Integrand size = 21, antiderivative size = 69 \[ \int \frac {1}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=\frac {1}{2} \arctan \left (\frac {1-\sqrt {1-x^2}}{x \sqrt [4]{1-x^2}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x \sqrt [4]{1-x^2}}{1+\sqrt {1-x^2}}\right ) \] Output:
1/2*arctan((1-(-x^2+1)^(1/2))/x/(-x^2+1)^(1/4))+1/2*arctanh(x*(-x^2+1)^(1/ 4)/(1+(-x^2+1)^(1/2)))
Time = 0.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.52 \[ \int \frac {1}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=\frac {1}{4} \arctan \left (\frac {\sqrt [4]{1-x^2}}{-x+\sqrt [4]{1-x^2}}\right )-\frac {1}{4} \arctan \left (\frac {\sqrt [4]{1-x^2}}{x+\sqrt [4]{1-x^2}}\right )+\frac {1}{4} \text {arctanh}\left (\frac {2 x \sqrt [4]{1-x^2}}{x^2+2 \sqrt {1-x^2}}\right ) \] Input:
Integrate[1/((1 - x^2)^(1/4)*(2 - x^2)),x]
Output:
ArcTan[(1 - x^2)^(1/4)/(-x + (1 - x^2)^(1/4))]/4 - ArcTan[(1 - x^2)^(1/4)/ (x + (1 - x^2)^(1/4))]/4 + ArcTanh[(2*x*(1 - x^2)^(1/4))/(x^2 + 2*Sqrt[1 - x^2])]/4
Time = 0.16 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {308}
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.
\(\displaystyle \int \frac {1}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx\) |
\(\Big \downarrow \) 308 |
\(\displaystyle \frac {1}{2} \arctan \left (\frac {1-\sqrt {1-x^2}}{x \sqrt [4]{1-x^2}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {1-x^2}+1}{x \sqrt [4]{1-x^2}}\right )\) |
Input:
Int[1/((1 - x^2)^(1/4)*(2 - x^2)),x]
Output:
ArcTan[(1 - Sqrt[1 - x^2])/(x*(1 - x^2)^(1/4))]/2 + ArcTanh[(1 + Sqrt[1 - x^2])/(x*(1 - x^2)^(1/4))]/2
Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit h[{q = Rt[b^2/a, 4]}, Simp[(-b/(2*a*d*q))*ArcTan[(b + q^2*Sqrt[a + b*x^2])/ (q^3*x*(a + b*x^2)^(1/4))], x] - Simp[(b/(2*a*d*q))*ArcTanh[(b - q^2*Sqrt[a + b*x^2])/(q^3*x*(a + b*x^2)^(1/4))], x]] /; FreeQ[{a, b, c, d}, x] && EqQ [b*c - 2*a*d, 0] && PosQ[b^2/a]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.97 (sec) , antiderivative size = 306, normalized size of antiderivative = 4.43
method | result | size |
trager | \(\frac {\ln \left (-\frac {4 \left (-x^{2}+1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )-2 \left (-x^{2}+1\right )^{\frac {3}{4}}-x \sqrt {-x^{2}+1}-4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{4}}-4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x +x}{x^{2}-2}\right )}{2}-\ln \left (-\frac {4 \left (-x^{2}+1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )-2 \left (-x^{2}+1\right )^{\frac {3}{4}}-x \sqrt {-x^{2}+1}-4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{4}}-4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x +x}{x^{2}-2}\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )+\operatorname {RootOf}\left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \ln \left (\frac {4 \left (-x^{2}+1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )+x \sqrt {-x^{2}+1}-4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{4}}-4 \operatorname {RootOf}\left (8 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x +2 \left (-x^{2}+1\right )^{\frac {1}{4}}+x}{x^{2}-2}\right )\) | \(306\) |
