Optimal. Leaf size=104 \[ \frac {\text {ArcTan}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2}}}{x \sqrt {1-x^4}}\right )}{2 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}+\frac {x^4}{\sqrt {2}}}{x \sqrt {1-x^4}}\right )}{2 \sqrt {2}} \]
[Out]
________________________________________________________________________________________
Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.29, antiderivative size = 139, normalized size of antiderivative = 1.34, number of steps
used = 16, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {6860, 415,
227, 418, 1227, 551} \begin {gather*} -\frac {1}{2} \left (1+i \sqrt {3}\right ) F(\text {ArcSin}(x)|-1)-\frac {1}{2} \left (1-i \sqrt {3}\right ) F(\text {ArcSin}(x)|-1)+\frac {1}{2} \Pi \left (\frac {1}{2} \left (-i-\sqrt {3}\right );\text {ArcSin}(x)|-1\right )+\frac {1}{2} \Pi \left (\frac {1}{2} \left (i-\sqrt {3}\right );\text {ArcSin}(x)|-1\right )+\frac {1}{2} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}};\text {ArcSin}(x)|-1\right )+\frac {1}{2} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}};\text {ArcSin}(x)|-1\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 227
Rule 415
Rule 418
Rule 551
Rule 1227
Rule 6860
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^4} \left (1+x^4\right )}{1-x^4+x^8} \, dx &=\int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt {1-x^4}}{-1-i \sqrt {3}+2 x^4}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {1-x^4}}{-1+i \sqrt {3}+2 x^4}\right ) \, dx\\ &=\left (1-i \sqrt {3}\right ) \int \frac {\sqrt {1-x^4}}{-1-i \sqrt {3}+2 x^4} \, dx+\left (1+i \sqrt {3}\right ) \int \frac {\sqrt {1-x^4}}{-1+i \sqrt {3}+2 x^4} \, dx\\ &=\frac {1}{2} \left (-1-i \sqrt {3}\right ) \int \frac {1}{\sqrt {1-x^4}} \, dx+\left (-1-i \sqrt {3}\right ) \int \frac {1}{\sqrt {1-x^4} \left (-1-i \sqrt {3}+2 x^4\right )} \, dx+\frac {1}{2} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1-x^4}} \, dx+\left (-1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1-x^4} \left (-1+i \sqrt {3}+2 x^4\right )} \, dx\\ &=-\frac {1}{2} \left (1-i \sqrt {3}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {1}{2} \left (1+i \sqrt {3}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{2} \int \frac {1}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \sqrt {1-x^4}} \, dx+\frac {1}{2} \int \frac {1}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right ) \sqrt {1-x^4}} \, dx+\frac {1}{2} \int \frac {1}{\left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \sqrt {1-x^4}} \, dx+\frac {1}{2} \int \frac {1}{\left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right ) \sqrt {1-x^4}} \, dx\\ &=-\frac {1}{2} \left (1-i \sqrt {3}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {1}{2} \left (1+i \sqrt {3}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{2} \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )} \, dx+\frac {1}{2} \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}}\right )} \, dx+\frac {1}{2} \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )} \, dx+\frac {1}{2} \int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+\frac {x^2}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}}\right )} \, dx\\ &=-\frac {1}{2} \left (1-i \sqrt {3}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac {1}{2} \left (1+i \sqrt {3}\right ) F\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{2} \Pi \left (\frac {1}{2} \left (-i-\sqrt {3}\right );\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{2} \Pi \left (\frac {1}{2} \left (i-\sqrt {3}\right );\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{2} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}};\left .\sin ^{-1}(x)\right |-1\right )+\frac {1}{2} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}};\left .\sin ^{-1}(x)\right |-1\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.24, size = 46, normalized size = 0.44 \begin {gather*} -\frac {1}{2} (-1)^{3/4} \left (\text {ArcTan}\left (\frac {(1+i) x}{\sqrt {2-2 x^4}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{-1} x}{\sqrt {1-x^4}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.04, size = 114, normalized size = 1.10
method | result | size |
default | \(\frac {\left (-\frac {\ln \left (\frac {-x^{4}+1}{x^{2}}-\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}+1\right )}{4}-\frac {\arctan \left (-1+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}\right )}{2}+\frac {\ln \left (\frac {-x^{4}+1}{x^{2}}+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}+1\right )}{4}-\frac {\arctan \left (1+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}\right )}{2}\right ) \sqrt {2}}{2}\) | \(114\) |
elliptic | \(\frac {\left (-\frac {\ln \left (\frac {-x^{4}+1}{x^{2}}-\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}+1\right )}{4}-\frac {\arctan \left (-1+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}\right )}{2}+\frac {\ln \left (\frac {-x^{4}+1}{x^{2}}+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}+1\right )}{4}-\frac {\arctan \left (1+\frac {\sqrt {-x^{4}+1}\, \sqrt {2}}{x}\right )}{2}\right ) \sqrt {2}}{2}\) | \(114\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{5} x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+2 \sqrt {-x^{4}+1}\, x}{\left (\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+1\right ) \left (\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-1\right )}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \sqrt {-x^{4}+1}\, x +\RootOf \left (\textit {\_Z}^{4}+1\right )}{\left (\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x +1\right ) \left (\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x +1\right )}\right )}{4}\) | \(217\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 466 vs.
\(2 (82) = 164\).
time = 0.46, size = 466, normalized size = 4.48 \begin {gather*} \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {x^{8} - x^{4} + 2 \, \sqrt {2} {\left (x^{5} + x^{3} - x\right )} \sqrt {-x^{4} + 1} - {\left (4 \, \sqrt {-x^{4} + 1} x^{3} - \sqrt {2} {\left (x^{8} + 2 \, x^{6} - 3 \, x^{4} - 2 \, x^{2} + 1\right )}\right )} \sqrt {\frac {x^{8} - 4 \, x^{6} - x^{4} + 2 \, \sqrt {2} {\left (x^{5} - x^{3} - x\right )} \sqrt {-x^{4} + 1} + 4 \, x^{2} + 1}{x^{8} - x^{4} + 1}} + 1}{x^{8} + 4 \, x^{6} - x^{4} - 4 \, x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {x^{8} - x^{4} - 2 \, \sqrt {2} {\left (x^{5} + x^{3} - x\right )} \sqrt {-x^{4} + 1} - {\left (4 \, \sqrt {-x^{4} + 1} x^{3} + \sqrt {2} {\left (x^{8} + 2 \, x^{6} - 3 \, x^{4} - 2 \, x^{2} + 1\right )}\right )} \sqrt {\frac {x^{8} - 4 \, x^{6} - x^{4} - 2 \, \sqrt {2} {\left (x^{5} - x^{3} - x\right )} \sqrt {-x^{4} + 1} + 4 \, x^{2} + 1}{x^{8} - x^{4} + 1}} + 1}{x^{8} + 4 \, x^{6} - x^{4} - 4 \, x^{2} + 1}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{8} - 4 \, x^{6} - x^{4} + 2 \, \sqrt {2} {\left (x^{5} - x^{3} - x\right )} \sqrt {-x^{4} + 1} + 4 \, x^{2} + 1\right )}}{x^{8} - x^{4} + 1}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{8} - 4 \, x^{6} - x^{4} - 2 \, \sqrt {2} {\left (x^{5} - x^{3} - x\right )} \sqrt {-x^{4} + 1} + 4 \, x^{2} + 1\right )}}{x^{8} - x^{4} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{4} + 1\right )}{x^{8} - x^{4} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {1-x^4}\,\left (x^4+1\right )}{x^8-x^4+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________