\(\int \frac {-1+x^2}{(1+x^2) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\) [920]

Optimal result

Integrand size = 37, antiderivative size = 70 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {2 a-c} x}{\sqrt {a}+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {2 a-c}} \]

[Out]

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)

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.96 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {2 a-c} x}{\sqrt {a} \left (1+x^2\right )-\sqrt {x \left (b+c x+b x^2\right )+a \left (1+x^4\right )}}\right )}{\sqrt {2 a-c}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {\ln \left (\frac {2 \sqrt {-2 a +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+b \,x^{2}+\left (-4 a +2 c \right ) x +b}{x^{2}+1}\right )}{\sqrt {-2 a +c}}\)	\(66\)
pseudoelliptic	\(-\frac {\ln \left (\frac {2 \sqrt {-2 a +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+b \,x^{2}+\left (-4 a +2 c \right ) x +b}{x^{2}+1}\right )}{\sqrt {-2 a +c}}\)	\(66\)
elliptic	\(\text {Expression too large to display}\)	\(90322\)

[In]

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

[Out]

-1/(-2*a+c)^(1/2)*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))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (60) = 120\).

Time = 0.45 (sec) , antiderivative size = 291, normalized size of antiderivative = 4.16 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\left [-\frac {\sqrt {-2 \, a + c} \log \left (-\frac {{\left (8 \, a^{2} - b^{2} - 4 \, a c\right )} x^{4} + 8 \, {\left (2 \, a b - b c\right )} x^{3} - 2 \, {\left (8 \, a^{2} + b^{2} - 12 \, a c + 4 \, c^{2}\right )} x^{2} + 8 \, a^{2} + 4 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (b x^{2} - 2 \, {\left (2 \, a - c\right )} x + b\right )} \sqrt {-2 \, a + c} - b^{2} - 4 \, a c + 8 \, {\left (2 \, a b - b c\right )} x}{x^{4} + 2 \, x^{2} + 1}\right )}{2 \, {\left (2 \, a - c\right )}}, -\frac {\arctan \left (-\frac {\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (b x^{2} - 2 \, {\left (2 \, a - c\right )} x + b\right )} \sqrt {2 \, a - c}}{2 \, {\left ({\left (2 \, a^{2} - a c\right )} x^{4} + {\left (2 \, a b - b c\right )} x^{3} + {\left (2 \, a c - c^{2}\right )} x^{2} + 2 \, a^{2} - a c + {\left (2 \, a b - b c\right )} x\right )}}\right )}{\sqrt {2 \, a - c}}\right ] \]

[In]

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

[Out]

[-1/2*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))/(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))/sqrt(2*a - c)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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