Integrand size = 35, antiderivative size = 193 \[ \int \frac {x}{\left (1-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {\sqrt {-2 a-2 b-c} \arctan \left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a}-2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{2 (2 a+2 b+c)}-\frac {\sqrt {-2 a+2 b-c} \arctan \left (\frac {\sqrt {-2 a+2 b-c} x}{\sqrt {a}+2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{2 (2 a-2 b+c)} \] Output:
(-2*a-2*b-c)^(1/2)*arctan((-2*a-2*b-c)^(1/2)*x/(a^(1/2)-2*a^(1/2)*x+a^(1/2 )*x^2-(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)))/(4*a+4*b+2*c)-(-2*a+2*b-c)^(1/2)*a rctan((-2*a+2*b-c)^(1/2)*x/(a^(1/2)+2*a^(1/2)*x+a^(1/2)*x^2-(a*x^4+b*x^3+c *x^2+b*x+a)^(1/2)))/(4*a-4*b+2*c)
Time = 1.18 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.78 \[ \int \frac {x}{\left (1-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {1}{2} \left (-\frac {\arctan \left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a} (-1+x)^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {-2 a-2 b-c}}+\frac {\arctan \left (\frac {\sqrt {-2 a+2 b-c} x}{\sqrt {a} (1+x)^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {-2 a+2 b-c}}\right ) \] Input:
Integrate[x/((1 - x^2)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]
Output:
(-(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
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.
| \(\displaystyle \int \frac {x}{\left (1-x^2\right ) \sqrt {a x^4+a+b x^3+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (-\frac {1}{2 (x-1) \sqrt {a x^4+a+b x^3+b x+c x^2}}-\frac {1}{2 (x+1) \sqrt {a x^4+a+b x^3+b x+c x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{2} \int \frac {1}{(x-1) \sqrt {a x^4+b x^3+c x^2+b x+a}}dx-\frac {1}{2} \int \frac {1}{(x+1) \sqrt {a x^4+b x^3+c x^2+b x+a}}dx\) |
Input:
Int[x/((1 - x^2)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]
Output:
$Aborted
Time = 4.17 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.94
| method | result | size |
| default | \(-\frac {\ln \left (\frac {2 \sqrt {2 a -2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (b -4 a \right ) x^{2}+\left (-4 a -2 b +2 c \right ) x -4 a +b}{\left (1+x \right )^{2}}\right ) \sqrt {2 a +2 b +c}-\ln \left (\frac {2 \sqrt {2 a +2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (b +4 a \right ) x^{2}+\left (-4 a +2 b +2 c \right ) x +4 a +b}{\left (-1+x \right )^{2}}\right ) \sqrt {2 a -2 b +c}}{4 \sqrt {2 a +2 b +c}\, \sqrt {2 a -2 b +c}}\) | \(181\) |
| pseudoelliptic | \(\frac {-\ln \left (\frac {2 \sqrt {2 a -2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (b -4 a \right ) x^{2}+\left (-4 a -2 b +2 c \right ) x -4 a +b}{\left (1+x \right )^{2}}\right ) \sqrt {2 a +2 b +c}+\ln \left (\frac {2 \sqrt {2 a +2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (b +4 a \right ) x^{2}+\left (-4 a +2 b +2 c \right ) x +4 a +b}{\left (-1+x \right )^{2}}\right ) \sqrt {2 a -2 b +c}}{4 \sqrt {2 a +2 b +c}\, \sqrt {2 a -2 b +c}}\) | \(181\) |
| elliptic | \(\text {Expression too large to display}\) | \(78106\) |
Input:
int(x/(-x^2+1)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/4*(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+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-2*b+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. 354 vs. \(2 (165) = 330\).
Time = 0.38 (sec) , antiderivative size = 1661, normalized size of antiderivative = 8.61 \[ \int \frac {x}{\left (1-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\text {Too large to display} \] Input:
integrate(x/(-x^2+1)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="fricas" )
Output:
[1/8*((2*a + 2*b + c)*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)) + sqrt(2*a + 2*b + c)*(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)*sq rt(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*b^2 + 4*a*c + c^2), -1/8*(2*(2*a - 2*b + c)*sqrt(-2*a - 2*b - c)*arctan(1/2*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)/((2*a^2 + 2*a*b + a*c)*x^4 + (2*a*b + 2*b^2 + b *c)*x^3 + (2*(a + b)*c + c^2)*x^2 + 2*a^2 + 2*a*b + a*c + (2*a*b + 2*b^2 + b*c)*x)) - (2*a + 2*b + c)*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)*...
