\(\int \frac {1+x^2}{(-1+x^2) (2+x^2) \sqrt {-3+x^4}} \, dx\) [278]

Optimal result

Integrand size = 29, antiderivative size = 26 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \left (2+x^2\right ) \sqrt {-3+x^4}} \, dx=\frac {\arctan \left (\frac {x \sqrt {-3+x^4}}{\sqrt {2}}\right )}{3 \sqrt {2}} \]

[Out]

1/6*arctan(1/2*x*(x^4-3)^(1/2)*2^(1/2))*2^(1/2)

Rubi [C] (verified)

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

Time = 0.49 (sec) , antiderivative size = 131, normalized size of antiderivative = 5.04, number of steps used = 18, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {6857, 1229, 229, 1471, 554, 259, 552, 551} \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \left (2+x^2\right ) \sqrt {-3+x^4}} \, dx=\frac {\sqrt {\sqrt {3}-x^2} \sqrt {\sqrt {3} x^2+3} \operatorname {EllipticPi}\left (-\frac {\sqrt {3}}{2},\arcsin \left (\frac {x}{\sqrt [4]{3}}\right ),-1\right )}{6 \sqrt {3} \sqrt {x^4-3}}-\frac {2 \sqrt {\sqrt {3}-x^2} \sqrt {\sqrt {3} x^2+3} \operatorname {EllipticPi}\left (\sqrt {3},\arcsin \left (\frac {x}{\sqrt [4]{3}}\right ),-1\right )}{3 \sqrt {3} \sqrt {x^4-3}} \]

[In]

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

[Out]

(Sqrt[Sqrt[3] - x^2]*Sqrt[3 + Sqrt[3]*x^2]*EllipticPi[-1/2*Sqrt[3], ArcSin[x/3^(1/4)], -1])/(6*Sqrt[3]*Sqrt[-3
+ x^4]) - (2*Sqrt[Sqrt[3] - x^2]*Sqrt[3 + Sqrt[3]*x^2]*EllipticPi[Sqrt[3], ArcSin[x/3^(1/4)], -1])/(3*Sqrt[3]
*Sqrt[-3 + x^4])

Rule 229

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[(-a)*b, 2]}, Simp[Sqrt[(a - q*x^2)/(a + q*x^2)]*(Sq
rt[(a + q*x^2)/q]/(Sqrt[2]*Sqrt[a + b*x^4]*Sqrt[a/(a + q*x^2)]))*EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]],
1/2], x]] /; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]

Rule 259

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_)*((a2_.) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a1 + b1*x^n)^FracPar
t[p]*((a2 + b2*x^n)^FracPart[p]/(a1*a2 + b1*b2*x^(2*n))^FracPart[p]), Int[(a1*a2 + b1*b2*x^(2*n))^p, x], x] /;
FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] &&  !IntegerQ[p]

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 552

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

Rule 554

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

Rule 1229

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

Rule 1471

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Dist[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]), Int[(d + e*x^n)^(p
+ q)*(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, c, d, e, f, g, n, p, q, r}, x] && EqQ[n2, 2*n] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p]

