\(\int \frac {e^{\arctan (a x)}}{(c+a^2 c x^2)^5} \, dx\) [266]

Optimal result

Integrand size = 19, antiderivative size = 149 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^5} \, dx=\frac {8064 e^{\arctan (a x)}}{40885 a c^5}+\frac {e^{\arctan (a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac {56 e^{\arctan (a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac {336 e^{\arctan (a x)} (1+4 a x)}{8177 a c^5 \left (1+a^2 x^2\right )^2}+\frac {4032 e^{\arctan (a x)} (1+2 a x)}{40885 a c^5 \left (1+a^2 x^2\right )} \] Output:

8064/40885*exp(arctan(a*x))/a/c^5+1/65*exp(arctan(a*x))*(8*a*x+1)/a/c^5/(a 
^2*x^2+1)^4+56/2405*exp(arctan(a*x))*(6*a*x+1)/a/c^5/(a^2*x^2+1)^3+336/817 
7*exp(arctan(a*x))*(4*a*x+1)/a/c^5/(a^2*x^2+1)^2+4032/40885*exp(arctan(a*x 
))*(2*a*x+1)/a/c^5/(a^2*x^2+1)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^5} \, dx=\frac {629 e^{\arctan (a x)} (1+8 a x)+\frac {56 \left (c+a^2 c x^2\right ) \left (17 c e^{\arctan (a x)} (1+6 a x)+6 \left (c+a^2 c x^2\right ) \left (5 e^{\arctan (a x)} (1+4 a x)+12 (1-i a x)^{\frac {i}{2}} (1+i a x)^{-\frac {i}{2}} (-i+a x) (i+a x) \left (3+2 a x+2 a^2 x^2\right )\right )\right )}{c^2}}{40885 a c \left (c+a^2 c x^2\right )^4} \] Input:

Integrate[E^ArcTan[a*x]/(c + a^2*c*x^2)^5,x]
 

Output:

(629*E^ArcTan[a*x]*(1 + 8*a*x) + (56*(c + a^2*c*x^2)*(17*c*E^ArcTan[a*x]*( 
1 + 6*a*x) + 6*(c + a^2*c*x^2)*(5*E^ArcTan[a*x]*(1 + 4*a*x) + (12*(1 - I*a 
*x)^(I/2)*(-I + a*x)*(I + a*x)*(3 + 2*a*x + 2*a^2*x^2))/(1 + I*a*x)^(I/2)) 
))/c^2)/(40885*a*c*(c + a^2*c*x^2)^4)
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5593, 27, 5593, 5593, 5593, 5594}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^{\arctan (a x)}}{\left (a^2 c x^2+c\right )^5} \, dx\)

\(\Big \downarrow \) 5593

\(\displaystyle \frac {56 \int \frac {e^{\arctan (a x)}}{c^4 \left (a^2 x^2+1\right )^4}dx}{65 c}+\frac {(8 a x+1) e^{\arctan (a x)}}{65 a c^5 \left (a^2 x^2+1\right )^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {56 \int \frac {e^{\arctan (a x)}}{\left (a^2 x^2+1\right )^4}dx}{65 c^5}+\frac {(8 a x+1) e^{\arctan (a x)}}{65 a c^5 \left (a^2 x^2+1\right )^4}\)

\(\Big \downarrow \) 5593

\(\displaystyle \frac {56 \left (\frac {30}{37} \int \frac {e^{\arctan (a x)}}{\left (a^2 x^2+1\right )^3}dx+\frac {(6 a x+1) e^{\arctan (a x)}}{37 a \left (a^2 x^2+1\right )^3}\right )}{65 c^5}+\frac {(8 a x+1) e^{\arctan (a x)}}{65 a c^5 \left (a^2 x^2+1\right )^4}\)

