\(\int \frac {1}{x+\sqrt {3-2 x-x^2}} \, dx\) [22]

Optimal result

Integrand size = 18, antiderivative size = 180 \[ \int \frac {1}{x+\sqrt {3-2 x-x^2}} \, dx=\arctan \left (\frac {\sqrt {3}-\sqrt {3-2 x-x^2}}{x}\right )-\frac {1}{2} \log \left (-\frac {3-x-\sqrt {3} \sqrt {3-2 x-x^2}}{x^2}\right )+\frac {1}{14} \left (7+\sqrt {7}\right ) \log \left (1+\sqrt {3}-\sqrt {7}-\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}\right )+\frac {1}{14} \left (7-\sqrt {7}\right ) \log \left (1+\sqrt {3}+\sqrt {7}-\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}\right ) \] Output:

arctan((3^(1/2)-(-x^2-2*x+3)^(1/2))/x)-1/2*ln(-(3-x-3^(1/2)*(-x^2-2*x+3)^( 
1/2))/x^2)+1/14*(7+7^(1/2))*ln(1+3^(1/2)-7^(1/2)-3^(1/2)*(3^(1/2)-(-x^2-2* 
x+3)^(1/2))/x)+1/14*(7-7^(1/2))*ln(1+3^(1/2)+7^(1/2)-3^(1/2)*(3^(1/2)-(-x^ 
2-2*x+3)^(1/2))/x)
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.62 \[ \int \frac {1}{x+\sqrt {3-2 x-x^2}} \, dx=\frac {1}{14} \left (-14 \arctan \left (\frac {\sqrt {3-2 x-x^2}}{3+x}\right )-7 \log (-1+x)-\left (-7+\sqrt {7}\right ) \log \left (-2+\sqrt {7} (-1+x)+2 x-\sqrt {3-2 x-x^2}\right )+\left (7+\sqrt {7}\right ) \log \left (2+\sqrt {7} (-1+x)-2 x+\sqrt {3-2 x-x^2}\right )\right ) \] Input:

Integrate[(x + Sqrt[3 - 2*x - x^2])^(-1),x]
 

Output:

(-14*ArcTan[Sqrt[3 - 2*x - x^2]/(3 + x)] - 7*Log[-1 + x] - (-7 + Sqrt[7])* 
Log[-2 + Sqrt[7]*(-1 + x) + 2*x - Sqrt[3 - 2*x - x^2]] + (7 + Sqrt[7])*Log 
[2 + Sqrt[7]*(-1 + x) - 2*x + Sqrt[3 - 2*x - x^2]])/14
 

Rubi [A] (warning: unable to verify)

Time = 0.52 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {7285, 2142, 27, 452, 216, 240, 1142, 27, 1083, 219, 1103}

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.

\(\displaystyle \int \frac {1}{\sqrt {-x^2-2 x+3}+x} \, dx\)

\(\Big \downarrow \) 7285

\(\displaystyle 2 \int \frac {-\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}+\frac {2 \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}+\sqrt {3}}{\left (\frac {\left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}+1\right ) \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )\)

\(\Big \downarrow \) 2142

\(\displaystyle 2 \left (\frac {1}{32} \int -\frac {16 \left (1-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )}{\frac {\left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}+1}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )+\frac {1}{32} \int \frac {16 \left (-\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}+\sqrt {3}+2\right )}{\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {1}{2} \int \frac {-\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}+\sqrt {3}+2}{\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )-\frac {1}{2} \int \frac {1-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}}{\frac {\left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}+1}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )\right )\)

\(\Big \downarrow \) 452

\(\displaystyle 2 \left (\frac {1}{2} \left (-\int \frac {1}{\frac {\left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}+1}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )-\int -\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x \left (\frac {\left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}+1\right )}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )\right )+\frac {1}{2} \int \frac {-\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}+\sqrt {3}+2}{\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )\right )\)

\(\Big \downarrow \) 216

\(\displaystyle 2 \left (\frac {1}{2} \left (\arctan \left (\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )-\int -\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x \left (\frac {\left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}+1\right )}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )\right )+\frac {1}{2} \int \frac {-\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}+\sqrt {3}+2}{\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )\right )\)