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 {2}{3 \left (-1+x^2\right ) \sqrt {-3+x^4}}+\frac {1}{3 \left (2+x^2\right ) \sqrt {-3+x^4}}\right ) \, dx \\ & = \frac {1}{3} \int \frac {1}{\left (2+x^2\right ) \sqrt {-3+x^4}} \, dx+\frac {2}{3} \int \frac {1}{\left (-1+x^2\right ) \sqrt {-3+x^4}} \, dx \\ & = \frac {2 \int \frac {1}{\sqrt {-3+x^4}} \, dx}{3 \left (-1+\sqrt {3}\right )}+\frac {2 \int \frac {\sqrt {3}-x^2}{\left (-1+x^2\right ) \sqrt {-3+x^4}} \, dx}{3 \left (-1+\sqrt {3}\right )}+\frac {\int \frac {1}{\sqrt {-3+x^4}} \, dx}{3 \left (2+\sqrt {3}\right )}+\frac {\int \frac {\sqrt {3}-x^2}{\left (2+x^2\right ) \sqrt {-3+x^4}} \, dx}{3 \left (2+\sqrt {3}\right )} \\ & = -\frac {\sqrt {2} \sqrt {\frac {\sqrt {3}+x^2}{\sqrt {3}-x^2}} \sqrt {-3+\sqrt {3} x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {-3+\sqrt {3} x^2}}\right ),\frac {1}{2}\right )}{3\ 3^{3/4} \left (1-\sqrt {3}\right ) \sqrt {\frac {1}{3-\sqrt {3} x^2}} \sqrt {-3+x^4}}+\frac {\sqrt {\frac {\sqrt {3}+x^2}{\sqrt {3}-x^2}} \sqrt {-3+\sqrt {3} x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {-3+\sqrt {3} x^2}}\right ),\frac {1}{2}\right )}{3 \sqrt {2} 3^{3/4} \left (2+\sqrt {3}\right ) \sqrt {\frac {1}{3-\sqrt {3} x^2}} \sqrt {-3+x^4}}+\frac {\left (2 \sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2}\right ) \int \frac {\sqrt {\sqrt {3}-x^2}}{\sqrt {-\sqrt {3}-x^2} \left (-1+x^2\right )} \, dx}{3 \left (-1+\sqrt {3}\right ) \sqrt {-3+x^4}}+\frac {\left (\sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2}\right ) \int \frac {\sqrt {\sqrt {3}-x^2}}{\sqrt {-\sqrt {3}-x^2} \left (2+x^2\right )} \, dx}{3 \left (2+\sqrt {3}\right ) \sqrt {-3+x^4}} \\ & = -\frac {\sqrt {2} \sqrt {\frac {\sqrt {3}+x^2}{\sqrt {3}-x^2}} \sqrt {-3+\sqrt {3} x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {-3+\sqrt {3} x^2}}\right ),\frac {1}{2}\right )}{3\ 3^{3/4} \left (1-\sqrt {3}\right ) \sqrt {\frac {1}{3-\sqrt {3} x^2}} \sqrt {-3+x^4}}+\frac {\sqrt {\frac {\sqrt {3}+x^2}{\sqrt {3}-x^2}} \sqrt {-3+\sqrt {3} x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {-3+\sqrt {3} x^2}}\right ),\frac {1}{2}\right )}{3 \sqrt {2} 3^{3/4} \left (2+\sqrt {3}\right ) \sqrt {\frac {1}{3-\sqrt {3} x^2}} \sqrt {-3+x^4}}+\frac {\left (\sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2} \left (2+x^2\right )} \, dx}{3 \sqrt {-3+x^4}}+\frac {\left (2 \sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2} \left (-1+x^2\right )} \, dx}{3 \sqrt {-3+x^4}}-\frac {\left (2 \sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2}} \, dx}{3 \left (-1+\sqrt {3}\right ) \sqrt {-3+x^4}}-\frac {\left (\sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2}\right ) \int \frac {1}{\sqrt {-\sqrt {3}-x^2} \sqrt {\sqrt {3}-x^2}} \, dx}{3 \left (2+\sqrt {3}\right ) \sqrt {-3+x^4}} \\ & = -\frac {\sqrt {2} \sqrt {\frac {\sqrt {3}+x^2}{\sqrt {3}-x^2}} \sqrt {-3+\sqrt {3} x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {-3+\sqrt {3} x^2}}\right ),\frac {1}{2}\right )}{3\ 3^{3/4} \left (1-\sqrt {3}\right ) \sqrt {\frac {1}{3-\sqrt {3} x^2}} \sqrt {-3+x^4}}+\frac {\sqrt {\frac {\sqrt {3}+x^2}{\sqrt {3}-x^2}} \sqrt {-3+\sqrt {3} x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {-3+\sqrt {3} x^2}}\right ),\frac {1}{2}\right )}{3 \sqrt {2} 3^{3/4} \left (2+\sqrt {3}\right ) \sqrt {\frac {1}{3-\sqrt {3} x^2}} \sqrt {-3+x^4}}-\frac {2 \int \frac {1}{\sqrt {-3+x^4}} \, dx}{3 \left (-1+\sqrt {3}\right )}-\frac {\int \frac {1}{\sqrt {-3+x^4}} \, dx}{3 \left (2+\sqrt {3}\right )}+\frac {\left (\sqrt {\sqrt {3}-x^2} \sqrt {1+\frac {x^2}{\sqrt {3}}}\right ) \int \frac {1}{\sqrt {\sqrt {3}-x^2} \left (2+x^2\right ) \sqrt {1+\frac {x^2}{\sqrt {3}}}} \, dx}{3 \sqrt {-3+x^4}}+\frac {\left (2 \sqrt {\sqrt {3}-x^2} \sqrt {1+\frac {x^2}{\sqrt {3}}}\right ) \int \frac {1}{\sqrt {\sqrt {3}-x^2} \left (-1+x^2\right ) \sqrt {1+\frac {x^2}{\sqrt {3}}}} \, dx}{3 \sqrt {-3+x^4}} \\ & = \frac {\sqrt {\sqrt {3}-x^2} \sqrt {3+\sqrt {3} x^2} \operatorname {EllipticPi}\left (-\frac {\sqrt {3}}{2},\arcsin \left (\frac {x}{\sqrt [4]{3}}\right ),-1\right )}{6 \sqrt {3} \sqrt {-3+x^4}}-\frac {2 \sqrt {\sqrt {3}-x^2} \sqrt {3+\sqrt {3} x^2} \operatorname {EllipticPi}\left (\sqrt {3},\arcsin \left (\frac {x}{\sqrt [4]{3}}\right ),-1\right )}{3 \sqrt {3} \sqrt {-3+x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.66 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \left (2+x^2\right ) \sqrt {-3+x^4}} \, dx=\frac {\arctan \left (\frac {x \sqrt {-3+x^4}}{\sqrt {2}}\right )}{3 \sqrt {2}} \]

