3.114 \(\int \frac{x}{4-x^2+\sqrt{4-x^2}} \, dx\)

Optimal. Leaf size=16 \[ -\log \left (\sqrt{4-x^2}+1\right ) \]

[Out]

-Log[1 + Sqrt[4 - x^2]]

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Rubi [A]  time = 0.0450431, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2155, 31} \[ -\log \left (\sqrt{4-x^2}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(4 - x^2 + Sqrt[4 - x^2]),x]

[Out]

-Log[1 + Sqrt[4 - x^2]]

Rule 2155

Int[(x_)^(m_.)/((c_) + (d_.)*(x_)^(n_) + (e_.)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Dist[1/n, Subst[Int
[x^((m + 1)/n - 1)/(c + d*x + e*Sqrt[a + b*x]), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[b*c
- a*d, 0] && IntegerQ[(m + 1)/n]

Rule 31

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

Rubi steps

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

Mathematica [A]  time = 0.0267052, size = 16, normalized size = 1. \[ -\log \left (\sqrt{4-x^2}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(4 - x^2 + Sqrt[4 - x^2]),x]

[Out]

-Log[1 + Sqrt[4 - x^2]]

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Maple [B]  time = 0.043, size = 266, normalized size = 16.6 \begin{align*} -{\frac{\ln \left ({x}^{2}-3 \right ) }{2}}+{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }{\it Artanh} \left ({\frac{2+2\,\sqrt{3} \left ( x+\sqrt{3} \right ) }{2}{\frac{1}{\sqrt{- \left ( x+\sqrt{3} \right ) ^{2}+2\,\sqrt{3} \left ( x+\sqrt{3} \right ) +1}}}} \right ) }-{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }\sqrt{- \left ( x+\sqrt{3} \right ) ^{2}+2\,\sqrt{3} \left ( x+\sqrt{3} \right ) +1}}+{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }\sqrt{- \left ( 2+x \right ) ^{2}+4\,x+8}}+{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }{\it Artanh} \left ({\frac{2-2\,\sqrt{3} \left ( x-\sqrt{3} \right ) }{2}{\frac{1}{\sqrt{- \left ( x-\sqrt{3} \right ) ^{2}-2\,\sqrt{3} \left ( x-\sqrt{3} \right ) +1}}}} \right ) }-{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }\sqrt{- \left ( x-\sqrt{3} \right ) ^{2}-2\,\sqrt{3} \left ( x-\sqrt{3} \right ) +1}}+{\frac{1}{ \left ( 4+2\,\sqrt{3} \right ) \left ( -2+\sqrt{3} \right ) }\sqrt{- \left ( -2+x \right ) ^{2}-4\,x+8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(4-x^2+(-x^2+4)^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.01516, size = 19, normalized size = 1.19 \begin{align*} -\log \left (\sqrt{-x^{2} + 4} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(4-x^2+(-x^2+4)^(1/2)),x, algorithm="maxima")

[Out]

-log(sqrt(-x^2 + 4) + 1)

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Fricas [B]  time = 0.824438, size = 144, normalized size = 9. \begin{align*} -\frac{1}{2} \, \log \left (x^{2} - 3\right ) + \frac{1}{2} \, \log \left (-\frac{x^{2} + 3 \, \sqrt{-x^{2} + 4} - 6}{x^{2}}\right ) - \frac{1}{2} \, \log \left (-\frac{x^{2} + \sqrt{-x^{2} + 4} - 2}{x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(4-x^2+(-x^2+4)^(1/2)),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 2.64185, size = 44, normalized size = 2.75 \begin{align*} \frac{\log{\left (2 \sqrt{4 - x^{2}} \right )}}{2} - \frac{\log{\left (2 \sqrt{4 - x^{2}} + 2 \right )}}{2} - \frac{\log{\left (x^{2} - \sqrt{4 - x^{2}} - 4 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(4-x**2+(-x**2+4)**(1/2)),x)

[Out]

log(2*sqrt(4 - x**2))/2 - log(2*sqrt(4 - x**2) + 2)/2 - log(x**2 - sqrt(4 - x**2) - 4)/2

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Giac [A]  time = 1.10885, size = 19, normalized size = 1.19 \begin{align*} -\log \left (\sqrt{-x^{2} + 4} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(4-x^2+(-x^2+4)^(1/2)),x, algorithm="giac")

[Out]

-log(sqrt(-x^2 + 4) + 1)