3.63.12 \(\int \frac {x^2+e^{\frac {-6+x+x^2}{x}} (6+x^2)+e^{x^2} (-x^2+2 e x^3-2 x^4)}{e^{\frac {-6+x+x^2}{x}} x^2+x^3+e^{x^2} (e x^2-x^3)} \, dx\)

Optimal. Leaf size=36 \[ \log \left (\frac {1}{5} \left (-e^{-\frac {6-x}{x}+x}-x+e^{x^2} (-e+x)\right )\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 30.73, 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 {x^2+e^{\frac {-6+x+x^2}{x}} \left (6+x^2\right )+e^{x^2} \left (-x^2+2 e x^3-2 x^4\right )}{e^{\frac {-6+x+x^2}{x}} x^2+x^3+e^{x^2} \left (e x^2-x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^2 + E^((-6 + x + x^2)/x)*(6 + x^2) + E^x^2*(-x^2 + 2*E*x^3 - 2*x^4))/(E^((-6 + x + x^2)/x)*x^2 + x^3 +
E^x^2*(E*x^2 - x^3)),x]

[Out]

x^2 + Log[E - x] + Defer[Int][E^(1 + 6/x)/(E^(1 + x) + E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x), x]/E
- Defer[Int][E^(2 + 6/x)/(E^(1 + x) + E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x), x]/E^2 + ((1 + E)*Defe
r[Int][E^(1 + x)/(E^(1 + x) + E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x), x])/E - ((1 + E)*Defer[Int][E^
(2 + x)/(E^(1 + x) + E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x), x])/E^2 + ((1 + 2*E)*Defer[Int][E^(3 +
x)/(E^(1 + x) + E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x), x])/E^2 - (2*Defer[Int][E^(4 + x)/(E^(1 + x)
+ E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x), x])/E^2 + Defer[Int][E^(3 + 6/x)/((E - x)*(E^(1 + x) + E^
(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x)), x]/E^2 + ((1 + E)*Defer[Int][E^(3 + x)/((E - x)*(E^(1 + x) +
E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x)), x])/E^2 - ((1 + 2*E)*Defer[Int][E^(4 + x)/((E - x)*(E^(1 +
x) + E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x)), x])/E^2 + (2*Defer[Int][E^(5 + x)/((E - x)*(E^(1 + x)
+ E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x)), x])/E^2 + (6*Defer[Int][E^(2 + x)/(x^2*(E^(1 + x) + E^(1
+ 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x)), x])/E - (6*Defer[Int][E^(1 + x)/(x*(E^(1 + x) + E^(1 + 6/x + x^2
) + E^(6/x)*x - E^(6/x + x^2)*x)), x])/E + (6*Defer[Int][E^(2 + x)/(x*(E^(1 + x) + E^(1 + 6/x + x^2) + E^(6/x)
*x - E^(6/x + x^2)*x)), x])/E^2 + Defer[Int][(E^(1 + 6/x)*x)/(E^(1 + x) + E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6
/x + x^2)*x), x]/E^2 + ((1 + E)*Defer[Int][(E^(1 + x)*x)/(E^(1 + x) + E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x +
x^2)*x), x])/E^2 - ((1 + 2*E)*Defer[Int][(E^(1 + x)*x)/(E^(1 + x) + E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x +
x^2)*x), x])/E + ((1 + 2*E)*Defer[Int][(E^(2 + x)*x)/(E^(1 + x) + E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2
)*x), x])/E^2 - (2*Defer[Int][(E^(3 + x)*x)/(E^(1 + x) + E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x), x])
/E^2 - (2*Defer[Int][(E^(1 + 6/x)*x^2)/(E^(1 + x) + E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x), x])/E +
