Integrand size = 95, antiderivative size = 99 \[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt {\frac {1-2 x^2}{1+2 x^2}}}{-1+x}\right )}{2 \sqrt [4]{2} 3^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt {\frac {1-2 x^2}{1+2 x^2}}}{-1+x}\right )}{2 \sqrt [4]{2} 3^{3/4}} \]
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)
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 \[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\text {Result too large to show} \]
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]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (2 x^2-1\right ) \left (4 x^4-4 x^2+4 x-1\right )}{\sqrt {\frac {1-2 x^2}{2 x^2+1}} \left (2 x^2+1\right ) \left (8 x^8-32 x^7+56 x^6-64 x^5+46 x^4-40 x^3+32 x^2-8 x-1\right )} \, dx\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {1-2 x^2} \int -\frac {\sqrt {1-2 x^2} \left (-4 x^4+4 x^2-4 x+1\right )}{\sqrt {2 x^2+1} \left (-8 x^8+32 x^7-56 x^6+64 x^5-46 x^4+40 x^3-32 x^2+8 x+1\right )}dx}{\sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {1-2 x^2} \int \frac {\sqrt {1-2 x^2} \left (-4 x^4+4 x^2-4 x+1\right )}{\sqrt {2 x^2+1} \left (-8 x^8+32 x^7-56 x^6+64 x^5-46 x^4+40 x^3-32 x^2+8 x+1\right )}dx}{\sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt {1-2 x^2} \int \left (\frac {4 \sqrt {1-2 x^2} x^4}{\sqrt {2 x^2+1} \left (8 x^8-32 x^7+56 x^6-64 x^5+46 x^4-40 x^3+32 x^2-8 x-1\right )}-\frac {4 \sqrt {1-2 x^2} x^2}{\sqrt {2 x^2+1} \left (8 x^8-32 x^7+56 x^6-64 x^5+46 x^4-40 x^3+32 x^2-8 x-1\right )}+\frac {4 \sqrt {1-2 x^2} x}{\sqrt {2 x^2+1} \left (8 x^8-32 x^7+56 x^6-64 x^5+46 x^4-40 x^3+32 x^2-8 x-1\right )}+\frac {\sqrt {1-2 x^2}}{\sqrt {2 x^2+1} \left (-8 x^8+32 x^7-56 x^6+64 x^5-46 x^4+40 x^3-32 x^2+8 x+1\right )}\right )dx}{\sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {1-2 x^2} \left (\int \frac {\sqrt {1-2 x^2}}{\sqrt {2 x^2+1} \left (-8 x^8+32 x^7-56 x^6+64 x^5-46 x^4+40 x^3-32 x^2+8 x+1\right )}dx+4 \int \frac {x \sqrt {1-2 x^2}}{\sqrt {2 x^2+1} \left (8 x^8-32 x^7+56 x^6-64 x^5+46 x^4-40 x^3+32 x^2-8 x-1\right )}dx-4 \int \frac {x^2 \sqrt {1-2 x^2}}{\sqrt {2 x^2+1} \left (8 x^8-32 x^7+56 x^6-64 x^5+46 x^4-40 x^3+32 x^2-8 x-1\right )}dx+4 \int \frac {x^4 \sqrt {1-2 x^2}}{\sqrt {2 x^2+1} \left (8 x^8-32 x^7+56 x^6-64 x^5+46 x^4-40 x^3+32 x^2-8 x-1\right )}dx\right )}{\sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}}\) |
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]
3.14.84.3.1 Defintions of rubi rules used
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
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
\[\text {Expression too large to display}\]
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)
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*_alph a^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*_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^6+56*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^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)*Ellipt icPi(x*2^(1/2),-32*_alpha^7+132*_alpha^6-240*_alpha^5+284*_alpha^4-216*_al pha^3+183*_alpha^2-148*_alpha+48,1/2*(-2)^(1/2)*2^(1/2))*_alpha^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*_alpha^6-240*_alpha^5+284*_alpha^4-216*_alpha^3+1...
Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 944, normalized size of antiderivative = 9.54 \[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\text {Too large to display} \]
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")
-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 - 4 2*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*(...
Timed out. \[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\text {Timed out} \]
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)
\[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\int { \frac {{\left (4 \, x^{4} - 4 \, x^{2} + 4 \, x - 1\right )} {\left (2 \, x^{2} - 1\right )}}{{\left (8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1\right )} {\left (2 \, x^{2} + 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}} \,d x } \]
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")
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)
\[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\int { \frac {{\left (4 \, x^{4} - 4 \, x^{2} + 4 \, x - 1\right )} {\left (2 \, x^{2} - 1\right )}}{{\left (8 \, x^{8} - 32 \, x^{7} + 56 \, x^{6} - 64 \, x^{5} + 46 \, x^{4} - 40 \, x^{3} + 32 \, x^{2} - 8 \, x - 1\right )} {\left (2 \, x^{2} + 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}} \,d x } \]
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")
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)
Timed out. \[ \int \frac {\left (-1+2 x^2\right ) \left (-1+4 x-4 x^2+4 x^4\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right ) \left (-1-8 x+32 x^2-40 x^3+46 x^4-64 x^5+56 x^6-32 x^7+8 x^8\right )} \, dx=\int -\frac {\left (2\,x^2-1\right )\,\left (4\,x^4-4\,x^2+4\,x-1\right )}{\left (2\,x^2+1\right )\,\sqrt {-\frac {2\,x^2-1}{2\,x^2+1}}\,\left (-8\,x^8+32\,x^7-56\,x^6+64\,x^5-46\,x^4+40\,x^3-32\,x^2+8\,x+1\right )} \,d x \]
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)