\(\int \frac {2+x}{(-3+x) \sqrt [4]{1-x^2} (1+x^2)} \, dx\) [2299]

Optimal result

Integrand size = 27, antiderivative size = 176 \[ \int \frac {2+x}{(-3+x) \sqrt [4]{1-x^2} \left (1+x^2\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}-\sqrt [4]{2} x+\sqrt [4]{1-x^2}}\right )}{2 \sqrt [4]{2}}+\frac {\arctan \left (\frac {\sqrt [4]{1-x^2}}{-\sqrt [4]{2}+\sqrt [4]{2} x+\sqrt [4]{1-x^2}}\right )}{2 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\left (-2 \sqrt [4]{2}+2 \sqrt [4]{2} x\right ) \sqrt [4]{1-x^2}}{\sqrt {2}-2 \sqrt {2} x+\sqrt {2} x^2+2 \sqrt {1-x^2}}\right )}{2 \sqrt [4]{2}} \]

[Out]

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.43 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.15, number of steps used = 32, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6857, 760, 408, 504, 1227, 551, 455, 65, 303, 1176, 631, 210, 1179, 642, 1024, 304, 209, 212} \[ \int \frac {2+x}{(-3+x) \sqrt [4]{1-x^2} \left (1+x^2\right )} \, dx=-\frac {\sqrt {x^2} \operatorname {EllipticPi}\left (-\frac {1}{\sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{2 \sqrt {2} x}+\frac {3 i \sqrt {x^2} \operatorname {EllipticPi}\left (-\frac {i}{2 \sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{4 \sqrt {2} x}-\frac {3 i \sqrt {x^2} \operatorname {EllipticPi}\left (\frac {i}{2 \sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{4 \sqrt {2} x}+\frac {\sqrt {x^2} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{2 \sqrt {2} x}-\frac {\arctan \left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}-\frac {\arctan \left (1-\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}\right )}{4 \sqrt [4]{2}}+\frac {\arctan \left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}+1\right )}{4 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}+\frac {\log \left (\sqrt {1-x^2}-2 \sqrt [4]{2} \sqrt [4]{1-x^2}+2 \sqrt {2}\right )}{8 \sqrt [4]{2}}-\frac {\log \left (\sqrt {1-x^2}+2 \sqrt [4]{2} \sqrt [4]{1-x^2}+2 \sqrt {2}\right )}{8 \sqrt [4]{2}} \]

[In]

Int[(2 + x)/((-3 + x)*(1 - x^2)^(1/4)*(1 + x^2)),x]

[Out]

-1/2*ArcTan[(1 - x^2)^(1/4)/2^(1/4)]/2^(1/4) - ArcTan[1 - (1 - x^2)^(1/4)/2^(1/4)]/(4*2^(1/4)) + ArcTan[1 + (1
- x^2)^(1/4)/2^(1/4)]/(4*2^(1/4)) + ArcTanh[(1 - x^2)^(1/4)/2^(1/4)]/(2*2^(1/4)) - (Sqrt[x^2]*EllipticPi[-(1/
Sqrt[2]), ArcSin[(1 - x^2)^(1/4)], -1])/(2*Sqrt[2]*x) + (((3*I)/4)*Sqrt[x^2]*EllipticPi[(-1/2*I)/Sqrt[2], ArcS
in[(1 - x^2)^(1/4)], -1])/(Sqrt[2]*x) - (((3*I)/4)*Sqrt[x^2]*EllipticPi[(I/2)/Sqrt[2], ArcSin[(1 - x^2)^(1/4)]
, -1])/(Sqrt[2]*x) + (Sqrt[x^2]*EllipticPi[1/Sqrt[2], ArcSin[(1 - x^2)^(1/4)], -1])/(2*Sqrt[2]*x) + Log[2*Sqrt
[2] - 2*2^(1/4)*(1 - x^2)^(1/4) + Sqrt[1 - x^2]]/(8*2^(1/4)) - Log[2*Sqrt[2] + 2*2^(1/4)*(1 - x^2)^(1/4) + Sqr
t[1 - x^2]]/(8*2^(1/4))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

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

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 212

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

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 408

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

Rule 455

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

Rule 504

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s
= Denominator[Rt[-a/b, 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), Int[1/
((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 760

