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

Optimal. Leaf size=172 \[ \frac{2 \left (\frac{3 \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-\sqrt{3}+4\right )}{7 \left (\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )^2}{x^2}-\frac{2 \left (1+\sqrt{3}\right ) \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-\sqrt{3}+2\right )}+\frac{8 \tanh ^{-1}\left (\frac{-\sqrt{3} \sqrt{-x^2-2 x+3}-\sqrt{3} x-x+3}{\sqrt{7} x}\right )}{7 \sqrt{7}} \]

[Out]

(2*(4 - Sqrt[3] + (3*(Sqrt[3] - Sqrt[3 - 2*x - x^2]))/x))/(7*(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)) + (8*ArcTanh[(3 - x - Sqrt[3]*x - Sqrt[3
]*Sqrt[3 - 2*x - x^2])/(Sqrt[7]*x)])/(7*Sqrt[7])

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Rubi [A]  time = 0.143687, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1660, 12, 618, 206} \[ \frac{2 \left (\frac{3 \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-\sqrt{3}+4\right )}{7 \left (\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )^2}{x^2}-\frac{2 \left (1+\sqrt{3}\right ) \left (\sqrt{3}-\sqrt{-x^2-2 x+3}\right )}{x}-\sqrt{3}+2\right )}+\frac{8 \tanh ^{-1}\left (\frac{-\sqrt{3} \sqrt{-x^2-2 x+3}-\sqrt{3} x-x+3}{\sqrt{7} x}\right )}{7 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[(x + Sqrt[3 - 2*x - x^2])^(-2),x]

[Out]

(2*(4 - Sqrt[3] + (3*(Sqrt[3] - Sqrt[3 - 2*x - x^2]))/x))/(7*(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)) + (8*ArcTanh[(3 - x - Sqrt[3]*x - Sqrt[3
]*Sqrt[3 - 2*x - x^2])/(Sqrt[7]*x)])/(7*Sqrt[7])