[In]

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

[Out]

ArcTan[(x*Sqrt[-3 + x^4])/Sqrt[2]]/(3*Sqrt[2])

Maple [C] (verified)

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

Time = 5.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.62

method	result	size
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {x^{4}-3}\, x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{\left (x^{2}+2\right ) \left (x -1\right )^{2} \left (1+x \right )^{2}}\right )}{12}\)	\(68\)
elliptic	\(-\frac {2 \sqrt {1+\frac {\sqrt {3}\, x^{2}}{3}}\, \sqrt {1-\frac {\sqrt {3}\, x^{2}}{3}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {\sqrt {3}}{3}}\, x , -\sqrt {3}, \frac {3^{\frac {3}{4}}}{3 \sqrt {-\frac {\sqrt {3}}{3}}}\right )}{3 \sqrt {-\frac {\sqrt {3}}{3}}\, \sqrt {x^{4}-3}}-\frac {\operatorname {EllipticPi}\left (\sqrt {-\frac {\sqrt {3}}{3}}\, x , \frac {\sqrt {3}}{2}, i\right ) \sqrt {3+\sqrt {3}\, x^{2}}\, \sqrt {3-\sqrt {3}\, x^{2}}}{18 \sqrt {-\frac {\sqrt {3}}{3}}\, \sqrt {x^{4}-3}}\)	\(125\)
default	\(-\frac {2 \sqrt {1+\frac {\sqrt {3}\, x^{2}}{3}}\, \sqrt {1-\frac {\sqrt {3}\, x^{2}}{3}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {\sqrt {3}}{3}}\, x , -\sqrt {3}, \frac {3^{\frac {3}{4}}}{3 \sqrt {-\frac {\sqrt {3}}{3}}}\right )}{3 \sqrt {-\frac {\sqrt {3}}{3}}\, \sqrt {x^{4}-3}}+\frac {\sqrt {1+\frac {\sqrt {3}\, x^{2}}{3}}\, \sqrt {1-\frac {\sqrt {3}\, x^{2}}{3}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {\sqrt {3}}{3}}\, x , \frac {\sqrt {3}}{2}, \frac {3^{\frac {3}{4}}}{3 \sqrt {-\frac {\sqrt {3}}{3}}}\right )}{6 \sqrt {-\frac {\sqrt {3}}{3}}\, \sqrt {x^{4}-3}}\)	\(136\)

[In]

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

[Out]

-1/12*RootOf(_Z^2+2)*ln((RootOf(_Z^2+2)*x^6-3*RootOf(_Z^2+2)*x^2+4*(x^4-3)^(1/2)*x-2*RootOf(_Z^2+2))/(x^2+2)/(
x-1)^2/(1+x)^2)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \left (2+x^2\right ) \sqrt {-3+x^4}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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