Integrand size = 12, antiderivative size = 73 \[ \int \frac {1}{-1-3 x^2+x^4} \, dx=-\sqrt {\frac {1}{26} \left (3+\sqrt {13}\right )} \arctan \left (\sqrt {\frac {2}{-3+\sqrt {13}}} x\right )-\sqrt {\frac {2}{13 \left (3+\sqrt {13}\right )}} \text {arctanh}\left (\sqrt {\frac {2}{3+\sqrt {13}}} x\right ) \] Output:
-1/13*arctanh(x*2^(1/2)/(3+13^(1/2))^(1/2))*26^(1/2)/(3+13^(1/2))^(1/2)-1/ 26*arctan(x*2^(1/2)/(-3+13^(1/2))^(1/2))*(78+26*13^(1/2))^(1/2)
Time = 0.02 (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[1 3]]*ArcTanh[Sqrt[2/(3 + Sqrt[13])]*x])/Sqrt[26])
Time = 0.15 (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 |
\(\displaystyle \frac {\int \frac {1}{x^2+\frac {1}{2} \left (-3-\sqrt {13}\right )}dx}{\sqrt {13}}-\sqrt {\frac {2}{13 \left (\sqrt {13}-3\right )}} \arctan \left (\sqrt {\frac {2}{\sqrt {13}-3}} x\right )\) |
\(\Big \downarrow \) 220 |
\(\displaystyle -\sqrt {\frac {2}{13 \left (\sqrt {13}-3\right )}} \arctan \left (\sqrt {\frac {2}{\sqrt {13}-3}} x\right )-\sqrt {\frac {2}{13 \left (3+\sqrt {13}\right )}} \text {arctanh}\left (\sqrt {\frac {2}{3+\sqrt {13}}} x\right )\) |
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 | \(-\frac {2 \sqrt {13}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {6+2 \sqrt {13}}}\right )}{13 \sqrt {6+2 \sqrt {13}}}-\frac {2 \sqrt {13}\, \arctan \left (\frac {2 x}{\sqrt {-6+2 \sqrt {13}}}\right )}{13 \sqrt {-6+2 \sqrt {13}}}\) | \(56\) |
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 = 0.08 (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)
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.33 \[ \int \frac {1}{-1-3 x^2+x^4} \, dx=\operatorname {RootSum} {\left (2704 t^{4} + 156 t^{2} - 1, \left ( t \mapsto t \log {\left (- 312 t^{3} - 22 t + x \right )} \right )\right )} \] Input:
integrate(1/(x**4-3*x**2-1),x)
Output:
RootSum(2704*_t**4 + 156*_t**2 - 1, Lambda(_t, _t*log(-312*_t**3 - 22*_t + x)))
\[ \int \frac {1}{-1-3 x^2+x^4} \, dx=\int { \frac {1}{x^{4} - 3 \, x^{2} - 1} \,d x } \] 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 = 0.14 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.27 \[ \int \frac {1}{-1-3 x^2+x^4} \, dx=-\frac {\sqrt {26}\,\mathrm {atanh}\left (\frac {\sqrt {26}\,x}{2\,\sqrt {\sqrt {13}-3}}-\frac {3\,\sqrt {13}\,\sqrt {26}\,x}{26\,\sqrt {\sqrt {13}-3}}\right )\,\sqrt {\sqrt {13}-3}}{26}-\frac {\sqrt {26}\,\mathrm {atanh}\left (\frac {\sqrt {26}\,x}{2\,\sqrt {-\sqrt {13}-3}}+\frac {3\,\sqrt {13}\,\sqrt {26}\,x}{26\,\sqrt {-\sqrt {13}-3}}\right )\,\sqrt {-\sqrt {13}-3}}{26} \] 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*(- 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
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)* sqrt(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*sqrt(sqrt(13) + 3)*sqrt(13)*log(sqrt(sqrt(13) + 3) + sqrt(2)*x) + 13*sqr t(sqrt(13) + 3)*log( - sqrt(sqrt(13) + 3) + sqrt(2)*x) - 13*sqrt(sqrt(13) + 3)*log(sqrt(sqrt(13) + 3) + sqrt(2)*x)))/104