\(\Big \downarrow \) 240

\(\displaystyle 2 \left (\frac {1}{2} \int \frac {-\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}+\sqrt {3}+2}{\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )+\frac {1}{2} \left (\arctan \left (\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )-\frac {1}{2} \log \left (\frac {\left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}+1\right )\right )\right )\)

\(\Big \downarrow \) 1142

\(\displaystyle 2 \left (\frac {1}{2} \left (\int \frac {1}{\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )+\frac {1}{2} \int \frac {2 \left (-\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}+\sqrt {3}+1\right )}{\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )\right )+\frac {1}{2} \left (\arctan \left (\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )-\frac {1}{2} \log \left (\frac {\left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}+1\right )\right )\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {1}{2} \left (\int \frac {1}{\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )+\int \frac {-\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}+\sqrt {3}+1}{\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )\right )+\frac {1}{2} \left (\arctan \left (\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )-\frac {1}{2} \log \left (\frac {\left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}+1\right )\right )\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle 2 \left (\frac {1}{2} \left (\int \frac {-\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}+\sqrt {3}+1}{\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )-2 \int \frac {1}{28-\frac {\left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}}d\left (2 \left (1+\sqrt {3}\right )-\frac {2 \sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}\right )\right )+\frac {1}{2} \left (\arctan \left (\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )-\frac {1}{2} \log \left (\frac {\left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}+1\right )\right )\right )\)

\(\Big \downarrow \) 219

\(\displaystyle 2 \left (\frac {1}{2} \left (\int \frac {-\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}+\sqrt {3}+1}{\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2}d\left (-\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{2 \sqrt {7} x}\right )}{\sqrt {7}}\right )+\frac {1}{2} \left (\arctan \left (\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )-\frac {1}{2} \log \left (\frac {\left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}+1\right )\right )\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle 2 \left (\frac {1}{2} \left (\arctan \left (\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right )-\frac {1}{2} \log \left (\frac {\left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}+1\right )\right )+\frac {1}{2} \left (\frac {\text {arctanh}\left (\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{2 \sqrt {7} x}\right )}{\sqrt {7}}+\frac {1}{2} \log \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )\right )\right )\)

Input:

Int[(x + Sqrt[3 - 2*x - x^2])^(-1),x]
 

Output:

2*((ArcTan[(Sqrt[3] - Sqrt[3 - 2*x - x^2])/x] - Log[1 + (Sqrt[3] - Sqrt[3 
- 2*x - x^2])^2/x^2]/2)/2 + (ArcTanh[(Sqrt[3] - Sqrt[3 - 2*x - x^2])/(2*Sq 
rt[7]*x)]/Sqrt[7] + Log[2 - Sqrt[3] - (2*(1 + Sqrt[3])*(Sqrt[3] - Sqrt[3 - 
 2*x - x^2]))/x + (Sqrt[3]*(Sqrt[3] - Sqrt[3 - 2*x - x^2])^2)/x^2]/2)/2)
 

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 452

Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[c   Int[1/ 
(a + b*x^2), x], x] + Simp[d   Int[x/(a + b*x^2), x], x] /; FreeQ[{a, b, c, 
 d}, x] && NeQ[b*c^2 + a*d^2, 0]
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 2142

Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (f_.)*(x_)^2)), x_Sym 
bol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2] 
, q = c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2}, Simp[1/q   Int[(A*c^2*d - a 
*c*C*d + A*b^2*f - a*b*B*f - a*A*c*f + a^2*C*f + c*(B*c*d - b*C*d + A*b*f - 
 a*B*f)*x)/(a + b*x + c*x^2), x], x] + Simp[1/q   Int[(c*C*d^2 + b*B*d*f - 
A*c*d*f - a*C*d*f + a*A*f^2 - f*(B*c*d - b*C*d + A*b*f - a*B*f)*x)/(d + f*x 
^2), x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, f}, x] && PolyQ[Px, x, 2]
 

rule 7285

Int[u_, x_Symbol] :> With[{lst = FunctionOfSquareRootOfQuadratic[u, x]}, Si 
mp[2   Subst[Int[lst[[1]], x], x, lst[[2]]], x] /;  !FalseQ[lst] && EqQ[lst 
[[3]], 1]] /; EulerIntegrandQ[u, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(358\) vs. \(2(142)=284\).

