3.12.28 \(\int \frac {(-1+x^4) \sqrt {1+x^2+x^4} (1+x^2+3 x^4+x^6+x^8)}{(1+x^4)^3 (1-x^2+x^4)} \, dx\)

Optimal. Leaf size=84 \[ \frac {31}{8} \tanh ^{-1}\left (\frac {x}{\sqrt {x^4+x^2+1}}\right )-3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+x^2+1}}\right )+\frac {\sqrt {x^4+x^2+1} \left (9 x^5+2 x^3+9 x\right )}{8 \left (x^4+1\right )^2} \]

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Rubi [F]  time = 3.46, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-1+x^4\right ) \sqrt {1+x^2+x^4} \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^4\right )^3 \left (1-x^2+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + x^4)*Sqrt[1 + x^2 + x^4]*(1 + x^2 + 3*x^4 + x^6 + x^8))/((1 + x^4)^3*(1 - x^2 + x^4)),x]

[Out]

((I/2)*x*Sqrt[1 + x^2 + x^4])/(I - x^2) + ((I/2)*x*Sqrt[1 + x^2 + x^4])/(I + x^2) + 4*ArcTanh[x/Sqrt[1 + x^2 +
x^4]] - 3*Sqrt[2]*ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^2 + x^4]] - (9*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*E
llipticF[2*ArcTan[x], 1/4])/(2*Sqrt[1 + x^2 + x^4]) - (3*(I - Sqrt[3])*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2
)^2]*EllipticF[2*ArcTan[x], 1/4])/((3*I - Sqrt[3])*Sqrt[1 + x^2 + x^4]) + (3*(5 - I*Sqrt[3])*(1 + x^2)*Sqrt[(1
+ x^2 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/4])/(4*Sqrt[1 + x^2 + x^4]) + (3*(5 + I*Sqrt[3])*(1 + x^2)
*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/4])/(4*Sqrt[1 + x^2 + x^4]) - (3*(I + Sqrt[3])*(1
+ x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/4])/((3*I + Sqrt[3])*Sqrt[1 + x^2 + x^4]) +
2*Defer[Int][(x^2*Sqrt[1 + x^2 + x^4])/(1 + x^4)^3, x] - Defer[Int][(x^2*Sqrt[1 + x^2 + x^4])/(1 + x^4)^2, x]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^4\right ) \sqrt {1+x^2+x^4} \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^4\right )^3 \left (1-x^2+x^4\right )} \, dx &=\int \left (\frac {2 x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^3}+\frac {\left (4-x^2\right ) \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^2}-\frac {2 \left (1+3 x^2\right ) \sqrt {1+x^2+x^4}}{1+x^4}+\frac {3 \left (-1+2 x^2\right ) \sqrt {1+x^2+x^4}}{1-x^2+x^4}\right ) \, dx\\ &=2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^3} \, dx-2 \int \frac {\left (1+3 x^2\right ) \sqrt {1+x^2+x^4}}{1+x^4} \, dx+3 \int \frac {\left (-1+2 x^2\right ) \sqrt {1+x^2+x^4}}{1-x^2+x^4} \, dx+\int \frac {\left (4-x^2\right ) \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^2} \, dx\\ &=2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^3} \, dx-2 \int \left (-\frac {\left (\frac {3}{2}-\frac {i}{2}\right ) \sqrt {1+x^2+x^4}}{i-x^2}+\frac {\left (\frac {3}{2}+\frac {i}{2}\right ) \sqrt {1+x^2+x^4}}{i+x^2}\right ) \, dx+3 \int \left (\frac {2 \sqrt {1+x^2+x^4}}{-1-i \sqrt {3}+2 x^2}+\frac {2 \sqrt {1+x^2+x^4}}{-1+i \sqrt {3}+2 x^2}\right ) \, dx+\int \left (\frac {4 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^2}-\frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^2}\right ) \, dx\\ &=-\left ((-3+i) \int \frac {\sqrt {1+x^2+x^4}}{i-x^2} \, dx\right )+2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^3} \, dx-(3+i) \int \frac {\sqrt {1+x^2+x^4}}{i+x^2} \, dx+4 \int \frac {\sqrt {1+x^2+x^4}}{\left (1+x^4\right )^2} \, dx+6 \int \frac {\sqrt {1+x^2+x^4}}{-1-i \sqrt {3}+2 x^2} \, dx+6 \int \frac {\sqrt {1+x^2+x^4}}{-1+i \sqrt {3}+2 x^2} \, dx-\int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^2} \, dx\\ &=-\left ((-3-i) \int \frac {(-1+i)-x^2}{\sqrt {1+x^2+x^4}} \, dx\right )-(-1-3 i) \int \frac {1}{\left (i-x^2\right ) \sqrt {1+x^2+x^4}} \, dx-(1-3 i) \int \frac {1}{\left (i+x^2\right ) \sqrt {1+x^2+x^4}} \, dx-\frac {3}{2} \int \frac {-3-i \sqrt {3}-2 x^2}{\sqrt {1+x^2+x^4}} \, dx-\frac {3}{2} \int \frac {-3+i \sqrt {3}-2 x^2}{\sqrt {1+x^2+x^4}} \, dx+2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^3} \, dx-(3-i) \int \frac {(1+i)+x^2}{\sqrt {1+x^2+x^4}} \, dx+4 \int \left (-\frac {\sqrt {1+x^2+x^4}}{4 \left (i-x^2\right )^2}-\frac {\sqrt {1+x^2+x^4}}{4 \left (i+x^2\right )^2}-\frac {\sqrt {1+x^2+x^4}}{2 \left (-1-x^4\right )}\right ) \, dx+\left (6 \left (1-i \sqrt {3}\right )\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {1+x^2+x^4}} \, dx+\left (6 \left (1+i \sqrt {3}\right )\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt {1+x^2+x^4}} \, dx-\int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^2} \, dx\\ &=-\left ((-3-i) \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx\right )-(-3+i) \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx-(-2-i) \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx-(-2-i) \int \frac {1+x^2}{\left (i-x^2\right ) \sqrt {1+x^2+x^4}} \, dx-(-2+i) \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx-2 \int \frac {\sqrt {1+x^2+x^4}}{-1-x^4} \, dx+2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^3} \, dx-(2-i) \int \frac {1+x^2}{\left (i+x^2\right ) \sqrt {1+x^2+x^4}} \, dx-2 \left (3 \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx\right )-(7-i) \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx-(7+i) \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx-\frac {\left (6 \left (i-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx}{3 i-\sqrt {3}}+\frac {\left (12 \left (i-\sqrt {3}\right )\right ) \int \frac {1+x^2}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt {1+x^2+x^4}} \, dx}{3 i-\sqrt {3}}+\frac {1}{2} \left (3 \left (5-i \sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx+\frac {1}{2} \left (3 \left (5+i \sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx-\frac {\left (6 \left (i+\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx}{3 i+\sqrt {3}}+\frac {\left (12 \left (i+\sqrt {3}\right )\right ) \int \frac {1+x^2}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {1+x^2+x^4}} \, dx}{3 i+\sqrt {3}}-\int \frac {\sqrt {1+x^2+x^4}}{\left (i-x^2\right )^2} \, dx-\int \frac {\sqrt {1+x^2+x^4}}{\left (i+x^2\right )^2} \, dx-\int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^2} \, dx\\ &=\frac {i x \sqrt {1+x^2+x^4}}{2 \left (i-x^2\right )}+\frac {i x \sqrt {1+x^2+x^4}}{2 \left (i+x^2\right )}-\frac {6 x \sqrt {1+x^2+x^4}}{1+x^2}+3 \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {3 \sqrt {\frac {3}{2}} \left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^2+x^4}}\right )}{3 i-\sqrt {3}}+\frac {3 \sqrt {\frac {3}{2}} \left (i-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^2+x^4}}\right )}{3-i \sqrt {3}}+\frac {6 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}-2 \left (-\frac {3 x \sqrt {1+x^2+x^4}}{1+x^2}+\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}\right )-\frac {5 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}-\frac {3 \left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\left (3 i-\sqrt {3}\right ) \sqrt {1+x^2+x^4}}+\frac {3 \left (5-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}}+\frac {3 \left (5+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}}-\frac {3 \left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\left (3 i+\sqrt {3}\right ) \sqrt {1+x^2+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{2};2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \sqrt {1+x^2+x^4}}-\frac {3 \left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^2+x^4}}-\frac {3 \left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \left (3 i+\sqrt {3}\right ) \sqrt {1+x^2+x^4}}+\frac {1}{2} i \int \frac {i-x^2}{\sqrt {1+x^2+x^4}} \, dx+\frac {1}{2} i \int \frac {i+x^2}{\sqrt {1+x^2+x^4}} \, dx+i \int \frac {1}{\left (i-x^2\right ) \sqrt {1+x^2+x^4}} \, dx+i \int \frac {1}{\left (i+x^2\right ) \sqrt {1+x^2+x^4}} \, dx+2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^3} \, dx-2 \int \left (-\frac {i \sqrt {1+x^2+x^4}}{2 \left (i-x^2\right )}-\frac {i \sqrt {1+x^2+x^4}}{2 \left (i+x^2\right )}\right ) \, dx-\int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^2} \, dx\\ &=\frac {i x \sqrt {1+x^2+x^4}}{2 \left (i-x^2\right )}+\frac {i x \sqrt {1+x^2+x^4}}{2 \left (i+x^2\right )}-\frac {6 x \sqrt {1+x^2+x^4}}{1+x^2}+3 \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {3 \sqrt {\frac {3}{2}} \left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^2+x^4}}\right )}{3 i-\sqrt {3}}+\frac {3 \sqrt {\frac {3}{2}} \left (i-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^2+x^4}}\right )}{3-i \sqrt {3}}+\frac {6 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}-2 \left (-\frac {3 x \sqrt {1+x^2+x^4}}{1+x^2}+\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}\right )-\frac {5 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}-\frac {3 \left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\left (3 i-\sqrt {3}\right ) \sqrt {1+x^2+x^4}}+\frac {3 \left (5-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}}+\frac {3 \left (5+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}}-\frac {3 \left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\left (3 i+\sqrt {3}\right ) \sqrt {1+x^2+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{2};2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \sqrt {1+x^2+x^4}}-\frac {3 \left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^2+x^4}}-\frac {3 \left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \left (3 i+\sqrt {3}\right ) \sqrt {1+x^2+x^4}}+\left (-\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx+\left (-\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx+\left (-\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1+x^2}{\left (i+x^2\right ) \sqrt {1+x^2+x^4}} \, dx+i \int \frac {\sqrt {1+x^2+x^4}}{i-x^2} \, dx+i \int \frac {\sqrt {1+x^2+x^4}}{i+x^2} \, dx+\left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1+x^2}{\left (i-x^2\right ) \sqrt {1+x^2+x^4}} \, dx+2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^3} \, dx-\int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^2} \, dx\\ &=\frac {i x \sqrt {1+x^2+x^4}}{2 \left (i-x^2\right )}+\frac {i x \sqrt {1+x^2+x^4}}{2 \left (i+x^2\right )}-\frac {6 x \sqrt {1+x^2+x^4}}{1+x^2}+4 \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {3 \sqrt {\frac {3}{2}} \left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^2+x^4}}\right )}{3 i-\sqrt {3}}+\frac {3 \sqrt {\frac {3}{2}} \left (i-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^2+x^4}}\right )}{3-i \sqrt {3}}+\frac {6 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}-2 \left (-\frac {3 x \sqrt {1+x^2+x^4}}{1+x^2}+\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}\right )-\frac {5 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}-\frac {3 \left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\left (3 i-\sqrt {3}\right ) \sqrt {1+x^2+x^4}}+\frac {3 \left (5-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}}+\frac {3 \left (5+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}}-\frac {3 \left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\left (3 i+\sqrt {3}\right ) \sqrt {1+x^2+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{2};2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \sqrt {1+x^2+x^4}}-\frac {3 \left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^2+x^4}}-\frac {3 \left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \left (3 i+\sqrt {3}\right ) \sqrt {1+x^2+x^4}}-i \int \frac {(-1+i)-x^2}{\sqrt {1+x^2+x^4}} \, dx-i \int \frac {(1+i)+x^2}{\sqrt {1+x^2+x^4}} \, dx+2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^3} \, dx-\int \frac {1}{\left (i-x^2\right ) \sqrt {1+x^2+x^4}} \, dx+\int \frac {1}{\left (i+x^2\right ) \sqrt {1+x^2+x^4}} \, dx-\int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^2} \, dx\\ &=\frac {i x \sqrt {1+x^2+x^4}}{2 \left (i-x^2\right )}+\frac {i x \sqrt {1+x^2+x^4}}{2 \left (i+x^2\right )}-\frac {6 x \sqrt {1+x^2+x^4}}{1+x^2}+4 \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {3 \sqrt {\frac {3}{2}} \left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^2+x^4}}\right )}{3 i-\sqrt {3}}+\frac {3 \sqrt {\frac {3}{2}} \left (i-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^2+x^4}}\right )}{3-i \sqrt {3}}+\frac {6 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}-2 \left (-\frac {3 x \sqrt {1+x^2+x^4}}{1+x^2}+\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}\right )-\frac {5 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}-\frac {3 \left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\left (3 i-\sqrt {3}\right ) \sqrt {1+x^2+x^4}}+\frac {3 \left (5-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}}+\frac {3 \left (5+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}}-\frac {3 \left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\left (3 i+\sqrt {3}\right ) \sqrt {1+x^2+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{2};2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \sqrt {1+x^2+x^4}}-\frac {3 \left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^2+x^4}}-\frac {3 \left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \left (3 i+\sqrt {3}\right ) \sqrt {1+x^2+x^4}}-(-1-2 i) \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx-(-1+2 i) \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx+\left (-\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx-\left (-\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1+x^2}{\left (i+x^2\right ) \sqrt {1+x^2+x^4}} \, dx+\left (-\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1+x^2}{\left (i-x^2\right ) \sqrt {1+x^2+x^4}} \, dx-\left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx+2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^3} \, dx-\int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^2} \, dx\\ &=\frac {i x \sqrt {1+x^2+x^4}}{2 \left (i-x^2\right )}+\frac {i x \sqrt {1+x^2+x^4}}{2 \left (i+x^2\right )}-\frac {6 x \sqrt {1+x^2+x^4}}{1+x^2}+4 \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )+\frac {3 \sqrt {\frac {3}{2}} \left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^2+x^4}}\right )}{3 i-\sqrt {3}}+\frac {3 \sqrt {\frac {3}{2}} \left (i-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^2+x^4}}\right )}{3-i \sqrt {3}}+\frac {6 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}-2 \left (-\frac {3 x \sqrt {1+x^2+x^4}}{1+x^2}+\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\sqrt {1+x^2+x^4}}\right )-\frac {9 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \sqrt {1+x^2+x^4}}-\frac {3 \left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\left (3 i-\sqrt {3}\right ) \sqrt {1+x^2+x^4}}+\frac {3 \left (5-i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}}+\frac {3 \left (5+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{4 \sqrt {1+x^2+x^4}}-\frac {3 \left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{\left (3 i+\sqrt {3}\right ) \sqrt {1+x^2+x^4}}-\frac {3 \left (i+\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \left (3 i-\sqrt {3}\right ) \sqrt {1+x^2+x^4}}-\frac {3 \left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {3}{4};2 \tan ^{-1}(x)|\frac {1}{4}\right )}{2 \left (3 i+\sqrt {3}\right ) \sqrt {1+x^2+x^4}}+2 \int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^3} \, dx-\int \frac {x^2 \sqrt {1+x^2+x^4}}{\left (1+x^4\right )^2} \, dx\\ \end {align*}