\[ \int \frac {x}{\left (1-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=- \int \frac {x}{x^{2} \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}} - \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx \] Input:
integrate(x/(-x**2+1)/(a*x**4+b*x**3+c*x**2+b*x+a)**(1/2),x)
Output:
-Integral(x/(x**2*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)
\[ \int \frac {x}{\left (1-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { -\frac {x}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (x^{2} - 1\right )}} \,d x } \] Input:
integrate(x/(-x^2+1)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="maxima" )
Output:
-integrate(x/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(x^2 - 1)), x)
\[ \int \frac {x}{\left (1-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { -\frac {x}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (x^{2} - 1\right )}} \,d x } \] Input:
integrate(x/(-x^2+1)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="giac")
Output:
integrate(-x/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(x^2 - 1)), x)
Timed out. \[ \int \frac {x}{\left (1-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=-\int \frac {x}{\left (x^2-1\right )\,\sqrt {a\,x^4+b\,x^3+c\,x^2+b\,x+a}} \,d x \] Input:
int(-x/((x^2 - 1)*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)),x)
Output:
-int(x/((x^2 - 1)*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)), x)
Time = 0.38 (sec) , antiderivative size = 595, normalized size of antiderivative = 3.08 \[ \int \frac {x}{\left (1-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {-2 \sqrt {2 a -2 b +c}\, \mathrm {log}\left (x^{2}+2 x +1\right ) a -2 \sqrt {2 a -2 b +c}\, \mathrm {log}\left (x^{2}+2 x +1\right ) b -\sqrt {2 a -2 b +c}\, \mathrm {log}\left (x^{2}+2 x +1\right ) c +2 \sqrt {2 a -2 b +c}\, \mathrm {log}\left (2 \sqrt {2 a -2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+4 a \,x^{2}+4 a x +4 a -b \,x^{2}+2 b x -b -2 c x \right ) a +2 \sqrt {2 a -2 b +c}\, \mathrm {log}\left (2 \sqrt {2 a -2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+4 a \,x^{2}+4 a x +4 a -b \,x^{2}+2 b x -b -2 c x \right ) b +\sqrt {2 a -2 b +c}\, \mathrm {log}\left (2 \sqrt {2 a -2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+4 a \,x^{2}+4 a x +4 a -b \,x^{2}+2 b x -b -2 c x \right ) c -2 \sqrt {2 a +2 b +c}\, \mathrm {log}\left (x^{2}-2 x +1\right ) a +2 \sqrt {2 a +2 b +c}\, \mathrm {log}\left (x^{2}-2 x +1\right ) b -\sqrt {2 a +2 b +c}\, \mathrm {log}\left (x^{2}-2 x +1\right ) c +2 \sqrt {2 a +2 b +c}\, \mathrm {log}\left (2 \sqrt {2 a +2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+4 a \,x^{2}-4 a x +4 a +b \,x^{2}+2 b x +b +2 c x \right ) a -2 \sqrt {2 a +2 b +c}\, \mathrm {log}\left (2 \sqrt {2 a +2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+4 a \,x^{2}-4 a x +4 a +b \,x^{2}+2 b x +b +2 c x \right ) b +\sqrt {2 a +2 b +c}\, \mathrm {log}\left (2 \sqrt {2 a +2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+4 a \,x^{2}-4 a x +4 a +b \,x^{2}+2 b x +b +2 c x \right ) c}{16 a^{2}+16 a c -16 b^{2}+4 c^{2}} \] Input:
int(x/(-x^2+1)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x)
Output:
( - 2*sqrt(2*a - 2*b + c)*log(x**2 + 2*x + 1)*a - 2*sqrt(2*a - 2*b + c)*lo g(x**2 + 2*x + 1)*b - sqrt(2*a - 2*b + c)*log(x**2 + 2*x + 1)*c + 2*sqrt(2 *a - 2*b + c)*log(2*sqrt(2*a - 2*b + c)*sqrt(a*x**4 + a + b*x**3 + b*x + c *x**2) + 4*a*x**2 + 4*a*x + 4*a - b*x**2 + 2*b*x - b - 2*c*x)*a + 2*sqrt(2 *a - 2*b + c)*log(2*sqrt(2*a - 2*b + c)*sqrt(a*x**4 + a + b*x**3 + b*x + c *x**2) + 4*a*x**2 + 4*a*x + 4*a - b*x**2 + 2*b*x - b - 2*c*x)*b + sqrt(2*a - 2*b + c)*log(2*sqrt(2*a - 2*b + c)*sqrt(a*x**4 + a + b*x**3 + b*x + c*x **2) + 4*a*x**2 + 4*a*x + 4*a - b*x**2 + 2*b*x - b - 2*c*x)*c - 2*sqrt(2*a + 2*b + c)*log(x**2 - 2*x + 1)*a + 2*sqrt(2*a + 2*b + c)*log(x**2 - 2*x + 1)*b - sqrt(2*a + 2*b + c)*log(x**2 - 2*x + 1)*c + 2*sqrt(2*a + 2*b + c)* log(2*sqrt(2*a + 2*b + c)*sqrt(a*x**4 + a + b*x**3 + b*x + c*x**2) + 4*a*x **2 - 4*a*x + 4*a + b*x**2 + 2*b*x + b + 2*c*x)*a - 2*sqrt(2*a + 2*b + c)* log(2*sqrt(2*a + 2*b + c)*sqrt(a*x**4 + a + b*x**3 + b*x + c*x**2) + 4*a*x **2 - 4*a*x + 4*a + b*x**2 + 2*b*x + b + 2*c*x)*b + sqrt(2*a + 2*b + c)*lo g(2*sqrt(2*a + 2*b + c)*sqrt(a*x**4 + a + b*x**3 + b*x + c*x**2) + 4*a*x** 2 - 4*a*x + 4*a + b*x**2 + 2*b*x + b + 2*c*x)*c)/(4*(4*a**2 + 4*a*c - 4*b* *2 + c**2))