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

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

[Out]

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

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Rubi [A]  time = 0.05, 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 = {2155, 31} \[ -\log \left (\sqrt {1-x^2}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(1 - x^2 + Sqrt[1 - x^2]),x]

[Out]

-Log[1 + Sqrt[1 - 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 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]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 16, normalized size = 1.00 \[ -\log \left (\sqrt {1-x^2}+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[1 + Sqrt[1 - x^2]]

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fricas [A]  time = 0.41, size = 21, normalized size = 1.31 \[ -\log \relax (x) + \log \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x) + log((sqrt(-x^2 + 1) - 1)/x)

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giac [A]  time = 0.82, size = 14, normalized size = 0.88 \[ -\log \left (\sqrt {-x^{2} + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(sqrt(-x^2 + 1) + 1)

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maple [B]  time = 0.02, size = 59, normalized size = 3.69 \[ -\arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )-\ln \relax (x )-\frac {\sqrt {2 x -\left (x +1\right )^{2}+2}}{2}-\frac {\sqrt {-2 x -\left (x -1\right )^{2}+2}}{2}+\sqrt {-x^{2}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.49, size = 14, normalized size = 0.88 \[ -\log \left (\sqrt {-x^{2} + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(sqrt(-x^2 + 1) + 1)

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mupad [B]  time = 0.13, size = 21, normalized size = 1.31 \[ \ln \left (\sqrt {\frac {1}{x^2}-1}-\sqrt {\frac {1}{x^2}}\right )-\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((1 - x^2)^(1/2) - x^2 + 1),x)

[Out]

log((1/x^2 - 1)^(1/2) - (1/x^2)^(1/2)) - log(x)

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sympy [B]  time = 5.47, size = 44, normalized size = 2.75 \[ \frac {\log {\left (2 \sqrt {1 - x^{2}} \right )}}{2} - \frac {\log {\left (2 \sqrt {1 - x^{2}} + 2 \right )}}{2} - \frac {\log {\left (x^{2} - \sqrt {1 - x^{2}} - 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(2*sqrt(1 - x**2))/2 - log(2*sqrt(1 - x**2) + 2)/2 - log(x**2 - sqrt(1 - x**2) - 1)/2

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