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

Optimal. Leaf size=180 \[ -\frac {1}{2} \log \left (-\frac {-\sqrt {3} \sqrt {-x^2-2 x+3}-x+3}{x^2}\right )+\frac {1}{14} \left (7+\sqrt {7}\right ) \log \left (-\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {7}+\sqrt {3}+1\right )+\frac {1}{14} \left (7-\sqrt {7}\right ) \log \left (-\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}+\sqrt {7}+\sqrt {3}+1\right )+\tan ^{-1}\left (\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.20, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1074, 632, 31, 635, 203, 260} \[ -\frac {1}{2} \log \left (-\frac {-\sqrt {3} \sqrt {-x^2-2 x+3}-x+3}{x^2}\right )+\frac {1}{14} \left (7+\sqrt {7}\right ) \log \left (-\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {7}+\sqrt {3}+1\right )+\frac {1}{14} \left (7-\sqrt {7}\right ) \log \left (-\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}+\sqrt {7}+\sqrt {3}+1\right )+\tan ^{-1}\left (\frac {\sqrt {3}-\sqrt {-x^2-2 x+3}}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(x + Sqrt[3 - 2*x - x^2])^(-1),x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 203

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 1074

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (f_.)*(x_)^2)), x_Symbol]
:> With[{q = c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2}, Dist[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] + Dist[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, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {1}{x+\sqrt {3-2 x-x^2}} \, dx &=2 \operatorname {Subst}\left (\int \frac {\sqrt {3}-2 x-\sqrt {3} x^2}{\left (1+x^2\right ) \left (2-\sqrt {3}+2 \left (1+\sqrt {3}\right ) x+\sqrt {3} x^2\right )} \, dx,x,\frac {-\sqrt {3}+\sqrt {3-2 x-x^2}}{x}\right )\\ &=\frac {1}{16} \operatorname {Subst}\left (\int \frac {-6+2 \sqrt {3} \left (2-\sqrt {3}\right )-4 \left (1+\sqrt {3}\right )-\left (-2 \sqrt {3}+2 \left (2-\sqrt {3}\right )+4 \sqrt {3} \left (1+\sqrt {3}\right )\right ) x}{1+x^2} \, dx,x,\frac {-\sqrt {3}+\sqrt {3-2 x-x^2}}{x}\right )+\frac {1}{16} \operatorname {Subst}\left (\int \frac {3 \sqrt {3}-\sqrt {3} \left (2-\sqrt {3}\right )^2+4 \left (2-\sqrt {3}\right ) \left (1+\sqrt {3}\right )+4 \sqrt {3} \left (1+\sqrt {3}\right )^2+\sqrt {3} \left (-2 \sqrt {3}+2 \left (2-\sqrt {3}\right )+4 \sqrt {3} \left (1+\sqrt {3}\right )\right ) x}{2-\sqrt {3}+2 \left (1+\sqrt {3}\right ) x+\sqrt {3} x^2} \, dx,x,\frac {-\sqrt {3}+\sqrt {3-2 x-x^2}}{x}\right )\\ &=-\left (\frac {1}{2} \left (\sqrt {\frac {3}{7}} \left (1-\sqrt {7}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {3}+\sqrt {7}+\sqrt {3} x} \, dx,x,\frac {-\sqrt {3}+\sqrt {3-2 x-x^2}}{x}\right )\right )+\frac {1}{2} \left (\sqrt {\frac {3}{7}} \left (1+\sqrt {7}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {3}-\sqrt {7}+\sqrt {3} x} \, dx,x,\frac {-\sqrt {3}+\sqrt {3-2 x-x^2}}{x}\right )-\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {-\sqrt {3}+\sqrt {3-2 x-x^2}}{x}\right )-\operatorname {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\frac {-\sqrt {3}+\sqrt {3-2 x-x^2}}{x}\right )\\ &=-\tan ^{-1}\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 )\\ \end {align*}