(2*Defer[Int][(E^(1 + x)*x^2)/(E^(1 + x) + E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x), x])/E - ((1 + 2*E
)*Defer[Int][(E^(1 + x)*x^2)/(E^(1 + x) + E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x), x])/E^2 - (2*Defer
[Int][(E^(2 + x)*x^2)/(E^(1 + x) + E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x), x])/E^2 - (4*Defer[Int][(
E^(1 + 6/x)*x^3)/(E^(1 + x) + E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x), x])/E^2 + (2*Defer[Int][(E^(1
+ x)*x^3)/(E^(1 + x) + E^(1 + 6/x + x^2) + E^(6/x)*x - E^(6/x + x^2)*x), x])/E^2 + Defer[Int][(E^(1 + 6/x)*x)/
(-E^(1 + x) - E^(1 + 6/x + x^2) - E^(6/x)*x + E^(6/x + x^2)*x), x]/E^2 + ((1 + E)*Defer[Int][(E^(1 + x)*x)/(-E
^(1 + x) - E^(1 + 6/x + x^2) - E^(6/x)*x + E^(6/x + x^2)*x), x])/E^2 - ((1 + 2*E)*Defer[Int][(E^(1 + x)*x^2)/(
-E^(1 + x) - E^(1 + 6/x + x^2) - E^(6/x)*x + E^(6/x + x^2)*x), x])/E^2 - (2*Defer[Int][(E^(1 + 6/x)*x^3)/(-E^(
1 + x) - E^(1 + 6/x + x^2) - E^(6/x)*x + E^(6/x + x^2)*x), x])/E^2 - (2*Defer[Int][(E^(6/x)*x^3)/(-E^(1 + x) -
E^(1 + 6/x + x^2) - E^(6/x)*x + E^(6/x + x^2)*x), x])/E + (2*Defer[Int][(E^(1 + x)*x^3)/(-E^(1 + x) - E^(1 +
6/x + x^2) - E^(6/x)*x + E^(6/x + x^2)*x), x])/E^2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-1+2 e x-2 x^2}{e-x}+\frac {6 e^{2+x}-6 e^{1+x} x+e^{1+\frac {6}{x}} x^2+e^{1+x} (1+e) x^2-e^{1+x} (1+2 e) x^3-2 e^{1+\frac {6}{x}} x^4+2 e^{1+x} x^4+2 e^{6/x} x^5}{(e-x) x^2 \left (e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x\right )}\right ) \, dx\\ &=\int \frac {-1+2 e x-2 x^2}{e-x} \, dx+\int \frac {6 e^{2+x}-6 e^{1+x} x+e^{1+\frac {6}{x}} x^2+e^{1+x} (1+e) x^2-e^{1+x} (1+2 e) x^3-2 e^{1+\frac {6}{x}} x^4+2 e^{1+x} x^4+2 e^{6/x} x^5}{(e-x) x^2 \left (e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x\right )} \, dx\\ &=\int \left (2 x+\frac {1}{-e+x}\right ) \, dx+\int \frac {2 e^{6/x} x^5+e^{2+x} \left (6+x^2-2 x^3\right )+e^{1+x} x \left (-6+x-x^2+2 x^3\right )+e^{1+\frac {6}{x}} \left (x^2-2 x^4\right )}{(e-x) x^2 \left (e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x\right )} \, dx\\ &=x^2+\log (e-x)+\int \left (\frac {6 e^{2+x}-6 e^{1+x} x+e^{1+\frac {6}{x}} x^2+e^{1+x} (1+e) x^2-e^{1+x} (1+2 e) x^3-2 e^{1+\frac {6}{x}} x^4+2 e^{1+x} x^4+2 e^{6/x} x^5}{e^2 (e-x) \left (e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x\right )}+\frac {6 e^{2+x}-6 e^{1+x} x+e^{1+\frac {6}{x}} x^2+e^{1+x} (1+e) x^2-e^{1+x} (1+2 e) x^3-2 e^{1+\frac {6}{x}} x^4+2 e^{1+x} x^4+2 e^{6/x} x^5}{e x^2 \left (e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x\right )}+\frac {6 e^{2+x}-6 e^{1+x} x+e^{1+\frac {6}{x}} x^2+e^{1+x} (1+e) x^2-e^{1+x} (1+2 e) x^3-2 e^{1+\frac {6}{x}} x^4+2 e^{1+x} x^4+2 e^{6/x} x^5}{e^2 x \left (e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x\right )}\right ) \, dx\\ &=x^2+\log (e-x)+\frac {\int \frac {6 e^{2+x}-6 e^{1+x} x+e^{1+\frac {6}{x}} x^2+e^{1+x} (1+e) x^2-e^{1+x} (1+2 e) x^3-2 e^{1+\frac {6}{x}} x^4+2 e^{1+x} x^4+2 e^{6/x} x^5}{(e-x) \left (e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x\right )} \, dx}{e^2}+\frac {\int \frac {6 e^{2+x}-6 e^{1+x} x+e^{1+\frac {6}{x}} x^2+e^{1+x} (1+e) x^2-e^{1+x} (1+2 e) x^3-2 e^{1+\frac {6}{x}} x^4+2 e^{1+x} x^4+2 e^{6/x} x^5}{x \left (e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x\right )} \, dx}{e^2}+\frac {\int \frac {6 e^{2+x}-6 e^{1+x} x+e^{1+\frac {6}{x}} x^2+e^{1+x} (1+e) x^2-e^{1+x} (1+2 e) x^3-2 e^{1+\frac {6}{x}} x^4+2 e^{1+x} x^4+2 e^{6/x} x^5}{x^2 \left (e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x\right )} \, dx}{e}\\ &=x^2+\log (e-x)+\frac {\int \left (-\frac {6 e^{1+x}}{e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x}+\frac {6 e^{2+x}}{x \left (e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x\right )}+\frac {e^{1+\frac {6}{x}} x}{e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x}+\frac {e^{1+x} (1+e) x}{e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x}-\frac {e^{1+x} (1+2 e) x^2}{e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x}-\frac {2 e^{1+\frac {6}{x}} x^3}{e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x}+\frac {2 e^{1+x} x^3}{e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x}-\frac {2 e^{6/x} x^4}{-e^{1+x}-e^{1+\frac {6}{x}+x^2}-e^{6/x} x+e^{\frac {6}{x}+x^2} x}\right ) \, dx}{e^2}+\frac {\int \frac {2 e^{6/x} x^5+e^{2+x} \left (6+x^2-2 x^3\right )+e^{1+x} x \left (-6+x-x^2+2 x^3\right )+e^{1+\frac {6}{x}} \left (x^2-2 x^4\right )}{(e-x) \left (e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x\right )} \, dx}{e^2}+\frac {\int \left (\frac {e^{1+\frac {6}{x}}}{e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x}+\frac {e^{1+x} (1+e)}{e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x}+\frac {6 e^{2+x}}{x^2 \left (e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x\right )}-\frac {6 e^{1+x}}{x \left (e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x\right )}-\frac {e^{1+x} (1+2 e) x}{e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x}-\frac {2 e^{1+\frac {6}{x}} x^2}{e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x}+\frac {2 e^{1+x} x^2}{e^{1+x}+e^{1+\frac {6}{x}+x^2}+e^{6/x} x-e^{\frac {6}{x}+x^2} x}-\frac {2 e^{6/x} x^3}{-e^{1+x}-e^{1+\frac {6}{x}+x^2}-e^{6/x} x+e^{\frac {6}{x}+x^2} x}\right ) \, dx}{e}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 4.29, size = 28, normalized size = 0.78 \begin {gather*} \log \left (e^{1-\frac {6}{x}+x}+e^{1+x^2}+x-e^{x^2} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2 + E^((-6 + x + x^2)/x)*(6 + x^2) + E^x^2*(-x^2 + 2*E*x^3 - 2*x^4))/(E^((-6 + x + x^2)/x)*x^2 +
x^3 + E^x^2*(E*x^2 - x^3)),x]

