\(\int \frac {x \sqrt {1-x^2}}{x-x^3+\sqrt {1-x^2}} \, dx\) [1022]

Optimal result

Integrand size = 33, antiderivative size = 42 \[ \int \frac {x \sqrt {1-x^2}}{x-x^3+\sqrt {1-x^2}} \, dx=\arcsin (x)-\frac {\arctan \left (\frac {1+4 x \sqrt {1-x^2}}{\sqrt {3} \left (1-2 x^2\right )}\right )}{\sqrt {3}} \]

[Out]

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.34 (sec) , antiderivative size = 149, normalized size of antiderivative = 3.55, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {6874, 1307, 222, 1188, 385, 211, 1265, 787, 632, 210} \[ \int \frac {x \sqrt {1-x^2}}{x-x^3+\sqrt {1-x^2}} \, dx=\arcsin (x)-\frac {\arctan \left (\frac {1-2 x^2}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {x}{\sqrt {-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt {1-x^2}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\sqrt {-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} x}{\sqrt {1-x^2}}\right )}{\sqrt {3}}-\frac {x^2}{2}+\frac {1}{4} (1-x)^2+\frac {1}{4} (x+1)^2 \]

[In]

Int[(x*Sqrt[1 - x^2])/(x - x^3 + Sqrt[1 - x^2]),x]

[Out]

(1 - x)^2/4 - x^2/2 + (1 + x)^2/4 + ArcSin[x] - ArcTan[(1 - 2*x^2)/Sqrt[3]]/Sqrt[3] - ArcTan[x/(Sqrt[-((I - Sq
rt[3])/(I + Sqrt[3]))]*Sqrt[1 - x^2])]/Sqrt[3] - ArcTan[(Sqrt[-((I - Sqrt[3])/(I + Sqrt[3]))]*x)/Sqrt[1 - x^2]
]/Sqrt[3]

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 211

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

Rule 222

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

Rule 385

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 787

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*g*(x/c
), x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1188

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]
}, Dist[2*(c/r), Int[(d + e*x^2)^q/(b - r + 2*c*x^2), x], x] - Dist[2*(c/r), Int[(d + e*x^2)^q/(b + r + 2*c*x^
2), x], x]] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !Integ
erQ[q]

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 1307

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Dist[e
*(f^2/c), Int[(f*x)^(m - 2)*(d + e*x^2)^(q - 1), x], x] - Dist[f^2/c, Int[(f*x)^(m - 2)*(d + e*x^2)^(q - 1)*(S
imp[a*e - (c*d - b*e)*x^2, x]/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c,
0] &&  !IntegerQ[q] && GtQ[q, 0] && GtQ[m, 1] && LeQ[m, 3]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} (-1+x)+\frac {1+x}{2}-\frac {x^2 \sqrt {1-x^2}}{1-x^2+x^4}+\frac {x^3 \left (1-x^2\right )}{1-x^2+x^4}\right ) \, dx \\ & = \frac {1}{4} (1-x)^2+\frac {1}{4} (1+x)^2-\int \frac {x^2 \sqrt {1-x^2}}{1-x^2+x^4} \, dx+\int \frac {x^3 \left (1-x^2\right )}{1-x^2+x^4} \, dx \\ & = \frac {1}{4} (1-x)^2+\frac {1}{4} (1+x)^2+\frac {1}{2} \text {Subst}\left (\int \frac {(1-x) x}{1-x+x^2} \, dx,x,x^2\right )+\int \frac {1}{\sqrt {1-x^2}} \, dx-\int \frac {1}{\sqrt {1-x^2} \left (1-x^2+x^4\right )} \, dx \\ & = \frac {1}{4} (1-x)^2-\frac {x^2}{2}+\frac {1}{4} (1+x)^2+\sin ^{-1}(x)+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^2\right )+\frac {(2 i) \int \frac {1}{\sqrt {1-x^2} \left (-1-i \sqrt {3}+2 x^2\right )} \, dx}{\sqrt {3}}-\frac {(2 i) \int \frac {1}{\sqrt {1-x^2} \left (-1+i \sqrt {3}+2 x^2\right )} \, dx}{\sqrt {3}} \\ & = \frac {1}{4} (1-x)^2-\frac {x^2}{2}+\frac {1}{4} (1+x)^2+\sin ^{-1}(x)-\frac {(2 i) \text {Subst}\left (\int \frac {1}{-1+i \sqrt {3}-\left (-1-i \sqrt {3}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )}{\sqrt {3}}+\frac {(2 i) \text {Subst}\left (\int \frac {1}{-1-i \sqrt {3}-\left (-1+i \sqrt {3}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )}{\sqrt {3}}-\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^2\right ) \\ & = \frac {1}{4} (1-x)^2-\frac {x^2}{2}+\frac {1}{4} (1+x)^2+\sin ^{-1}(x)-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt {1-x^2}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt {1-x^2}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {-1+2 x^2}{\sqrt {3}}\right )}{\sqrt {3}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.07 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.07 \[ \int \frac {x \sqrt {1-x^2}}{x-x^3+\sqrt {1-x^2}} \, dx=-2 \arctan \left (\frac {\sqrt {1-x^2}}{1+x}\right )+\text {RootSum}\left [1+2 \text {$\#$1}+2 \text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log (-1+x)+\log \left (\sqrt {1-x^2}+\text {$\#$1}-x \text {$\#$1}\right )-\log (-1+x) \text {$\#$1}^2+\log \left (\sqrt {1-x^2}+\text {$\#$1}-x \text {$\#$1}\right ) \text {$\#$1}^2}{1+2 \text {$\#$1}-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]

