∫ x_________ 2−√ _ 3 +(1+√

Optimal. Leaf size=72 \[ \sqrt {\frac {1}{23} \left (13+8 \sqrt {3}\right )} \tanh ^{-1}\left (\frac {1+\sqrt {3}+2 x}{\sqrt {2 \left (-2+3 \sqrt {3}\right )}}\right )+\frac {1}{2} \log \left (2-\sqrt {3}+\left (1+\sqrt {3}\right ) x+x^2\right ) \]

1/2*ln(2+x^2-3^(1/2)+x*(1+3^(1/2)))+1/23*arctanh((1+2*x+3^(1/2))/(-4+6*3^(1/2))^(1/2))*(299+184*3^(1/2))^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {648, 632, 212, 642} \begin {gather*} \frac {1}{2} \log \left (x^2+\left (1+\sqrt {3}\right ) x-\sqrt {3}+2\right )+\sqrt {\frac {1}{23} \left (13+8 \sqrt {3}\right )} \tanh ^{-1}\left (\frac {2 x+\sqrt {3}+1}{\sqrt {2 \left (3 \sqrt {3}-2\right )}}\right ) \end {gather*}

Sqrt[(13 + 8*Sqrt[3])/23]*ArcTanh[(1 + Sqrt[3] + 2*x)/Sqrt[2*(-2 + 3*Sqrt[3])]] + Log[2 - Sqrt[3] + (1 + Sqrt[

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] && (GtQ[a, 0] || LtQ[b, 0])

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]

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +

b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), 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]

\begin {align*} \int \frac {x}{2-\sqrt {3}+\left (1+\sqrt {3}\right ) x+x^2} \, dx &=\frac {1}{2} \int \frac {1+\sqrt {3}+2 x}{2-\sqrt {3}+\left (1+\sqrt {3}\right ) x+x^2} \, dx+\frac {1}{2} \left (-1-\sqrt {3}\right ) \int \frac {1}{2-\sqrt {3}+\left (1+\sqrt {3}\right ) x+x^2} \, dx\\ &=\frac {1}{2} \log \left (2-\sqrt {3}+\left (1+\sqrt {3}\right ) x+x^2\right )+\left (1+\sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{-2 \left (2-3 \sqrt {3}\right )-x^2} \, dx,x,1+\sqrt {3}+2 x\right )\\ &=\sqrt {\frac {1}{23} \left (13+8 \sqrt {3}\right )} \tanh ^{-1}\left (\frac {1+\sqrt {3}+2 x}{\sqrt {2 \left (-2+3 \sqrt {3}\right )}}\right )+\frac {1}{2} \log \left (2-\sqrt {3}+\left (1+\sqrt {3}\right ) x+x^2\right )\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 72, normalized size = 1.00 \begin {gather*} \frac {\left (1+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {1+\sqrt {3}+2 x}{\sqrt {-4+6 \sqrt {3}}}\right )}{\sqrt {-4+6 \sqrt {3}}}+\frac {1}{2} \log \left (2-\sqrt {3}+x+\sqrt {3} x+x^2\right ) \end {gather*}

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

((1 + Sqrt[3])*ArcTanh[(1 + Sqrt[3] + 2*x)/Sqrt[-4 + 6*Sqrt[3]]])/Sqrt[-4 + 6*Sqrt[3]] + Log[2 - Sqrt[3] + x +

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method	result	size

default	\(\frac {\ln \left (x \sqrt {3}+x^{2}-\sqrt {3}+x +2\right )}{2}-\frac {2 \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right ) \arctanh \left (\frac {1+2 x +\sqrt {3}}{\sqrt {-4+6 \sqrt {3}}}\right )}{\sqrt {-4+6 \sqrt {3}}}\)	\(58\)

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

1/2*ln(x*3^(1/2)+x^2-3^(1/2)+x+2)-2*(-1/2-1/2*3^(1/2))/(-4+6*3^(1/2))^(1/2)*arctanh((1+2*x+3^(1/2))/(-4+6*3^(1

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Maxima [A]
time = 0.52, size = 77, normalized size = 1.07 \begin {gather*} -\frac {{\left (\sqrt {3} + 1\right )} \log \left (\frac {2 \, x + \sqrt {3} - \sqrt {6 \, \sqrt {3} - 4} + 1}{2 \, x + \sqrt {3} + \sqrt {6 \, \sqrt {3} - 4} + 1}\right )}{2 \, \sqrt {6 \, \sqrt {3} - 4}} + \frac {1}{2} \, \log \left (x^{2} + x {\left (\sqrt {3} + 1\right )} - \sqrt {3} + 2\right ) \end {gather*}

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

-1/2*(sqrt(3) + 1)*log((2*x + sqrt(3) - sqrt(6*sqrt(3) - 4) + 1)/(2*x + sqrt(3) + sqrt(6*sqrt(3) - 4) + 1))/sq

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (51) = 102\).
time = 0.34, size = 131, normalized size = 1.82 \begin {gather*} \frac {1}{46} \, \sqrt {23} \sqrt {8 \, \sqrt {3} + 13} \log \left (\frac {23 \, x^{4} + 46 \, x^{3} + \sqrt {23} {\left (11 \, x^{3} + 24 \, x^{2} - \sqrt {3} {\left (5 \, x^{3} + 13 \, x^{2} - 6 \, x - 4\right )} - 4 \, x + 5\right )} \sqrt {8 \, \sqrt {3} + 13} + 23 \, \sqrt {3} {\left (3 \, x^{2} + 5 \, x + 4\right )} - 23 \, x + 138}{x^{4} + 2 \, x^{3} + 2 \, x^{2} + 10 \, x + 1}\right ) + \frac {1}{2} \, \log \left (x^{2} + \sqrt {3} {\left (x - 1\right )} + x + 2\right ) \end {gather*}

