3.3.89 \(\int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx\) [289]

Optimal. Leaf size=51 \[ \frac {1}{2} \sqrt {\frac {1-x^2}{1+x^2}} \left (1+x^2\right )-\tan ^{-1}\left (\sqrt {\frac {1-x^2}{1+x^2}}\right ) \]

[Out]

-arctan(((-x^2+1)/(x^2+1))^(1/2))+1/2*(x^2+1)*((-x^2+1)/(x^2+1))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1981, 1979, 294, 210} \begin {gather*} \frac {1}{2} \sqrt {\frac {1-x^2}{x^2+1}} \left (x^2+1\right )-\text {ArcTan}\left (\sqrt {\frac {1-x^2}{x^2+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[(1 - x^2)/(1 + x^2)],x]

[Out]

(Sqrt[(1 - x^2)/(1 + x^2)]*(1 + x^2))/2 - ArcTan[Sqrt[(1 - x^2)/(1 + x^2)]]

Rule 210

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

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 1979

Int[(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> With[{q = Denominator[p]
}, Dist[q*e*((b*c - a*d)/n), Subst[Int[x^(q*(p + 1) - 1)*(((-a)*e + c*x^q)^(1/n - 1)/(b*e - d*x^q)^(1/n + 1)),
x], x, (e*((a + b*x^n)/(c + d*x^n)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && FractionQ[p] && IntegerQ[1/n
]

Rule 1981

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.10, size = 85, normalized size = 1.67 \begin {gather*} \frac {\sqrt {\frac {1-x^2}{1+x^2}} \left (\sqrt {1-x^2} \left (1+x^2\right )-2 \sqrt {1+x^2} \tan ^{-1}\left (\frac {\sqrt {1-x^2}}{\sqrt {1+x^2}}\right )\right )}{2 \sqrt {1-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(1 - x^2)/(1 + x^2)]*(Sqrt[1 - x^2]*(1 + x^2) - 2*Sqrt[1 + x^2]*ArcTan[Sqrt[1 - x^2]/Sqrt[1 + x^2]]))/(2
*Sqrt[1 - x^2])

________________________________________________________________________________________

Maple [A]
time = 0.25, size = 52, normalized size = 1.02

method	result	size
default	\(\frac {\sqrt {-\frac {x^{2}-1}{x^{2}+1}}\, \left (x^{2}+1\right ) \left (\arcsin \left (x^{2}\right )+\sqrt {-x^{4}+1}\right )}{2 \sqrt {-\left (x^{2}+1\right ) \left (x^{2}-1\right )}}\)	\(52\)
risch	\(\frac {\left (x^{2}+1\right ) \sqrt {-\frac {x^{2}-1}{x^{2}+1}}}{2}-\frac {\arcsin \left (x^{2}\right ) \sqrt {-\frac {x^{2}-1}{x^{2}+1}}\, \sqrt {-\left (x^{2}+1\right ) \left (x^{2}-1\right )}}{2 \left (x^{2}-1\right )}\)	\(68\)
trager	\(\left (\frac {x^{2}}{2}+\frac {1}{2}\right ) \sqrt {-\frac {x^{2}-1}{x^{2}+1}}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {x^{2}-1}{x^{2}+1}}\, x^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {x^{2}-1}{x^{2}+1}}+x^{2}\right )}{2}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 55, normalized size = 1.08 \begin {gather*} \frac {1}{2} \, {\left (x^{2} + 1\right )} \sqrt {-\frac {x^{2} - 1}{x^{2} + 1}} - \arctan \left (\frac {{\left (x^{2} + 1\right )} \sqrt {-\frac {x^{2} - 1}{x^{2} + 1}} - 1}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x^2 + 1)*sqrt(-(x^2 - 1)/(x^2 + 1)) - arctan(((x^2 + 1)*sqrt(-(x^2 - 1)/(x^2 + 1)) - 1)/x^2)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sqrt {- \frac {\left (x - 1\right ) \left (x + 1\right )}{x^{2} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*sqrt(-(x - 1)*(x + 1)/(x**2 + 1)), x)

________________________________________________________________________________________

Giac [A]
time = 5.55, size = 18, normalized size = 0.35 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{4} + 1} + \frac {1}{2} \, \arcsin \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-x^4 + 1) + 1/2*arcsin(x^2)

________________________________________________________________________________________

Mupad [B]
time = 2.67, size = 55, normalized size = 1.08 \begin {gather*} -\mathrm {atan}\left (\sqrt {-\frac {x^2-1}{x^2+1}}\right )-\frac {\sqrt {-\frac {x^2-1}{x^2+1}}}{\frac {x^2-1}{x^2+1}-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(-(x^2 - 1)/(x^2 + 1))^(1/2),x)

[Out]

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

________________________________________________________________________________________