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Mathematica [C]  time = 1.46, size = 345, normalized size = 4.11 \begin {gather*} \frac {\frac {x \left (x^4+x^2+1\right ) \left (9 x^4+2 x^2+9\right )}{\left (x^4+1\right )^2}+\sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} \left (\frac {17}{2} i \left (\sqrt {3}+i\right ) F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-48 \left (\sqrt [3]{-1}-1\right ) \Pi \left (-1;i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-96 (-1)^{2/3} \Pi \left (-(-1)^{2/3};i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+48 \sqrt [3]{-1} \Pi \left (-(-1)^{2/3};i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-48 \Pi \left (-(-1)^{2/3};i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+31 \sqrt [3]{-1} \Pi \left (-(-1)^{5/6};i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-31 \Pi \left (-(-1)^{5/6};i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+31 \sqrt [3]{-1} \Pi \left ((-1)^{5/6};i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )-31 \Pi \left ((-1)^{5/6};i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )\right )}{8 \sqrt {x^4+x^2+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((-1 + x^4)*Sqrt[1 + x^2 + x^4]*(1 + x^2 + 3*x^4 + x^6 + x^8))/((1 + x^4)^3*(1 - x^2 + x^4)),x]

[Out]

((x*(1 + x^2 + x^4)*(9 + 2*x^2 + 9*x^4))/(1 + x^4)^2 + Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(((17
*I)/2)*(I + Sqrt[3])*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] - 48*(-1 + (-1)^(1/3))*EllipticPi[-1, I*Ar
cSinh[(-1)^(5/6)*x], (-1)^(2/3)] - 48*EllipticPi[-(-1)^(2/3), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + 48*(-1)^(
1/3)*EllipticPi[-(-1)^(2/3), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] - 96*(-1)^(2/3)*EllipticPi[-(-1)^(2/3), I*Ar
cSinh[(-1)^(5/6)*x], (-1)^(2/3)] - 31*EllipticPi[-(-1)^(5/6), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + 31*(-1)^(
1/3)*EllipticPi[-(-1)^(5/6), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] - 31*EllipticPi[(-1)^(5/6), I*ArcSinh[(-1)^(
5/6)*x], (-1)^(2/3)] + 31*(-1)^(1/3)*EllipticPi[(-1)^(5/6), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]))/(8*Sqrt[1 +
x^2 + x^4])

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IntegrateAlgebraic [A]  time = 1.49, size = 84, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^2+x^4} \left (9 x+2 x^3+9 x^5\right )}{8 \left (1+x^4\right )^2}+\frac {31}{8} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^4}}\right )-3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^2+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^4)*Sqrt[1 + x^2 + x^4]*(1 + x^2 + 3*x^4 + x^6 + x^8))/((1 + x^4)^3*(1 - x^2 + x^4)
),x]

[Out]

(Sqrt[1 + x^2 + x^4]*(9*x + 2*x^3 + 9*x^5))/(8*(1 + x^4)^2) + (31*ArcTanh[x/Sqrt[1 + x^2 + x^4]])/8 - 3*Sqrt[2
]*ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^2 + x^4]]

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fricas [B]  time = 0.59, size = 171, normalized size = 2.04 \begin {gather*} \frac {12 \, \sqrt {2} {\left (x^{8} + 2 \, x^{4} + 1\right )} \log \left (-\frac {x^{8} + 14 \, x^{6} + 19 \, x^{4} - 4 \, \sqrt {2} {\left (x^{5} + 3 \, x^{3} + x\right )} \sqrt {x^{4} + x^{2} + 1} + 14 \, x^{2} + 1}{x^{8} - 2 \, x^{6} + 3 \, x^{4} - 2 \, x^{2} + 1}\right ) + 31 \, {\left (x^{8} + 2 \, x^{4} + 1\right )} \log \left (-\frac {x^{4} + 2 \, x^{2} + 2 \, \sqrt {x^{4} + x^{2} + 1} x + 1}{x^{4} + 1}\right ) + 2 \, {\left (9 \, x^{5} + 2 \, x^{3} + 9 \, x\right )} \sqrt {x^{4} + x^{2} + 1}}{16 \, {\left (x^{8} + 2 \, x^{4} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)*(x^4+x^2+1)^(1/2)*(x^8+x^6+3*x^4+x^2+1)/(x^4+1)^3/(x^4-x^2+1),x, algorithm="fricas")

