\(\int \frac {1}{-1+3 x^2+x^4} \, dx\) [39]

Optimal result

Integrand size = 12, antiderivative size = 73 \[ \int \frac {1}{-1+3 x^2+x^4} \, dx=-\sqrt {\frac {2}{13 \left (3+\sqrt {13}\right )}} \arctan \left (\sqrt {\frac {2}{3+\sqrt {13}}} x\right )-\sqrt {\frac {1}{26} \left (3+\sqrt {13}\right )} \text {arctanh}\left (\sqrt {\frac {2}{-3+\sqrt {13}}} x\right ) \]

[Out]

-1/13*arctan(x*2^(1/2)/(3+13^(1/2))^(1/2))*26^(1/2)/(3+13^(1/2))^(1/2)-1/26*arctanh(x*2^(1/2)/(-3+13^(1/2))^(1
/2))*(78+26*13^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1107, 213, 209} \[ \int \frac {1}{-1+3 x^2+x^4} \, dx=-\sqrt {\frac {2}{13 \left (3+\sqrt {13}\right )}} \arctan \left (\sqrt {\frac {2}{3+\sqrt {13}}} x\right )-\sqrt {\frac {1}{26} \left (3+\sqrt {13}\right )} \text {arctanh}\left (\sqrt {\frac {2}{\sqrt {13}-3}} x\right ) \]

[In]

Int[(-1 + 3*x^2 + x^4)^(-1),x]

[Out]

-(Sqrt[2/(13*(3 + Sqrt[13]))]*ArcTan[Sqrt[2/(3 + Sqrt[13])]*x]) - Sqrt[(3 + Sqrt[13])/26]*ArcTanh[Sqrt[2/(-3 +
Sqrt[13])]*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 213

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

Rule 1107

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

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{\frac {3}{2}-\frac {\sqrt {13}}{2}+x^2} \, dx}{\sqrt {13}}-\frac {\int \frac {1}{\frac {3}{2}+\frac {\sqrt {13}}{2}+x^2} \, dx}{\sqrt {13}} \\ & = -\sqrt {\frac {2}{13 \left (3+\sqrt {13}\right )}} \arctan \left (\sqrt {\frac {2}{3+\sqrt {13}}} x\right )-\sqrt {\frac {1}{26} \left (3+\sqrt {13}\right )} \text {arctanh}\left (\sqrt {\frac {2}{-3+\sqrt {13}}} x\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.93 \[ \int \frac {1}{-1+3 x^2+x^4} \, dx=-\frac {\sqrt {-3+\sqrt {13}} \arctan \left (\sqrt {\frac {2}{3+\sqrt {13}}} x\right )+\sqrt {3+\sqrt {13}} \text {arctanh}\left (\sqrt {\frac {2}{-3+\sqrt {13}}} x\right )}{\sqrt {26}} \]

[In]

Integrate[(-1 + 3*x^2 + x^4)^(-1),x]

[Out]

-((Sqrt[-3 + Sqrt[13]]*ArcTan[Sqrt[2/(3 + Sqrt[13])]*x] + Sqrt[3 + Sqrt[13]]*ArcTanh[Sqrt[2/(-3 + Sqrt[13])]*x
])/Sqrt[26])

Maple [C] (verified)

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

Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.48

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}-1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}+3 \textit {\_R}}\right )}{2}\)	\(35\)
default	\(-\frac {2 \sqrt {13}\, \arctan \left (\frac {2 x}{\sqrt {6+2 \sqrt {13}}}\right )}{13 \sqrt {6+2 \sqrt {13}}}-\frac {2 \sqrt {13}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {-6+2 \sqrt {13}}}\right )}{13 \sqrt {-6+2 \sqrt {13}}}\)	\(56\)

[In]

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

[Out]

1/2*sum(1/(2*_R^3+3*_R)*ln(x-_R),_R=RootOf(_Z^4+3*_Z^2-1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (50) = 100\).

Time = 0.25 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.07 \[ \int \frac {1}{-1+3 x^2+x^4} \, dx=\frac {1}{52} \, \sqrt {26} \sqrt {\sqrt {13} + 3} \log \left (\sqrt {26} {\left (3 \, \sqrt {13} - 13\right )} \sqrt {\sqrt {13} + 3} + 52 \, x\right ) - \frac {1}{52} \, \sqrt {26} \sqrt {\sqrt {13} + 3} \log \left (-\sqrt {26} {\left (3 \, \sqrt {13} - 13\right )} \sqrt {\sqrt {13} + 3} + 52 \, x\right ) - \frac {1}{52} \, \sqrt {26} \sqrt {-\sqrt {13} + 3} \log \left (\sqrt {26} {\left (3 \, \sqrt {13} + 13\right )} \sqrt {-\sqrt {13} + 3} + 52 \, x\right ) + \frac {1}{52} \, \sqrt {26} \sqrt {-\sqrt {13} + 3} \log \left (-\sqrt {26} {\left (3 \, \sqrt {13} + 13\right )} \sqrt {-\sqrt {13} + 3} + 52 \, x\right ) \]

[In]

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

[Out]

1/52*sqrt(26)*sqrt(sqrt(13) + 3)*log(sqrt(26)*(3*sqrt(13) - 13)*sqrt(sqrt(13) + 3) + 52*x) - 1/52*sqrt(26)*sqr
t(sqrt(13) + 3)*log(-sqrt(26)*(3*sqrt(13) - 13)*sqrt(sqrt(13) + 3) + 52*x) - 1/52*sqrt(26)*sqrt(-sqrt(13) + 3)
*log(sqrt(26)*(3*sqrt(13) + 13)*sqrt(-sqrt(13) + 3) + 52*x) + 1/52*sqrt(26)*sqrt(-sqrt(13) + 3)*log(-sqrt(26)*
(3*sqrt(13) + 13)*sqrt(-sqrt(13) + 3) + 52*x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (68) = 136\).

