\(\int \frac {(-1+2 x^2) (-1+4 x-4 x^2+4 x^4)}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} (1+2 x^2) (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8)} \, dx\) [1384]

Optimal result

Integrand size = 95, antiderivative size = 99 \[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt {\frac {1-2 x^2}{1+2 x^2}}}{-1+x}\right )}{2 \sqrt [4]{2} 3^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt {\frac {1-2 x^2}{1+2 x^2}}}{-1+x}\right )}{2 \sqrt [4]{2} 3^{3/4}} \]

[Out]

1/12*arctan(1/2*3^(1/4)*2^(3/4)*((-2*x^2+1)/(2*x^2+1))^(1/2)/(-1+x))*2^(3/4)*3^(1/4)-1/12*arctanh(1/2*3^(1/4)*
2^(3/4)*((-2*x^2+1)/(2*x^2+1))^(1/2)/(-1+x))*2^(3/4)*3^(1/4)

Rubi [F]

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

[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*} \text {integral}& = \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*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 74.45 (sec) , antiderivative size = 64371, normalized size of antiderivative = 650.21 \[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\text {Result too large to show} \]

[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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.68 (sec) , antiderivative size = 867, normalized size of antiderivative = 8.76

\[\text {Expression too large to display}\]

[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(x*2^(1/2),-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(x*2^(1/2),-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(x*2^(1/2),-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(x*2^(
1/2),-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(x*2^(1/2),-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(x*2^(1/2),-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(x*2^(1/2),-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(x*2^(1/2),-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)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.53 (sec) , antiderivative size = 944, normalized size of antiderivative = 9.54 \[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\text {Too large to display} \]

[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]

-1/432*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 + 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*
I*54^(3/4)*log((54^(3/4)*(8*I*x^8 - 32*I*x^7 + 8*I*x^6 + 32*I*x^5 + 22*I*x^4 - 40*I*x^3 + 20*I*x^2 - 32*I*x +
17*I) - 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*I*x^8 - 32*I*x^7 + 48*I*x^
6 - 48*I*x^5 + 62*I*x^4 - 40*I*x^3 + 10*I*x^2 - 12*I*x + 7*I))/(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*I*54^(3/4)*log((54^(3/4)*(-8*I*x^8 + 32*I*x^7 - 8*I*x^6 - 32*I*x^5 - 22*I*x^
4 + 40*I*x^3 - 20*I*x^2 + 32*I*x - 17*I) - 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*I*x^8 + 32*I*x^7 - 48*I*x^6 + 48*I*x^5 - 62*I*x^4 + 40*I*x^3 - 10*I*x^2 + 12*I*x - 7*I))/(8*x^8 - 32
*x^7 + 56*x^6 - 64*x^5 + 46*x^4 - 40*x^3 + 32*x^2 - 8*x - 1))

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\text {Timed out} \]

[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

Maxima [F]

\[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\int { \frac {{\left (4 \, x^{4} - 4 \, x^{2} + 4 \, x - 1\right )} {\left (2 \, x^{2} - 1\right )}}{{\left (8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1\right )} {\left (2 \, x^{2} + 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\int { \frac {{\left (4 \, x^{4} - 4 \, x^{2} + 4 \, x - 1\right )} {\left (2 \, x^{2} - 1\right )}}{{\left (8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1\right )} {\left (2 \, x^{2} + 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\int -\frac {\left (2\,x^2-1\right )\,\left (4\,x^4-4\,x^2+4\,x-1\right )}{\left (2\,x^2+1\right )\,\sqrt {-\frac {2\,x^2-1}{2\,x^2+1}}\,\left (-8\,x^8+32\,x^7-56\,x^6+64\,x^5-46\,x^4+40\,x^3-32\,x^2+8\,x+1\right )} \,d x \]

[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)