Optimal. Leaf size=107 \[ \frac {\sqrt {x^4+3 x^2+1}}{4 \left (x^2-x+1\right )}-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{x^2+\sqrt {x^4+3 x^2+1}-x+1}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{x^2+\sqrt {x^4+3 x^2+1}+x+1}\right )}{2 \sqrt {2}} \]
________________________________________________________________________________________
Rubi [C] time = 10.76, antiderivative size = 5419, normalized size of antiderivative = 50.64, number of steps used = 136, number of rules used = 18, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6688, 6728, 1099, 6742, 1726, 1741, 12, 1247, 724, 204, 1716, 1189, 1135, 1214, 1456, 539, 1724, 2}
result too large to display
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 2
Rule 12
Rule 204
Rule 539
Rule 724
Rule 1099
Rule 1135
Rule 1189
Rule 1214
Rule 1247
Rule 1456
Rule 1716
Rule 1724
Rule 1726
Rule 1741
Rule 6688
Rule 6728
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right ) \left (1-x+x^2-x^3+x^4\right )}{\left (1-x+x^2\right )^2 \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx &=\int \frac {-1+x-x^5+x^6}{\left (1-x+x^2\right )^2 \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx\\ &=\int \left (\frac {1}{\sqrt {1+3 x^2+x^4}}+\frac {1+x}{2 \left (1-x+x^2\right )^2 \sqrt {1+3 x^2+x^4}}+\frac {-8+x}{4 \left (1-x+x^2\right ) \sqrt {1+3 x^2+x^4}}+\frac {-2-x}{4 \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}}\right ) \, dx\\ &=\frac {1}{4} \int \frac {-8+x}{\left (1-x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{4} \int \frac {-2-x}{\left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{2} \int \frac {1+x}{\left (1-x+x^2\right )^2 \sqrt {1+3 x^2+x^4}} \, dx+\int \frac {1}{\sqrt {1+3 x^2+x^4}} \, dx\\ &=\text {rest of steps removed due to Latex formating problem} \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.71, size = 767, normalized size = 7.17 \begin {gather*} \frac {-40 i \left (1+\sqrt [3]{-1}\right ) \left (x^2-x+1\right ) \sqrt {2 x^2-\sqrt {5}+3} \sqrt {2 x^2+\sqrt {5}+3} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )|\frac {7}{2}+\frac {3 \sqrt {5}}{2}\right )+40 (-1)^{5/6} \left (x^2-x+1\right ) \sqrt {2 x^2-\sqrt {5}+3} \sqrt {2 x^2+\sqrt {5}+3} \Pi \left (\frac {1}{2} \sqrt [3]{-1} \left (3+\sqrt {5}\right );i \sinh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )|\frac {1}{2} \left (7+3 \sqrt {5}\right )\right )+40 i \left (x^2-x+1\right ) \sqrt {2 x^2-\sqrt {5}+3} \sqrt {2 x^2+\sqrt {5}+3} \Pi \left (\frac {1}{2} \sqrt [3]{-1} \left (3+\sqrt {5}\right );i \sinh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )|\frac {1}{2} \left (7+3 \sqrt {5}\right )\right )+40 (-1)^{5/6} \left (x^2-x+1\right ) \sqrt {2 x^2-\sqrt {5}+3} \sqrt {2 x^2+\sqrt {5}+3} \Pi \left (-\frac {1}{2} (-1)^{2/3} \left (3+\sqrt {5}\right );i \sinh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )|\frac {1}{2} \left (7+3 \sqrt {5}\right )\right )+40 i \left (x^2-x+1\right ) \sqrt {2 x^2-\sqrt {5}+3} \sqrt {2 x^2+\sqrt {5}+3} \Pi \left (-\frac {1}{2} (-1)^{2/3} \left (3+\sqrt {5}\right );i \sinh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )|\frac {1}{2} \left (7+3 \sqrt {5}\right )\right )+8 \left (1+\sqrt [3]{-1}\right ) \sqrt {6-2 \sqrt {5}} \left (x^4+3 x^2+1\right )+3 \left (1+\sqrt [3]{-1}\right ) \sqrt {6-2 \sqrt {5}} \left (x^2-x+1\right ) \sqrt {x^4+3 x^2+1} \left (\left (1-i \sqrt {3}\right )^{3/2} \tan ^{-1}\left (\frac {\left (4-2 i \sqrt {3}\right ) x^2-3 i \sqrt {3}+1}{4 \sqrt {1+i \sqrt {3}} \sqrt {x^4+3 x^2+1}}\right )+\left (1+i \sqrt {3}\right )^{3/2} \tan ^{-1}\left (\frac {\left (4+2 i \sqrt {3}\right ) x^2+3 i \sqrt {3}+1}{4 \sqrt {1-i \sqrt {3}} \sqrt {x^4+3 x^2+1}}\right )\right )}{32 \left (1+\sqrt [3]{-1}\right ) \sqrt {6-2 \sqrt {5}} \left (x^2-x+1\right ) \sqrt {x^4+3 x^2+1}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 2.10, size = 107, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+3 x^2+x^4}}{4 \left (1-x+x^2\right )}-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{1-x+x^2+\sqrt {1+3 x^2+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{1+x+x^2+\sqrt {1+3 x^2+x^4}}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.54, size = 184, normalized size = 1.72 \begin {gather*} \frac {\sqrt {2} {\left (x^{2} - x + 1\right )} \log \left (\frac {3 \, x^{4} - 2 \, x^{3} + 2 \, \sqrt {2} \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} - x + 1\right )} + 9 \, x^{2} - 2 \, x + 3}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) + 4 \, \sqrt {2} {\left (x^{2} - x + 1\right )} \log \left (\frac {3 \, x^{4} + 2 \, x^{3} - 2 \, \sqrt {2} \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + x + 1\right )} + 9 \, x^{2} + 2 \, x + 3}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + 4 \, \sqrt {x^{4} + 3 \, x^{2} + 1}}{16 \, {\left (x^{2} - x + 1\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*} \int \frac {{\left (x^{4} - x^{3} + x^{2} - x + 1\right )} {\left (x^{2} - 1\right )}}{\sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + x + 1\right )} {\left (x^{2} - x + 1\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 1.65, size = 320, normalized size = 2.99