[In]

Integrate[(x*Sqrt[1 - x^2])/(x - x^3 + Sqrt[1 - x^2]),x]

[Out]

-2*ArcTan[Sqrt[1 - x^2]/(1 + x)] + RootSum[1 + 2*#1 + 2*#1^2 - 2*#1^3 + #1^4 & , (-Log[-1 + x] + Log[Sqrt[1 -
x^2] + #1 - x*#1] - Log[-1 + x]*#1^2 + Log[Sqrt[1 - x^2] + #1 - x*#1]*#1^2)/(1 + 2*#1 - 3*#1^2 + 2*#1^3) & ]

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.12

method	result	size
trager	\(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{2}+3 x \sqrt {-x^{2}+1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{2}-x^{2}+2}\right )}{3}\)	\(89\)
default	\(\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}-1\right ) \sqrt {3}}{3}\right )}{3}+\left (\frac {1}{4}+\frac {i \sqrt {3}}{12}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )+\left (-\frac {i \sqrt {3}}{12}+\frac {1}{4}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )+\left (-\frac {1}{4}+\frac {i \sqrt {3}}{12}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )+\left (-\frac {1}{4}-\frac {i \sqrt {3}}{12}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )-\left (\frac {1}{4}+\frac {i \sqrt {3}}{12}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )-\left (-\frac {i \sqrt {3}}{12}+\frac {1}{4}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )+\left (\frac {1}{4}+\frac {i \sqrt {3}}{12}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )+\left (-\frac {i \sqrt {3}}{12}+\frac {1}{4}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )-2 \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x}\right )\)	\(456\)

[In]

int(x*(-x^2+1)^(1/2)/(x-x^3+(-x^2+1)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

RootOf(_Z^2+1)*ln(RootOf(_Z^2+1)*(-x^2+1)^(1/2)+x)+1/3*RootOf(_Z^2+3)*ln((2*RootOf(_Z^2+3)*x^2+3*x*(-x^2+1)^(1
/2)-RootOf(_Z^2+3))/(RootOf(_Z^2+3)*x^2-x^2+2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.74 \[ \int \frac {x \sqrt {1-x^2}}{x-x^3+\sqrt {1-x^2}} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} - 1\right )} \sqrt {-x^{2} + 1}}{3 \, {\left (x^{3} - x\right )}}\right ) - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]

[In]

integrate(x*(-x^2+1)^(1/2)/(x-x^3+(-x^2+1)^(1/2)),x, algorithm="fricas")

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 - 1)) + 1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 - 1)*sqrt(-x^2 + 1)/(x^3 -
x)) - 2*arctan((sqrt(-x^2 + 1) - 1)/x)

Sympy [F]

\[ \int \frac {x \sqrt {1-x^2}}{x-x^3+\sqrt {1-x^2}} \, dx=- \int \frac {x \sqrt {1 - x^{2}}}{x^{3} - x - \sqrt {1 - x^{2}}}\, dx \]