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

Rule 1024

Int[((g_) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Dist[g, Int[(a + c
*x^2)^p*(d + f*x^2)^q, x], x] + Dist[h, Int[x*(a + c*x^2)^p*(d + f*x^2)^q, x], x] /; FreeQ[{a, c, d, f, g, h,
p, q}, x]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1227

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[Sqrt[-c],
Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c,
0]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2 (-3+x) \sqrt [4]{1-x^2}}+\frac {-1-x}{2 \sqrt [4]{1-x^2} \left (1+x^2\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {1}{(-3+x) \sqrt [4]{1-x^2}} \, dx+\frac {1}{2} \int \frac {-1-x}{\sqrt [4]{1-x^2} \left (1+x^2\right )} \, dx \\ & = -\left (\frac {1}{2} \int \frac {x}{\sqrt [4]{1-x^2} \left (9-x^2\right )} \, dx\right )-\frac {1}{2} \int \frac {1}{\sqrt [4]{1-x^2} \left (1+x^2\right )} \, dx-\frac {1}{2} \int \frac {x}{\sqrt [4]{1-x^2} \left (1+x^2\right )} \, dx-\frac {3}{2} \int \frac {1}{\sqrt [4]{1-x^2} \left (9-x^2\right )} \, dx \\ & = -\left (\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-x} (9-x)} \, dx,x,x^2\right )\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-x} (1+x)} \, dx,x,x^2\right )-\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^4} \left (-2+x^4\right )} \, dx,x,\sqrt [4]{1-x^2}\right )}{x}-\frac {\left (3 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-8-x^4\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1-x^2}\right )}{x} \\ & = \frac {\sqrt {x^2} \text {Subst}\left (\int \frac {1}{\left (\sqrt {2}-x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1-x^2}\right )}{2 x}-\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {1}{\left (\sqrt {2}+x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1-x^2}\right )}{2 x}-\frac {\left (3 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 i \sqrt {2}-x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1-x^2}\right )}{2 x}+\frac {\left (3 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 i \sqrt {2}+x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{1-x^2}\right )}{2 x}+\text {Subst}\left (\int \frac {x^2}{2-x^4} \, dx,x,\sqrt [4]{1-x^2}\right )+\text {Subst}\left (\int \frac {x^2}{8+x^4} \, dx,x,\sqrt [4]{1-x^2}\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {2}-x^2} \, dx,x,\sqrt [4]{1-x^2}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {2}+x^2} \, dx,x,\sqrt [4]{1-x^2}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {2 \sqrt {2}-x^2}{8+x^4} \, dx,x,\sqrt [4]{1-x^2}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {2 \sqrt {2}+x^2}{8+x^4} \, dx,x,\sqrt [4]{1-x^2}\right )+\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (\sqrt {2}-x^2\right ) \sqrt {1+x^2}} \, dx,x,\sqrt [4]{1-x^2}\right )}{2 x}-\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {2}+x^2\right )} \, dx,x,\sqrt [4]{1-x^2}\right )}{2 x}-\frac {\left (3 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (2 i \sqrt {2}-x^2\right ) \sqrt {1+x^2}} \, dx,x,\sqrt [4]{1-x^2}\right )}{2 x}+\frac {\left (3 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (2 i \sqrt {2}+x^2\right )} \, dx,x,\sqrt [4]{1-x^2}\right )}{2 x} \\ & = -\frac {\arctan \left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}-\frac {\sqrt {x^2} \operatorname {EllipticPi}\left (-\frac {1}{\sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{2 \sqrt {2} x}+\frac {3 i \sqrt {x^2} \operatorname {EllipticPi}\left (-\frac {i}{2 \sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{4 \sqrt {2} x}-\frac {3 i \sqrt {x^2} \operatorname {EllipticPi}\left (\frac {i}{2 \sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{4 \sqrt {2} x}+\frac {\sqrt {x^2} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{2 \sqrt {2} x}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{2 \sqrt {2}-2 \sqrt [4]{2} x+x^2} \, dx,x,\sqrt [4]{1-x^2}\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{2 \sqrt {2}+2 \sqrt [4]{2} x+x^2} \, dx,x,\sqrt [4]{1-x^2}\right )+\frac {\text {Subst}\left (\int \frac {2 \sqrt [4]{2}+2 x}{-2 \sqrt {2}-2 \sqrt [4]{2} x-x^2} \, dx,x,\sqrt [4]{1-x^2}\right )}{8 \sqrt [4]{2}}+\frac {\text {Subst}\left (\int \frac {2 \sqrt [4]{2}-2 x}{-2 \sqrt {2}+2 \sqrt [4]{2} x-x^2} \, dx,x,\sqrt [4]{1-x^2}\right )}{8 \sqrt [4]{2}} \\ & = -\frac {\arctan \left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}-\frac {\sqrt {x^2} \operatorname {EllipticPi}\left (-\frac {1}{\sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{2 \sqrt {2} x}+\frac {3 i \sqrt {x^2} \operatorname {EllipticPi}\left (-\frac {i}{2 \sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{4 \sqrt {2} x}-\frac {3 i \sqrt {x^2} \operatorname {EllipticPi}\left (\frac {i}{2 \sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{4 \sqrt {2} x}+\frac {\sqrt {x^2} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{2 \sqrt {2} x}+\frac {\log \left (2 \sqrt {2}-2 \sqrt [4]{2} \sqrt [4]{1-x^2}+\sqrt {1-x^2}\right )}{8 \sqrt [4]{2}}-\frac {\log \left (2 \sqrt {2}+2 \sqrt [4]{2} \sqrt [4]{1-x^2}+\sqrt {1-x^2}\right )}{8 \sqrt [4]{2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}\right )}{4 \sqrt [4]{2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}\right )}{4 \sqrt [4]{2}} \\ & = -\frac {\arctan \left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}-\frac {\arctan \left (1-\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}\right )}{4 \sqrt [4]{2}}+\frac {\arctan \left (1+\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}\right )}{4 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}}\right )}{2 \sqrt [4]{2}}-\frac {\sqrt {x^2} \operatorname {EllipticPi}\left (-\frac {1}{\sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{2 \sqrt {2} x}+\frac {3 i \sqrt {x^2} \operatorname {EllipticPi}\left (-\frac {i}{2 \sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{4 \sqrt {2} x}-\frac {3 i \sqrt {x^2} \operatorname {EllipticPi}\left (\frac {i}{2 \sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{4 \sqrt {2} x}+\frac {\sqrt {x^2} \operatorname {EllipticPi}\left (\frac {1}{\sqrt {2}},\arcsin \left (\sqrt [4]{1-x^2}\right ),-1\right )}{2 \sqrt {2} x}+\frac {\log \left (2 \sqrt {2}-2 \sqrt [4]{2} \sqrt [4]{1-x^2}+\sqrt {1-x^2}\right )}{8 \sqrt [4]{2}}-\frac {\log \left (2 \sqrt {2}+2 \sqrt [4]{2} \sqrt [4]{1-x^2}+\sqrt {1-x^2}\right )}{8 \sqrt [4]{2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.84 \[ \int \frac {2+x}{(-3+x) \sqrt [4]{1-x^2} \left (1+x^2\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}-\sqrt [4]{2} x-\sqrt [4]{1-x^2}}\right )+\arctan \left (\frac {\sqrt [4]{1-x^2}}{\sqrt [4]{2}-\sqrt [4]{2} x+\sqrt [4]{1-x^2}}\right )+\text {arctanh}\left (\frac {2 (-1+x) \sqrt [4]{2-2 x^2}}{\sqrt {2}-2 \sqrt {2} x+\sqrt {2} x^2+2 \sqrt {1-x^2}}\right )}{2 \sqrt [4]{2}} \]

