3.14.84 \(\int \frac {(-1+2 x^2) (-1+4 x-4 x^2+4 x^4)}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} (1+2 x^2) (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8)} \, dx\)

Optimal. Leaf size=99 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt {\frac {1-2 x^2}{2 x^2+1}}}{x-1}\right )}{2 \sqrt [4]{2} 3^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt {\frac {1-2 x^2}{2 x^2+1}}}{x-1}\right )}{2 \sqrt [4]{2} 3^{3/4}} \]

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Rubi [F]  time = 8.16, 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+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + 2*x^2)*(-1 + 4*x - 4*x^2 + 4*x^4))/(Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*(1 + 2*x^2)*(-1 - 8*x + 32*x^2 -
40*x^3 + 46*x^4 - 64*x^5 + 56*x^6 - 32*x^7 + 8*x^8)),x]

[Out]

-((Sqrt[1 - 2*x^2]*Defer[Int][Sqrt[1 - 2*x^2]/(Sqrt[1 + 2*x^2]*(1 + 8*x - 32*x^2 + 40*x^3 - 46*x^4 + 64*x^5 -
56*x^6 + 32*x^7 - 8*x^8)), x])/(Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*Sqrt[1 + 2*x^2])) - (4*Sqrt[1 - 2*x^2]*Defer[Int
][(x*Sqrt[1 - 2*x^2])/(Sqrt[1 + 2*x^2]*(-1 - 8*x + 32*x^2 - 40*x^3 + 46*x^4 - 64*x^5 + 56*x^6 - 32*x^7 + 8*x^8
)), x])/(Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*Sqrt[1 + 2*x^2]) + (4*Sqrt[1 - 2*x^2]*Defer[Int][(x^2*Sqrt[1 - 2*x^2])/
(Sqrt[1 + 2*x^2]*(-1 - 8*x + 32*x^2 - 40*x^3 + 46*x^4 - 64*x^5 + 56*x^6 - 32*x^7 + 8*x^8)), x])/(Sqrt[(1 - 2*x
^2)/(1 + 2*x^2)]*Sqrt[1 + 2*x^2]) - (4*Sqrt[1 - 2*x^2]*Defer[Int][(x^4*Sqrt[1 - 2*x^2])/(Sqrt[1 + 2*x^2]*(-1 -
8*x + 32*x^2 - 40*x^3 + 46*x^4 - 64*x^5 + 56*x^6 - 32*x^7 + 8*x^8)), x])/(Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*Sqrt[
1 + 2*x^2])

Rubi steps

\begin {align*} \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx &=\frac {\sqrt {1-2 x^2} \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {1-2 x^2} \sqrt {1+2 x^2} \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}\\ &=-\frac {\sqrt {1-2 x^2} \int \frac {\sqrt {1-2 x^2} \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {1+2 x^2} \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}\\ &=-\frac {\sqrt {1-2 x^2} \int \left (\frac {\sqrt {1-2 x^2}}{\sqrt {1+2 x^2} \left (1+8 x-32 x^2+40 x^3-46 x^4+64 x^5-56 x^6+32 x^7-8 x^8\right )}+\frac {4 x \sqrt {1-2 x^2}}{\sqrt {1+2 x^2} \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )}-\frac {4 x^2 \sqrt {1-2 x^2}}{\sqrt {1+2 x^2} \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )}+\frac {4 x^4 \sqrt {1-2 x^2}}{\sqrt {1+2 x^2} \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )}\right ) \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}\\ &=-\frac {\sqrt {1-2 x^2} \int \frac {\sqrt {1-2 x^2}}{\sqrt {1+2 x^2} \left (1+8 x-32 x^2+40 x^3-46 x^4+64 x^5-56 x^6+32 x^7-8 x^8\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (4 \sqrt {1-2 x^2}\right ) \int \frac {x \sqrt {1-2 x^2}}{\sqrt {1+2 x^2} \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (4 \sqrt {1-2 x^2}\right ) \int \frac {x^2 \sqrt {1-2 x^2}}{\sqrt {1+2 x^2} \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (4 \sqrt {1-2 x^2}\right ) \int \frac {x^4 \sqrt {1-2 x^2}}{\sqrt {1+2 x^2} \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}\\ \end {align*}