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{1}{\left (x+\sqrt{3-2 x-x^2}\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{-\sqrt{3}+2 x+\sqrt{3} x^2}{\left (2-\sqrt{3}+2 \left (1+\sqrt{3}\right ) x+\sqrt{3} x^2\right )^2} \, dx,x,\frac{-\sqrt{3}+\sqrt{3-2 x-x^2}}{x}\right )\\ &=\frac{2 \left (4-\sqrt{3}+\frac{3 \left (\sqrt{3}-\sqrt{3-2 x-x^2}\right )}{x}\right )}{7 \left (2-\sqrt{3}-\frac{2 \left (1+\sqrt{3}\right ) \left (\sqrt{3}-\sqrt{3-2 x-x^2}\right )}{x}+\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{3-2 x-x^2}\right )^2}{x^2}\right )}-\frac{1}{14} \operatorname{Subst}\left (\int -\frac{16}{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 )\\ &=\frac{2 \left (4-\sqrt{3}+\frac{3 \left (\sqrt{3}-\sqrt{3-2 x-x^2}\right )}{x}\right )}{7 \left (2-\sqrt{3}-\frac{2 \left (1+\sqrt{3}\right ) \left (\sqrt{3}-\sqrt{3-2 x-x^2}\right )}{x}+\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{3-2 x-x^2}\right )^2}{x^2}\right )}+\frac{8}{7} \operatorname{Subst}\left (\int \frac{1}{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 )\\ &=\frac{2 \left (4-\sqrt{3}+\frac{3 \left (\sqrt{3}-\sqrt{3-2 x-x^2}\right )}{x}\right )}{7 \left (2-\sqrt{3}-\frac{2 \left (1+\sqrt{3}\right ) \left (\sqrt{3}-\sqrt{3-2 x-x^2}\right )}{x}+\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{3-2 x-x^2}\right )^2}{x^2}\right )}-\frac{16}{7} \operatorname{Subst}\left (\int \frac{1}{28-x^2} \, dx,x,\frac{2 \left (-3+x+\sqrt{3} x+\sqrt{3} \sqrt{3-2 x-x^2}\right )}{x}\right )\\ &=\frac{2 \left (4-\sqrt{3}+\frac{3 \left (\sqrt{3}-\sqrt{3-2 x-x^2}\right )}{x}\right )}{7 \left (2-\sqrt{3}-\frac{2 \left (1+\sqrt{3}\right ) \left (\sqrt{3}-\sqrt{3-2 x-x^2}\right )}{x}+\frac{\sqrt{3} \left (\sqrt{3}-\sqrt{3-2 x-x^2}\right )^2}{x^2}\right )}+\frac{8 \tanh ^{-1}\left (\frac{3-x-\sqrt{3} x-\sqrt{3} \sqrt{3-2 x-x^2}}{\sqrt{7} x}\right )}{7 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.465425, size = 306, normalized size = 1.78 \[ \frac{1}{98} \left (\frac{7 (3-8 x)}{2 x^2+2 x-3}-\frac{14 (x-3) \sqrt{-x^2-2 x+3}}{2 x^2+2 x-3}-2 \left (1+\sqrt{7}\right ) \sqrt{\frac{14}{4+\sqrt{7}}} \log \left (\sqrt{14 \left (4+\sqrt{7}\right )} \sqrt{-x^2-2 x+3}-\sqrt{7} x+7 x+7 \sqrt{7}+7\right )-\frac{2}{3} \left (\sqrt{7}-1\right ) \sqrt{14 \left (4+\sqrt{7}\right )} \log \left (-\sqrt{14} \sqrt{\left (\sqrt{7}-4\right ) \left (x^2+2 x-3\right )}+\left (7+\sqrt{7}\right ) x-7 \sqrt{7}+7\right )-4 \sqrt{7} \log \left (-2 x+\sqrt{7}-1\right )+\frac{2}{3} \left (\sqrt{7}-1\right ) \sqrt{14 \left (4+\sqrt{7}\right )} \log \left (2 x-\sqrt{7}+1\right )+2 \left (1+\sqrt{7}\right ) \sqrt{\frac{14}{4+\sqrt{7}}} \log \left (2 x+\sqrt{7}+1\right )+4 \sqrt{7} \log \left (2 x+\sqrt{7}+1\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((7*(3 - 8*x))/(-3 + 2*x + 2*x^2) - (14*(-3 + x)*Sqrt[3 - 2*x - x^2])/(-3 + 2*x + 2*x^2) - 4*Sqrt[7]*Log[-1 +
Sqrt[7] - 2*x] + (2*(-1 + Sqrt[7])*Sqrt[14*(4 + Sqrt[7])]*Log[1 - Sqrt[7] + 2*x])/3 + 4*Sqrt[7]*Log[1 + Sqrt[7
] + 2*x] + 2*(1 + Sqrt[7])*Sqrt[14/(4 + Sqrt[7])]*Log[1 + Sqrt[7] + 2*x] - 2*(1 + Sqrt[7])*Sqrt[14/(4 + Sqrt[7
])]*Log[7 + 7*Sqrt[7] + 7*x - Sqrt[7]*x + Sqrt[14*(4 + Sqrt[7])]*Sqrt[3 - 2*x - x^2]] - (2*(-1 + Sqrt[7])*Sqrt
[14*(4 + Sqrt[7])]*Log[7 - 7*Sqrt[7] + (7 + Sqrt[7])*x - Sqrt[14]*Sqrt[(-4 + Sqrt[7])*(-3 + 2*x + x^2)]])/3)/9
8

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Maple [B]  time = 0.027, size = 1066, normalized size = 6.2 \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))^2,x)

[Out]

