Integrand size = 22, antiderivative size = 79 \[ \int \frac {x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=\frac {\arctan \left (\frac {1-\sqrt {1-x^2}}{\sqrt {2} \sqrt [4]{1-x^2}}\right )}{\sqrt {2}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1-x^2}}{1+\sqrt {1-x^2}}\right )}{\sqrt {2}} \] Output:
1/2*arctan(1/2*(1-(-x^2+1)^(1/2))/(-x^2+1)^(1/4)*2^(1/2))*2^(1/2)+1/2*arct anh(2^(1/2)*(-x^2+1)^(1/4)/(1+(-x^2+1)^(1/2)))*2^(1/2)
Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.92 \[ \int \frac {x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=\frac {-\arctan \left (\frac {-1+\sqrt {1-x^2}}{\sqrt {2} \sqrt [4]{1-x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1-x^2}}{1+\sqrt {1-x^2}}\right )}{\sqrt {2}} \] Input:
Integrate[x/((1 - x^2)^(1/4)*(2 - x^2)),x]
Output:
(-ArcTan[(-1 + Sqrt[1 - x^2])/(Sqrt[2]*(1 - x^2)^(1/4))] + ArcTanh[(Sqrt[2 ]*(1 - x^2)^(1/4))/(1 + Sqrt[1 - x^2])])/Sqrt[2]
Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.97, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {348}
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.
\(\displaystyle \int \frac {x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx\) |
\(\Big \downarrow \) 348 |
\(\displaystyle \frac {\arctan \left (\frac {1-\sqrt {1-x^2}}{\sqrt {2} \sqrt [4]{1-x^2}}\right )}{\sqrt {2}}+\frac {\text {arctanh}\left (\frac {\sqrt {1-x^2}+1}{\sqrt {2} \sqrt [4]{1-x^2}}\right )}{\sqrt {2}}\) |
Input:
Int[x/((1 - x^2)^(1/4)*(2 - x^2)),x]
Output:
ArcTan[(1 - Sqrt[1 - x^2])/(Sqrt[2]*(1 - x^2)^(1/4))]/Sqrt[2] + ArcTanh[(1 + Sqrt[1 - x^2])/(Sqrt[2]*(1 - x^2)^(1/4))]/Sqrt[2]
Int[(x_)/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Simp[(-(Sqrt[2]*Rt[a, 4]*d)^(-1))*ArcTan[(Rt[a, 4]^2 - Sqrt[a + b*x^2])/(Sq rt[2]*Rt[a, 4]*(a + b*x^2)^(1/4))], x] - Simp[(1/(Sqrt[2]*Rt[a, 4]*d))*ArcT anh[(Rt[a, 4]^2 + Sqrt[a + b*x^2])/(Sqrt[2]*Rt[a, 4]*(a + b*x^2)^(1/4))], x ] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && PosQ[a]
Time = 1.73 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.22
method | result | size |
pseudoelliptic | \(-\frac {\sqrt {2}\, \left (\ln \left (\frac {\sqrt {-x^{2}+1}-\left (-x^{2}+1\right )^{\frac {1}{4}} \sqrt {2}+1}{\sqrt {-x^{2}+1}+\left (-x^{2}+1\right )^{\frac {1}{4}} \sqrt {2}+1}\right )+2 \arctan \left (\left (-x^{2}+1\right )^{\frac {1}{4}} \sqrt {2}+1\right )+2 \arctan \left (\left (-x^{2}+1\right )^{\frac {1}{4}} \sqrt {2}-1\right )\right )}{4}\) | \(96\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (-x^{2}+1\right )^{\frac {3}{4}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {-x^{2}+1}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (-x^{2}+1\right )^{\frac {1}{4}}-x^{2}}{x^{2}-2}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (-x^{2}+1\right )^{\frac {3}{4}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {-x^{2}+1}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (-x^{2}+1\right )^{\frac {1}{4}}-x^{2}}{x^{2}-2}\right )}{2}\) | \(160\) |
Input:
int(x/(-x^2+1)^(1/4)/(-x^2+2),x,method=_RETURNVERBOSE)
Output:
-1/4*2^(1/2)*(ln(((-x^2+1)^(1/2)-(-x^2+1)^(1/4)*2^(1/2)+1)/((-x^2+1)^(1/2) +(-x^2+1)^(1/4)*2^(1/2)+1))+2*arctan((-x^2+1)^(1/4)*2^(1/2)+1)+2*arctan((- x^2+1)^(1/4)*2^(1/2)-1))
Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.32 \[ \int \frac {x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} - 1\right ) + \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} + \sqrt {-x^{2} + 1} + 1\right ) - \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} + \sqrt {-x^{2} + 1} + 1\right ) \] Input:
integrate(x/(-x^2+1)^(1/4)/(-x^2+2),x, algorithm="fricas")
Output:
-1/2*sqrt(2)*arctan(sqrt(2)*(-x^2 + 1)^(1/4) + 1) - 1/2*sqrt(2)*arctan(sqr t(2)*(-x^2 + 1)^(1/4) - 1) + 1/4*sqrt(2)*log(sqrt(2)*(-x^2 + 1)^(1/4) + sq rt(-x^2 + 1) + 1) - 1/4*sqrt(2)*log(-sqrt(2)*(-x^2 + 1)^(1/4) + sqrt(-x^2 + 1) + 1)
\[ \int \frac {x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=- \int \frac {x}{x^{2} \sqrt [4]{1 - x^{2}} - 2 \sqrt [4]{1 - x^{2}}}\, dx \] Input:
integrate(x/(-x**2+1)**(1/4)/(-x**2+2),x)
Output:
-Integral(x/(x**2*(1 - x**2)**(1/4) - 2*(1 - x**2)**(1/4)), x)
Time = 0.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.44 \[ \int \frac {x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} + \sqrt {-x^{2} + 1} + 1\right ) - \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} + \sqrt {-x^{2} + 1} + 1\right ) \] Input:
integrate(x/(-x^2+1)^(1/4)/(-x^2+2),x, algorithm="maxima")
Output:
-1/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(-x^2 + 1)^(1/4))) - 1/2*sqrt (2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(-x^2 + 1)^(1/4))) + 1/4*sqrt(2)*log( sqrt(2)*(-x^2 + 1)^(1/4) + sqrt(-x^2 + 1) + 1) - 1/4*sqrt(2)*log(-sqrt(2)* (-x^2 + 1)^(1/4) + sqrt(-x^2 + 1) + 1)
Time = 0.12 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.44 \[ \int \frac {x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} + \sqrt {-x^{2} + 1} + 1\right ) - \frac {1}{4} \, \sqrt {2} \log \left (-\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} + \sqrt {-x^{2} + 1} + 1\right ) \] Input:
integrate(x/(-x^2+1)^(1/4)/(-x^2+2),x, algorithm="giac")
Output:
-1/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(-x^2 + 1)^(1/4))) - 1/2*sqrt (2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(-x^2 + 1)^(1/4))) + 1/4*sqrt(2)*log( sqrt(2)*(-x^2 + 1)^(1/4) + sqrt(-x^2 + 1) + 1) - 1/4*sqrt(2)*log(-sqrt(2)* (-x^2 + 1)^(1/4) + sqrt(-x^2 + 1) + 1)
Time = 1.56 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.62 \[ \int \frac {x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (1-x^2\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (1-x^2\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right ) \] Input:
int(-x/((1 - x^2)^(1/4)*(x^2 - 2)),x)
Output:
- 2^(1/2)*atan(2^(1/2)*(1 - x^2)^(1/4)*(1/2 - 1i/2))*(1/2 - 1i/2) - 2^(1/2 )*atan(2^(1/2)*(1 - x^2)^(1/4)*(1/2 + 1i/2))*(1/2 + 1i/2)
\[ \int \frac {x}{\sqrt [4]{1-x^2} \left (2-x^2\right )} \, dx=-\left (\int \frac {x}{\left (-x^{2}+1\right )^{\frac {1}{4}} x^{2}-2 \left (-x^{2}+1\right )^{\frac {1}{4}}}d x \right ) \] Input:
int(x/(-x^2+1)^(1/4)/(-x^2+2),x)
Output:
- int(x/(( - x**2 + 1)**(1/4)*x**2 - 2*( - x**2 + 1)**(1/4)),x)