\(\Big \downarrow \) 5593

\(\displaystyle \frac {56 \left (\frac {30}{37} \left (\frac {12}{17} \int \frac {e^{\arctan (a x)}}{\left (a^2 x^2+1\right )^2}dx+\frac {(4 a x+1) e^{\arctan (a x)}}{17 a \left (a^2 x^2+1\right )^2}\right )+\frac {(6 a x+1) e^{\arctan (a x)}}{37 a \left (a^2 x^2+1\right )^3}\right )}{65 c^5}+\frac {(8 a x+1) e^{\arctan (a x)}}{65 a c^5 \left (a^2 x^2+1\right )^4}\)

\(\Big \downarrow \) 5593

\(\displaystyle \frac {56 \left (\frac {30}{37} \left (\frac {12}{17} \left (\frac {2}{5} \int \frac {e^{\arctan (a x)}}{a^2 x^2+1}dx+\frac {(2 a x+1) e^{\arctan (a x)}}{5 a \left (a^2 x^2+1\right )}\right )+\frac {(4 a x+1) e^{\arctan (a x)}}{17 a \left (a^2 x^2+1\right )^2}\right )+\frac {(6 a x+1) e^{\arctan (a x)}}{37 a \left (a^2 x^2+1\right )^3}\right )}{65 c^5}+\frac {(8 a x+1) e^{\arctan (a x)}}{65 a c^5 \left (a^2 x^2+1\right )^4}\)

\(\Big \downarrow \) 5594

\(\displaystyle \frac {(8 a x+1) e^{\arctan (a x)}}{65 a c^5 \left (a^2 x^2+1\right )^4}+\frac {56 \left (\frac {(6 a x+1) e^{\arctan (a x)}}{37 a \left (a^2 x^2+1\right )^3}+\frac {30}{37} \left (\frac {(4 a x+1) e^{\arctan (a x)}}{17 a \left (a^2 x^2+1\right )^2}+\frac {12}{17} \left (\frac {(2 a x+1) e^{\arctan (a x)}}{5 a \left (a^2 x^2+1\right )}+\frac {2 e^{\arctan (a x)}}{5 a}\right )\right )\right )}{65 c^5}\)

Input:

Int[E^ArcTan[a*x]/(c + a^2*c*x^2)^5,x]
 

Output:

(E^ArcTan[a*x]*(1 + 8*a*x))/(65*a*c^5*(1 + a^2*x^2)^4) + (56*((E^ArcTan[a* 
x]*(1 + 6*a*x))/(37*a*(1 + a^2*x^2)^3) + (30*((E^ArcTan[a*x]*(1 + 4*a*x))/ 
(17*a*(1 + a^2*x^2)^2) + (12*((2*E^ArcTan[a*x])/(5*a) + (E^ArcTan[a*x]*(1 
+ 2*a*x))/(5*a*(1 + a^2*x^2))))/17))/37))/(65*c^5)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 5593

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> S 
imp[(n - 2*a*(p + 1)*x)*(c + d*x^2)^(p + 1)*(E^(n*ArcTan[a*x])/(a*c*(n^2 + 
4*(p + 1)^2))), x] + Simp[2*(p + 1)*((2*p + 3)/(c*(n^2 + 4*(p + 1)^2)))   I 
nt[(c + d*x^2)^(p + 1)*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] 
&& EqQ[d, a^2*c] && LtQ[p, -1] &&  !IntegerQ[I*n] && NeQ[n^2 + 4*(p + 1)^2, 
 0] && IntegerQ[2*p]
 

rule 5594

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[E 
^(n*ArcTan[a*x])/(a*c*n), x] /; FreeQ[{a, c, d, n}, x] && EqQ[d, a^2*c]
 

Maple [A] (verified)