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Mathematica [C]  time = 60.49, size = 64755, normalized size = 654.09 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((-1 + 2*x^2)*(-1 + 4*x - 4*x^2 + 4*x^4))/(Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*(1 + 2*x^2)*(-1 - 8*x + 32*
x^2 - 40*x^3 + 46*x^4 - 64*x^5 + 56*x^6 - 32*x^7 + 8*x^8)),x]

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.89, size = 99, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt {\frac {1-2 x^2}{1+2 x^2}}}{-1+x}\right )}{2 \sqrt [4]{2} 3^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt {\frac {1-2 x^2}{1+2 x^2}}}{-1+x}\right )}{2 \sqrt [4]{2} 3^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + 2*x^2)*(-1 + 4*x - 4*x^2 + 4*x^4))/(Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*(1 + 2*x^2)*(-1 -
8*x + 32*x^2 - 40*x^3 + 46*x^4 - 64*x^5 + 56*x^6 - 32*x^7 + 8*x^8)),x]

[Out]

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

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fricas [B]  time = 1.79, size = 708, normalized size = 7.15 \begin {gather*} \frac {1}{108} \cdot 54^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (54^{\frac {3}{4}} {\left (8 \, x^{8} - 32 \, x^{7} + 48 \, x^{6} - 48 \, x^{5} + 62 \, x^{4} - 40 \, x^{3} + 10 \, x^{2} - 12 \, x + 7\right )} + 3 \cdot 54^{\frac {1}{4}} {\left (8 \, x^{8} - 32 \, x^{7} + 8 \, x^{6} + 32 \, x^{5} + 22 \, x^{4} - 40 \, x^{3} + 20 \, x^{2} - 32 \, x + 17\right )}\right )} \sqrt {7 \, \sqrt {6} - 12} + 20 \, {\left (54^{\frac {3}{4}} {\left (4 \, x^{7} - 12 \, x^{6} + 16 \, x^{5} - 16 \, x^{4} + 13 \, x^{3} - 7 \, x^{2} + 3 \, x - 1\right )} - 9 \cdot 54^{\frac {1}{4}} {\left (4 \, x^{5} - 4 \, x^{4} - x + 1\right )}\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}}{90 \, {\left (8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1\right )}}\right ) - \frac {1}{432} \cdot 54^{\frac {3}{4}} \log \left (-\frac {54^{\frac {3}{4}} {\left (8 \, x^{8} - 32 \, x^{7} + 8 \, x^{6} + 32 \, x^{5} + 22 \, x^{4} - 40 \, x^{3} + 20 \, x^{2} - 32 \, x + 17\right )} + 36 \, {\left (24 \, x^{7} - 72 \, x^{6} + 84 \, x^{5} - 84 \, x^{4} + 78 \, x^{3} - 42 \, x^{2} + \sqrt {6} {\left (4 \, x^{7} - 12 \, x^{6} + 4 \, x^{5} - 4 \, x^{4} + 13 \, x^{3} - 7 \, x^{2} + 6 \, x - 4\right )} + 21 \, x - 9\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}} + 18 \cdot 54^{\frac {1}{4}} {\left (8 \, x^{8} - 32 \, x^{7} + 48 \, x^{6} - 48 \, x^{5} + 62 \, x^{4} - 40 \, x^{3} + 10 \, x^{2} - 12 \, x + 7\right )}}{8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1}\right ) + \frac {1}{432} \cdot 54^{\frac {3}{4}} \log \left (\frac {54^{\frac {3}{4}} {\left (8 \, x^{8} - 32 \, x^{7} + 8 \, x^{6} + 32 \, x^{5} + 22 \, x^{4} - 40 \, x^{3} + 20 \, x^{2} - 32 \, x + 17\right )} - 36 \, {\left (24 \, x^{7} - 72 \, x^{6} + 84 \, x^{5} - 84 \, x^{4} + 78 \, x^{3} - 42 \, x^{2} + \sqrt {6} {\left (4 \, x^{7} - 12 \, x^{6} + 4 \, x^{5} - 4 \, x^{4} + 13 \, x^{3} - 7 \, x^{2} + 6 \, x - 4\right )} + 21 \, x - 9\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}} + 18 \cdot 54^{\frac {1}{4}} {\left (8 \, x^{8} - 32 \, x^{7} + 48 \, x^{6} - 48 \, x^{5} + 62 \, x^{4} - 40 \, x^{3} + 10 \, x^{2} - 12 \, x + 7\right )}}{8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-1)*(4*x^4-4*x^2+4*x-1)/((-2*x^2+1)/(2*x^2+1))^(1/2)/(2*x^2+1)/(8*x^8-32*x^7+56*x^6-64*x^5+46*
x^4-40*x^3+32*x^2-8*x-1),x, algorithm="fricas")

