3.17.21 \(\int \frac {(-1+x^4) (1+x^2+3 x^4+x^6+x^8)}{(1+x^2+x^4)^{3/2} (1+3 x^2+5 x^4+3 x^6+x^8)} \, dx\)

Optimal. Leaf size=110 \[ -\frac {x}{\sqrt {x^4+x^2+1}}+\sqrt {\frac {1}{3} \left (-2+2 i \sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x}{\left (\sqrt {3}-i\right ) \sqrt {x^4+x^2+1}}\right )+\sqrt {\frac {1}{3} \left (-2-2 i \sqrt {3}\right )} \tan ^{-1}\left (\frac {2 x}{\left (\sqrt {3}+i\right ) \sqrt {x^4+x^2+1}}\right ) \]

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Rubi [F]  time = 1.62, 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 ) \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + x^4)*(1 + x^2 + 3*x^4 + x^6 + x^8))/((1 + x^2 + x^4)^(3/2)*(1 + 3*x^2 + 5*x^4 + 3*x^6 + x^8)),x]

[Out]

(x*(1 - x^2))/Sqrt[1 + x^2 + x^4] - (x*(2 + x^2))/(3*Sqrt[1 + x^2 + x^4]) - (2*x*(1 + 2*x^2))/(3*Sqrt[1 + x^2
+ x^4]) + (8*x*Sqrt[1 + x^2 + x^4])/(3*(1 + x^2)) - (8*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticE[2
*ArcTan[x], 1/4])/(3*Sqrt[1 + x^2 + x^4]) + (3*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[
x], 1/4])/Sqrt[1 + x^2 + x^4] - 4*Defer[Int][1/((1 + x^2 + x^4)^(3/2)*(1 + 3*x^2 + 5*x^4 + 3*x^6 + x^8)), x] -
8*Defer[Int][x^2/((1 + x^2 + x^4)^(3/2)*(1 + 3*x^2 + 5*x^4 + 3*x^6 + x^8)), x] - 12*Defer[Int][x^4/((1 + x^2
+ x^4)^(3/2)*(1 + 3*x^2 + 5*x^4 + 3*x^6 + x^8)), x] - 2*Defer[Int][x^6/((1 + x^2 + x^4)^(3/2)*(1 + 3*x^2 + 5*x
^4 + 3*x^6 + x^8)), x]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^4\right ) \left (1+x^2+3 x^4+x^6+x^8\right )}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx &=\int \left (\frac {3}{\left (1+x^2+x^4\right )^{3/2}}-\frac {2 x^2}{\left (1+x^2+x^4\right )^{3/2}}+\frac {x^4}{\left (1+x^2+x^4\right )^{3/2}}-\frac {2 \left (2+4 x^2+6 x^4+x^6\right )}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )}\right ) \, dx\\ &=-\left (2 \int \frac {x^2}{\left (1+x^2+x^4\right )^{3/2}} \, dx\right )-2 \int \frac {2+4 x^2+6 x^4+x^6}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx+3 \int \frac {1}{\left (1+x^2+x^4\right )^{3/2}} \, dx+\int \frac {x^4}{\left (1+x^2+x^4\right )^{3/2}} \, dx\\ &=\frac {x \left (1-x^2\right )}{\sqrt {1+x^2+x^4}}-\frac {x \left (2+x^2\right )}{3 \sqrt {1+x^2+x^4}}-\frac {2 x \left (1+2 x^2\right )}{3 \sqrt {1+x^2+x^4}}+\frac {1}{3} \int \frac {2+x^2}{\sqrt {1+x^2+x^4}} \, dx+\frac {2}{3} \int \frac {1+2 x^2}{\sqrt {1+x^2+x^4}} \, dx-2 \int \left (\frac {2}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )}+\frac {4 x^2}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )}+\frac {6 x^4}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )}+\frac {x^6}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )}\right ) \, dx+\int \frac {2+x^2}{\sqrt {1+x^2+x^4}} \, dx\\ &=\frac {x \left (1-x^2\right )}{\sqrt {1+x^2+x^4}}-\frac {x \left (2+x^2\right )}{3 \sqrt {1+x^2+x^4}}-\frac {2 x \left (1+2 x^2\right )}{3 \sqrt {1+x^2+x^4}}-\frac {1}{3} \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx-\frac {4}{3} \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx+2 \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx-2 \int \frac {x^6}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx+3 \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx-4 \int \frac {1}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx-8 \int \frac {x^2}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx-12 \int \frac {x^4}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx+\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx-\int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx\\ &=\frac {x \left (1-x^2\right )}{\sqrt {1+x^2+x^4}}-\frac {x \left (2+x^2\right )}{3 \sqrt {1+x^2+x^4}}-\frac {2 x \left (1+2 x^2\right )}{3 \sqrt {1+x^2+x^4}}+\frac {8 x \sqrt {1+x^2+x^4}}{3 \left (1+x^2\right )}-\frac {8 \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 )}{3 \sqrt {1+x^2+x^4}}+\frac {3 \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}}-2 \int \frac {x^6}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx-4 \int \frac {1}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx-8 \int \frac {x^2}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx-12 \int \frac {x^4}{\left (1+x^2+x^4\right )^{3/2} \left (1+3 x^2+5 x^4+3 x^6+x^8\right )} \, dx\\ \end {align*}

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Mathematica [C]  time = 2.29, size = 1525, normalized size = 13.86

result too large to display

Warning: Unable to verify antiderivative.