Time = 0.54 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.99

method	result	size
default	\(\frac {\sqrt {7}\, \left (-\frac {\sqrt {-4 \left (x +\frac {1}{2}-\frac {\sqrt {7}}{2}\right )^{2}+4 \left (-1-\sqrt {7}\right ) \left (x +\frac {1}{2}-\frac {\sqrt {7}}{2}\right )+8-2 \sqrt {7}}}{4}-\frac {\left (-1-\sqrt {7}\right ) \arcsin \left (\frac {x +1}{\sqrt {2-\frac {\sqrt {7}}{2}+\frac {\left (-1-\sqrt {7}\right )^{2}}{4}}}\right )}{4}+\frac {\left (2-\frac {\sqrt {7}}{2}\right ) \operatorname {arctanh}\left (\frac {4-\sqrt {7}+\left (-1-\sqrt {7}\right ) \left (x +\frac {1}{2}-\frac {\sqrt {7}}{2}\right )}{\left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) \sqrt {-4 \left (x +\frac {1}{2}-\frac {\sqrt {7}}{2}\right )^{2}+4 \left (-1-\sqrt {7}\right ) \left (x +\frac {1}{2}-\frac {\sqrt {7}}{2}\right )+8-2 \sqrt {7}}}\right )}{-1+\sqrt {7}}\right )}{7}+\frac {\sqrt {7}\, \left (\frac {\sqrt {-4 \left (x +\frac {1}{2}+\frac {\sqrt {7}}{2}\right )^{2}+4 \left (-1+\sqrt {7}\right ) \left (x +\frac {1}{2}+\frac {\sqrt {7}}{2}\right )+8+2 \sqrt {7}}}{4}+\frac {\left (-1+\sqrt {7}\right ) \arcsin \left (\frac {x +1}{\sqrt {2+\frac {\sqrt {7}}{2}+\frac {\left (-1+\sqrt {7}\right )^{2}}{4}}}\right )}{4}-\frac {\left (2+\frac {\sqrt {7}}{2}\right ) \operatorname {arctanh}\left (\frac {4+\sqrt {7}+\left (-1+\sqrt {7}\right ) \left (x +\frac {1}{2}+\frac {\sqrt {7}}{2}\right )}{\left (\frac {1}{2}+\frac {\sqrt {7}}{2}\right ) \sqrt {-4 \left (x +\frac {1}{2}+\frac {\sqrt {7}}{2}\right )^{2}+4 \left (-1+\sqrt {7}\right ) \left (x +\frac {1}{2}+\frac {\sqrt {7}}{2}\right )+8+2 \sqrt {7}}}\right )}{2 \left (\frac {1}{2}+\frac {\sqrt {7}}{2}\right )}\right )}{7}+\frac {\ln \left (2 x^{2}+2 x -3\right )}{4}+\frac {\sqrt {7}\, \operatorname {arctanh}\left (\frac {\left (4 x +2\right ) \sqrt {7}}{14}\right )}{14}\)	\(359\)
trager	\(\text {Expression too large to display}\)	\(854\)

Input:

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

Output:

1/7*7^(1/2)*(-1/4*(-4*(x+1/2-1/2*7^(1/2))^2+4*(-1-7^(1/2))*(x+1/2-1/2*7^(1 
/2))+8-2*7^(1/2))^(1/2)-1/4*(-1-7^(1/2))*arcsin(1/(2-1/2*7^(1/2)+1/4*(-1-7 
^(1/2))^2)^(1/2)*(x+1))+1/2*(2-1/2*7^(1/2))/(-1/2+1/2*7^(1/2))*arctanh((4- 
7^(1/2)+(-1-7^(1/2))*(x+1/2-1/2*7^(1/2)))/(-1/2+1/2*7^(1/2))/(-4*(x+1/2-1/ 
2*7^(1/2))^2+4*(-1-7^(1/2))*(x+1/2-1/2*7^(1/2))+8-2*7^(1/2))^(1/2)))+1/7*7 
^(1/2)*(1/4*(-4*(x+1/2+1/2*7^(1/2))^2+4*(-1+7^(1/2))*(x+1/2+1/2*7^(1/2))+8 
+2*7^(1/2))^(1/2)+1/4*(-1+7^(1/2))*arcsin(1/(2+1/2*7^(1/2)+1/4*(-1+7^(1/2) 
)^2)^(1/2)*(x+1))-1/2*(2+1/2*7^(1/2))/(1/2+1/2*7^(1/2))*arctanh((4+7^(1/2) 
+(-1+7^(1/2))*(x+1/2+1/2*7^(1/2)))/(1/2+1/2*7^(1/2))/(-4*(x+1/2+1/2*7^(1/2 
))^2+4*(-1+7^(1/2))*(x+1/2+1/2*7^(1/2))+8+2*7^(1/2))^(1/2)))+1/4*ln(2*x^2+ 
2*x-3)+1/14*7^(1/2)*arctanh(1/14*(4*x+2)*7^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (136) = 272\).