[In]

integrate(x*(-x**2+1)**(1/2)/(x-x**3+(-x**2+1)**(1/2)),x)

[Out]

-Integral(x*sqrt(1 - x**2)/(x**3 - x - sqrt(1 - x**2)), x)

Maxima [F]

\[ \int \frac {x \sqrt {1-x^2}}{x-x^3+\sqrt {1-x^2}} \, dx=\int { -\frac {\sqrt {-x^{2} + 1} x}{x^{3} - x - \sqrt {-x^{2} + 1}} \,d x } \]

[In]

integrate(x*(-x^2+1)^(1/2)/(x-x^3+(-x^2+1)^(1/2)),x, algorithm="maxima")

[Out]

1/2*x^2 + integrate(-(x^4 - x^2)/(x^3 - x - sqrt(x + 1)*sqrt(-x + 1)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (37) = 74\).

Time = 0.35 (sec) , antiderivative size = 193, normalized size of antiderivative = 4.60 \[ \int \frac {x \sqrt {1-x^2}}{x-x^3+\sqrt {1-x^2}} \, dx=\frac {1}{2} \, \pi \mathrm {sgn}\left (x\right ) - \frac {1}{6} \, \sqrt {3} {\left (\pi \mathrm {sgn}\left (x\right ) + 2 \, \arctan \left (-\frac {\sqrt {3} x {\left (\frac {\sqrt {-x^{2} + 1} - 1}{x} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right )\right )} - \frac {1}{6} \, \sqrt {3} {\left (\pi \mathrm {sgn}\left (x\right ) + 2 \, \arctan \left (\frac {\sqrt {3} x {\left (\frac {\sqrt {-x^{2} + 1} - 1}{x} - \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right )\right )} + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) + \arctan \left (-\frac {x {\left (\frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right ) \]

[In]

integrate(x*(-x^2+1)^(1/2)/(x-x^3+(-x^2+1)^(1/2)),x, algorithm="giac")

[Out]

1/2*pi*sgn(x) - 1/6*sqrt(3)*(pi*sgn(x) + 2*arctan(-1/3*sqrt(3)*x*((sqrt(-x^2 + 1) - 1)/x + (sqrt(-x^2 + 1) - 1
)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1))) - 1/6*sqrt(3)*(pi*sgn(x) + 2*arctan(1/3*sqrt(3)*x*((sqrt(-x^2 + 1) - 1)/x
- (sqrt(-x^2 + 1) - 1)^2/x^2 + 1)/(sqrt(-x^2 + 1) - 1))) + 1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 - 1)) + arcta
n(-1/2*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1))

Mupad [B] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 549, normalized size of antiderivative = 13.07 \[ \int \frac {x \sqrt {1-x^2}}{x-x^3+\sqrt {1-x^2}} \, dx=\mathrm {asin}\left (x\right )-\frac {\ln \left (\frac {\frac {\left (x\,\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )-1\right )\,1{}\mathrm {i}}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^2}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{\frac {\sqrt {3}}{2}-x+\frac {1}{2}{}\mathrm {i}}\right )}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^2}\,\left (\sqrt {3}-4\,{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}\right )}+\frac {\ln \left (\frac {\frac {\left (x\,\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )-1\right )\,1{}\mathrm {i}}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^2}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}}\right )}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^2}\,\left (-\sqrt {3}+4\,{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}\right )}-\frac {\ln \left (\frac {\frac {\left (x\,\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )+1\right )\,1{}\mathrm {i}}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^2}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}}\right )}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^2}\,\left (-\sqrt {3}+4\,{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}\right )}-\frac {\ln \left (x-\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}{\sqrt {3}-4\,{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}}-\frac {\ln \left (x+\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}{\sqrt {3}-4\,{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}}+\frac {\ln \left (\frac {\frac {\left (x\,\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )+1\right )\,1{}\mathrm {i}}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^2}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}}\right )}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^2}\,\left (\sqrt {3}-4\,{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}\right )}+\frac {\ln \left (x-\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}{-\sqrt {3}+4\,{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}}+\frac {\ln \left (x+\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}{-\sqrt {3}+4\,{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}} \]

[In]

int((x*(1 - x^2)^(1/2))/(x - x^3 + (1 - x^2)^(1/2)),x)

[Out]

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