-3/28*(4*x+2)/(2*x^2+2*x-3)+4/49*7^(1/2)*arctanh(1/14*(4*x+2)*7^(1/2))+1/14*(-2*x+6)/(2*x^2+2*x-3)-2*(-1/14+1/
14*7^(1/2))*(-1/4/(2-1/2*7^(1/2))/(x+1/2-1/2*7^(1/2))*(-(x+1/2-1/2*7^(1/2))^2+(-1-7^(1/2))*(x+1/2-1/2*7^(1/2))
+2-1/2*7^(1/2))^(3/2)+1/8*(-1-7^(1/2))/(2-1/2*7^(1/2))*(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)+1/2*(-1-7^(1/2))*arcsin(1/(2-1/2*7^(1/2)+1/4*(-1-7^(1/2))^2)^(1/2)*(1+x))-(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)))-1/2/(2-1/2*7^(1/2))*(-1/4*(-2*x-2)*(-
(x+1/2-1/2*7^(1/2))^2+(-1-7^(1/2))*(x+1/2-1/2*7^(1/2))+2-1/2*7^(1/2))^(1/2)-1/8*(-8+2*7^(1/2)-(-1-7^(1/2))^2)*
arcsin(1/(2-1/2*7^(1/2)+1/4*(-1-7^(1/2))^2)^(1/2)*(1+x))))+1/49*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)*(1+x))-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)))-1/49*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)*(1+x))-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)))-2*(-1/14-1/14*7^(1/2))*(-1/4/(2+1/2*7^(1/2))/(x+1/2+1/2*7^(1/2))*(-(x+1/2+1/2*7^(1/2))^2+(-1+
7^(1/2))*(x+1/2+1/2*7^(1/2))+2+1/2*7^(1/2))^(3/2)+1/8*(-1+7^(1/2))/(2+1/2*7^(1/2))*(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)+1/2*(-1+7^(1/2))*arcsin(1/(2+1/2*7^(1/2)+1/4*(-1+7^(
1/2))^2)^(1/2)*(1+x))-(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)))-1/2/(2+1/2*
7^(1/2))*(-1/4*(-2*x-2)*(-(x+1/2+1/2*7^(1/2))^2+(-1+7^(1/2))*(x+1/2+1/2*7^(1/2))+2+1/2*7^(1/2))^(1/2)-1/8*(-8-
2*7^(1/2)-(-1+7^(1/2))^2)*arcsin(1/(2+1/2*7^(1/2)+1/4*(-1+7^(1/2))^2)^(1/2)*(1+x))))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x + \sqrt{-x^{2} - 2 \, x + 3}\right )}^{2}}\,{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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.78358, size = 440, normalized size = 2.56 \begin{align*} \frac{2 \, \sqrt{7}{\left (2 \, x^{2} + 2 \, x - 3\right )} \log \left (\frac{x^{4} + 44 \, x^{3} - \sqrt{7}{\left (3 \, x^{3} + x^{2} - 45 \, x + 45\right )} \sqrt{-x^{2} - 2 \, x + 3} + 26 \, x^{2} - 276 \, x + 207}{4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9}\right ) + 4 \, \sqrt{7}{\left (2 \, x^{2} + 2 \, x - 3\right )} \log \left (\frac{2 \, x^{2} + \sqrt{7}{\left (2 \, x + 1\right )} + 2 \, x + 4}{2 \, x^{2} + 2 \, x - 3}\right ) - 14 \, \sqrt{-x^{2} - 2 \, x + 3}{\left (x - 3\right )} - 56 \, x + 21}{98 \,{\left (2 \, x^{2} + 2 \, x - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/98*(2*sqrt(7)*(2*x^2 + 2*x - 3)*log((x^4 + 44*x^3 - sqrt(7)*(3*x^3 + x^2 - 45*x + 45)*sqrt(-x^2 - 2*x + 3) +
26*x^2 - 276*x + 207)/(4*x^4 + 8*x^3 - 8*x^2 - 12*x + 9)) + 4*sqrt(7)*(2*x^2 + 2*x - 3)*log((2*x^2 + sqrt(7)*
(2*x + 1) + 2*x + 4)/(2*x^2 + 2*x - 3)) - 14*sqrt(-x^2 - 2*x + 3)*(x - 3) - 56*x + 21)/(2*x^2 + 2*x - 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.23232, size = 473, normalized size = 2.75 \begin{align*} -\frac{2}{49} \, \sqrt{7} \log \left (\frac{{\left | 4 \, x - 2 \, \sqrt{7} + 2 \right |}}{{\left | 4 \, x + 2 \, \sqrt{7} + 2 \right |}}\right ) + \frac{2}{49} \, \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{2}{49} \, \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{8 \, x - 3}{14 \,{\left (2 \, x^{2} + 2 \, x - 3\right )}} - \frac{8 \,{\left (\frac{5 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac{26 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac{11 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{3}}{{\left (x + 1\right )}^{3}} - 6\right )}}{21 \,{\left (\frac{8 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac{26 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac{8 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{3}}{{\left (x + 1\right )}^{3}} - \frac{3 \,{\left (\sqrt{-x^{2} - 2 \, x + 3} - 2\right )}^{4}}{{\left (x + 1\right )}^{4}} - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/49*sqrt(7)*log(abs(4*x - 2*sqrt(7) + 2)/abs(4*x + 2*sqrt(7) + 2)) + 2/49*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)) - 2/49*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)) - 1/14*(8*x - 3)/(2*x^2 + 2*x - 3) - 8/21*(5*(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) + 26*(sqrt(-x^2 -
2*x + 3) - 2)^2/(x + 1)^2 + 11*(sqrt(-x^2 - 2*x + 3) - 2)^3/(x + 1)^3 - 6)/(8*(sqrt(-x^2 - 2*x + 3) - 2)/(x +
1) + 26*(sqrt(-x^2 - 2*x + 3) - 2)^2/(x + 1)^2 + 8*(sqrt(-x^2 - 2*x + 3) - 2)^3/(x + 1)^3 - 3*(sqrt(-x^2 - 2*
x + 3) - 2)^4/(x + 1)^4 - 3)