\(\int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx\) [68]

Optimal result

Integrand size = 21, antiderivative size = 43 \[ \int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx=\frac {1}{2} \left (1+x^2\right ) \sqrt {-1+\frac {2}{1+x^2}}-\arctan \left (\sqrt {-1+\frac {2}{1+x^2}}\right ) \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(95\) vs. \(2(43)=86\).

Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.21 \[ \int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx=\frac {\sqrt {\frac {1-x^2}{1+x^2}} \left (\sqrt {1-x^2} \left (1+x^2\right )+4 \sqrt {1+x^2} \arctan \left (\frac {\sqrt {1-x^2}}{\sqrt {2}-\sqrt {1+x^2}}\right )\right )}{2 \sqrt {1-x^2}} \] Input:

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

Output:

(Sqrt[(1 - x^2)/(1 + x^2)]*(Sqrt[1 - x^2]*(1 + x^2) + 4*Sqrt[1 + x^2]*ArcT 
an[Sqrt[1 - x^2]/(Sqrt[2] - Sqrt[1 + x^2])]))/(2*Sqrt[1 - x^2])
 

Rubi [A] (warning: unable to verify)

Time = 0.34 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.33, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2053, 2051, 252, 216}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2053

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

\(\Big \downarrow \) 2051

\(\displaystyle -2 \int \frac {x^4}{\left (x^4+1\right )^2}d\sqrt {\frac {1-x^2}{x^2+1}}\)

\(\Big \downarrow \) 252

\(\displaystyle -2 \left (\frac {1}{2} \int \frac {1}{x^4+1}d\sqrt {\frac {1-x^2}{x^2+1}}-\frac {\sqrt {\frac {1-x^2}{x^2+1}}}{2 \left (x^4+1\right )}\right )\)

\(\Big \downarrow \) 216

\(\displaystyle -2 \left (\frac {1}{2} \arctan \left (\sqrt {\frac {1-x^2}{x^2+1}}\right )-\frac {\sqrt {\frac {1-x^2}{x^2+1}}}{2 \left (x^4+1\right )}\right )\)

Input:

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

Output:

-2*(-1/2*Sqrt[(1 - x^2)/(1 + x^2)]/(1 + x^4) + ArcTan[Sqrt[(1 - x^2)/(1 + 
x^2)]]/2)
 

Defintions of rubi rules used

rule 216

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

rule 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

rule 2051

Int[(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_ 
Symbol] :> With[{q = Denominator[p]}, Simp[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 2053

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) 
))^(p_), x_Symbol] :> Simp[1/n   Subst[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]]
 

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.21

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.28 \[ \int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx=\frac {1}{2} \, {\left (x^{2} + 1\right )} \sqrt {-\frac {x^{2} - 1}{x^{2} + 1}} - \arctan \left (\frac {{\left (x^{2} + 1\right )} \sqrt {-\frac {x^{2} - 1}{x^{2} + 1}} - 1}{x^{2}}\right ) \] Input:

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

Output:

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]

\[ \int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx=\int x \sqrt {- \frac {\left (x - 1\right ) \left (x + 1\right )}{x^{2} + 1}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx=\int { x \sqrt {-\frac {x^{2} - 1}{x^{2} + 1}} \,d x } \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.42 \[ \int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx=\frac {1}{2} \, \sqrt {-x^{4} + 1} + \frac {1}{2} \, \arcsin \left (x^{2}\right ) \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 9.32 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.28 \[ \int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx=-\mathrm {atan}\left (\sqrt {-\frac {x^2-1}{x^2+1}}\right )-\frac {\sqrt {-\frac {x^2-1}{x^2+1}}}{\frac {x^2-1}{x^2+1}-1} \] Input:

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

Output:

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

Reduce [F]

\[ \int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx=\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}}{2}-\left (\int \frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, x}{x^{4}-1}d x \right ) \] Input:

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

Output:

(sqrt( - x**2 + 1)*sqrt(x**2 + 1) - 2*int((sqrt( - x**2 + 1)*sqrt(x**2 + 1 
)*x)/(x**4 - 1),x))/2