Time = 100.00 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.58

method	result	size
gosper	\(\frac {{\mathrm e}^{\arctan \left (a x \right )} \left (8064 a^{8} x^{8}+8064 a^{7} x^{7}+36288 x^{6} a^{6}+30912 a^{5} x^{5}+62160 a^{4} x^{4}+43344 a^{3} x^{3}+48664 a^{2} x^{2}+25528 a x +15357\right )}{40885 \left (a^{2} x^{2}+1\right )^{4} c^{5} a}\)	\(87\)
orering	\(\frac {\left (8064 a^{8} x^{8}+8064 a^{7} x^{7}+36288 x^{6} a^{6}+30912 a^{5} x^{5}+62160 a^{4} x^{4}+43344 a^{3} x^{3}+48664 a^{2} x^{2}+25528 a x +15357\right ) \left (a^{2} x^{2}+1\right ) {\mathrm e}^{\arctan \left (a x \right )}}{40885 a \left (a^{2} c \,x^{2}+c \right )^{5}}\)	\(94\)
parallelrisch	\(\frac {8064 a^{8} {\mathrm e}^{\arctan \left (a x \right )} x^{8}+8064 a^{7} {\mathrm e}^{\arctan \left (a x \right )} x^{7}+36288 a^{6} {\mathrm e}^{\arctan \left (a x \right )} x^{6}+30912 a^{5} {\mathrm e}^{\arctan \left (a x \right )} x^{5}+62160 a^{4} {\mathrm e}^{\arctan \left (a x \right )} x^{4}+43344 a^{3} {\mathrm e}^{\arctan \left (a x \right )} x^{3}+48664 a^{2} {\mathrm e}^{\arctan \left (a x \right )} x^{2}+25528 \,{\mathrm e}^{\arctan \left (a x \right )} a x +15357 \,{\mathrm e}^{\arctan \left (a x \right )}}{40885 c^{5} \left (a^{2} x^{2}+1\right )^{4} a}\)	\(128\)

Input:

int(exp(arctan(a*x))/(a^2*c*x^2+c)^5,x,method=_RETURNVERBOSE)
 

Output:

1/40885*exp(arctan(a*x))*(8064*a^8*x^8+8064*a^7*x^7+36288*a^6*x^6+30912*a^ 
5*x^5+62160*a^4*x^4+43344*a^3*x^3+48664*a^2*x^2+25528*a*x+15357)/(a^2*x^2+ 
1)^4/c^5/a
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.81 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^5} \, dx=\frac {{\left (8064 \, a^{8} x^{8} + 8064 \, a^{7} x^{7} + 36288 \, a^{6} x^{6} + 30912 \, a^{5} x^{5} + 62160 \, a^{4} x^{4} + 43344 \, a^{3} x^{3} + 48664 \, a^{2} x^{2} + 25528 \, a x + 15357\right )} e^{\left (\arctan \left (a x\right )\right )}}{40885 \, {\left (a^{9} c^{5} x^{8} + 4 \, a^{7} c^{5} x^{6} + 6 \, a^{5} c^{5} x^{4} + 4 \, a^{3} c^{5} x^{2} + a c^{5}\right )}} \] Input:

integrate(exp(arctan(a*x))/(a^2*c*x^2+c)^5,x, algorithm="fricas")
 

Output:

1/40885*(8064*a^8*x^8 + 8064*a^7*x^7 + 36288*a^6*x^6 + 30912*a^5*x^5 + 621 
60*a^4*x^4 + 43344*a^3*x^3 + 48664*a^2*x^2 + 25528*a*x + 15357)*e^(arctan( 
a*x))/(a^9*c^5*x^8 + 4*a^7*c^5*x^6 + 6*a^5*c^5*x^4 + 4*a^3*c^5*x^2 + a*c^5 
)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (139) = 278\).

Time = 23.45 (sec) , antiderivative size = 620, normalized size of antiderivative = 4.16 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^5} \, dx=\begin {cases} \frac {8064 a^{8} x^{8} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {8064 a^{7} x^{7} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {36288 a^{6} x^{6} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {30912 a^{5} x^{5} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {62160 a^{4} x^{4} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {43344 a^{3} x^{3} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {48664 a^{2} x^{2} e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {25528 a x e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} + \frac {15357 e^{\operatorname {atan}{\left (a x \right )}}}{40885 a^{9} c^{5} x^{8} + 163540 a^{7} c^{5} x^{6} + 245310 a^{5} c^{5} x^{4} + 163540 a^{3} c^{5} x^{2} + 40885 a c^{5}} & \text {for}\: a \neq 0 \\\frac {x}{c^{5}} & \text {otherwise} \end {cases} \] Input:

integrate(exp(atan(a*x))/(a**2*c*x**2+c)**5,x)
 

