Integrand size = 37, antiderivative size = 265 \[ \int \frac {x^2}{\left (1-x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt {2 a-c} x}{\sqrt {a}+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{2 \sqrt {2 a-c}}+\frac {\sqrt {-2 a-2 b-c} \arctan \left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a}-2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{4 (2 a+2 b+c)}+\frac {\sqrt {-2 a+2 b-c} \arctan \left (\frac {\sqrt {-2 a+2 b-c} x}{\sqrt {a}+2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{4 (2 a-2 b+c)} \]
1/2*arctan((2*a-c)^(1/2)*x/(a^(1/2)+a^(1/2)*x^2-(a*x^4+b*x^3+c*x^2+b*x+a)^ (1/2)))/(2*a-c)^(1/2)+(-2*a-2*b-c)^(1/2)*arctan((-2*a-2*b-c)^(1/2)*x/(a^(1 /2)-2*x*a^(1/2)+a^(1/2)*x^2-(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)))/(8*a+8*b+4*c )+(-2*a+2*b-c)^(1/2)*arctan((-2*a+2*b-c)^(1/2)*x/(a^(1/2)+2*x*a^(1/2)+a^(1 /2)*x^2-(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)))/(8*a-8*b+4*c)
Time = 1.58 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.82 \[ \int \frac {x^2}{\left (1-x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {1}{4} \left (-\frac {\arctan \left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a} (-1+x)^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {-2 a-2 b-c}}-\frac {\arctan \left (\frac {\sqrt {-2 a+2 b-c} x}{\sqrt {a} (1+x)^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {-2 a+2 b-c}}+\frac {2 \arctan \left (\frac {\sqrt {2 a-c} x}{\sqrt {a} \left (1+x^2\right )-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {2 a-c}}\right ) \]
(-(ArcTan[(Sqrt[-2*a - 2*b - c]*x)/(Sqrt[a]*(-1 + x)^2 - Sqrt[a + b*x + c* x^2 + b*x^3 + a*x^4])]/Sqrt[-2*a - 2*b - c]) - ArcTan[(Sqrt[-2*a + 2*b - c ]*x)/(Sqrt[a]*(1 + x)^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])]/Sqrt[-2* a + 2*b - c] + (2*ArcTan[(Sqrt[2*a - c]*x)/(Sqrt[a]*(1 + x^2) - Sqrt[a + b *x + c*x^2 + b*x^3 + a*x^4])])/Sqrt[2*a - c])/4
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.
| \(\displaystyle \int \frac {x^2}{\left (1-x^4\right ) \sqrt {a x^4+a+b x^3+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (-\frac {1}{2 \left (x^2-1\right ) \sqrt {a x^4+a+b x^3+b x+c x^2}}-\frac {1}{2 \left (x^2+1\right ) \sqrt {a x^4+a+b x^3+b x+c x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{4} i \int \frac {1}{(i-x) \sqrt {a x^4+b x^3+c x^2+b x+a}}dx+\frac {1}{4} \int \frac {1}{(1-x) \sqrt {a x^4+b x^3+c x^2+b x+a}}dx-\frac {1}{4} i \int \frac {1}{(x+i) \sqrt {a x^4+b x^3+c x^2+b x+a}}dx+\frac {1}{4} \int \frac {1}{(x+1) \sqrt {a x^4+b x^3+c x^2+b x+a}}dx\) |
3.28.72.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 3.26 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.02
| method | result | size |
| pseudoelliptic | \(\frac {\left (\ln \left (\frac {2 \sqrt {2 a +2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (4 a +b \right ) x^{2}+\left (-4 a +2 b +2 c \right ) x +4 a +b}{\left (-1+x \right )^{2}}\right ) \sqrt {-2 a +c}-2 \ln \left (\frac {2 \sqrt {-2 a +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+b \,x^{2}+\left (-4 a +2 c \right ) x +b}{x^{2}+1}\right ) \sqrt {2 a +2 b +c}\right ) \sqrt {2 a -2 b +c}+\ln \left (\frac {2 \sqrt {2 a -2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (b -4 a \right ) x^{2}+\left (-4 a -2 b +2 c \right ) x -4 a +b}{\left (1+x \right )^{2}}\right ) \sqrt {-2 a +c}\, \sqrt {2 a +2 b +c}}{8 \sqrt {-2 a +c}\, \sqrt {2 a +2 b +c}\, \sqrt {2 a -2 b +c}}\) | \(271\) |
| default | \(-\frac {-\ln \left (\frac {2 \sqrt {2 a +2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (4 a +b \right ) x^{2}+\left (-4 a +2 b +2 c \right ) x +4 a +b}{\left (-1+x \right )^{2}}\right ) \sqrt {-2 a +c}\, \sqrt {2 a -2 b +c}+2 \ln \left (\frac {2 \sqrt {-2 a +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+b \,x^{2}+\left (-4 a +2 c \right ) x +b}{x^{2}+1}\right ) \sqrt {2 a +2 b +c}\, \sqrt {2 a -2 b +c}-\ln \left (\frac {2 \sqrt {2 a -2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (b -4 a \right ) x^{2}+\left (-4 a -2 b +2 c \right ) x -4 a +b}{\left (1+x \right )^{2}}\right ) \sqrt {-2 a +c}\, \sqrt {2 a +2 b +c}}{8 \sqrt {-2 a +c}\, \sqrt {2 a -2 b +c}\, \sqrt {2 a +2 b +c}}\) | \(281\) |
1/8*((ln((2*(2*a+2*b+c)^(1/2)*(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)+(4*a+b)*x^2+ (-4*a+2*b+2*c)*x+4*a+b)/(-1+x)^2)*(-2*a+c)^(1/2)-2*ln((2*(-2*a+c)^(1/2)*(a *x^4+b*x^3+c*x^2+b*x+a)^(1/2)+b*x^2+(-4*a+2*c)*x+b)/(x^2+1))*(2*a+2*b+c)^( 1/2))*(2*a-2*b+c)^(1/2)+ln((2*(2*a-2*b+c)^(1/2)*(a*x^4+b*x^3+c*x^2+b*x+a)^ (1/2)+(b-4*a)*x^2+(-4*a-2*b+2*c)*x-4*a+b)/(1+x)^2)*(-2*a+c)^(1/2)*(2*a+2*b +c)^(1/2))/(-2*a+c)^(1/2)/(2*a+2*b+c)^(1/2)/(2*a-2*b+c)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (225) = 450\).
