Optimal. Leaf size=125 \[ \frac {2}{3} \left (-1+x^4\right )^{3/4}+\text {ArcTan}\left (\frac {\sqrt [4]{-1+x^4}}{x}\right )+\frac {\text {ArcTan}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {\sqrt {-1+x^4}}{\sqrt {2}}}{\sqrt [4]{-1+x^4}}\right )}{2 \sqrt {2}}-\tanh ^{-1}\left (\frac {\sqrt [4]{-1+x^4}}{x}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 170, normalized size of antiderivative = 1.36, number of steps
used = 18, number of rules used = 14, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {1847, 246,
218, 212, 209, 457, 81, 65, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\text {ArcTan}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\text {ArcTan}\left (1-\sqrt {2} \sqrt [4]{x^4-1}\right )}{2 \sqrt {2}}+\frac {\text {ArcTan}\left (\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{2 \sqrt {2}}+\frac {2}{3} \left (x^4-1\right )^{3/4}+\frac {\log \left (\sqrt {x^4-1}-\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{4 \sqrt {2}}-\frac {\log \left (\sqrt {x^4-1}+\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{4 \sqrt {2}}-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 81
Rule 209
Rule 210
Rule 212
Rule 218
Rule 246
Rule 303
Rule 457
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1847
Rubi steps
\begin {align*} \int \frac {1-2 x+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx &=\int \left (-\frac {2}{\sqrt [4]{-1+x^4}}+\frac {1+2 x^4}{x \sqrt [4]{-1+x^4}}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\sqrt [4]{-1+x^4}} \, dx\right )+\int \frac {1+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx\\ &=\frac {1}{4} \text {Subst}\left (\int \frac {1+2 x}{\sqrt [4]{-1+x} x} \, dx,x,x^4\right )-2 \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {2}{3} \left (-1+x^4\right )^{3/4}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x} x} \, dx,x,x^4\right )-\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {2}{3} \left (-1+x^4\right )^{3/4}-\tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^4}\right )\\ &=\frac {2}{3} \left (-1+x^4\right )^{3/4}-\tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^4}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^4}\right )\\ &=\frac {2}{3} \left (-1+x^4\right )^{3/4}-\tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )+\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )}{4 \sqrt {2}}\\ &=\frac {2}{3} \left (-1+x^4\right )^{3/4}-\tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+x^4}+\sqrt {-1+x^4}\right )}{4 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+x^4}+\sqrt {-1+x^4}\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [4]{-1+x^4}\right )}{2 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [4]{-1+x^4}\right )}{2 \sqrt {2}}\\ &=\frac {2}{3} \left (-1+x^4\right )^{3/4}-\tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{-1+x^4}\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (1+\sqrt {2} \sqrt [4]{-1+x^4}\right )}{2 \sqrt {2}}-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+x^4}+\sqrt {-1+x^4}\right )}{4 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+x^4}+\sqrt {-1+x^4}\right )}{4 \sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 5.32, size = 120, normalized size = 0.96 \begin {gather*} \frac {2}{3} \left (-1+x^4\right )^{3/4}+\text {ArcTan}\left (\frac {\sqrt [4]{-1+x^4}}{x}\right )-\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{-1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}}-\tanh ^{-1}\left (\frac {\sqrt [4]{-1+x^4}}{x}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-1+x^4}}{1+\sqrt {-1+x^4}}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 8.22, size = 142, normalized size = 1.14
method | result | size |
meijerg | \(\frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \left (\frac {\pi \sqrt {2}\, x^{4} \hypergeom \left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], x^{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \left (2\right )-\frac {\pi }{2}+4 \ln \left (x \right )+i \pi \right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{8 \pi \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}+\frac {\left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} x^{4} \hypergeom \left (\left [\frac {1}{4}, 1\right ], \left [2\right ], x^{4}\right )}{2 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}-\frac {2 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], x^{4}\right )}{\mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}\) | \(142\) |
trager | \(\frac {2 \left (x^{4}-1\right )^{\frac {3}{4}}}{3}-\frac {\ln \left (2 \left (x^{4}-1\right )^{\frac {3}{4}} x +2 x^{2} \sqrt {x^{4}-1}+2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}+2 x^{4}-1\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 \sqrt {x^{4}-1}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}} x -2 x^{3} \left (x^{4}-1\right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{4}-1}+2 \left (x^{4}-1\right )^{\frac {3}{4}}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{4}-1\right )^{\frac {1}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (x^{4}-1\right )^{\frac {3}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right ) \sqrt {x^{4}-1}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{4}-1\right )^{\frac {1}{4}}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{x^{4}}\right )}{4}\) | \(331\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.47, size = 155, normalized size = 1.24 \begin {gather*} \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) + \frac {2}{3} \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} + \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + 1\right ) + \frac {1}{2} \, \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} - 1\right ) \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. 517 vs.
\(2 (96) = 192\).
time = 7.88, size = 517, normalized size = 4.14 \begin {gather*} \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {x^{8} + 4 \, \sqrt {x^{4} - 1} x^{4} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} {\left (x^{4} - 4\right )} + 2 \, \sqrt {2} {\left (3 \, x^{4} - 4\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}} - {\left (4 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{4} + 2 \, \sqrt {2} \sqrt {x^{4} - 1} {\left (x^{4} - 4\right )} + \sqrt {2} {\left (x^{8} - 10 \, x^{4} + 8\right )} + 16 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}}\right )} \sqrt {\frac {x^{4} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1}}{x^{4}}}}{x^{8} - 16 \, x^{4} + 16}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {x^{8} + 4 \, \sqrt {x^{4} - 1} x^{4} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} {\left (x^{4} - 4\right )} - 2 \, \sqrt {2} {\left (3 \, x^{4} - 4\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}} - {\left (4 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{4} - 2 \, \sqrt {2} \sqrt {x^{4} - 1} {\left (x^{4} - 4\right )} - \sqrt {2} {\left (x^{8} - 10 \, x^{4} + 8\right )} + 16 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}}\right )} \sqrt {\frac {x^{4} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1}}{x^{4}}}}{x^{8} - 16 \, x^{4} + 16}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{4} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1}\right )}}{x^{4}}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{4} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1}\right )}}{x^{4}}\right ) + \frac {2}{3} \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} + \frac {1}{2} \, \arctan \left (2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x\right ) + \frac {1}{2} \, \log \left (-2 \, x^{4} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 2 \, \sqrt {x^{4} - 1} x^{2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 2.01, size = 71, normalized size = 0.57 \begin {gather*} - \frac {x e^{- \frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {x^{4}} \right )}}{2 \Gamma \left (\frac {5}{4}\right )} + \frac {2 \left (x^{4} - 1\right )^{\frac {3}{4}}}{3} - \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{4 x \Gamma \left (\frac {5}{4}\right )} \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 [B]
time = 1.30, size = 81, normalized size = 0.65 \begin {gather*} \frac {2\,{\left (x^4-1\right )}^{3/4}}{3}-\frac {2\,x\,{\left (1-x^4\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ x^4\right )}{{\left (x^4-1\right )}^{1/4}}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^4-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^4-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________