[Out]

1/16*(12*sqrt(2)*(x^8 + 2*x^4 + 1)*log(-(x^8 + 14*x^6 + 19*x^4 - 4*sqrt(2)*(x^5 + 3*x^3 + x)*sqrt(x^4 + x^2 +
1) + 14*x^2 + 1)/(x^8 - 2*x^6 + 3*x^4 - 2*x^2 + 1)) + 31*(x^8 + 2*x^4 + 1)*log(-(x^4 + 2*x^2 + 2*sqrt(x^4 + x^
2 + 1)*x + 1)/(x^4 + 1)) + 2*(9*x^5 + 2*x^3 + 9*x)*sqrt(x^4 + x^2 + 1))/(x^8 + 2*x^4 + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} + x^{6} + 3 \, x^{4} + x^{2} + 1\right )} \sqrt {x^{4} + x^{2} + 1} {\left (x^{4} - 1\right )}}{{\left (x^{4} - x^{2} + 1\right )} {\left (x^{4} + 1\right )}^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)*(x^4+x^2+1)^(1/2)*(x^8+x^6+3*x^4+x^2+1)/(x^4+1)^3/(x^4-x^2+1),x, algorithm="giac")

[Out]

integrate((x^8 + x^6 + 3*x^4 + x^2 + 1)*sqrt(x^4 + x^2 + 1)*(x^4 - 1)/((x^4 - x^2 + 1)*(x^4 + 1)^3), x)

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maple [A]  time = 0.85, size = 128, normalized size = 1.52

method	result	size
elliptic	\(\frac {\left (3 \ln \left (-1+\frac {\sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{2 x}\right )-3 \ln \left (1+\frac {\sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{2 x}\right )-\frac {8 \left (-\frac {9 \left (x^{4}+x^{2}+1\right )^{\frac {3}{2}} \sqrt {2}}{64 x^{3}}+\frac {7 \sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{64 x}\right )}{\left (\frac {x^{4}+x^{2}+1}{x^{2}}-1\right )^{2}}+\frac {31 \sqrt {2}\, \arctanh \left (\frac {\sqrt {x^{4}+x^{2}+1}}{x}\right )}{8}\right ) \sqrt {2}}{2}\)	\(128\)
trager	\(\frac {\left (9 x^{4}+2 x^{2}+9\right ) x \sqrt {x^{4}+x^{2}+1}}{8 \left (x^{4}+1\right )^{2}}-\frac {3 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{4}+3 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+4 \sqrt {x^{4}+x^{2}+1}\, x +\RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{4}-x^{2}+1}\right )}{2}-\frac {31 \ln \left (-\frac {-x^{4}+2 \sqrt {x^{4}+x^{2}+1}\, x -2 x^{2}-1}{x^{4}+1}\right )}{16}\)	\(135\)
risch	\(\frac {\left (9 x^{4}+2 x^{2}+9\right ) x \sqrt {x^{4}+x^{2}+1}}{8 \left (x^{4}+1\right )^{2}}+\frac {17 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{4 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {31 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+5 x^{2}+4\right )}{10 \sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+x^{2}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {\sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {x^{2}+2-i \sqrt {3}\, x^{2}}\, \sqrt {x^{2}+2+i \sqrt {3}\, x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , \frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}\right )\right )}{32}+\frac {3 \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+7 x^{2}+8\right ) \sqrt {2}}{28 \sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+x^{2}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {2 \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x^{2}+2-i \sqrt {3}\, x^{2}}\, \sqrt {x^{2}+2+i \sqrt {3}\, x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , \frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}-\frac {1}{2}-\frac {i \sqrt {3}}{2}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}\right )\right )}{4}\)	\(468\)
default	\(\frac {17 \sqrt {1-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2+2 i \sqrt {3}}}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{4 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}+\frac {3 \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+7 x^{2}+8\right ) \sqrt {2}}{28 \sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+x^{2}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {2 \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x^{2}+2-i \sqrt {3}\, x^{2}}\, \sqrt {x^{2}+2+i \sqrt {3}\, x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , \frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}-\frac {1}{2}-\frac {i \sqrt {3}}{2}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}\right )\right )}{4}+\frac {x^{3} \sqrt {x^{4}+x^{2}+1}}{4 \left (x^{4}+1\right )^{2}}-\frac {2 \left (-\frac {1}{8} x^{3}-\frac {1}{16} x \right ) \sqrt {x^{4}+x^{2}+1}}{x^{4}+1}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}-1\right ) \left (-\frac {\arctanh \left (\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+5 x^{2}+4\right )}{10 \sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+x^{2}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {\sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {x^{2}+2-i \sqrt {3}\, x^{2}}\, \sqrt {x^{2}+2+i \sqrt {3}\, x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , \frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}\right )\right )}{32}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-3\right ) \left (-\frac {\arctanh \left (\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+5 x^{2}+4\right )}{10 \sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+x^{2}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {\sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {x^{2}+2-i \sqrt {3}\, x^{2}}\, \sqrt {x^{2}+2+i \sqrt {3}\, x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , \frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}\right )\right )}{4}+\frac {\left (-\frac {1}{4} x^{3}+x \right ) \sqrt {x^{4}+x^{2}+1}}{x^{4}+1}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (3 \underline {\hspace {1.25 ex}}\alpha ^{2}+4\right ) \left (-\frac {\arctanh \left (\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}+1\right ) \left (-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+5 x^{2}+4\right )}{10 \sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {x^{4}+x^{2}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {\sqrt {2}\, \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {x^{2}+2-i \sqrt {3}\, x^{2}}\, \sqrt {x^{2}+2+i \sqrt {3}\, x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, x , \frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}, \frac {\sqrt {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}{\sqrt {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}\right )\right )}{16}\)	\(862\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-1)*(x^4+x^2+1)^(1/2)*(x^8+x^6+3*x^4+x^2+1)/(x^4+1)^3/(x^4-x^2+1),x,method=_RETURNVERBOSE)