Time = 2.11 (sec) , antiderivative size = 4886, normalized size of antiderivative = 18.44 \[ \int \frac {x^2}{\left (1-x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\text {Too large to display} \]
[1/16*((4*a^2 + 4*a*b - 2*b*c - c^2)*sqrt(2*a - 2*b + c)*log(((24*a^2 - 16 *a*b + b^2 + 4*a*c)*x^4 + 4*(8*a^2 + 4*a*b - 3*b^2 - 2*(2*a - b)*c)*x^3 + 2*(24*a^2 + 3*b^2 - 4*(a + 2*b)*c + 4*c^2)*x^2 - 4*sqrt(a*x^4 + b*x^3 + c* x^2 + b*x + a)*((4*a - b)*x^2 + 2*(2*a + b - c)*x + 4*a - b)*sqrt(2*a - 2* b + c) + 24*a^2 - 16*a*b + b^2 + 4*a*c + 4*(8*a^2 + 4*a*b - 3*b^2 - 2*(2*a - b)*c)*x)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1)) + (4*a^2 - 4*a*b + 2*b*c - c^ 2)*sqrt(2*a + 2*b + c)*log(((24*a^2 + 16*a*b + b^2 + 4*a*c)*x^4 - 4*(8*a^2 - 4*a*b - 3*b^2 - 2*(2*a + b)*c)*x^3 + 2*(24*a^2 + 3*b^2 - 4*(a - 2*b)*c + 4*c^2)*x^2 + 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a + b)*x^2 - 2* (2*a - b - c)*x + 4*a + b)*sqrt(2*a + 2*b + c) + 24*a^2 + 16*a*b + b^2 + 4 *a*c - 4*(8*a^2 - 4*a*b - 3*b^2 - 2*(2*a + b)*c)*x)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)) - 2*(4*a^2 - 4*b^2 + 4*a*c + c^2)*sqrt(-2*a + c)*log(-((8*a^2 - b^2 - 4*a*c)*x^4 + 8*(2*a*b - b*c)*x^3 - 2*(8*a^2 + b^2 - 12*a*c + 4*c^2) *x^2 + 8*a^2 + 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(b*x^2 - 2*(2*a - c )*x + b)*sqrt(-2*a + c) - b^2 - 4*a*c + 8*(2*a*b - b*c)*x)/(x^4 + 2*x^2 + 1)))/(8*a^3 - 8*a*b^2 - 2*a*c^2 - c^3 + 4*(a^2 + b^2)*c), -1/16*(4*(4*a^2 - 4*b^2 + 4*a*c + c^2)*sqrt(2*a - c)*arctan(-1/2*sqrt(a*x^4 + b*x^3 + c*x^ 2 + b*x + a)*(b*x^2 - 2*(2*a - c)*x + b)*sqrt(2*a - c)/((2*a^2 - a*c)*x^4 + (2*a*b - b*c)*x^3 + (2*a*c - c^2)*x^2 + 2*a^2 - a*c + (2*a*b - b*c)*x)) - (4*a^2 + 4*a*b - 2*b*c - c^2)*sqrt(2*a - 2*b + c)*log(((24*a^2 - 16*a...
\[ \int \frac {x^2}{\left (1-x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=- \int \frac {x^{2}}{x^{4} \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}} - \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx \]
-Integral(x**2/(x**4*sqrt(a*x**4 + a + b*x**3 + b*x + c*x**2) - sqrt(a*x** 4 + a + b*x**3 + b*x + c*x**2)), x)
\[ \int \frac {x^2}{\left (1-x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { -\frac {x^{2}}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (x^{4} - 1\right )}} \,d x } \]
\[ \int \frac {x^2}{\left (1-x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { -\frac {x^{2}}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (x^{4} - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\left (1-x^4\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=-\int \frac {x^2}{\left (x^4-1\right )\,\sqrt {a\,x^4+b\,x^3+c\,x^2+b\,x+a}} \,d x \]