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Mathematica [A]  time = 0.38, size = 197, normalized size = 1.09 \[ \frac {1}{28} \left (-\sqrt {14 \left (4+\sqrt {7}\right )} \tanh ^{-1}\left (\frac {\left (\sqrt {7}-1\right ) x+\sqrt {7}+7}{\sqrt {2 \left (4+\sqrt {7}\right )} \sqrt {-x^2-2 x+3}}\right )-\sqrt {56-14 \sqrt {7}} \tanh ^{-1}\left (\frac {\sqrt {7} x+x+\sqrt {7}-7}{\sqrt {2} \sqrt {\left (\sqrt {7}-4\right ) \left (x^2+2 x-3\right )}}\right )-\sqrt {7} \log \left (2 x-\sqrt {7}+1\right )+7 \log \left (2 x-\sqrt {7}+1\right )+\sqrt {7} \log \left (2 x+\sqrt {7}+1\right )+7 \log \left (2 x+\sqrt {7}+1\right )+14 \sin ^{-1}\left (\frac {x+1}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(14*ArcSin[(1 + x)/2] - Sqrt[14*(4 + Sqrt[7])]*ArcTanh[(7 + Sqrt[7] + (-1 + Sqrt[7])*x)/(Sqrt[2*(4 + Sqrt[7])]
*Sqrt[3 - 2*x - x^2])] - Sqrt[56 - 14*Sqrt[7]]*ArcTanh[(-7 + Sqrt[7] + x + Sqrt[7]*x)/(Sqrt[2]*Sqrt[(-4 + Sqrt
[7])*(-3 + 2*x + x^2)])] + 7*Log[1 - Sqrt[7] + 2*x] - Sqrt[7]*Log[1 - Sqrt[7] + 2*x] + 7*Log[1 + Sqrt[7] + 2*x
] + Sqrt[7]*Log[1 + Sqrt[7] + 2*x])/28

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fricas [B]  time = 0.46, size = 372, normalized size = 2.07 \[ \frac {1}{56} \, \sqrt {7} \log \left (\frac {24 \, x^{4} + 62 \, x^{3} - 153 \, x^{2} + 2 \, \sqrt {7} {\left (3 \, x^{4} + x^{3} - 45 \, x^{2} + 45 \, x\right )} - {\left (14 \, x^{3} - 84 \, x^{2} + \sqrt {7} {\left (8 \, x^{3} - 30 \, x^{2} + 27 \, x - 27\right )} + 126 \, x\right )} \sqrt {-x^{2} - 2 \, x + 3} + 180 \, x - 135}{4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9}\right ) + \frac {1}{56} \, \sqrt {7} \log \left (\frac {24 \, x^{4} + 62 \, x^{3} - 153 \, x^{2} - 2 \, \sqrt {7} {\left (3 \, x^{4} + x^{3} - 45 \, x^{2} + 45 \, x\right )} + {\left (14 \, x^{3} - 84 \, x^{2} - \sqrt {7} {\left (8 \, x^{3} - 30 \, x^{2} + 27 \, x - 27\right )} + 126 \, x\right )} \sqrt {-x^{2} - 2 \, x + 3} + 180 \, x - 135}{4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9}\right ) + \frac {1}{28} \, \sqrt {7} \log \left (\frac {2 \, x^{2} + \sqrt {7} {\left (2 \, x + 1\right )} + 2 \, x + 4}{2 \, x^{2} + 2 \, x - 3}\right ) - \frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{2} - 2 \, x + 3} {\left (x + 1\right )}}{x^{2} + 2 \, x - 3}\right ) + \frac {1}{4} \, \log \left (2 \, x^{2} + 2 \, x - 3\right ) - \frac {1}{8} \, \log \left (\frac {2 \, \sqrt {-x^{2} - 2 \, x + 3} x + 2 \, x - 3}{x^{2}}\right ) + \frac {1}{8} \, \log \left (-\frac {2 \, \sqrt {-x^{2} - 2 \, x + 3} x - 2 \, x + 3}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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 + sqr
t(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)) + 1/56*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)