Output:

Piecewise((8064*a**8*x**8*exp(atan(a*x))/(40885*a**9*c**5*x**8 + 163540*a* 
*7*c**5*x**6 + 245310*a**5*c**5*x**4 + 163540*a**3*c**5*x**2 + 40885*a*c** 
5) + 8064*a**7*x**7*exp(atan(a*x))/(40885*a**9*c**5*x**8 + 163540*a**7*c** 
5*x**6 + 245310*a**5*c**5*x**4 + 163540*a**3*c**5*x**2 + 40885*a*c**5) + 3 
6288*a**6*x**6*exp(atan(a*x))/(40885*a**9*c**5*x**8 + 163540*a**7*c**5*x** 
6 + 245310*a**5*c**5*x**4 + 163540*a**3*c**5*x**2 + 40885*a*c**5) + 30912* 
a**5*x**5*exp(atan(a*x))/(40885*a**9*c**5*x**8 + 163540*a**7*c**5*x**6 + 2 
45310*a**5*c**5*x**4 + 163540*a**3*c**5*x**2 + 40885*a*c**5) + 62160*a**4* 
x**4*exp(atan(a*x))/(40885*a**9*c**5*x**8 + 163540*a**7*c**5*x**6 + 245310 
*a**5*c**5*x**4 + 163540*a**3*c**5*x**2 + 40885*a*c**5) + 43344*a**3*x**3* 
exp(atan(a*x))/(40885*a**9*c**5*x**8 + 163540*a**7*c**5*x**6 + 245310*a**5 
*c**5*x**4 + 163540*a**3*c**5*x**2 + 40885*a*c**5) + 48664*a**2*x**2*exp(a 
tan(a*x))/(40885*a**9*c**5*x**8 + 163540*a**7*c**5*x**6 + 245310*a**5*c**5 
*x**4 + 163540*a**3*c**5*x**2 + 40885*a*c**5) + 25528*a*x*exp(atan(a*x))/( 
40885*a**9*c**5*x**8 + 163540*a**7*c**5*x**6 + 245310*a**5*c**5*x**4 + 163 
540*a**3*c**5*x**2 + 40885*a*c**5) + 15357*exp(atan(a*x))/(40885*a**9*c**5 
*x**8 + 163540*a**7*c**5*x**6 + 245310*a**5*c**5*x**4 + 163540*a**3*c**5*x 
**2 + 40885*a*c**5), Ne(a, 0)), (x/c**5, True))
 

Maxima [F]

\[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^5} \, dx=\int { \frac {e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{5}} \,d x } \] Input:

integrate(exp(arctan(a*x))/(a^2*c*x^2+c)^5,x, algorithm="maxima")
 

Output:

integrate(e^(arctan(a*x))/(a^2*c*x^2 + c)^5, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (134) = 268\).

Time = 0.21 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.62 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^5} \, dx =\text {Too large to display} \] Input:

integrate(exp(arctan(a*x))/(a^2*c*x^2+c)^5,x, algorithm="giac")
 

Output:

1/40885*(15357*e^(arctan(a*x))*tan(1/2*arctan(a*x))^16 - 51056*e^(arctan(a 
*x))*tan(1/2*arctan(a*x))^15 + 71800*e^(arctan(a*x))*tan(1/2*arctan(a*x))^ 
14 + 10640*e^(arctan(a*x))*tan(1/2*arctan(a*x))^13 + 256620*e^(arctan(a*x) 
)*tan(1/2*arctan(a*x))^12 - 327600*e^(arctan(a*x))*tan(1/2*arctan(a*x))^11 
 + 404040*e^(arctan(a*x))*tan(1/2*arctan(a*x))^10 + 254800*e^(arctan(a*x)) 
*tan(1/2*arctan(a*x))^9 + 568750*e^(arctan(a*x))*tan(1/2*arctan(a*x))^8 - 
254800*e^(arctan(a*x))*tan(1/2*arctan(a*x))^7 + 404040*e^(arctan(a*x))*tan 
(1/2*arctan(a*x))^6 + 327600*e^(arctan(a*x))*tan(1/2*arctan(a*x))^5 + 2566 
20*e^(arctan(a*x))*tan(1/2*arctan(a*x))^4 - 10640*e^(arctan(a*x))*tan(1/2* 
arctan(a*x))^3 + 71800*e^(arctan(a*x))*tan(1/2*arctan(a*x))^2 + 51056*e^(a 
rctan(a*x))*tan(1/2*arctan(a*x)) + 15357*e^(arctan(a*x)))/(a*c^5*tan(1/2*a 
rctan(a*x))^16 + 8*a*c^5*tan(1/2*arctan(a*x))^14 + 28*a*c^5*tan(1/2*arctan 
(a*x))^12 + 56*a*c^5*tan(1/2*arctan(a*x))^10 + 70*a*c^5*tan(1/2*arctan(a*x 
))^8 + 56*a*c^5*tan(1/2*arctan(a*x))^6 + 28*a*c^5*tan(1/2*arctan(a*x))^4 + 
 8*a*c^5*tan(1/2*arctan(a*x))^2 + a*c^5)
 

Mupad [B] (verification not implemented)

Time = 23.50 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^5} \, dx=\frac {{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (\frac {15357}{40885\,a^9\,c^5}+\frac {25528\,x}{40885\,a^8\,c^5}+\frac {8064\,x^8}{40885\,a\,c^5}+\frac {8064\,x^7}{40885\,a^2\,c^5}+\frac {36288\,x^6}{40885\,a^3\,c^5}+\frac {30912\,x^5}{40885\,a^4\,c^5}+\frac {336\,x^4}{221\,a^5\,c^5}+\frac {43344\,x^3}{40885\,a^6\,c^5}+\frac {48664\,x^2}{40885\,a^7\,c^5}\right )}{\frac {1}{a^8}+x^8+\frac {4\,x^6}{a^2}+\frac {6\,x^4}{a^4}+\frac {4\,x^2}{a^6}} \] Input:

int(exp(atan(a*x))/(c + a^2*c*x^2)^5,x)
 

Output:

(exp(atan(a*x))*(15357/(40885*a^9*c^5) + (25528*x)/(40885*a^8*c^5) + (8064 
*x^8)/(40885*a*c^5) + (8064*x^7)/(40885*a^2*c^5) + (36288*x^6)/(40885*a^3* 
c^5) + (30912*x^5)/(40885*a^4*c^5) + (336*x^4)/(221*a^5*c^5) + (43344*x^3) 
/(40885*a^6*c^5) + (48664*x^2)/(40885*a^7*c^5)))/(1/a^8 + x^8 + (4*x^6)/a^ 
2 + (6*x^4)/a^4 + (4*x^2)/a^6)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.74 \[ \int \frac {e^{\arctan (a x)}}{\left (c+a^2 c x^2\right )^5} \, dx=\frac {e^{\mathit {atan} \left (a x \right )} \left (8064 a^{8} x^{8}+8064 a^{7} x^{7}+36288 a^{6} x^{6}+30912 a^{5} x^{5}+62160 a^{4} x^{4}+43344 a^{3} x^{3}+48664 a^{2} x^{2}+25528 a x +15357\right )}{40885 a \,c^{5} \left (a^{8} x^{8}+4 a^{6} x^{6}+6 a^{4} x^{4}+4 a^{2} x^{2}+1\right )} \] Input:

int(exp(atan(a*x))/(a^2*c*x^2+c)^5,x)
 

Output:

(e**atan(a*x)*(8064*a**8*x**8 + 8064*a**7*x**7 + 36288*a**6*x**6 + 30912*a 
**5*x**5 + 62160*a**4*x**4 + 43344*a**3*x**3 + 48664*a**2*x**2 + 25528*a*x 
 + 15357))/(40885*a*c**5*(a**8*x**8 + 4*a**6*x**6 + 6*a**4*x**4 + 4*a**2*x 
**2 + 1))