exp(-2*exp(x))/(-60*exp(1/2) + E + 900))/x

Maxima [F]

\[ \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx=\int { -\frac {4 \, {\left (8 \, x e^{x} + {\left (e - 60 \, e^{\frac {1}{2}} + 900\right )} e^{\left (2 \, e^{x}\right )}\right )} e^{\left (\frac {4 \, e^{\left (-2 \, e^{x}\right )}}{e - 60 \, e^{\frac {1}{2}} + 900} - 2 \, e^{x}\right )}}{x^{2} e - 60 \, x^{2} e^{\frac {1}{2}} + 900 \, x^{2}} \,d x } \]

[In]

integrate(((-4*exp(1/4)^4+240*exp(1/4)^2-3600)*exp(exp(x))^2-32*exp(x)*x)*exp(4/(exp(1/4)^4-60*exp(1/4)^2+900)
/exp(exp(x))^2)/(x^2*exp(1/4)^4-60*x^2*exp(1/4)^2+900*x^2)/exp(exp(x))^2,x, algorithm="maxima")

[Out]

-4*integrate((8*x*e^x + (e - 60*e^(1/2) + 900)*e^(2*e^x))*e^(4*e^(-2*e^x)/(e - 60*e^(1/2) + 900) - 2*e^x)/(x^2
*e - 60*x^2*e^(1/2) + 900*x^2), x)

Giac [F]

\[ \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx=\int { -\frac {4 \, {\left (8 \, x e^{x} + {\left (e - 60 \, e^{\frac {1}{2}} + 900\right )} e^{\left (2 \, e^{x}\right )}\right )} e^{\left (\frac {4 \, e^{\left (-2 \, e^{x}\right )}}{e - 60 \, e^{\frac {1}{2}} + 900} - 2 \, e^{x}\right )}}{x^{2} e - 60 \, x^{2} e^{\frac {1}{2}} + 900 \, x^{2}} \,d x } \]

[In]

integrate(((-4*exp(1/4)^4+240*exp(1/4)^2-3600)*exp(exp(x))^2-32*exp(x)*x)*exp(4/(exp(1/4)^4-60*exp(1/4)^2+900)
/exp(exp(x))^2)/(x^2*exp(1/4)^4-60*x^2*exp(1/4)^2+900*x^2)/exp(exp(x))^2,x, algorithm="giac")

[Out]

integrate(-4*(8*x*e^x + (e - 60*e^(1/2) + 900)*e^(2*e^x))*e^(4*e^(-2*e^x)/(e - 60*e^(1/2) + 900) - 2*e^x)/(x^2
*e - 60*x^2*e^(1/2) + 900*x^2), x)

Mupad [B] (verification not implemented)

Time = 15.97 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx=\frac {4\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{-2\,{\mathrm {e}}^x}}{\mathrm {e}-60\,\sqrt {\mathrm {e}}+900}}}{x} \]

[In]

int(-(exp((4*exp(-2*exp(x)))/(exp(1) - 60*exp(1/2) + 900))*exp(-2*exp(x))*(32*x*exp(x) + exp(2*exp(x))*(4*exp(
1) - 240*exp(1/2) + 3600)))/(x^2*exp(1) - 60*x^2*exp(1/2) + 900*x^2),x)

[Out]

(4*exp((4*exp(-2*exp(x)))/(exp(1) - 60*exp(1/2) + 900)))/x