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giac [B]  time = 0.52, size = 287, normalized size = 1.59 \[ -\frac {1}{28} \, \sqrt {7} \log \left (\frac {{\left | 4 \, x - 2 \, \sqrt {7} + 2 \right |}}{{\left | 4 \, x + 2 \, \sqrt {7} + 2 \right |}}\right ) + \frac {1}{28} \, \sqrt {7} \log \left (\frac {{\left | -2 \, \sqrt {7} + \frac {6 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}{{\left | 2 \, \sqrt {7} + \frac {6 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}\right ) - \frac {1}{28} \, \sqrt {7} \log \left (\frac {{\left | -2 \, \sqrt {7} + \frac {2 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}{{\left | 2 \, \sqrt {7} + \frac {2 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}\right ) + \frac {1}{2} \, \arcsin \left (\frac {1}{2} \, x + \frac {1}{2}\right ) + \frac {1}{4} \, \log \left ({\left | 2 \, x^{2} + 2 \, x - 3 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | \frac {4 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac {3 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} - 1 \right |}\right ) - \frac {1}{4} \, \log \left ({\left | -\frac {4 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac {{\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/28*sqrt(7)*log(abs(4*x - 2*sqrt(7) + 2)/abs(4*x + 2*sqrt(7) + 2)) + 1/28*sqrt(7)*log(abs(-2*sqrt(7) + 6*(sq
rt(-x^2 - 2*x + 3) - 2)/(x + 1) + 4)/abs(2*sqrt(7) + 6*(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) + 4)) - 1/28*sqrt(7)
*log(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))

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maple [B]  time = 0.07, size = 551, normalized size = 3.06 \[ \frac {\sqrt {7}\, \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 )}{-\frac {7}{2}+\frac {7 \sqrt {7}}{2}}-\frac {\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 )}{4 \left (-\frac {1}{2}+\frac {\sqrt {7}}{2}\right )}-\frac {\sqrt {7}\, \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 )}{7 \left (\frac {1}{2}+\frac {\sqrt {7}}{2}\right )}-\frac {\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 )}{4 \left (\frac {1}{2}+\frac {\sqrt {7}}{2}\right )}+\frac {\sqrt {7}\, \arctanh \left (\frac {\left (4 x +2\right ) \sqrt {7}}{14}\right )}{14}+\frac {\sqrt {7}\, \arcsin \left (\frac {x +1}{\sqrt {2-\frac {\sqrt {7}}{2}+\frac {\left (-1-\sqrt {7}\right )^{2}}{4}}}\right )}{28}+\frac {\arcsin \left (\frac {x +1}{\sqrt {2-\frac {\sqrt {7}}{2}+\frac {\left (-1-\sqrt {7}\right )^{2}}{4}}}\right )}{4}-\frac {\sqrt {7}\, \arcsin \left (\frac {x +1}{\sqrt {2+\frac {\sqrt {7}}{2}+\frac {\left (-1+\sqrt {7}\right )^{2}}{4}}}\right )}{28}+\frac {\arcsin \left (\frac {x +1}{\sqrt {2+\frac {\sqrt {7}}{2}+\frac {\left (-1+\sqrt {7}\right )^{2}}{4}}}\right )}{4}+\frac {\ln \left (2 x^{2}+2 x -3\right )}{4}+\frac {\sqrt {7}\, \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}}}{28}-\frac {\sqrt {7}\, \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}}}{28} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28*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)-1/28*arcsin(1/(2+
1/2*7^(1/2)+1/4*(-1+7^(1/2))^2)^(1/2)*(x+1))*7^(1/2)+1/4*arcsin(1/(2+1/2*7^(1/2)+1/4*(-1+7^(1/2))^2)^(1/2)*(x+
1))-1/7/(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))*7^(1/2)-1/4/(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))-1/28*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)+1/28*arcsin(1/(2-1/2*7^(1/2)+1/4*(-1-7^(1/2))^2)^(1/2)*(x+1))*7^(1/2)+1/4*arcsin(1/(2-1/2*7^(1/2)
+1/4*(-1-7^(1/2))^2)^(1/2)*(x+1))+1/7/(-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))*7^(1/2)-1/
4/(-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))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x + \sqrt {-x^{2} - 2 \, x + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x+\sqrt {-x^2-2\,x+3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x + (3 - x^2 - 2*x)^(1/2)),x)

[Out]

int(1/(x + (3 - x^2 - 2*x)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x + \sqrt {- x^{2} - 2 x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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