3.2.14 \(\int \frac {x}{4-x^2+\sqrt {4-x^2}} \, dx\) [114]

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

[Out]

-ln(1+(-x^2+4)^(1/2))

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Rubi [A]
time = 0.03, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 = {2186, 31} \begin {gather*} -\log \left (\sqrt {4-x^2}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(4 - x^2 + Sqrt[4 - x^2]),x]

[Out]

-Log[1 + Sqrt[4 - x^2]]

Rule 31

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

Rule 2186

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]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} -\log \left (1+\sqrt {4-x^2}\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(270\) vs. \(2(14)=28\).
time = 0.07, size = 271, normalized size = 16.94

method	result	size
trager	\(-\ln \left (-1-\sqrt {-x^{2}+4}\right )\)	\(17\)
default	\(-\frac {\ln \left (x^{2}-3\right )}{2}+\frac {\sqrt {-\left (-2+x \right )^{2}-4 x +8}-2 \arcsin \left (\frac {x}{2}\right )}{2 \left (2+\sqrt {3}\right ) \left (-2+\sqrt {3}\right )}-\frac {\sqrt {-\left (x +\sqrt {3}\right )^{2}+2 \sqrt {3}\, \left (x +\sqrt {3}\right )+1}+\sqrt {3}\, \arcsin \left (\frac {x}{2}\right )-\arctanh \left (\frac {2+2 \sqrt {3}\, \left (x +\sqrt {3}\right )}{2 \sqrt {-\left (x +\sqrt {3}\right )^{2}+2 \sqrt {3}\, \left (x +\sqrt {3}\right )+1}}\right )}{2 \left (2+\sqrt {3}\right ) \left (-2+\sqrt {3}\right )}-\frac {\sqrt {-\left (x -\sqrt {3}\right )^{2}-2 \sqrt {3}\, \left (x -\sqrt {3}\right )+1}-\sqrt {3}\, \arcsin \left (\frac {x}{2}\right )-\arctanh \left (\frac {2-2 \sqrt {3}\, \left (x -\sqrt {3}\right )}{2 \sqrt {-\left (x -\sqrt {3}\right )^{2}-2 \sqrt {3}\, \left (x -\sqrt {3}\right )+1}}\right )}{2 \left (2+\sqrt {3}\right ) \left (-2+\sqrt {3}\right )}+\frac {\sqrt {-\left (2+x \right )^{2}+4 x +8}+2 \arcsin \left (\frac {x}{2}\right )}{2 \left (2+\sqrt {3}\right ) \left (-2+\sqrt {3}\right )}\)	\(271\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 4.42, size = 14, normalized size = 0.88 \begin {gather*} -\log \left (\sqrt {-x^{2} + 4} + 1\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (14) = 28\).
time = 0.78, size = 55, normalized size = 3.44 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (12) = 24\).
time = 2.10, size = 44, normalized size = 2.75 \begin {gather*} \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 {gather*}

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 = 0.45, size = 14, normalized size = 0.88 \begin {gather*} -\log \left (\sqrt {-x^{2} + 4} + 1\right ) \end {gather*}

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)

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Mupad [B]
time = 0.16, size = 87, normalized size = 5.44 \begin {gather*} -\frac {\ln \left (x-\sqrt {3}\right )}{2}-\frac {\ln \left (\frac {\sqrt {3}\,x\,1{}\mathrm {i}+\sqrt {4-x^2}\,1{}\mathrm {i}+4{}\mathrm {i}}{x+\sqrt {3}}\right )}{2}-\frac {\ln \left (x+\sqrt {3}\right )}{2}-\frac {\ln \left (\frac {-\sqrt {3}\,x\,1{}\mathrm {i}+\sqrt {4-x^2}\,1{}\mathrm {i}+4{}\mathrm {i}}{x-\sqrt {3}}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((4 - x^2)^(1/2) - x^2 + 4),x)

[Out]

- log(x - 3^(1/2))/2 - log((3^(1/2)*x*1i + (4 - x^2)^(1/2)*1i + 4i)/(x + 3^(1/2)))/2 - log(x + 3^(1/2))/2 - lo
g(((4 - x^2)^(1/2)*1i - 3^(1/2)*x*1i + 4i)/(x - 3^(1/2)))/2

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