[Out]

1/108*54^(3/4)*arctan(1/90*(sqrt(2)*(54^(3/4)*(8*x^8 - 32*x^7 + 48*x^6 - 48*x^5 + 62*x^4 - 40*x^3 + 10*x^2 - 1
2*x + 7) + 3*54^(1/4)*(8*x^8 - 32*x^7 + 8*x^6 + 32*x^5 + 22*x^4 - 40*x^3 + 20*x^2 - 32*x + 17))*sqrt(7*sqrt(6)
- 12) + 20*(54^(3/4)*(4*x^7 - 12*x^6 + 16*x^5 - 16*x^4 + 13*x^3 - 7*x^2 + 3*x - 1) - 9*54^(1/4)*(4*x^5 - 4*x^
4 - x + 1))*sqrt(-(2*x^2 - 1)/(2*x^2 + 1)))/(8*x^8 - 32*x^7 + 56*x^6 - 64*x^5 + 46*x^4 - 40*x^3 + 32*x^2 - 8*x
- 1)) - 1/432*54^(3/4)*log(-(54^(3/4)*(8*x^8 - 32*x^7 + 8*x^6 + 32*x^5 + 22*x^4 - 40*x^3 + 20*x^2 - 32*x + 17
) + 36*(24*x^7 - 72*x^6 + 84*x^5 - 84*x^4 + 78*x^3 - 42*x^2 + sqrt(6)*(4*x^7 - 12*x^6 + 4*x^5 - 4*x^4 + 13*x^3
- 7*x^2 + 6*x - 4) + 21*x - 9)*sqrt(-(2*x^2 - 1)/(2*x^2 + 1)) + 18*54^(1/4)*(8*x^8 - 32*x^7 + 48*x^6 - 48*x^5
+ 62*x^4 - 40*x^3 + 10*x^2 - 12*x + 7))/(8*x^8 - 32*x^7 + 56*x^6 - 64*x^5 + 46*x^4 - 40*x^3 + 32*x^2 - 8*x -
1)) + 1/432*54^(3/4)*log((54^(3/4)*(8*x^8 - 32*x^7 + 8*x^6 + 32*x^5 + 22*x^4 - 40*x^3 + 20*x^2 - 32*x + 17) -
36*(24*x^7 - 72*x^6 + 84*x^5 - 84*x^4 + 78*x^3 - 42*x^2 + sqrt(6)*(4*x^7 - 12*x^6 + 4*x^5 - 4*x^4 + 13*x^3 - 7
*x^2 + 6*x - 4) + 21*x - 9)*sqrt(-(2*x^2 - 1)/(2*x^2 + 1)) + 18*54^(1/4)*(8*x^8 - 32*x^7 + 48*x^6 - 48*x^5 + 6
2*x^4 - 40*x^3 + 10*x^2 - 12*x + 7))/(8*x^8 - 32*x^7 + 56*x^6 - 64*x^5 + 46*x^4 - 40*x^3 + 32*x^2 - 8*x - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x^{4} - 4 \, x^{2} + 4 \, x - 1\right )} {\left (2 \, x^{2} - 1\right )}}{{\left (8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1\right )} {\left (2 \, x^{2} + 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-1)*(4*x^4-4*x^2+4*x-1)/((-2*x^2+1)/(2*x^2+1))^(1/2)/(2*x^2+1)/(8*x^8-32*x^7+56*x^6-64*x^5+46*
x^4-40*x^3+32*x^2-8*x-1),x, algorithm="giac")

[Out]

integrate((4*x^4 - 4*x^2 + 4*x - 1)*(2*x^2 - 1)/((8*x^8 - 32*x^7 + 56*x^6 - 64*x^5 + 46*x^4 - 40*x^3 + 32*x^2
- 8*x - 1)*(2*x^2 + 1)*sqrt(-(2*x^2 - 1)/(2*x^2 + 1))), x)