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

1/46*sqrt(23)*sqrt(8*sqrt(3) + 13)*log((23*x^4 + 46*x^3 + sqrt(23)*(11*x^3 + 24*x^2 - sqrt(3)*(5*x^3 + 13*x^2

- 6*x - 4) - 4*x + 5)*sqrt(8*sqrt(3) + 13) + 23*sqrt(3)*(3*x^2 + 5*x + 4) - 23*x + 138)/(x^4 + 2*x^3 + 2*x^2 +

10*x + 1)) + 1/2*log(x^2 + sqrt(3)*(x - 1) + x + 2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (58) = 116\).
time = 0.75, size = 202, normalized size = 2.81 \begin {gather*} \left (\frac {\sqrt {5 + 4 \sqrt {3}}}{2 \cdot \left (2 - 3 \sqrt {3}\right )} + \frac {1}{2}\right ) \log {\left (x - \frac {5 \sqrt {3}}{5 + 4 \sqrt {3}} + \left (\frac {\sqrt {5 + 4 \sqrt {3}}}{2 \cdot \left (2 - 3 \sqrt {3}\right )} + \frac {1}{2}\right ) \left (\frac {47}{22 + 13 \sqrt {3}} + \frac {33 \sqrt {3}}{22 + 13 \sqrt {3}}\right ) + \frac {11}{5 + 4 \sqrt {3}} \right )} + \left (\frac {1}{2} - \frac {\sqrt {5 + 4 \sqrt {3}}}{2 \cdot \left (2 - 3 \sqrt {3}\right )}\right ) \log {\left (x - \frac {5 \sqrt {3}}{5 + 4 \sqrt {3}} + \frac {11}{5 + 4 \sqrt {3}} + \left (\frac {1}{2} - \frac {\sqrt {5 + 4 \sqrt {3}}}{2 \cdot \left (2 - 3 \sqrt {3}\right )}\right ) \left (\frac {47}{22 + 13 \sqrt {3}} + \frac {33 \sqrt {3}}{22 + 13 \sqrt {3}}\right ) \right )} \end {gather*}

(sqrt(5 + 4*sqrt(3))/(2*(2 - 3*sqrt(3))) + 1/2)*log(x - 5*sqrt(3)/(5 + 4*sqrt(3)) + (sqrt(5 + 4*sqrt(3))/(2*(2

- 3*sqrt(3))) + 1/2)*(47/(22 + 13*sqrt(3)) + 33*sqrt(3)/(22 + 13*sqrt(3))) + 11/(5 + 4*sqrt(3))) + (1/2 - sqr

t(5 + 4*sqrt(3))/(2*(2 - 3*sqrt(3))))*log(x - 5*sqrt(3)/(5 + 4*sqrt(3)) + 11/(5 + 4*sqrt(3)) + (1/2 - sqrt(5 +

4*sqrt(3))/(2*(2 - 3*sqrt(3))))*(47/(22 + 13*sqrt(3)) + 33*sqrt(3)/(22 + 13*sqrt(3))))

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Giac [A]
time = 5.07, size = 80, normalized size = 1.11 \begin {gather*} -\frac {{\left (\sqrt {3} + 1\right )} \log \left (\frac {{\left | 2 \, x + \sqrt {3} - \sqrt {6 \, \sqrt {3} - 4} + 1 \right |}}{{\left | 2 \, x + \sqrt {3} + \sqrt {6 \, \sqrt {3} - 4} + 1 \right |}}\right )}{2 \, \sqrt {6 \, \sqrt {3} - 4}} + \frac {1}{2} \, \log \left ({\left | x^{2} + x {\left (\sqrt {3} + 1\right )} - \sqrt {3} + 2 \right |}\right ) \end {gather*}

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

-1/2*(sqrt(3) + 1)*log(abs(2*x + sqrt(3) - sqrt(6*sqrt(3) - 4) + 1)/abs(2*x + sqrt(3) + sqrt(6*sqrt(3) - 4) +

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

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Mupad [B]
time = 4.23, size = 233, normalized size = 3.24 \begin {gather*} \ln \left (x-\left (\frac {\frac {\sqrt {\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}}{2}+\frac {\sqrt {3}\,\sqrt {\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}}{2}}{\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}+\frac {1}{2}\right )\,\left (2\,x+\sqrt {3}+1\right )\right )\,\left (\frac {\frac {\sqrt {\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}}{2}+\frac {\sqrt {3}\,\sqrt {\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}}{2}}{\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}+\frac {1}{2}\right )-\ln \left (x+\left (\frac {\frac {\sqrt {\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}}{2}+\frac {\sqrt {3}\,\sqrt {\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}}{2}}{\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}-\frac {1}{2}\right )\,\left (2\,x+\sqrt {3}+1\right )\right )\,\left (\frac {\frac {\sqrt {\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}}{2}+\frac {\sqrt {3}\,\sqrt {\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}}{2}}{\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+7\right )}-\frac {1}{2}\right ) \end {gather*}

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

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

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

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

1))*((((3^(1/2) - 1)*(3^(1/2) + 7))^(1/2)/2 + (3^(1/2)*((3^(1/2) - 1)*(3^(1/2) + 7))^(1/2))/2)/((3^(1/2) - 1)*

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3.10.59 \(\int \frac {x}{2-\sqrt {3}+(1+\sqrt {3}) x+x^2} \, dx\) [959]