Input:
int(1/(-x^2+1)^(1/4)/(-x^2+2),x,method=_RETURNVERBOSE)
Output:
1/2*ln(-(4*(-x^2+1)^(3/4)*RootOf(8*_Z^2-4*_Z+1)-2*(-x^2+1)^(3/4)-x*(-x^2+1 )^(1/2)-4*RootOf(8*_Z^2-4*_Z+1)*(-x^2+1)^(1/4)-4*RootOf(8*_Z^2-4*_Z+1)*x+x )/(x^2-2))-ln(-(4*(-x^2+1)^(3/4)*RootOf(8*_Z^2-4*_Z+1)-2*(-x^2+1)^(3/4)-x* (-x^2+1)^(1/2)-4*RootOf(8*_Z^2-4*_Z+1)*(-x^2+1)^(1/4)-4*RootOf(8*_Z^2-4*_Z +1)*x+x)/(x^2-2))*RootOf(8*_Z^2-4*_Z+1)+RootOf(8*_Z^2-4*_Z+1)*ln((4*(-x^2+ 1)^(3/4)*RootOf(8*_Z^2-4*_Z+1)+x*(-x^2+1)^(1/2)-4*RootOf(8*_Z^2-4*_Z+1)*(- x^2+1)^(1/4)-4*RootOf(8*_Z^2-4*_Z+1)*x+2*(-x^2+1)^(1/4)+x)/(x^2-2))
Time = 1.69 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.55 \[ \int \frac {1}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=-\frac {1}{2} \, \arctan \left (\frac {x^{4} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{4}} x^{3} - 2 \, x^{2} + 4 \, {\left (-x^{2} + 1\right )}^{\frac {3}{4}} x + 2 \, {\left (x^{2} - 2\right )} \sqrt {-x^{2} + 1}}{x^{4} + 4 \, x^{2} - 4}\right ) + \frac {1}{4} \, \log \left (-\frac {x^{2} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{4}} x + 2 \, \sqrt {-x^{2} + 1}}{x^{2} - 2}\right ) \] Input:
integrate(1/(-x^2+1)^(1/4)/(-x^2+2),x, algorithm="fricas")
Output:
-1/2*arctan((x^4 + 2*(-x^2 + 1)^(1/4)*x^3 - 2*x^2 + 4*(-x^2 + 1)^(3/4)*x + 2*(x^2 - 2)*sqrt(-x^2 + 1))/(x^4 + 4*x^2 - 4)) + 1/4*log(-(x^2 + 2*(-x^2 + 1)^(1/4)*x + 2*sqrt(-x^2 + 1))/(x^2 - 2))
\[ \int \frac {1}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=- \int \frac {1}{x^{2} \sqrt [4]{1 - x^{2}} - 2 \sqrt [4]{1 - x^{2}}}\, dx \] Input:
integrate(1/(-x**2+1)**(1/4)/(-x**2+2),x)
Output:
-Integral(1/(x**2*(1 - x**2)**(1/4) - 2*(1 - x**2)**(1/4)), x)
\[ \int \frac {1}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=\int { -\frac {1}{{\left (x^{2} - 2\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/(-x^2+1)^(1/4)/(-x^2+2),x, algorithm="maxima")
Output:
-integrate(1/((x^2 - 2)*(-x^2 + 1)^(1/4)), x)
\[ \int \frac {1}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=\int { -\frac {1}{{\left (x^{2} - 2\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}}} \,d x } \] Input:
integrate(1/(-x^2+1)^(1/4)/(-x^2+2),x, algorithm="giac")
Output:
integrate(-1/((x^2 - 2)*(-x^2 + 1)^(1/4)), x)
Timed out. \[ \int \frac {1}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=-\int \frac {1}{{\left (1-x^2\right )}^{1/4}\,\left (x^2-2\right )} \,d x \] Input:
int(-1/((1 - x^2)^(1/4)*(x^2 - 2)),x)
Output:
-int(1/((1 - x^2)^(1/4)*(x^2 - 2)), x)
\[ \int \frac {1}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=-\left (\int \frac {1}{\left (-x^{2}+1\right )^{\frac {1}{4}} x^{2}-2 \left (-x^{2}+1\right )^{\frac {1}{4}}}d x \right ) \] Input:
int(1/(-x^2+1)^(1/4)/(-x^2+2),x)
Output:
- int(1/(( - x**2 + 1)**(1/4)*x**2 - 2*( - x**2 + 1)**(1/4)),x)