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maple [C]  time = 0.08, size = 867, normalized size = 8.76 \[\frac {\left (2 x^{2}-1\right ) \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (8 \textit {\_Z}^{8}-32 \textit {\_Z}^{7}+56 \textit {\_Z}^{6}-64 \textit {\_Z}^{5}+46 \textit {\_Z}^{4}-40 \textit {\_Z}^{3}+32 \textit {\_Z}^{2}-8 \textit {\_Z} -1\right )}{\sum }\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{3}-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha -1\right ) \left (8 \sqrt {2}\, \sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (\sqrt {2}\, x , -32 \underline {\hspace {1.25 ex}}\alpha ^{7}+132 \underline {\hspace {1.25 ex}}\alpha ^{6}-240 \underline {\hspace {1.25 ex}}\alpha ^{5}+284 \underline {\hspace {1.25 ex}}\alpha ^{4}-216 \underline {\hspace {1.25 ex}}\alpha ^{3}+183 \underline {\hspace {1.25 ex}}\alpha ^{2}-148 \underline {\hspace {1.25 ex}}\alpha +48, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{7}-32 \sqrt {2}\, \sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (\sqrt {2}\, x , -32 \underline {\hspace {1.25 ex}}\alpha ^{7}+132 \underline {\hspace {1.25 ex}}\alpha ^{6}-240 \underline {\hspace {1.25 ex}}\alpha ^{5}+284 \underline {\hspace {1.25 ex}}\alpha ^{4}-216 \underline {\hspace {1.25 ex}}\alpha ^{3}+183 \underline {\hspace {1.25 ex}}\alpha ^{2}-148 \underline {\hspace {1.25 ex}}\alpha +48, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{6}+56 \sqrt {2}\, \sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (\sqrt {2}\, x , -32 \underline {\hspace {1.25 ex}}\alpha ^{7}+132 \underline {\hspace {1.25 ex}}\alpha ^{6}-240 \underline {\hspace {1.25 ex}}\alpha ^{5}+284 \underline {\hspace {1.25 ex}}\alpha ^{4}-216 \underline {\hspace {1.25 ex}}\alpha ^{3}+183 \underline {\hspace {1.25 ex}}\alpha ^{2}-148 \underline {\hspace {1.25 ex}}\alpha +48, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{5}-64 \sqrt {2}\, \sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (\sqrt {2}\, x , -32 \underline {\hspace {1.25 ex}}\alpha ^{7}+132 \underline {\hspace {1.25 ex}}\alpha ^{6}-240 \underline {\hspace {1.25 ex}}\alpha ^{5}+284 \underline {\hspace {1.25 ex}}\alpha ^{4}-216 \underline {\hspace {1.25 ex}}\alpha ^{3}+183 \underline {\hspace {1.25 ex}}\alpha ^{2}-148 \underline {\hspace {1.25 ex}}\alpha +48, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{4}+46 \sqrt {2}\, \sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (\sqrt {2}\, x , -32 \underline {\hspace {1.25 ex}}\alpha ^{7}+132 \underline {\hspace {1.25 ex}}\alpha ^{6}-240 \underline {\hspace {1.25 ex}}\alpha ^{5}+284 \underline {\hspace {1.25 ex}}\alpha ^{4}-216 \underline {\hspace {1.25 ex}}\alpha ^{3}+183 \underline {\hspace {1.25 ex}}\alpha ^{2}-148 \underline {\hspace {1.25 ex}}\alpha +48, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{3}-40 \sqrt {2}\, \sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (\sqrt {2}\, x , -32 \underline {\hspace {1.25 ex}}\alpha ^{7}+132 \underline {\hspace {1.25 ex}}\alpha ^{6}-240 \underline {\hspace {1.25 ex}}\alpha ^{5}+284 \underline {\hspace {1.25 ex}}\alpha ^{4}-216 \underline {\hspace {1.25 ex}}\alpha ^{3}+183 \underline {\hspace {1.25 ex}}\alpha ^{2}-148 \underline {\hspace {1.25 ex}}\alpha +48, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{2}+32 \sqrt {2}\, \sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (\sqrt {2}\, x , -32 \underline {\hspace {1.25 ex}}\alpha ^{7}+132 \underline {\hspace {1.25 ex}}\alpha ^{6}-240 \underline {\hspace {1.25 ex}}\alpha ^{5}+284 \underline {\hspace {1.25 ex}}\alpha ^{4}-216 \underline {\hspace {1.25 ex}}\alpha ^{3}+183 \underline {\hspace {1.25 ex}}\alpha ^{2}-148 \underline {\hspace {1.25 ex}}\alpha +48, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha -8 \sqrt {2}\, \sqrt {-2 x^{2}+1}\, \sqrt {2 x^{2}+1}\, \EllipticPi \left (\sqrt {2}\, x , -32 \underline {\hspace {1.25 ex}}\alpha ^{7}+132 \underline {\hspace {1.25 ex}}\alpha ^{6}-240 \underline {\hspace {1.25 ex}}\alpha ^{5}+284 \underline {\hspace {1.25 ex}}\alpha ^{4}-216 \underline {\hspace {1.25 ex}}\alpha ^{3}+183 \underline {\hspace {1.25 ex}}\alpha ^{2}-148 \underline {\hspace {1.25 ex}}\alpha +48, \frac {\sqrt {-2}\, \sqrt {2}}{2}\right ) \sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}-\arctanh \left (\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{2} \left (32 \underline {\hspace {1.25 ex}}\alpha ^{7}-132 \underline {\hspace {1.25 ex}}\alpha ^{6}+240 \underline {\hspace {1.25 ex}}\alpha ^{5}-284 \underline {\hspace {1.25 ex}}\alpha ^{4}+216 \underline {\hspace {1.25 ex}}\alpha ^{3}-183 \underline {\hspace {1.25 ex}}\alpha ^{2}+2 x^{2}+148 \underline {\hspace {1.25 ex}}\alpha -48\right )}{\sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-4 x^{4}+1}}\right ) \sqrt {-4 x^{4}+1}\right )}{\sqrt {-4 \underline {\hspace {1.25 ex}}\alpha ^{4}+1}\, \sqrt {-4 x^{4}+1}}\right )}{24 \sqrt {-\frac {2 x^{2}-1}{2 x^{2}+1}}\, \sqrt {-\left (2 x^{2}+1\right ) \left (2 x^{2}-1\right )}}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2-1)*(4*x^4-4*x^2+4*x-1)/((-2*x^2+1)/(2*x^2+1))^(1/2)/(2*x^2+1)/(8*x^8-32*x^7+56*x^6-64*x^5+46*x^4-40
*x^3+32*x^2-8*x-1),x)