Time = 0.26 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.00 \[ \int \frac {1}{-1+3 x^2+x^4} \, dx=\sqrt {\frac {3}{104} + \frac {\sqrt {13}}{104}} \log {\left (x - 22 \sqrt {\frac {3}{104} + \frac {\sqrt {13}}{104}} + 312 \left (\frac {3}{104} + \frac {\sqrt {13}}{104}\right )^{\frac {3}{2}} \right )} - \sqrt {\frac {3}{104} + \frac {\sqrt {13}}{104}} \log {\left (x - 312 \left (\frac {3}{104} + \frac {\sqrt {13}}{104}\right )^{\frac {3}{2}} + 22 \sqrt {\frac {3}{104} + \frac {\sqrt {13}}{104}} \right )} - 2 \sqrt {- \frac {3}{104} + \frac {\sqrt {13}}{104}} \operatorname {atan}{\left (\frac {2 \sqrt {2} x}{3 \sqrt {-3 + \sqrt {13}} + \sqrt {13} \sqrt {-3 + \sqrt {13}}} \right )} \]

[In]

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

[Out]

sqrt(3/104 + sqrt(13)/104)*log(x - 22*sqrt(3/104 + sqrt(13)/104) + 312*(3/104 + sqrt(13)/104)**(3/2)) - sqrt(3
/104 + sqrt(13)/104)*log(x - 312*(3/104 + sqrt(13)/104)**(3/2) + 22*sqrt(3/104 + sqrt(13)/104)) - 2*sqrt(-3/10
4 + sqrt(13)/104)*atan(2*sqrt(2)*x/(3*sqrt(-3 + sqrt(13)) + sqrt(13)*sqrt(-3 + sqrt(13))))

Maxima [F]

\[ \int \frac {1}{-1+3 x^2+x^4} \, dx=\int { \frac {1}{x^{4} + 3 \, x^{2} - 1} \,d x } \]

[In]

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

[Out]

integrate(1/(x^4 + 3*x^2 - 1), x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01 \[ \int \frac {1}{-1+3 x^2+x^4} \, dx=-\frac {1}{26} \, \sqrt {26 \, \sqrt {13} - 78} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {13} + \frac {3}{2}}}\right ) - \frac {1}{52} \, \sqrt {26 \, \sqrt {13} + 78} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {13} - \frac {3}{2}} \right |}\right ) + \frac {1}{52} \, \sqrt {26 \, \sqrt {13} + 78} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {13} - \frac {3}{2}} \right |}\right ) \]

[In]

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

[Out]

-1/26*sqrt(26*sqrt(13) - 78)*arctan(x/sqrt(1/2*sqrt(13) + 3/2)) - 1/52*sqrt(26*sqrt(13) + 78)*log(abs(x + sqrt
(1/2*sqrt(13) - 3/2))) + 1/52*sqrt(26*sqrt(13) + 78)*log(abs(x - sqrt(1/2*sqrt(13) - 3/2)))

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.27 \[ \int \frac {1}{-1+3 x^2+x^4} \, dx=-\frac {\sqrt {26}\,\mathrm {atanh}\left (\frac {\sqrt {26}\,x}{2\,\sqrt {\sqrt {13}+3}}+\frac {3\,\sqrt {13}\,\sqrt {26}\,x}{26\,\sqrt {\sqrt {13}+3}}\right )\,\sqrt {\sqrt {13}+3}}{26}-\frac {\sqrt {26}\,\mathrm {atanh}\left (\frac {\sqrt {26}\,x}{2\,\sqrt {3-\sqrt {13}}}-\frac {3\,\sqrt {13}\,\sqrt {26}\,x}{26\,\sqrt {3-\sqrt {13}}}\right )\,\sqrt {3-\sqrt {13}}}{26} \]

[In]

int(1/(3*x^2 + x^4 - 1),x)

[Out]

- (26^(1/2)*atanh((26^(1/2)*x)/(2*(13^(1/2) + 3)^(1/2)) + (3*13^(1/2)*26^(1/2)*x)/(26*(13^(1/2) + 3)^(1/2)))*(
13^(1/2) + 3)^(1/2))/26 - (26^(1/2)*atanh((26^(1/2)*x)/(2*(3 - 13^(1/2))^(1/2)) - (3*13^(1/2)*26^(1/2)*x)/(26*
(3 - 13^(1/2))^(1/2)))*(3 - 13^(1/2))^(1/2))/26