Integrand size = 29, antiderivative size = 62 \[ \int \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx=-\frac {2 \arctan \left (\frac {a x}{1+3 x+3 x^2+x^3+\sqrt {1+6 x+\left (15-a^2\right ) x^2+20 x^3+15 x^4+6 x^5+x^6}}\right )}{a} \]
[Out]
\[ \int \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx=\int \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2}{\sqrt {-a^2 x^2+(1+x)^6}}-\frac {3}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}}\right ) \, dx \\ & = 2 \int \frac {1}{\sqrt {-a^2 x^2+(1+x)^6}} \, dx-3 \int \frac {1}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx=-\frac {2 \arctan \left (\frac {a x}{1+3 x+3 x^2+x^3+\sqrt {1+6 x+\left (15-a^2\right ) x^2+20 x^3+15 x^4+6 x^5+x^6}}\right )}{a} \]
[In]
[Out]
Time = 1.56 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.15
method | result | size |
pseudoelliptic | \(-\frac {\ln \left (2\right )+\ln \left (\frac {-a^{2} x +\sqrt {-a^{2}}\, \sqrt {-\left (-x^{3}-3 x^{2}+\left (-3+a \right ) x -1\right ) \left (x^{3}+3 x^{2}+\left (3+a \right ) x +1\right )}}{\left (1+x \right )^{3}}\right )}{\sqrt {-a^{2}}}\) | \(71\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.76 \[ \int \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx=\frac {\arctan \left (\frac {\sqrt {x^{6} + 6 \, x^{5} + 15 \, x^{4} - {\left (a^{2} - 15\right )} x^{2} + 20 \, x^{3} + 6 \, x + 1}}{a x}\right )}{a} \]
[In]
[Out]
\[ \int \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx=\int \frac {2 x - 1}{\sqrt {\left (- a x + x^{3} + 3 x^{2} + 3 x + 1\right ) \left (a x + x^{3} + 3 x^{2} + 3 x + 1\right )} \left (x + 1\right )}\, dx \]
[In]
[Out]
\[ \int \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx=\int { \frac {2 \, x - 1}{\sqrt {{\left (x + 1\right )}^{6} - a^{2} x^{2}} {\left (x + 1\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx=\int { \frac {2 \, x - 1}{\sqrt {{\left (x + 1\right )}^{6} - a^{2} x^{2}} {\left (x + 1\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx=\int \frac {2\,x-1}{\sqrt {{\left (x+1\right )}^6-a^2\,x^2}\,\left (x+1\right )} \,d x \]
[In]
[Out]