[Out]

1/24*(2*x^2-1)*sum((2*_alpha^3-2*_alpha^2+_alpha-1)*(8*2^(1/2)*(-4*_alpha^4+1)^(1/2)*(-2*x^2+1)^(1/2)*(2*x^2+1
)^(1/2)*EllipticPi(2^(1/2)*x,-32*_alpha^7+132*_alpha^6-240*_alpha^5+284*_alpha^4-216*_alpha^3+183*_alpha^2-148
*_alpha+48,1/2*(-2)^(1/2)*2^(1/2))*_alpha^7-32*2^(1/2)*(-4*_alpha^4+1)^(1/2)*(-2*x^2+1)^(1/2)*(2*x^2+1)^(1/2)*
EllipticPi(2^(1/2)*x,-32*_alpha^7+132*_alpha^6-240*_alpha^5+284*_alpha^4-216*_alpha^3+183*_alpha^2-148*_alpha+
48,1/2*(-2)^(1/2)*2^(1/2))*_alpha^6+56*2^(1/2)*(-4*_alpha^4+1)^(1/2)*(-2*x^2+1)^(1/2)*(2*x^2+1)^(1/2)*Elliptic
Pi(2^(1/2)*x,-32*_alpha^7+132*_alpha^6-240*_alpha^5+284*_alpha^4-216*_alpha^3+183*_alpha^2-148*_alpha+48,1/2*(
-2)^(1/2)*2^(1/2))*_alpha^5-64*2^(1/2)*(-4*_alpha^4+1)^(1/2)*(-2*x^2+1)^(1/2)*(2*x^2+1)^(1/2)*EllipticPi(2^(1/
2)*x,-32*_alpha^7+132*_alpha^6-240*_alpha^5+284*_alpha^4-216*_alpha^3+183*_alpha^2-148*_alpha+48,1/2*(-2)^(1/2
)*2^(1/2))*_alpha^4+46*2^(1/2)*(-4*_alpha^4+1)^(1/2)*(-2*x^2+1)^(1/2)*(2*x^2+1)^(1/2)*EllipticPi(2^(1/2)*x,-32
*_alpha^7+132*_alpha^6-240*_alpha^5+284*_alpha^4-216*_alpha^3+183*_alpha^2-148*_alpha+48,1/2*(-2)^(1/2)*2^(1/2
))*_alpha^3-40*2^(1/2)*(-4*_alpha^4+1)^(1/2)*(-2*x^2+1)^(1/2)*(2*x^2+1)^(1/2)*EllipticPi(2^(1/2)*x,-32*_alpha^
7+132*_alpha^6-240*_alpha^5+284*_alpha^4-216*_alpha^3+183*_alpha^2-148*_alpha+48,1/2*(-2)^(1/2)*2^(1/2))*_alph
a^2+32*2^(1/2)*(-4*_alpha^4+1)^(1/2)*(-2*x^2+1)^(1/2)*(2*x^2+1)^(1/2)*EllipticPi(2^(1/2)*x,-32*_alpha^7+132*_a
lpha^6-240*_alpha^5+284*_alpha^4-216*_alpha^3+183*_alpha^2-148*_alpha+48,1/2*(-2)^(1/2)*2^(1/2))*_alpha-8*2^(1
/2)*(-2*x^2+1)^(1/2)*(2*x^2+1)^(1/2)*EllipticPi(2^(1/2)*x,-32*_alpha^7+132*_alpha^6-240*_alpha^5+284*_alpha^4-
216*_alpha^3+183*_alpha^2-148*_alpha+48,1/2*(-2)^(1/2)*2^(1/2))*(-4*_alpha^4+1)^(1/2)-arctanh(2*_alpha^2*(32*_
