Integrand size = 12, antiderivative size = 73 \[ \int \frac {1}{-1+3 x^2+x^4} \, dx=-\sqrt {\frac {2}{13 \left (3+\sqrt {13}\right )}} \arctan \left (\sqrt {\frac {2}{3+\sqrt {13}}} x\right )-\sqrt {\frac {1}{26} \left (3+\sqrt {13}\right )} \text {arctanh}\left (\sqrt {\frac {2}{-3+\sqrt {13}}} x\right ) \] Output:
-1/13*arctan(x*2^(1/2)/(3+13^(1/2))^(1/2))*26^(1/2)/(3+13^(1/2))^(1/2)-1/2 6*arctanh(x*2^(1/2)/(-3+13^(1/2))^(1/2))*(78+26*13^(1/2))^(1/2)
Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.93 \[ \int \frac {1}{-1+3 x^2+x^4} \, dx=-\frac {\sqrt {-3+\sqrt {13}} \arctan \left (\sqrt {\frac {2}{3+\sqrt {13}}} x\right )+\sqrt {3+\sqrt {13}} \text {arctanh}\left (\sqrt {\frac {2}{-3+\sqrt {13}}} x\right )}{\sqrt {26}} \] Input:
Integrate[(-1 + 3*x^2 + x^4)^(-1),x]
Output:
-((Sqrt[-3 + Sqrt[13]]*ArcTan[Sqrt[2/(3 + Sqrt[13])]*x] + Sqrt[3 + Sqrt[13 ]]*ArcTanh[Sqrt[2/(-3 + Sqrt[13])]*x])/Sqrt[26])
Time = 0.17 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1406, 216, 220}
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.
\(\displaystyle \int \frac {1}{x^4+3 x^2-1} \, dx\) |
\(\Big \downarrow \) 1406 |
\(\displaystyle \frac {\int \frac {1}{x^2+\frac {1}{2} \left (3-\sqrt {13}\right )}dx}{\sqrt {13}}-\frac {\int \frac {1}{x^2+\frac {1}{2} \left (3+\sqrt {13}\right )}dx}{\sqrt {13}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\int \frac {1}{x^2+\frac {1}{2} \left (3-\sqrt {13}\right )}dx}{\sqrt {13}}-\sqrt {\frac {2}{13 \left (3+\sqrt {13}\right )}} \arctan \left (\sqrt {\frac {2}{3+\sqrt {13}}} x\right )\) |
\(\Big \downarrow \) 220 |
\(\displaystyle -\sqrt {\frac {2}{13 \left (3+\sqrt {13}\right )}} \arctan \left (\sqrt {\frac {2}{3+\sqrt {13}}} x\right )-\sqrt {\frac {2}{13 \left (\sqrt {13}-3\right )}} \text {arctanh}\left (\sqrt {\frac {2}{\sqrt {13}-3}} x\right )\) |
Input:
Int[(-1 + 3*x^2 + x^4)^(-1),x]
Output:
-(Sqrt[2/(13*(3 + Sqrt[13]))]*ArcTan[Sqrt[2/(3 + Sqrt[13])]*x]) - Sqrt[2/( 13*(-3 + Sqrt[13]))]*ArcTanh[Sqrt[2/(-3 + Sqrt[13])]*x]
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])
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])
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q I nt[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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (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 | \(-\frac {2 \sqrt {13}\, \arctan \left (\frac {2 x}{\sqrt {6+2 \sqrt {13}}}\right )}{13 \sqrt {6+2 \sqrt {13}}}-\frac {2 \sqrt {13}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {-6+2 \sqrt {13}}}\right )}{13 \sqrt {-6+2 \sqrt {13}}}\) | \(56\) |
Input:
int(1/(x^4+3*x^2-1),x,method=_RETURNVERBOSE)
Output:
1/2*sum(1/(2*_R^3+3*_R)*ln(x-_R),_R=RootOf(_Z^4+3*_Z^2-1))
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.29 \[ \int \frac {1}{-1+3 x^2+x^4} \, dx=-\sqrt {\frac {1}{26} \, \sqrt {13} - \frac {3}{26}} \arctan \left (\sqrt {13} x \sqrt {\frac {1}{26} \, \sqrt {13} - \frac {3}{26}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{26} \, \sqrt {13} + \frac {3}{26}} \log \left ({\left (3 \, \sqrt {13} - 13\right )} \sqrt {\frac {1}{26} \, \sqrt {13} + \frac {3}{26}} + 2 \, x\right ) - \frac {1}{2} \, \sqrt {\frac {1}{26} \, \sqrt {13} + \frac {3}{26}} \log \left (-{\left (3 \, \sqrt {13} - 13\right )} \sqrt {\frac {1}{26} \, \sqrt {13} + \frac {3}{26}} + 2 \, x\right ) \] Input:
integrate(1/(x^4+3*x^2-1),x, algorithm="fricas")
Output:
-sqrt(1/26*sqrt(13) - 3/26)*arctan(sqrt(13)*x*sqrt(1/26*sqrt(13) - 3/26)) + 1/2*sqrt(1/26*sqrt(13) + 3/26)*log((3*sqrt(13) - 13)*sqrt(1/26*sqrt(13) + 3/26) + 2*x) - 1/2*sqrt(1/26*sqrt(13) + 3/26)*log(-(3*sqrt(13) - 13)*sqr t(1/26*sqrt(13) + 3/26) + 2*x)
Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (68) = 136\).
Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.00 \[ \int \frac {1}{-1+3 x^2+x^4} \, dx=\sqrt {\frac {3}{104} + \frac {\sqrt {13}}{104}} \log {\left (x - 22 \sqrt {\frac {3}{104} + \frac {\sqrt {13}}{104}} + 312 \left (\frac {3}{104} + \frac {\sqrt {13}}{104}\right )^{\frac {3}{2}} \right )} - \sqrt {\frac {3}{104} + \frac {\sqrt {13}}{104}} \log {\left (x - 312 \left (\frac {3}{104} + \frac {\sqrt {13}}{104}\right )^{\frac {3}{2}} + 22 \sqrt {\frac {3}{104} + \frac {\sqrt {13}}{104}} \right )} - 2 \sqrt {- \frac {3}{104} + \frac {\sqrt {13}}{104}} \operatorname {atan}{\left (\frac {2 \sqrt {2} x}{3 \sqrt {-3 + \sqrt {13}} + \sqrt {13} \sqrt {-3 + \sqrt {13}}} \right )} \] Input:
integrate(1/(x**4+3*x**2-1),x)
Output:
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*(3/10 4 + sqrt(13)/104)**(3/2) + 22*sqrt(3/104 + sqrt(13)/104)) - 2*sqrt(-3/104 + sqrt(13)/104)*atan(2*sqrt(2)*x/(3*sqrt(-3 + sqrt(13)) + sqrt(13)*sqrt(-3 + sqrt(13))))
\[ \int \frac {1}{-1+3 x^2+x^4} \, dx=\int { \frac {1}{x^{4} + 3 \, x^{2} - 1} \,d x } \] Input:
integrate(1/(x^4+3*x^2-1),x, algorithm="maxima")
Output:
integrate(1/(x^4 + 3*x^2 - 1), x)
Time = 0.14 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01 \[ \int \frac {1}{-1+3 x^2+x^4} \, dx=-\frac {1}{26} \, \sqrt {26 \, \sqrt {13} - 78} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {13} + \frac {3}{2}}}\right ) - \frac {1}{52} \, \sqrt {26 \, \sqrt {13} + 78} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {13} - \frac {3}{2}} \right |}\right ) + \frac {1}{52} \, \sqrt {26 \, \sqrt {13} + 78} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {13} - \frac {3}{2}} \right |}\right ) \] Input:
integrate(1/(x^4+3*x^2-1),x, algorithm="giac")
Output:
-1/26*sqrt(26*sqrt(13) - 78)*arctan(x/sqrt(1/2*sqrt(13) + 3/2)) - 1/52*sqr t(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)))
Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.27 \[ \int \frac {1}{-1+3 x^2+x^4} \, dx=-\frac {\sqrt {26}\,\mathrm {atanh}\left (\frac {\sqrt {26}\,x}{2\,\sqrt {\sqrt {13}+3}}+\frac {3\,\sqrt {13}\,\sqrt {26}\,x}{26\,\sqrt {\sqrt {13}+3}}\right )\,\sqrt {\sqrt {13}+3}}{26}-\frac {\sqrt {26}\,\mathrm {atanh}\left (\frac {\sqrt {26}\,x}{2\,\sqrt {3-\sqrt {13}}}-\frac {3\,\sqrt {13}\,\sqrt {26}\,x}{26\,\sqrt {3-\sqrt {13}}}\right )\,\sqrt {3-\sqrt {13}}}{26} \] Input:
int(1/(3*x^2 + x^4 - 1),x)
Output:
- (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)*at anh((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
Time = 0.15 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.79 \[ \int \frac {1}{-1+3 x^2+x^4} \, dx=\frac {\sqrt {2}\, \left (6 \sqrt {\sqrt {13}+3}\, \sqrt {13}\, \mathit {atan} \left (\frac {2 x}{\sqrt {\sqrt {13}+3}\, \sqrt {2}}\right )-26 \sqrt {\sqrt {13}+3}\, \mathit {atan} \left (\frac {2 x}{\sqrt {\sqrt {13}+3}\, \sqrt {2}}\right )+3 \sqrt {\sqrt {13}-3}\, \sqrt {13}\, \mathrm {log}\left (-\sqrt {\sqrt {13}-3}+\sqrt {2}\, x \right )-3 \sqrt {\sqrt {13}-3}\, \sqrt {13}\, \mathrm {log}\left (\sqrt {\sqrt {13}-3}+\sqrt {2}\, x \right )+13 \sqrt {\sqrt {13}-3}\, \mathrm {log}\left (-\sqrt {\sqrt {13}-3}+\sqrt {2}\, x \right )-13 \sqrt {\sqrt {13}-3}\, \mathrm {log}\left (\sqrt {\sqrt {13}-3}+\sqrt {2}\, x \right )\right )}{104} \] Input:
int(1/(x^4+3*x^2-1),x)
Output:
(sqrt(2)*(6*sqrt(sqrt(13) + 3)*sqrt(13)*atan((2*x)/(sqrt(sqrt(13) + 3)*sqr t(2))) - 26*sqrt(sqrt(13) + 3)*atan((2*x)/(sqrt(sqrt(13) + 3)*sqrt(2))) + 3*sqrt(sqrt(13) - 3)*sqrt(13)*log( - sqrt(sqrt(13) - 3) + sqrt(2)*x) - 3*s qrt(sqrt(13) - 3)*sqrt(13)*log(sqrt(sqrt(13) - 3) + sqrt(2)*x) + 13*sqrt(s qrt(13) - 3)*log( - sqrt(sqrt(13) - 3) + sqrt(2)*x) - 13*sqrt(sqrt(13) - 3 )*log(sqrt(sqrt(13) - 3) + sqrt(2)*x)))/104