[In]

Integrate[((-1 + x^4)*(1 + x^2 + 3*x^4 + x^6 + x^8))/((1 + x^2 + x^4)^(3/2)*(1 + 3*x^2 + 5*x^4 + 3*x^6 + x^8))
,x]

[Out]

(-x - 2*(-1)^(2/3)*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*(-((EllipticPi[-((-1)^(1/3)/Root[1 + 3*#1
+ 5*#1^2 + 3*#1^3 + #1^4 & , 1, 0]), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]*(-1 + Root[1 + 3*#1 + 5*#1^2 + 3*#1
^3 + #1^4 & , 1, 0]^2))/((Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 1, 0] - Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 +
#1^4 & , 2, 0])*(Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 1, 0] - Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 &
, 3, 0])*(Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 1, 0] - Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 4, 0])
)) + (EllipticPi[-((-1)^(1/3)/Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 2, 0]), I*ArcSinh[(-1)^(5/6)*x], (-1)
^(2/3)]*(-1 + Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 2, 0]^2))/((Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 &
, 1, 0] - Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 2, 0])*(Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 2, 0]
- Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 3, 0])*(Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 2, 0] - Root[1
+ 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 4, 0])) + EllipticPi[-((-1)^(1/3)/Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4
& , 3, 0]), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]/((-Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 1, 0] + Root[1
+ 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 3, 0])*(-Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 2, 0] + Root[1 + 3*#1
+ 5*#1^2 + 3*#1^3 + #1^4 & , 3, 0])*(Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 3, 0] - Root[1 + 3*#1 + 5*#1^2
+ 3*#1^3 + #1^4 & , 4, 0])) - (EllipticPi[-((-1)^(1/3)/Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 3, 0]), I*A
rcSinh[(-1)^(5/6)*x], (-1)^(2/3)]*Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 3, 0]^2)/((-Root[1 + 3*#1 + 5*#1^
2 + 3*#1^3 + #1^4 & , 1, 0] + Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 3, 0])*(-Root[1 + 3*#1 + 5*#1^2 + 3*#
1^3 + #1^4 & , 2, 0] + Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 3, 0])*(Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1
^4 & , 3, 0] - Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 4, 0])) + EllipticPi[-((-1)^(1/3)/Root[1 + 3*#1 + 5*
#1^2 + 3*#1^3 + #1^4 & , 4, 0]), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]/((-Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^
4 & , 1, 0] + Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 4, 0])*(-Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 2
, 0] + Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 4, 0])*(-Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 3, 0] +
Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 4, 0])) - (EllipticPi[-((-1)^(1/3)/Root[1 + 3*#1 + 5*#1^2 + 3*#1^3
+ #1^4 & , 4, 0]), I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)]*Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 4, 0]^2)/((
-Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 1, 0] + Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 4, 0])*(-Root[1
+ 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 2, 0] + Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 4, 0])*(-Root[1 + 3*#1
+ 5*#1^2 + 3*#1^3 + #1^4 & , 3, 0] + Root[1 + 3*#1 + 5*#1^2 + 3*#1^3 + #1^4 & , 4, 0]))))/Sqrt[1 + x^2 + x^4]

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IntegrateAlgebraic [A]  time = 1.15, size = 73, normalized size = 0.66 \begin {gather*} -\frac {x}{\sqrt {1+x^2+x^4}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x \sqrt {1+x^2+x^4}}{1+x^4}\right )}{\sqrt {3}}-\tanh ^{-1}\left (\frac {x \sqrt {1+x^2+x^4}}{\left (1+x^2\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^4)*(1 + x^2 + 3*x^4 + x^6 + x^8))/((1 + x^2 + x^4)^(3/2)*(1 + 3*x^2 + 5*x^4 + 3*x^
6 + x^8)),x]

[Out]

-(x/Sqrt[1 + x^2 + x^4]) + ArcTan[(Sqrt[3]*x*Sqrt[1 + x^2 + x^4])/(1 + x^4)]/Sqrt[3] - ArcTanh[(x*Sqrt[1 + x^2
+ x^4])/(1 + x^2)^2]