method | result | size |
trager | \(\frac {\sqrt {x^{4}+3 x^{2}+1}}{4 x^{2}-4 x +4}+\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-7 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{10}-29 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{9}+10 \sqrt {x^{4}+3 x^{2}+1}\, x^{8}-89 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{8}+40 \sqrt {x^{4}+3 x^{2}+1}\, x^{7}-190 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{7}+116 \sqrt {x^{4}+3 x^{2}+1}\, x^{6}-283 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{6}+192 \sqrt {x^{4}+3 x^{2}+1}\, x^{5}-363 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{5}+270 \sqrt {x^{4}+3 x^{2}+1}\, x^{4}-283 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{4}+192 \sqrt {x^{4}+3 x^{2}+1}\, x^{3}-190 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{3}+116 \sqrt {x^{4}+3 x^{2}+1}\, x^{2}-89 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+40 \sqrt {x^{4}+3 x^{2}+1}\, x -29 \RootOf \left (\textit {\_Z}^{2}-2\right ) x +10 \sqrt {x^{4}+3 x^{2}+1}-7 \RootOf \left (\textit {\_Z}^{2}-2\right )}{\left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )^{4}}\right )}{8}\) | \(320\) |
elliptic | \(-\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \sqrt {2}}{4}\right )+\sqrt {6}\, \arctan \left (\frac {2 \sqrt {6}\, \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (x^{2}-1\right )}{3 \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5\right ) \left (-x^{2}-1\right )}\right )\right )}{12 \sqrt {-\frac {\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5}{\left (\frac {x^{2}-1}{-x^{2}-1}+1\right )^{2}}}\, \left (\frac {x^{2}-1}{-x^{2}-1}+1\right )}-\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (\frac {3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \sqrt {2}}{4}\right ) \left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-\frac {2 \sqrt {6}\, \arctan \left (\frac {2 \sqrt {6}\, \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (x^{2}-1\right )}{3 \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5\right ) \left (-x^{2}-1\right )}\right ) \left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+9 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \sqrt {2}}{4}\right )-6 \sqrt {6}\, \arctan \left (\frac {2 \sqrt {6}\, \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (x^{2}-1\right )}{3 \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5\right ) \left (-x^{2}-1\right )}\right )-12 \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\right )}{24 \sqrt {-\frac {\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5}{\left (\frac {x^{2}-1}{-x^{2}-1}+1\right )^{2}}}\, \left (\frac {x^{2}-1}{-x^{2}-1}+1\right ) \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+3\right )}+\frac {\left (\frac {1}{-8+\frac {4 \sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{x}}+\frac {5 \ln \left (-1+\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}\right )}{8}+\frac {1}{8+\frac {4 \sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{x}}-\frac {5 \ln \left (1+\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}\right )}{8}\right ) \sqrt {2}}{2}\) | \(661\) |
risch | \(\frac {\sqrt {x^{4}+3 x^{2}+1}}{4 x^{2}-4 x +4}+\frac {5 \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticF \left (x \left (\frac {i \sqrt {5}}{2}-\frac {i}{2}\right ), \frac {3}{2}+\frac {\sqrt {5}}{2}\right )}{4 \left (\frac {i \sqrt {5}}{2}-\frac {i}{2}\right ) \sqrt {x^{4}+3 x^{2}+1}}+\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\left (-2+i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}-\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1-i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1-i \sqrt {3}}}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , -\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}-\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{4}+\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\left (-2-i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}+\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1+i \sqrt {3}}}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}+\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{4}+\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (2+i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}+\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1+i \sqrt {3}}}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , \frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}+\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )+\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (2-i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}-\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1-i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1-i \sqrt {3}}}-\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , \frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}-\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )\) | \(836\) |
default | \(\frac {5 \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticF \left (x \left (\frac {i \sqrt {5}}{2}-\frac {i}{2}\right ), \frac {3}{2}+\frac {\sqrt {5}}{2}\right )}{4 \left (\frac {i \sqrt {5}}{2}-\frac {i}{2}\right ) \sqrt {x^{4}+3 x^{2}+1}}+\frac {\left (\frac {1}{2}+\frac {5 i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (2+i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}+\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1+i \sqrt {3}}}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , \frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}+\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{4}+\frac {\left (\frac {1}{2}-\frac {5 i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (2-i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}-\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1-i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1-i \sqrt {3}}}-\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , \frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}-\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{4}+\frac {\sqrt {x^{4}+3 x^{2}+1}}{4 x^{2}-4 x +4}+\frac {\left (\frac {3}{4}-\frac {i \sqrt {3}}{4}\right ) \left (-\frac {\arctanh \left (\frac {\left (2+i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}+\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1+i \sqrt {3}}}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , \frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}+\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2}+\frac {\left (\frac {3}{4}+\frac {i \sqrt {3}}{4}\right ) \left (-\frac {\arctanh \left (\frac {\left (2-i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}-\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1-i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1-i \sqrt {3}}}-\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , \frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}-\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2}+\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\left (-2+i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}-\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1-i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1-i \sqrt {3}}}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , -\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}-\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{4}+\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\left (-2-i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}+\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1+i \sqrt {3}}}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}+\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{4}\) | \(1206\) |
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*} \int \frac {{\left (x^{4} - x^{3} + x^{2} - x + 1\right )} {\left (x^{2} - 1\right )}}{\sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + x + 1\right )} {\left (x^{2} - x + 1\right )}^{2}}\,{d x} \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 {\left (x^2-1\right )\,\left (x^4-x^3+x^2-x+1\right )}{{\left (x^2-x+1\right )}^2\,\sqrt {x^4+3\,x^2+1}\,\left (x^2+x+1\right )} \,d x \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 {\left (x - 1\right ) \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}{\left (x^{2} - x + 1\right )^{2} \left (x^{2} + x + 1\right ) \sqrt {x^{4} + 3 x^{2} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________