\(\int \frac {144+144 x+48 x^2+48 x^3+4 x^4+4 x^5+e^{\frac {49+14 e+e^2}{6+x^2}} (-36+86 x^2+28 e x^2+2 e^2 x^2-x^4)}{36+12 x^2+x^4} \, dx\) [7818]

Optimal result

Integrand size = 83, antiderivative size = 28 \[ \int \frac {144+144 x+48 x^2+48 x^3+4 x^4+4 x^5+e^{\frac {49+14 e+e^2}{6+x^2}} \left (-36+86 x^2+28 e x^2+2 e^2 x^2-x^4\right )}{36+12 x^2+x^4} \, dx=2 x \left (\frac {1}{2} \left (4-e^{\frac {(7+e)^2}{6+x^2}}\right )+x\right ) \]

[Out]

2*x*(x-1/2*exp((7+exp(1))^2/(x^2+6))+2)

Rubi [F]

\[ \int \frac {144+144 x+48 x^2+48 x^3+4 x^4+4 x^5+e^{\frac {49+14 e+e^2}{6+x^2}} \left (-36+86 x^2+28 e x^2+2 e^2 x^2-x^4\right )}{36+12 x^2+x^4} \, dx=\int \frac {144+144 x+48 x^2+48 x^3+4 x^4+4 x^5+e^{\frac {49+14 e+e^2}{6+x^2}} \left (-36+86 x^2+28 e x^2+2 e^2 x^2-x^4\right )}{36+12 x^2+x^4} \, dx \]

[In]

Int[(144 + 144*x + 48*x^2 + 48*x^3 + 4*x^4 + 4*x^5 + E^((49 + 14*E + E^2)/(6 + x^2))*(-36 + 86*x^2 + 28*E*x^2
+ 2*E^2*x^2 - x^4))/(36 + 12*x^2 + x^4),x]

[Out]

6*x + 2*x^2 - (12*x)/(6 + x^2) - (2*x^3)/(6 + x^2) - Defer[Int][E^((7 + E)^2/(6 + x^2)), x] + ((7 + E)^2*Defer
[Int][E^((7 + E)^2/(6 + x^2))/(I*Sqrt[6] - x)^2, x])/2 + ((I/2)*(7 + E)^2*Defer[Int][E^((7 + E)^2/(6 + x^2))/(
I*Sqrt[6] - x), x])/Sqrt[6] + ((7 + E)^2*Defer[Int][E^((7 + E)^2/(6 + x^2))/(I*Sqrt[6] + x)^2, x])/2 + ((I/2)*
(7 + E)^2*Defer[Int][E^((7 + E)^2/(6 + x^2))/(I*Sqrt[6] + x), x])/Sqrt[6]