[Out]

Log[E^(1 - 6/x + x) + E^(1 + x^2) + x - E^x^2*x]

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 47, normalized size = 1.31 \begin {gather*} \log \left (x - e\right ) + \log \left (-\frac {{\left (x - e\right )} e^{\left (x^{2}\right )} - x - e^{\left (\frac {x^{2} + x - 6}{x}\right )}}{x - e}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3*exp(1)-2*x^4-x^2)*exp(x^2)+(x^2+6)*exp((x^2+x-6)/x)+x^2)/((x^2*exp(1)-x^3)*exp(x^2)+x^2*exp(
(x^2+x-6)/x)+x^3),x, algorithm="fricas")

[Out]

log(x - e) + log(-((x - e)*e^(x^2) - x - e^((x^2 + x - 6)/x))/(x - e))

________________________________________________________________________________________

giac [A]  time = 0.18, size = 27, normalized size = 0.75 \begin {gather*} \log \left (-x e^{\left (x^{2}\right )} + x + e^{\left (x^{2} + 1\right )} + e^{\left (\frac {x^{2} + x - 6}{x}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3*exp(1)-2*x^4-x^2)*exp(x^2)+(x^2+6)*exp((x^2+x-6)/x)+x^2)/((x^2*exp(1)-x^3)*exp(x^2)+x^2*exp(
(x^2+x-6)/x)+x^3),x, algorithm="giac")

[Out]

log(-x*e^(x^2) + x + e^(x^2 + 1) + e^((x^2 + x - 6)/x))

________________________________________________________________________________________

maple [A]  time = 0.29, size = 29, normalized size = 0.81

method	result	size
norman	\(\ln \left ({\mathrm e} \,{\mathrm e}^{x^{2}}-{\mathrm e}^{x^{2}} x +{\mathrm e}^{\frac {x^{2}+x -6}{x}}+x \right )\)	\(29\)
risch	\(x -\frac {6}{x}-\frac {x^{2}+x -6}{x}+\ln \left ({\mathrm e}^{x^{2}+1}-{\mathrm e}^{x^{2}} x +{\mathrm e}^{\frac {\left (3+x \right ) \left (x -2\right )}{x}}+x \right )\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^3*exp(1)-2*x^4-x^2)*exp(x^2)+(x^2+6)*exp((x^2+x-6)/x)+x^2)/((x^2*exp(1)-x^3)*exp(x^2)+x^2*exp((x^2+x
-6)/x)+x^3),x,method=_RETURNVERBOSE)

[Out]

ln(exp(1)*exp(x^2)-exp(x^2)*x+exp((x^2+x-6)/x)+x)

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 57, normalized size = 1.58 \begin {gather*} \log \left (x - e\right ) + \log \left (\frac {{\left ({\left (x - e\right )} e^{\left (x^{2} + \frac {6}{x}\right )} - x e^{\frac {6}{x}} - e^{\left (x + 1\right )}\right )} e^{\left (-\frac {6}{x}\right )}}{x - e}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3*exp(1)-2*x^4-x^2)*exp(x^2)+(x^2+6)*exp((x^2+x-6)/x)+x^2)/((x^2*exp(1)-x^3)*exp(x^2)+x^2*exp(
(x^2+x-6)/x)+x^3),x, algorithm="maxima")

[Out]

log(x - e) + log(((x - e)*e^(x^2 + 6/x) - x*e^(6/x) - e^(x + 1))*e^(-6/x)/(x - e))

________________________________________________________________________________________

mupad [B]  time = 4.21, size = 25, normalized size = 0.69 \begin {gather*} \ln \left (x+{\mathrm {e}}^{x-\frac {6}{x}+1}+{\mathrm {e}}^{x^2+1}-x\,{\mathrm {e}}^{x^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - exp(x^2)*(x^2 - 2*x^3*exp(1) + 2*x^4) + exp((x + x^2 - 6)/x)*(x^2 + 6))/(x^2*exp((x + x^2 - 6)/x) +
exp(x^2)*(x^2*exp(1) - x^3) + x^3),x)

[Out]

log(x + exp(x - 6/x + 1) + exp(x^2 + 1) - x*exp(x^2))

________________________________________________________________________________________

sympy [A]  time = 0.46, size = 27, normalized size = 0.75 \begin {gather*} \log {\left (- x e^{x^{2}} + x + e e^{x^{2}} + e^{\frac {x^{2} + x - 6}{x}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**3*exp(1)-2*x**4-x**2)*exp(x**2)+(x**2+6)*exp((x**2+x-6)/x)+x**2)/((x**2*exp(1)-x**3)*exp(x**2
)+x**2*exp((x**2+x-6)/x)+x**3),x)

[Out]

log(-x*exp(x**2) + x + E*exp(x**2) + exp((x**2 + x - 6)/x))

________________________________________________________________________________________