[Out]

1/2*(3*ln(-1+1/2*(x^4+x^2+1)^(1/2)*2^(1/2)/x)-3*ln(1+1/2*(x^4+x^2+1)^(1/2)*2^(1/2)/x)-8*(-9/64*(x^4+x^2+1)^(3/
2)*2^(1/2)/x^3+7/64*(x^4+x^2+1)^(1/2)*2^(1/2)/x)/((x^4+x^2+1)/x^2-1)^2+31/8*2^(1/2)*arctanh(1/x*(x^4+x^2+1)^(1
/2)))*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} + x^{6} + 3 \, x^{4} + x^{2} + 1\right )} \sqrt {x^{4} + x^{2} + 1} {\left (x^{4} - 1\right )}}{{\left (x^{4} - x^{2} + 1\right )} {\left (x^{4} + 1\right )}^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)*(x^4+x^2+1)^(1/2)*(x^8+x^6+3*x^4+x^2+1)/(x^4+1)^3/(x^4-x^2+1),x, algorithm="maxima")

[Out]

integrate((x^8 + x^6 + 3*x^4 + x^2 + 1)*sqrt(x^4 + x^2 + 1)*(x^4 - 1)/((x^4 - x^2 + 1)*(x^4 + 1)^3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^4-1\right )\,\sqrt {x^4+x^2+1}\,\left (x^8+x^6+3\,x^4+x^2+1\right )}{{\left (x^4+1\right )}^3\,\left (x^4-x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 - 1)*(x^2 + x^4 + 1)^(1/2)*(x^2 + 3*x^4 + x^6 + x^8 + 1))/((x^4 + 1)^3*(x^4 - x^2 + 1)),x)

[Out]

int(((x^4 - 1)*(x^2 + x^4 + 1)^(1/2)*(x^2 + 3*x^4 + x^6 + x^8 + 1))/((x^4 + 1)^3*(x^4 - x^2 + 1)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-1)*(x**4+x**2+1)**(1/2)*(x**8+x**6+3*x**4+x**2+1)/(x**4+1)**3/(x**4-x**2+1),x)

[Out]

Timed out

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