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fricas [A]  time = 0.62, size = 146, normalized size = 1.33 \begin {gather*} -\frac {2 \, \sqrt {3} {\left (x^{4} + x^{2} + 1\right )} \arctan \left (\frac {\sqrt {3} \sqrt {x^{4} + x^{2} + 1} {\left (x^{4} + 1\right )}}{3 \, {\left (x^{5} + x^{3} + x\right )}}\right ) - 3 \, {\left (x^{4} + x^{2} + 1\right )} \log \left (\frac {x^{8} + 5 \, x^{6} + 7 \, x^{4} + 5 \, x^{2} - 2 \, {\left (x^{5} + 2 \, x^{3} + x\right )} \sqrt {x^{4} + x^{2} + 1} + 1}{x^{8} + 3 \, x^{6} + 5 \, x^{4} + 3 \, x^{2} + 1}\right ) + 6 \, \sqrt {x^{4} + x^{2} + 1} x}{6 \, {\left (x^{4} + x^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)*(x^8+x^6+3*x^4+x^2+1)/(x^4+x^2+1)^(3/2)/(x^8+3*x^6+5*x^4+3*x^2+1),x, algorithm="fricas")

[Out]

-1/6*(2*sqrt(3)*(x^4 + x^2 + 1)*arctan(1/3*sqrt(3)*sqrt(x^4 + x^2 + 1)*(x^4 + 1)/(x^5 + x^3 + x)) - 3*(x^4 + x
^2 + 1)*log((x^8 + 5*x^6 + 7*x^4 + 5*x^2 - 2*(x^5 + 2*x^3 + x)*sqrt(x^4 + x^2 + 1) + 1)/(x^8 + 3*x^6 + 5*x^4 +
3*x^2 + 1)) + 6*sqrt(x^4 + x^2 + 1)*x)/(x^4 + x^2 + 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 )} {\left (x^{4} - 1\right )}}{{\left (x^{8} + 3 \, x^{6} + 5 \, x^{4} + 3 \, x^{2} + 1\right )} {\left (x^{4} + x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)*(x^8+x^6+3*x^4+x^2+1)/(x^4+x^2+1)^(3/2)/(x^8+3*x^6+5*x^4+3*x^2+1),x, algorithm="giac")

[Out]

integrate((x^8 + x^6 + 3*x^4 + x^2 + 1)*(x^4 - 1)/((x^8 + 3*x^6 + 5*x^4 + 3*x^2 + 1)*(x^4 + x^2 + 1)^(3/2)), x
)

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maple [A]  time = 1.85, size = 152, normalized size = 1.38