Time = 0.12 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.07 \[ \int \frac {1}{x+\sqrt {3-2 x-x^2}} \, dx=\frac {1}{56} \, \sqrt {7} \log \left (\frac {24 \, x^{4} + 62 \, x^{3} - 153 \, x^{2} + 2 \, \sqrt {7} {\left (3 \, x^{4} + x^{3} - 45 \, x^{2} + 45 \, x\right )} - {\left (14 \, x^{3} - 84 \, x^{2} + \sqrt {7} {\left (8 \, x^{3} - 30 \, x^{2} + 27 \, x - 27\right )} + 126 \, x\right )} \sqrt {-x^{2} - 2 \, x + 3} + 180 \, x - 135}{4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9}\right ) + \frac {1}{56} \, \sqrt {7} \log \left (\frac {24 \, x^{4} + 62 \, x^{3} - 153 \, x^{2} - 2 \, \sqrt {7} {\left (3 \, x^{4} + x^{3} - 45 \, x^{2} + 45 \, x\right )} + {\left (14 \, x^{3} - 84 \, x^{2} - \sqrt {7} {\left (8 \, x^{3} - 30 \, x^{2} + 27 \, x - 27\right )} + 126 \, x\right )} \sqrt {-x^{2} - 2 \, x + 3} + 180 \, x - 135}{4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9}\right ) + \frac {1}{28} \, \sqrt {7} \log \left (\frac {2 \, x^{2} + \sqrt {7} {\left (2 \, x + 1\right )} + 2 \, x + 4}{2 \, x^{2} + 2 \, x - 3}\right ) - \frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{2} - 2 \, x + 3} {\left (x + 1\right )}}{x^{2} + 2 \, x - 3}\right ) + \frac {1}{4} \, \log \left (2 \, x^{2} + 2 \, x - 3\right ) - \frac {1}{8} \, \log \left (\frac {2 \, \sqrt {-x^{2} - 2 \, x + 3} x + 2 \, x - 3}{x^{2}}\right ) + \frac {1}{8} \, \log \left (-\frac {2 \, \sqrt {-x^{2} - 2 \, x + 3} x - 2 \, x + 3}{x^{2}}\right ) \] Input:

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

Output:

1/56*sqrt(7)*log((24*x^4 + 62*x^3 - 153*x^2 + 2*sqrt(7)*(3*x^4 + x^3 - 45* 
x^2 + 45*x) - (14*x^3 - 84*x^2 + sqrt(7)*(8*x^3 - 30*x^2 + 27*x - 27) + 12 
6*x)*sqrt(-x^2 - 2*x + 3) + 180*x - 135)/(4*x^4 + 8*x^3 - 8*x^2 - 12*x + 9 
)) + 1/56*sqrt(7)*log((24*x^4 + 62*x^3 - 153*x^2 - 2*sqrt(7)*(3*x^4 + x^3 
- 45*x^2 + 45*x) + (14*x^3 - 84*x^2 - sqrt(7)*(8*x^3 - 30*x^2 + 27*x - 27) 
 + 126*x)*sqrt(-x^2 - 2*x + 3) + 180*x - 135)/(4*x^4 + 8*x^3 - 8*x^2 - 12* 
x + 9)) + 1/28*sqrt(7)*log((2*x^2 + sqrt(7)*(2*x + 1) + 2*x + 4)/(2*x^2 + 
2*x - 3)) - 1/2*arctan(sqrt(-x^2 - 2*x + 3)*(x + 1)/(x^2 + 2*x - 3)) + 1/4 
*log(2*x^2 + 2*x - 3) - 1/8*log((2*sqrt(-x^2 - 2*x + 3)*x + 2*x - 3)/x^2) 
+ 1/8*log(-(2*sqrt(-x^2 - 2*x + 3)*x - 2*x + 3)/x^2)
 