alpha^7-132*_alpha^6+240*_alpha^5-284*_alpha^4+216*_alpha^3-183*_alpha^2+2*x^2+148*_alpha-48)/(-4*_alpha^4+1)^
(1/2)/(-4*x^4+1)^(1/2))*(-4*x^4+1)^(1/2))/(-4*_alpha^4+1)^(1/2)/(-4*x^4+1)^(1/2),_alpha=RootOf(8*_Z^8-32*_Z^7+
56*_Z^6-64*_Z^5+46*_Z^4-40*_Z^3+32*_Z^2-8*_Z-1))/(-(2*x^2-1)/(2*x^2+1))^(1/2)/(-(2*x^2+1)*(2*x^2-1))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x^{4} - 4 \, x^{2} + 4 \, x - 1\right )} {\left (2 \, x^{2} - 1\right )}}{{\left (8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1\right )} {\left (2 \, x^{2} + 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-1)*(4*x^4-4*x^2+4*x-1)/((-2*x^2+1)/(2*x^2+1))^(1/2)/(2*x^2+1)/(8*x^8-32*x^7+56*x^6-64*x^5+46*
x^4-40*x^3+32*x^2-8*x-1),x, algorithm="maxima")

[Out]

integrate((4*x^4 - 4*x^2 + 4*x - 1)*(2*x^2 - 1)/((8*x^8 - 32*x^7 + 56*x^6 - 64*x^5 + 46*x^4 - 40*x^3 + 32*x^2
- 8*x - 1)*(2*x^2 + 1)*sqrt(-(2*x^2 - 1)/(2*x^2 + 1))), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (2\,x^2-1\right )\,\left (4\,x^4-4\,x^2+4\,x-1\right )}{\left (2\,x^2+1\right )\,\sqrt {-\frac {2\,x^2-1}{2\,x^2+1}}\,\left (-8\,x^8+32\,x^7-56\,x^6+64\,x^5-46\,x^4+40\,x^3-32\,x^2+8\,x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x^2 - 1)*(4*x - 4*x^2 + 4*x^4 - 1))/((2*x^2 + 1)*(-(2*x^2 - 1)/(2*x^2 + 1))^(1/2)*(8*x - 32*x^2 + 40*
x^3 - 46*x^4 + 64*x^5 - 56*x^6 + 32*x^7 - 8*x^8 + 1)),x)

[Out]

int(-((2*x^2 - 1)*(4*x - 4*x^2 + 4*x^4 - 1))/((2*x^2 + 1)*(-(2*x^2 - 1)/(2*x^2 + 1))^(1/2)*(8*x - 32*x^2 + 40*
x^3 - 46*x^4 + 64*x^5 - 56*x^6 + 32*x^7 - 8*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((2*x**2-1)*(4*x**4-4*x**2+4*x-1)/((-2*x**2+1)/(2*x**2+1))**(1/2)/(2*x**2+1)/(8*x**8-32*x**7+56*x**6-
64*x**5+46*x**4-40*x**3+32*x**2-8*x-1),x)

[Out]

Timed out

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