Integrand size = 33, antiderivative size = 94 \[ \int \frac {-1+x}{(1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=-\frac {2 \sqrt {-2 a+2 b-c} \arctan \left (\frac {\sqrt {-2 a+2 b-c} x}{\sqrt {a}+2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{2 a-2 b+c} \]
[Out]
\[ \int \frac {-1+x}{(1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int \frac {-1+x}{(1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \]
[In]
[Out]
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}{(1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx \\ & = -\left (2 \int \frac {1}{(1+x) \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 \\ \end{align*}
Time = 0.68 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.78 \[ \int \frac {-1+x}{(1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {-2 a+2 b-c} x}{\sqrt {a} (1+x)^2-\sqrt {x \left (b+c x+b x^2\right )+a \left (1+x^4\right )}}\right )}{\sqrt {-2 a+2 b-c}} \]
[In]
[Out]
Time = 2.15 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.85
method | result | size |
default | \(-\frac {\ln \left (\frac {2 \sqrt {2 a -2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (b -4 a \right ) x^{2}+\left (-4 a -2 b +2 c \right ) x -4 a +b}{\left (1+x \right )^{2}}\right )}{\sqrt {2 a -2 b +c}}\) | \(80\) |
pseudoelliptic | \(-\frac {\ln \left (\frac {2 \sqrt {2 a -2 b +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+\left (b -4 a \right ) x^{2}+\left (-4 a -2 b +2 c \right ) x -4 a +b}{\left (1+x \right )^{2}}\right )}{\sqrt {2 a -2 b +c}}\) | \(80\) |
elliptic | \(\text {Expression too large to display}\) | \(2813\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (82) = 164\).
Time = 0.45 (sec) , antiderivative size = 392, normalized size of antiderivative = 4.17 \[ \int \frac {-1+x}{(1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\left [\frac {\log \left (\frac {{\left (24 \, a^{2} - 16 \, a b + b^{2} + 4 \, a c\right )} x^{4} + 4 \, {\left (8 \, a^{2} + 4 \, a b - 3 \, b^{2} - 2 \, {\left (2 \, a - b\right )} c\right )} x^{3} + 2 \, {\left (24 \, a^{2} + 3 \, b^{2} - 4 \, {\left (a + 2 \, b\right )} c + 4 \, c^{2}\right )} x^{2} + 4 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left ({\left (4 \, a - b\right )} x^{2} + 2 \, {\left (2 \, a + b - c\right )} x + 4 \, a - b\right )} \sqrt {2 \, a - 2 \, b + c} + 24 \, a^{2} - 16 \, a b + b^{2} + 4 \, a c + 4 \, {\left (8 \, a^{2} + 4 \, a b - 3 \, b^{2} - 2 \, {\left (2 \, a - b\right )} c\right )} x}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1}\right )}{2 \, \sqrt {2 \, a - 2 \, b + c}}, \frac {\sqrt {-2 \, a + 2 \, b - c} \arctan \left (-\frac {\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left ({\left (4 \, a - b\right )} x^{2} + 2 \, {\left (2 \, a + b - c\right )} x + 4 \, a - b\right )} \sqrt {-2 \, a + 2 \, b - c}}{2 \, {\left ({\left (2 \, a^{2} - 2 \, a b + a c\right )} x^{4} + {\left (2 \, a b - 2 \, b^{2} + b c\right )} x^{3} + {\left (2 \, {\left (a - b\right )} c + c^{2}\right )} x^{2} + 2 \, a^{2} - 2 \, a b + a c + {\left (2 \, a b - 2 \, b^{2} + b c\right )} x\right )}}\right )}{2 \, a - 2 \, b + c}\right ] \]
[In]
[Out]
\[ \int \frac {-1+x}{(1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int \frac {x - 1}{\left (x + 1\right ) \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {-1+x}{(1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { \frac {x - 1}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (x + 1\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {-1+x}{(1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {-1+x}{(1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int \frac {x-1}{\left (x+1\right )\,\sqrt {a\,x^4+b\,x^3+c\,x^2+b\,x+a}} \,d x \]
[In]
[Out]