Rubi steps \begin{align*} \text {integral}& = \int \frac {144+144 x+48 x^2+48 x^3+4 x^4+4 x^5+e^{\frac {49+14 e+e^2}{6+x^2}} \left (-36+86 x^2+28 e x^2+2 e^2 x^2-x^4\right )}{\left (6+x^2\right )^2} \, dx \\ & = \int \left (\frac {144}{\left (6+x^2\right )^2}+\frac {144 x}{\left (6+x^2\right )^2}+\frac {48 x^2}{\left (6+x^2\right )^2}+\frac {48 x^3}{\left (6+x^2\right )^2}+\frac {4 x^4}{\left (6+x^2\right )^2}+\frac {4 x^5}{\left (6+x^2\right )^2}+\frac {e^{\frac {(7+e)^2}{6+x^2}} \left (-36+2 \left (43+14 e+e^2\right ) x^2-x^4\right )}{\left (6+x^2\right )^2}\right ) \, dx \\ & = 4 \int \frac {x^4}{\left (6+x^2\right )^2} \, dx+4 \int \frac {x^5}{\left (6+x^2\right )^2} \, dx+48 \int \frac {x^2}{\left (6+x^2\right )^2} \, dx+48 \int \frac {x^3}{\left (6+x^2\right )^2} \, dx+144 \int \frac {1}{\left (6+x^2\right )^2} \, dx+144 \int \frac {x}{\left (6+x^2\right )^2} \, dx+\int \frac {e^{\frac {(7+e)^2}{6+x^2}} \left (-36+2 \left (43+14 e+e^2\right ) x^2-x^4\right )}{\left (6+x^2\right )^2} \, dx \\ & = -\frac {72}{6+x^2}-\frac {12 x}{6+x^2}-\frac {2 x^3}{6+x^2}+2 \text {Subst}\left (\int \frac {x^2}{(6+x)^2} \, dx,x,x^2\right )+6 \int \frac {x^2}{6+x^2} \, dx+12 \int \frac {1}{6+x^2} \, dx+24 \int \frac {1}{6+x^2} \, dx+24 \text {Subst}\left (\int \frac {x}{(6+x)^2} \, dx,x,x^2\right )+\int \left (-e^{\frac {(7+e)^2}{6+x^2}}-\frac {12 e^{\frac {(7+e)^2}{6+x^2}} (7+e)^2}{\left (6+x^2\right )^2}+\frac {2 e^{\frac {(7+e)^2}{6+x^2}} (7+e)^2}{6+x^2}\right ) \, dx \\ & = 6 x-\frac {72}{6+x^2}-\frac {12 x}{6+x^2}-\frac {2 x^3}{6+x^2}+6 \sqrt {6} \arctan \left (\frac {x}{\sqrt {6}}\right )+2 \text {Subst}\left (\int \left (1+\frac {36}{(6+x)^2}-\frac {12}{6+x}\right ) \, dx,x,x^2\right )+24 \text {Subst}\left (\int \left (-\frac {6}{(6+x)^2}+\frac {1}{6+x}\right ) \, dx,x,x^2\right )-36 \int \frac {1}{6+x^2} \, dx+\left (2 (7+e)^2\right ) \int \frac {e^{\frac {(7+e)^2}{6+x^2}}}{6+x^2} \, dx-\left (12 (7+e)^2\right ) \int \frac {e^{\frac {(7+e)^2}{6+x^2}}}{\left (6+x^2\right )^2} \, dx-\int e^{\frac {(7+e)^2}{6+x^2}} \, dx \\ & = 6 x+2 x^2-\frac {12 x}{6+x^2}-\frac {2 x^3}{6+x^2}+\left (2 (7+e)^2\right ) \int \left (\frac {i e^{\frac {(7+e)^2}{6+x^2}}}{2 \sqrt {6} \left (i \sqrt {6}-x\right )}+\frac {i e^{\frac {(7+e)^2}{6+x^2}}}{2 \sqrt {6} \left (i \sqrt {6}+x\right )}\right ) \, dx-\left (12 (7+e)^2\right ) \int \left (-\frac {e^{\frac {(7+e)^2}{6+x^2}}}{24 \left (i \sqrt {6}-x\right )^2}-\frac {e^{\frac {(7+e)^2}{6+x^2}}}{24 \left (i \sqrt {6}+x\right )^2}-\frac {e^{\frac {(7+e)^2}{6+x^2}}}{12 \left (-6-x^2\right )}\right ) \, dx-\int e^{\frac {(7+e)^2}{6+x^2}} \, dx \\ & = 6 x+2 x^2-\frac {12 x}{6+x^2}-\frac {2 x^3}{6+x^2}+\frac {1}{2} (7+e)^2 \int \frac {e^{\frac {(7+e)^2}{6+x^2}}}{\left (i \sqrt {6}-x\right )^2} \, dx+\frac {1}{2} (7+e)^2 \int \frac {e^{\frac {(7+e)^2}{6+x^2}}}{\left (i \sqrt {6}+x\right )^2} \, dx+(7+e)^2 \int \frac {e^{\frac {(7+e)^2}{6+x^2}}}{-6-x^2} \, dx+\frac {\left (i (7+e)^2\right ) \int \frac {e^{\frac {(7+e)^2}{6+x^2}}}{i \sqrt {6}-x} \, dx}{\sqrt {6}}+\frac {\left (i (7+e)^2\right ) \int \frac {e^{\frac {(7+e)^2}{6+x^2}}}{i \sqrt {6}+x} \, dx}{\sqrt {6}}-\int e^{\frac {(7+e)^2}{6+x^2}} \, dx \\ & = 6 x+2 x^2-\frac {12 x}{6+x^2}-\frac {2 x^3}{6+x^2}+\frac {1}{2} (7+e)^2 \int \frac {e^{\frac {(7+e)^2}{6+x^2}}}{\left (i \sqrt {6}-x\right )^2} \, dx+\frac {1}{2} (7+e)^2 \int \frac {e^{\frac {(7+e)^2}{6+x^2}}}{\left (i \sqrt {6}+x\right )^2} \, dx+(7+e)^2 \int \left (-\frac {i e^{\frac {(7+e)^2}{6+x^2}}}{2 \sqrt {6} \left (i \sqrt {6}-x\right )}-\frac {i e^{\frac {(7+e)^2}{6+x^2}}}{2 \sqrt {6} \left (i \sqrt {6}+x\right )}\right ) \, dx+\frac {\left (i (7+e)^2\right ) \int \frac {e^{\frac {(7+e)^2}{6+x^2}}}{i \sqrt {6}-x} \, dx}{\sqrt {6}}+\frac {\left (i (7+e)^2\right ) \int \frac {e^{\frac {(7+e)^2}{6+x^2}}}{i \sqrt {6}+x} \, dx}{\sqrt {6}}-\int e^{\frac {(7+e)^2}{6+x^2}} \, dx \\ & = 6 x+2 x^2-\frac {12 x}{6+x^2}-\frac {2 x^3}{6+x^2}+\frac {1}{2} (7+e)^2 \int \frac {e^{\frac {(7+e)^2}{6+x^2}}}{\left (i \sqrt {6}-x\right )^2} \, dx+\frac {1}{2} (7+e)^2 \int \frac {e^{\frac {(7+e)^2}{6+x^2}}}{\left (i \sqrt {6}+x\right )^2} \, dx-\frac {\left (i (7+e)^2\right ) \int \frac {e^{\frac {(7+e)^2}{6+x^2}}}{i \sqrt {6}-x} \, dx}{2 \sqrt {6}}-\frac {\left (i (7+e)^2\right ) \int \frac {e^{\frac {(7+e)^2}{6+x^2}}}{i \sqrt {6}+x} \, dx}{2 \sqrt {6}}+\frac {\left (i (7+e)^2\right ) \int \frac {e^{\frac {(7+e)^2}{6+x^2}}}{i \sqrt {6}-x} \, dx}{\sqrt {6}}+\frac {\left (i (7+e)^2\right ) \int \frac {e^{\frac {(7+e)^2}{6+x^2}}}{i \sqrt {6}+x} \, dx}{\sqrt {6}}-\int e^{\frac {(7+e)^2}{6+x^2}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {144+144 x+48 x^2+48 x^3+4 x^4+4 x^5+e^{\frac {49+14 e+e^2}{6+x^2}} \left (-36+86 x^2+28 e x^2+2 e^2 x^2-x^4\right )}{36+12 x^2+x^4} \, dx=x \left (4-e^{\frac {(7+e)^2}{6+x^2}}+2 x\right ) \]