Sympy [F]

\[ \int \frac {1}{x+\sqrt {3-2 x-x^2}} \, dx=\int \frac {1}{x + \sqrt {- x^{2} - 2 x + 3}}\, dx \] Input:

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

Output:

Integral(1/(x + sqrt(-x**2 - 2*x + 3)), x)
 

Maxima [F]

\[ \int \frac {1}{x+\sqrt {3-2 x-x^2}} \, dx=\int { \frac {1}{x + \sqrt {-x^{2} - 2 \, x + 3}} \,d x } \] Input:

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

Output:

integrate(1/(x + sqrt(-x^2 - 2*x + 3)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (136) = 272\).

Time = 0.16 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.59 \[ \int \frac {1}{x+\sqrt {3-2 x-x^2}} \, dx=-\frac {1}{28} \, \sqrt {7} \log \left (\frac {{\left | 4 \, x - 2 \, \sqrt {7} + 2 \right |}}{{\left | 4 \, x + 2 \, \sqrt {7} + 2 \right |}}\right ) + \frac {1}{28} \, \sqrt {7} \log \left (\frac {{\left | -2 \, \sqrt {7} + \frac {6 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}{{\left | 2 \, \sqrt {7} + \frac {6 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}\right ) - \frac {1}{28} \, \sqrt {7} \log \left (\frac {{\left | -2 \, \sqrt {7} + \frac {2 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}{{\left | 2 \, \sqrt {7} + \frac {2 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}\right ) + \frac {1}{2} \, \arcsin \left (\frac {1}{2} \, x + \frac {1}{2}\right ) + \frac {1}{4} \, \log \left ({\left | 2 \, x^{2} + 2 \, x - 3 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | \frac {4 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac {3 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} - 1 \right |}\right ) - \frac {1}{4} \, \log \left ({\left | -\frac {4 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac {{\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} - 3 \right |}\right ) \] Input:

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

Output:

-1/28*sqrt(7)*log(abs(4*x - 2*sqrt(7) + 2)/abs(4*x + 2*sqrt(7) + 2)) + 1/2 
8*sqrt(7)*log(abs(-2*sqrt(7) + 6*(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) + 4)/a 
bs(2*sqrt(7) + 6*(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) + 4)) - 1/28*sqrt(7)*l 
og(abs(-2*sqrt(7) + 2*(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) - 4)/abs(2*sqrt(7 
) + 2*(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) - 4)) + 1/2*arcsin(1/2*x + 1/2) + 
 1/4*log(abs(2*x^2 + 2*x - 3)) + 1/4*log(abs(4*(sqrt(-x^2 - 2*x + 3) - 2)/ 
(x + 1) + 3*(sqrt(-x^2 - 2*x + 3) - 2)^2/(x + 1)^2 - 1)) - 1/4*log(abs(-4* 
(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) + (sqrt(-x^2 - 2*x + 3) - 2)^2/(x + 1)^ 
2 - 3))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x+\sqrt {3-2 x-x^2}} \, dx=\int \frac {1}{x+\sqrt {-x^2-2\,x+3}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.37 \[ \int \frac {1}{x+\sqrt {3-2 x-x^2}} \, dx=\frac {\mathit {asin} \left (\frac {x}{2}+\frac {1}{2}\right )}{2}-\frac {\sqrt {7}\, \mathrm {log}\left (-\sqrt {7}+3 \tan \left (\frac {\mathit {asin} \left (\frac {x}{2}+\frac {1}{2}\right )}{2}\right )-2\right )}{14}+\frac {\sqrt {7}\, \mathrm {log}\left (\sqrt {7}+3 \tan \left (\frac {\mathit {asin} \left (\frac {x}{2}+\frac {1}{2}\right )}{2}\right )-2\right )}{14}+\frac {\mathrm {log}\left (\sqrt {-x^{2}-2 x +3}+x \right )}{2} \] Input:

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

Output:

(7*asin((x + 1)/2) - sqrt(7)*log( - sqrt(7) + 3*tan(asin((x + 1)/2)/2) - 2 
) + sqrt(7)*log(sqrt(7) + 3*tan(asin((x + 1)/2)/2) - 2) + 7*log(sqrt( - x* 
*2 - 2*x + 3) + x))/14