Integrand size = 43, antiderivative size = 236 \[ \int \frac {1+a k x+k x^2}{\left (-1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\frac {\arctan \left (\frac {\left (1-2 i \sqrt {k}-k\right ) x}{1+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{\left (-1+\sqrt {k}\right ) \left (1+\sqrt {k}\right )}+\frac {\arctan \left (\frac {\left (1+2 i \sqrt {k}-k\right ) x}{1+k x^2+\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{\left (-1+\sqrt {k}\right ) \left (1+\sqrt {k}\right )}+\frac {a \sqrt {k} \arctan \left (\frac {\left (2 \sqrt {k}-2 k^{3/2}\right ) x^2}{1-2 k x^2+k^2 x^4+\left (1+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{2 \left (-1+\sqrt {k}\right ) \left (1+\sqrt {k}\right )} \]
[Out]
Time = 0.20 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.44, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {1976, 1701, 1712, 210, 12, 1261, 738} \[ \int \frac {1+a k x+k x^2}{\left (-1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=-\frac {a \sqrt {k} \arctan \left (\frac {(1-k) \left (k x^2+1\right )}{2 \sqrt {k} \sqrt {k^2 x^4-\left (k^2+1\right ) x^2+1}}\right )}{2 (1-k)}-\frac {\arctan \left (\frac {(1-k) x}{\sqrt {k^2 x^4+\left (-k^2-1\right ) x^2+1}}\right )}{1-k} \]
[In]
[Out]
Rule 12
Rule 210
Rule 738
Rule 1261
Rule 1701
Rule 1712
Rule 1976
Rubi steps \begin{align*} \text {integral}& = \int \frac {1+a k x+k x^2}{\left (-1+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \int \frac {a k x}{\left (-1+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+\int \frac {1+k x^2}{\left (-1+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = (a k) \int \frac {x}{\left (-1+k x^2\right ) \sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx+\text {Subst}\left (\int \frac {1}{-1-\left (1-2 k+k^2\right ) x^2} \, dx,x,\frac {x}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \\ & = -\frac {\arctan \left (\frac {(1-k) x}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{1-k}+\frac {1}{2} (a k) \text {Subst}\left (\int \frac {1}{(-1+k x) \sqrt {1+\left (-1-k^2\right ) x+k^2 x^2}} \, dx,x,x^2\right ) \\ & = -\frac {\arctan \left (\frac {(1-k) x}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{1-k}-(a k) \text {Subst}\left (\int \frac {1}{8 k^2+4 k \left (-1-k^2\right )-x^2} \, dx,x,\frac {-1+2 k-k^2-(1-k)^2 k x^2}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right ) \\ & = -\frac {\arctan \left (\frac {(1-k) x}{\sqrt {1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{1-k}-\frac {a \sqrt {k} \arctan \left (\frac {(1-k) \left (1+k x^2\right )}{2 \sqrt {k} \sqrt {1-\left (1+k^2\right ) x^2+k^2 x^4}}\right )}{2 (1-k)} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 3.98 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.67 \[ \int \frac {1+a k x+k x^2}{\left (-1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\frac {-a \sqrt {k} \sqrt {-1+x^2} \sqrt {-1+k^2 x^2} \arctan \left (\frac {\sqrt {-1+k^2 x^2}}{\sqrt {k} \sqrt {-1+x^2}}\right )+(-1+k) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \operatorname {EllipticF}\left (\arcsin (x),k^2\right )-2 (-1+k) \sqrt {1-x^2} \sqrt {1-k^2 x^2} \operatorname {EllipticPi}\left (k,\arcsin (x),k^2\right )}{(-1+k) \sqrt {\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}} \]
[In]
[Out]
Time = 2.21 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.72
method | result | size |
elliptic | \(-\frac {a \ln \left (\frac {-\frac {2 \left (k^{2}-2 k +1\right )}{k}+\left (-k^{2}+2 k -1\right ) \left (x^{2}-\frac {1}{k}\right )+2 \sqrt {-\frac {k^{2}-2 k +1}{k}}\, \sqrt {k^{2} \left (x^{2}-\frac {1}{k}\right )^{2}+\left (-k^{2}+2 k -1\right ) \left (x^{2}-\frac {1}{k}\right )-\frac {k^{2}-2 k +1}{k}}}{x^{2}-\frac {1}{k}}\right )}{2 \sqrt {-\frac {k^{2}-2 k +1}{k}}}+\frac {\arctan \left (\frac {\sqrt {\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )}}{x \left (-1+k \right )}\right )}{-1+k}\) | \(171\) |
default | \(\frac {\left (a k -2 \sqrt {k}\right ) \ln \left (\frac {\sqrt {-\left (-1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}-2 k^{\frac {3}{2}} x^{2}-2 \sqrt {k}+\left (-k^{2}-2 k -1\right ) x}{1+2 \sqrt {k}\, x +k \,x^{2}}\right )+\left (-a k -2 \sqrt {k}\right ) \ln \left (\frac {\sqrt {-\left (-1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}+2 k^{\frac {3}{2}} x^{2}+2 \sqrt {k}+\left (-k^{2}-2 k -1\right ) x}{1-2 \sqrt {k}\, x +k \,x^{2}}\right )-4 \sqrt {k}\, \ln \left (2\right )}{4 \sqrt {-\left (-1+k \right )^{2}}\, \sqrt {k}}\) | \(184\) |
pseudoelliptic | \(\frac {\left (a k -2 \sqrt {k}\right ) \ln \left (\frac {\sqrt {-\left (-1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}-2 k^{\frac {3}{2}} x^{2}-2 \sqrt {k}+\left (-k^{2}-2 k -1\right ) x}{1+2 \sqrt {k}\, x +k \,x^{2}}\right )+\left (-a k -2 \sqrt {k}\right ) \ln \left (\frac {\sqrt {-\left (-1+k \right )^{2}}\, \sqrt {\left (x^{2}-1\right ) \left (k^{2} x^{2}-1\right )}+2 k^{\frac {3}{2}} x^{2}+2 \sqrt {k}+\left (-k^{2}-2 k -1\right ) x}{1-2 \sqrt {k}\, x +k \,x^{2}}\right )-4 \sqrt {k}\, \ln \left (2\right )}{4 \sqrt {-\left (-1+k \right )^{2}}\, \sqrt {k}}\) | \(184\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1793 vs. \(2 (196) = 392\).
Time = 1.13 (sec) , antiderivative size = 1793, normalized size of antiderivative = 7.60 \[ \int \frac {1+a k x+k x^2}{\left (-1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {1+a k x+k x^2}{\left (-1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int \frac {a k x + k x^{2} + 1}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (k x - 1\right ) \left (k x + 1\right )} \left (k x^{2} - 1\right )}\, dx \]
[In]
[Out]
\[ \int \frac {1+a k x+k x^2}{\left (-1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int { \frac {a k x + k x^{2} + 1}{{\left (k x^{2} - 1\right )} \sqrt {{\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1+a k x+k x^2}{\left (-1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int { \frac {a k x + k x^{2} + 1}{{\left (k x^{2} - 1\right )} \sqrt {{\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1+a k x+k x^2}{\left (-1+k x^2\right ) \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}} \, dx=\int \frac {k\,x^2+a\,k\,x+1}{\left (k\,x^2-1\right )\,\sqrt {\left (x^2-1\right )\,\left (k^2\,x^2-1\right )}} \,d x \]
[In]
[Out]