[In]

Integrate[(144 + 144*x + 48*x^2 + 48*x^3 + 4*x^4 + 4*x^5 + E^((49 + 14*E + E^2)/(6 + x^2))*(-36 + 86*x^2 + 28*
E*x^2 + 2*E^2*x^2 - x^4))/(36 + 12*x^2 + x^4),x]

[Out]

x*(4 - E^((7 + E)^2/(6 + x^2)) + 2*x)

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07

method	result	size
risch	\(2 x^{2}-x \,{\mathrm e}^{\frac {{\mathrm e}^{2}+14 \,{\mathrm e}+49}{x^{2}+6}}+4 x\)	\(30\)
parallelrisch	\(2 x^{2}-x \,{\mathrm e}^{\frac {{\mathrm e}^{2}+14 \,{\mathrm e}+49}{x^{2}+6}}+4 x -48\)	\(33\)
parts	\(4 x +\frac {-6 x \,{\mathrm e}^{\frac {{\mathrm e}^{2}+14 \,{\mathrm e}+49}{x^{2}+6}}-{\mathrm e}^{\frac {{\mathrm e}^{2}+14 \,{\mathrm e}+49}{x^{2}+6}} x^{3}}{x^{2}+6}+2 x^{2}\)	\(65\)
norman	\(\frac {24 x +4 x^{3}+2 x^{4}-6 x \,{\mathrm e}^{\frac {{\mathrm e}^{2}+14 \,{\mathrm e}+49}{x^{2}+6}}-{\mathrm e}^{\frac {{\mathrm e}^{2}+14 \,{\mathrm e}+49}{x^{2}+6}} x^{3}-72}{x^{2}+6}\)	\(70\)

[In]

int(((2*x^2*exp(1)^2+28*x^2*exp(1)-x^4+86*x^2-36)*exp((exp(1)^2+14*exp(1)+49)/(x^2+6))+4*x^5+4*x^4+48*x^3+48*x
^2+144*x+144)/(x^4+12*x^2+36),x,method=_RETURNVERBOSE)

[Out]

2*x^2-x*exp((exp(2)+14*exp(1)+49)/(x^2+6))+4*x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {144+144 x+48 x^2+48 x^3+4 x^4+4 x^5+e^{\frac {49+14 e+e^2}{6+x^2}} \left (-36+86 x^2+28 e x^2+2 e^2 x^2-x^4\right )}{36+12 x^2+x^4} \, dx=2 \, x^{2} - x e^{\left (\frac {e^{2} + 14 \, e + 49}{x^{2} + 6}\right )} + 4 \, x \]

