\(\int \frac {x}{(1-x^2) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\) [2400]

Optimal result

Integrand size = 35, antiderivative size = 193 \[ \int \frac {x}{\left (1-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {\sqrt {-2 a-2 b-c} \arctan \left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a}-2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{2 (2 a+2 b+c)}-\frac {\sqrt {-2 a+2 b-c} \arctan \left (\frac {\sqrt {-2 a+2 b-c} x}{\sqrt {a}+2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{2 (2 a-2 b+c)} \]

[Out]

(-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
)))/(4*a+4*b+2*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)))/(4*a-4*b+2*c)

Rubi [F]

\[ \int \frac {x}{\left (1-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int \frac {x}{\left (1-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \]

[In]

Int[x/((1 - x^2)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]

[Out]

-1/2*Defer[Int][1/((-1 + x)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]), x] - Defer[Int][1/((1 + x)*Sqrt[a + b*x +
c*x^2 + b*x^3 + a*x^4]), x]/2

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{2 (-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}}-\frac {1}{2 (1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {1}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\right )-\frac {1}{2} \int \frac {1}{(1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.78 \[ \int \frac {x}{\left (1-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {1}{2} \left (-\frac {\arctan \left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a} (-1+x)^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {-2 a-2 b-c}}+\frac {\arctan \left (\frac {\sqrt {-2 a+2 b-c} x}{\sqrt {a} (1+x)^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {-2 a+2 b-c}}\right ) \]

[In]

Integrate[x/((1 - x^2)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]

[Out]

(-(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

Maple [A] (verified)

Time = 2.12 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {\ln \left (\frac {2 \sqrt {2 a -2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (b -4 a \right ) x^{2}+\left (-4 a -2 b +2 c \right ) x -4 a +b}{\left (1+x \right )^{2}}\right ) \sqrt {2 a +2 b +c}-\ln \left (\frac {2 \sqrt {2 a +2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (4 a +b \right ) x^{2}+\left (-4 a +2 b +2 c \right ) x +4 a +b}{\left (-1+x \right )^{2}}\right ) \sqrt {2 a -2 b +c}}{4 \sqrt {2 a -2 b +c}\, \sqrt {2 a +2 b +c}}\)	\(181\)
pseudoelliptic	\(\frac {-\ln \left (\frac {2 \sqrt {2 a -2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (b -4 a \right ) x^{2}+\left (-4 a -2 b +2 c \right ) x -4 a +b}{\left (1+x \right )^{2}}\right ) \sqrt {2 a +2 b +c}+\ln \left (\frac {2 \sqrt {2 a +2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (4 a +b \right ) x^{2}+\left (-4 a +2 b +2 c \right ) x +4 a +b}{\left (-1+x \right )^{2}}\right ) \sqrt {2 a -2 b +c}}{4 \sqrt {2 a -2 b +c}\, \sqrt {2 a +2 b +c}}\)	\(181\)
elliptic	\(\text {Expression too large to display}\)	\(78106\)

[In]

int(x/(-x^2+1)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-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)+(4*a+b)*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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (165) = 330\).

Time = 0.61 (sec) , antiderivative size = 1661, normalized size of antiderivative = 8.61 \[ \int \frac {x}{\left (1-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\text {Too large to display} \]

[In]

integrate(x/(-x^2+1)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[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)*lo
g(((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*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)*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)) + 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)*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*b^2 + 4*a*c + c^2), 1/4*((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)*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)))/(4*a^2 - 4*b^2
+ 4*a*c + c^2)]

Sympy [F]

\[ \int \frac {x}{\left (1-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=- \int \frac {x}{x^{2} \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}} - \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx \]

[In]

integrate(x/(-x**2+1)/(a*x**4+b*x**3+c*x**2+b*x+a)**(1/2),x)

[Out]

-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)

Maxima [F]

\[ \int \frac {x}{\left (1-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { -\frac {x}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (x^{2} - 1\right )}} \,d x } \]

[In]

integrate(x/(-x^2+1)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

-integrate(x/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(x^2 - 1)), x)

Giac [F]

\[ \int \frac {x}{\left (1-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { -\frac {x}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (x^{2} - 1\right )}} \,d x } \]

[In]

integrate(x/(-x^2+1)/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

integrate(-x/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(x^2 - 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\left (1-x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=-\int \frac {x}{\left (x^2-1\right )\,\sqrt {a\,x^4+b\,x^3+c\,x^2+b\,x+a}} \,d x \]

[In]

int(-x/((x^2 - 1)*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)),x)

[Out]

-int(x/((x^2 - 1)*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)), x)