[In]

Integrate[(2 + x)/((-3 + x)*(1 - x^2)^(1/4)*(1 + x^2)),x]

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 4.68 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.32

method	result	size
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \ln \left (-\frac {2 \sqrt {-x^{2}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3} x -2 \sqrt {-x^{2}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3}+2 \left (-x^{2}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} x^{2}-4 \left (-x^{2}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) x^{3}+2 \left (-x^{2}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) x^{2}-4 \left (-x^{2}+1\right )^{\frac {3}{4}}+5 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )}{\left (-3+x \right ) \left (x^{2}+1\right )}\right )}{4}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {-x^{2}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} x -2 \sqrt {-x^{2}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}-2 \left (-x^{2}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} x^{2}+4 \left (-x^{2}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{3}-2 \left (-x^{2}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}-4 \left (-x^{2}+1\right )^{\frac {3}{4}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right )}{\left (-3+x \right ) \left (x^{2}+1\right )}\right )}{4}\)	\(408\)

[In]

int((2+x)/(-3+x)/(-x^2+1)^(1/4)/(x^2+1),x,method=_RETURNVERBOSE)

[Out]

-1/4*RootOf(_Z^4+2)*ln(-(2*(-x^2+1)^(1/2)*RootOf(_Z^4+2)^3*x-2*(-x^2+1)^(1/2)*RootOf(_Z^4+2)^3+2*(-x^2+1)^(1/4
)*RootOf(_Z^4+2)^2*x^2-4*(-x^2+1)^(1/4)*RootOf(_Z^4+2)^2*x+RootOf(_Z^4+2)*x^3+2*(-x^2+1)^(1/4)*RootOf(_Z^4+2)^
2-3*RootOf(_Z^4+2)*x^2-4*(-x^2+1)^(3/4)+5*RootOf(_Z^4+2)*x+RootOf(_Z^4+2))/(-3+x)/(x^2+1))+1/4*RootOf(_Z^2+Roo
tOf(_Z^4+2)^2)*ln(-(2*(-x^2+1)^(1/2)*RootOf(_Z^2+RootOf(_Z^4+2)^2)*RootOf(_Z^4+2)^2*x-2*(-x^2+1)^(1/2)*RootOf(
_Z^2+RootOf(_Z^4+2)^2)*RootOf(_Z^4+2)^2-2*(-x^2+1)^(1/4)*RootOf(_Z^4+2)^2*x^2+4*(-x^2+1)^(1/4)*RootOf(_Z^4+2)^
2*x-RootOf(_Z^2+RootOf(_Z^4+2)^2)*x^3-2*(-x^2+1)^(1/4)*RootOf(_Z^4+2)^2-4*(-x^2+1)^(3/4)+3*RootOf(_Z^2+RootOf(
_Z^4+2)^2)*x^2-5*RootOf(_Z^2+RootOf(_Z^4+2)^2)*x-RootOf(_Z^2+RootOf(_Z^4+2)^2))/(-3+x)/(x^2+1))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.08 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.35 \[ \int \frac {2+x}{(-3+x) \sqrt [4]{1-x^2} \left (1+x^2\right )} \, dx=\left (\frac {1}{64} i + \frac {1}{64}\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \log \left (\frac {8^{\frac {3}{4}} \sqrt {2} {\left (\left (i + 1\right ) \, x^{3} - \left (3 i + 3\right ) \, x^{2} + \left (5 i + 5\right ) \, x + i + 1\right )} - 8 \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {-x^{2} + 1} {\left (-\left (i - 1\right ) \, x + i - 1\right )} - 16 \, \sqrt {2} {\left (i \, x^{2} - 2 i \, x + i\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} + 32 \, {\left (-x^{2} + 1\right )}^{\frac {3}{4}}}{x^{3} - 3 \, x^{2} + x - 3}\right ) - \left (\frac {1}{64} i - \frac {1}{64}\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \log \left (\frac {8^{\frac {3}{4}} \sqrt {2} {\left (-\left (i - 1\right ) \, x^{3} + \left (3 i - 3\right ) \, x^{2} - \left (5 i - 5\right ) \, x - i + 1\right )} - 8 \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {-x^{2} + 1} {\left (\left (i + 1\right ) \, x - i - 1\right )} - 16 \, \sqrt {2} {\left (-i \, x^{2} + 2 i \, x - i\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} + 32 \, {\left (-x^{2} + 1\right )}^{\frac {3}{4}}}{x^{3} - 3 \, x^{2} + x - 3}\right ) + \left (\frac {1}{64} i - \frac {1}{64}\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \log \left (\frac {8^{\frac {3}{4}} \sqrt {2} {\left (\left (i - 1\right ) \, x^{3} - \left (3 i - 3\right ) \, x^{2} + \left (5 i - 5\right ) \, x + i - 1\right )} - 8 \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {-x^{2} + 1} {\left (-\left (i + 1\right ) \, x + i + 1\right )} - 16 \, \sqrt {2} {\left (-i \, x^{2} + 2 i \, x - i\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} + 32 \, {\left (-x^{2} + 1\right )}^{\frac {3}{4}}}{x^{3} - 3 \, x^{2} + x - 3}\right ) - \left (\frac {1}{64} i + \frac {1}{64}\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \log \left (\frac {8^{\frac {3}{4}} \sqrt {2} {\left (-\left (i + 1\right ) \, x^{3} + \left (3 i + 3\right ) \, x^{2} - \left (5 i + 5\right ) \, x - i - 1\right )} - 8 \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {-x^{2} + 1} {\left (\left (i - 1\right ) \, x - i + 1\right )} - 16 \, \sqrt {2} {\left (i \, x^{2} - 2 i \, x + i\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} + 32 \, {\left (-x^{2} + 1\right )}^{\frac {3}{4}}}{x^{3} - 3 \, x^{2} + x - 3}\right ) \]