method	result	size
risch	\(-\frac {x}{\sqrt {x^{4}+x^{2}+1}}-\frac {\ln \left (\frac {x^{4}+x^{2}+1}{x^{2}}+\frac {\sqrt {x^{4}+x^{2}+1}}{x}+1\right )}{2}-\frac {\sqrt {2}\, \sqrt {6}\, \arctan \left (\frac {\left (\frac {2 \sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{x}+\sqrt {2}\right ) \sqrt {6}}{6}\right )}{6}+\frac {\ln \left (\frac {x^{4}+x^{2}+1}{x^{2}}-\frac {\sqrt {x^{4}+x^{2}+1}}{x}+1\right )}{2}-\frac {\sqrt {2}\, \sqrt {6}\, \arctan \left (\frac {\left (\frac {2 \sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{x}-\sqrt {2}\right ) \sqrt {6}}{6}\right )}{6}\)	\(152\)
elliptic	\(\frac {\left (-\frac {\sqrt {2}\, \ln \left (\frac {x^{4}+x^{2}+1}{x^{2}}+\frac {\sqrt {x^{4}+x^{2}+1}}{x}+1\right )}{2}-\frac {\sqrt {6}\, \arctan \left (\frac {\left (\frac {2 \sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{x}+\sqrt {2}\right ) \sqrt {6}}{6}\right )}{3}+\frac {\sqrt {2}\, \ln \left (\frac {x^{4}+x^{2}+1}{x^{2}}-\frac {\sqrt {x^{4}+x^{2}+1}}{x}+1\right )}{2}-\frac {\sqrt {6}\, \arctan \left (\frac {\left (\frac {2 \sqrt {x^{4}+x^{2}+1}\, \sqrt {2}}{x}-\sqrt {2}\right ) \sqrt {6}}{6}\right )}{3}-\frac {\sqrt {2}\, x}{\sqrt {x^{4}+x^{2}+1}}\right ) \sqrt {2}}{2}\)	\(160\)
trager	\(-\frac {x}{\sqrt {x^{4}+x^{2}+1}}-2 \ln \left (-\frac {x^{4}+12 \sqrt {x^{4}+x^{2}+1}\, \RootOf \left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -6 \RootOf \left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-4 \sqrt {x^{4}+x^{2}+1}\, x +2 x^{2}+1}{x^{4}+6 \RootOf \left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+1}\right ) \RootOf \left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+2 \RootOf \left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \ln \left (\frac {-x^{4}+12 \sqrt {x^{4}+x^{2}+1}\, \RootOf \left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -6 \RootOf \left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-2 \sqrt {x^{4}+x^{2}+1}\, x +x^{2}-1}{-x^{4}+6 \RootOf \left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-3 x^{2}-1}\right )+\ln \left (-\frac {x^{4}+12 \sqrt {x^{4}+x^{2}+1}\, \RootOf \left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -6 \RootOf \left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-4 \sqrt {x^{4}+x^{2}+1}\, x +2 x^{2}+1}{x^{4}+6 \RootOf \left (12 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+1}\right )\)	\(314\)
default	\(-\frac {2 \left (\frac {1}{6} x^{3}+\frac {1}{3} x \right )}{\sqrt {x^{4}+x^{2}+1}}+\frac {-\frac {4}{3} x^{3}-\frac {2}{3} x}{\sqrt {x^{4}+x^{2}+1}}-\frac {6 \left (-\frac {1}{6} x +\frac {1}{6} x^{3}\right )}{\sqrt {x^{4}+x^{2}+1}}+\frac {\frac {8}{3} x^{3}-\frac {2}{3} x}{\sqrt {x^{4}+x^{2}+1}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{8}+3 \textit {\_Z}^{6}+5 \textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{6}+6 \underline {\hspace {1.25 ex}}\alpha ^{4}+8 \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 (-12 \underline {\hspace {1.25 ex}}\alpha ^{6}-30 \underline {\hspace {1.25 ex}}\alpha ^{4}-45 \underline {\hspace {1.25 ex}}\alpha ^{2}+7 x^{2}-10\right )}{14 \sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{2}+1}\, \sqrt {x^{4}+x^{2}+1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{2}+1}}-\frac {\sqrt {2}\, \left (-\underline {\hspace {1.25 ex}}\alpha ^{7}-3 \underline {\hspace {1.25 ex}}\alpha ^{5}-5 \underline {\hspace {1.25 ex}}\alpha ^{3}-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 \underline {\hspace {1.25 ex}}\alpha ^{6} \sqrt {3}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{6}}{2}+\frac {3}{2}+\frac {3 i \underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {3}}{2}+\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{4}}{2}+\frac {5 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {5 \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}+\frac {3 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 )}{6}\)	\(356\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-1)*(x^8+x^6+3*x^4+x^2+1)/(x^4+x^2+1)^(3/2)/(x^8+3*x^6+5*x^4+3*x^2+1),x,method=_RETURNVERBOSE)

[Out]

-x/(x^4+x^2+1)^(1/2)-1/2*ln((x^4+x^2+1)/x^2+1/x*(x^4+x^2+1)^(1/2)+1)-1/6*2^(1/2)*6^(1/2)*arctan(1/6*(2*(x^4+x^
2+1)^(1/2)*2^(1/2)/x+2^(1/2))*6^(1/2))+1/2*ln((x^4+x^2+1)/x^2-1/x*(x^4+x^2+1)^(1/2)+1)-1/6*2^(1/2)*6^(1/2)*arc
tan(1/6*(2*(x^4+x^2+1)^(1/2)*2^(1/2)/x-2^(1/2))*6^(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 )} {\left (x^{4} - 1\right )}}{{\left (x^{8} + 3 \, x^{6} + 5 \, x^{4} + 3 \, x^{2} + 1\right )} {\left (x^{4} + x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)*(x^8+x^6+3*x^4+x^2+1)/(x^4+x^2+1)^(3/2)/(x^8+3*x^6+5*x^4+3*x^2+1),x, algorithm="maxima")

[Out]

integrate((x^8 + x^6 + 3*x^4 + x^2 + 1)*(x^4 - 1)/((x^8 + 3*x^6 + 5*x^4 + 3*x^2 + 1)*(x^4 + x^2 + 1)^(3/2)), 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 )\,\left (x^8+x^6+3\,x^4+x^2+1\right )}{{\left (x^4+x^2+1\right )}^{3/2}\,\left (x^8+3\,x^6+5\,x^4+3\,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 + 3*x^4 + x^6 + x^8 + 1))/((x^2 + x^4 + 1)^(3/2)*(3*x^2 + 5*x^4 + 3*x^6 + x^8 + 1)),x)

[Out]

int(((x^4 - 1)*(x^2 + 3*x^4 + x^6 + x^8 + 1))/((x^2 + x^4 + 1)^(3/2)*(3*x^2 + 5*x^4 + 3*x^6 + x^8 + 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**8+x**6+3*x**4+x**2+1)/(x**4+x**2+1)**(3/2)/(x**8+3*x**6+5*x**4+3*x**2+1),x)

[Out]

Timed out

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