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[(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

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Rubi [A]  time = 0.195066, antiderivative size = 180, normalized size of antiderivative = 1., 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 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]

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 31

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

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 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]

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*}

Mathematica [A]  time = 0.388879, 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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Maple [B]  time = 0.059, size = 551, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

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)*(1+x))*7^(1/2)+1/4*arcsin(1/(2+1/2*7^(1/2)+1/4*(-1+7^(1/2))^2)^(1/2)*(1+
x))-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)*(1+x))*7^(1/2)+1/4*arcsin(1/(2-1/2*7^(1/2)
+1/4*(-1-7^(1/2))^2)^(1/2)*(1+x))+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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x + \sqrt{-x^{2} - 2 \, x + 3}}\,{d x} \end{align*}

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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Fricas [B]  time = 1.84741, size = 972, normalized size = 5.4 \begin{align*} \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 ) \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x + \sqrt{- x^{2} - 2 x + 3}}\, dx \end{align*}

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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Giac [B]  time = 1.29647, size = 387, normalized size = 2.15 \begin{align*} -\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 ) \end{align*}

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))