[In]

integrate(((2*x^2*exp(1)^2+28*x^2*exp(1)-x^4+86*x^2-36)*exp((exp(1)^2+14*exp(1)+49)/(x^2+6))+4*x^5+4*x^4+48*x^
3+48*x^2+144*x+144)/(x^4+12*x^2+36),x, algorithm="fricas")

[Out]

2*x^2 - x*e^((e^2 + 14*e + 49)/(x^2 + 6)) + 4*x

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {144+144 x+48 x^2+48 x^3+4 x^4+4 x^5+e^{\frac {49+14 e+e^2}{6+x^2}} \left (-36+86 x^2+28 e x^2+2 e^2 x^2-x^4\right )}{36+12 x^2+x^4} \, dx=2 x^{2} - x e^{\frac {e^{2} + 14 e + 49}{x^{2} + 6}} + 4 x \]

[In]

integrate(((2*x**2*exp(1)**2+28*x**2*exp(1)-x**4+86*x**2-36)*exp((exp(1)**2+14*exp(1)+49)/(x**2+6))+4*x**5+4*x
**4+48*x**3+48*x**2+144*x+144)/(x**4+12*x**2+36),x)

[Out]

2*x**2 - x*exp((exp(2) + 14*E + 49)/(x**2 + 6)) + 4*x

Maxima [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {144+144 x+48 x^2+48 x^3+4 x^4+4 x^5+e^{\frac {49+14 e+e^2}{6+x^2}} \left (-36+86 x^2+28 e x^2+2 e^2 x^2-x^4\right )}{36+12 x^2+x^4} \, dx=2 \, x^{2} - x e^{\left (\frac {e^{2}}{x^{2} + 6} + \frac {14 \, e}{x^{2} + 6} + \frac {49}{x^{2} + 6}\right )} + 4 \, x \]

[In]

integrate(((2*x^2*exp(1)^2+28*x^2*exp(1)-x^4+86*x^2-36)*exp((exp(1)^2+14*exp(1)+49)/(x^2+6))+4*x^5+4*x^4+48*x^
3+48*x^2+144*x+144)/(x^4+12*x^2+36),x, algorithm="maxima")

[Out]

2*x^2 - x*e^(e^2/(x^2 + 6) + 14*e/(x^2 + 6) + 49/(x^2 + 6)) + 4*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).

Time = 0.33 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {144+144 x+48 x^2+48 x^3+4 x^4+4 x^5+e^{\frac {49+14 e+e^2}{6+x^2}} \left (-36+86 x^2+28 e x^2+2 e^2 x^2-x^4\right )}{36+12 x^2+x^4} \, dx=2 \, x^{2} - x e^{\left (-\frac {x^{2} e^{2} + 14 \, x^{2} e + 49 \, x^{2}}{6 \, {\left (x^{2} + 6\right )}} + \frac {1}{6} \, e^{2} + \frac {7}{3} \, e + \frac {49}{6}\right )} + 4 \, x \]

[In]

integrate(((2*x^2*exp(1)^2+28*x^2*exp(1)-x^4+86*x^2-36)*exp((exp(1)^2+14*exp(1)+49)/(x^2+6))+4*x^5+4*x^4+48*x^
3+48*x^2+144*x+144)/(x^4+12*x^2+36),x, algorithm="giac")

[Out]

2*x^2 - x*e^(-1/6*(x^2*e^2 + 14*x^2*e + 49*x^2)/(x^2 + 6) + 1/6*e^2 + 7/3*e + 49/6) + 4*x

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {144+144 x+48 x^2+48 x^3+4 x^4+4 x^5+e^{\frac {49+14 e+e^2}{6+x^2}} \left (-36+86 x^2+28 e x^2+2 e^2 x^2-x^4\right )}{36+12 x^2+x^4} \, dx=4\,x+2\,x^2-x\,{\mathrm {e}}^{\frac {49}{x^2+6}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^2}{x^2+6}}\,{\mathrm {e}}^{\frac {14\,\mathrm {e}}{x^2+6}} \]

[In]

int((144*x + exp((14*exp(1) + exp(2) + 49)/(x^2 + 6))*(28*x^2*exp(1) + 2*x^2*exp(2) + 86*x^2 - x^4 - 36) + 48*
x^2 + 48*x^3 + 4*x^4 + 4*x^5 + 144)/(12*x^2 + x^4 + 36),x)

[Out]

4*x + 2*x^2 - x*exp(49/(x^2 + 6))*exp(exp(2)/(x^2 + 6))*exp((14*exp(1))/(x^2 + 6))