[In]

integrate((2+x)/(-3+x)/(-x^2+1)^(1/4)/(x^2+1),x, algorithm="fricas")

[Out]

(1/64*I + 1/64)*8^(3/4)*sqrt(2)*log((8^(3/4)*sqrt(2)*((I + 1)*x^3 - (3*I + 3)*x^2 + (5*I + 5)*x + I + 1) - 8*8
^(1/4)*sqrt(2)*sqrt(-x^2 + 1)*(-(I - 1)*x + I - 1) - 16*sqrt(2)*(I*x^2 - 2*I*x + I)*(-x^2 + 1)^(1/4) + 32*(-x^
2 + 1)^(3/4))/(x^3 - 3*x^2 + x - 3)) - (1/64*I - 1/64)*8^(3/4)*sqrt(2)*log((8^(3/4)*sqrt(2)*(-(I - 1)*x^3 + (3
*I - 3)*x^2 - (5*I - 5)*x - I + 1) - 8*8^(1/4)*sqrt(2)*sqrt(-x^2 + 1)*((I + 1)*x - I - 1) - 16*sqrt(2)*(-I*x^2
+ 2*I*x - I)*(-x^2 + 1)^(1/4) + 32*(-x^2 + 1)^(3/4))/(x^3 - 3*x^2 + x - 3)) + (1/64*I - 1/64)*8^(3/4)*sqrt(2)
*log((8^(3/4)*sqrt(2)*((I - 1)*x^3 - (3*I - 3)*x^2 + (5*I - 5)*x + I - 1) - 8*8^(1/4)*sqrt(2)*sqrt(-x^2 + 1)*(
-(I + 1)*x + I + 1) - 16*sqrt(2)*(-I*x^2 + 2*I*x - I)*(-x^2 + 1)^(1/4) + 32*(-x^2 + 1)^(3/4))/(x^3 - 3*x^2 + x
- 3)) - (1/64*I + 1/64)*8^(3/4)*sqrt(2)*log((8^(3/4)*sqrt(2)*(-(I + 1)*x^3 + (3*I + 3)*x^2 - (5*I + 5)*x - I
- 1) - 8*8^(1/4)*sqrt(2)*sqrt(-x^2 + 1)*((I - 1)*x - I + 1) - 16*sqrt(2)*(I*x^2 - 2*I*x + I)*(-x^2 + 1)^(1/4)
+ 32*(-x^2 + 1)^(3/4))/(x^3 - 3*x^2 + x - 3))

Sympy [F]

\[ \int \frac {2+x}{(-3+x) \sqrt [4]{1-x^2} \left (1+x^2\right )} \, dx=\int \frac {x + 2}{\sqrt [4]{- \left (x - 1\right ) \left (x + 1\right )} \left (x - 3\right ) \left (x^{2} + 1\right )}\, dx \]

[In]

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

[Out]

Integral((x + 2)/((-(x - 1)*(x + 1))**(1/4)*(x - 3)*(x**2 + 1)), x)

Maxima [F]

\[ \int \frac {2+x}{(-3+x) \sqrt [4]{1-x^2} \left (1+x^2\right )} \, dx=\int { \frac {x + 2}{{\left (x^{2} + 1\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} {\left (x - 3\right )}} \,d x } \]

[In]

integrate((2+x)/(-3+x)/(-x^2+1)^(1/4)/(x^2+1),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {2+x}{(-3+x) \sqrt [4]{1-x^2} \left (1+x^2\right )} \, dx=\int { \frac {x + 2}{{\left (x^{2} + 1\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} {\left (x - 3\right )}} \,d x } \]

[In]

integrate((2+x)/(-3+x)/(-x^2+1)^(1/4)/(x^2+1),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {2+x}{(-3+x) \sqrt [4]{1-x^2} \left (1+x^2\right )} \, dx=\int \frac {x+2}{{\left (1-x^2\right )}^{1/4}\,\left (x^2+1\right )\,\left (x-3\right )} \,d x \]

[In]

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

[Out]

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