Integrand size = 14, antiderivative size = 92 \[ \int \frac {1}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {x}{a^2}-\frac {\sqrt {b} (3 a+2 b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{2 a^2 (a+b)^{3/2} f}-\frac {b \tan (e+f x)}{2 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4213, 425, 536, 209, 211} \[ \int \frac {1}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=-\frac {\sqrt {b} (3 a+2 b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{2 a^2 f (a+b)^{3/2}}+\frac {x}{a^2}-\frac {b \tan (e+f x)}{2 a f (a+b) \left (a+b \tan ^2(e+f x)+b\right )} \]
[In]
[Out]
Rule 209
Rule 211
Rule 425
Rule 536
Rule 4213
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {b \tan (e+f x)}{2 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\text {Subst}\left (\int \frac {2 a+b-b x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{2 a (a+b) f} \\ & = -\frac {b \tan (e+f x)}{2 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{a^2 f}-\frac {(b (3 a+2 b)) \text {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{2 a^2 (a+b) f} \\ & = \frac {x}{a^2}-\frac {\sqrt {b} (3 a+2 b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{2 a^2 (a+b)^{3/2} f}-\frac {b \tan (e+f x)}{2 a (a+b) f \left (a+b+b \tan ^2(e+f x)\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.68 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.61 \[ \int \frac {1}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {(a+2 b+a \cos (2 (e+f x))) \sec ^4(e+f x) \left (2 x (a+2 b+a \cos (2 (e+f x)))+\frac {b (3 a+2 b) \arctan \left (\frac {\sec (f x) (\cos (2 e)-i \sin (2 e)) (-((a+2 b) \sin (f x))+a \sin (2 e+f x))}{2 \sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4}}\right ) (a+2 b+a \cos (2 (e+f x))) (\cos (2 e)-i \sin (2 e))}{(a+b)^{3/2} f \sqrt {b (\cos (e)-i \sin (e))^4}}+\frac {b ((a+2 b) \sin (2 e)-a \sin (2 f x))}{(a+b) f (\cos (e)-\sin (e)) (\cos (e)+\sin (e))}\right )}{8 a^2 \left (a+b \sec ^2(e+f x)\right )^2} \]
[In]
[Out]
Time = 0.41 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {-\frac {b \left (\frac {a \tan \left (f x +e \right )}{2 \left (a +b \right ) \left (a +b +b \tan \left (f x +e \right )^{2}\right )}+\frac {\left (3 a +2 b \right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{2 \left (a +b \right ) \sqrt {\left (a +b \right ) b}}\right )}{a^{2}}+\frac {\arctan \left (\tan \left (f x +e \right )\right )}{a^{2}}}{f}\) | \(90\) |
default | \(\frac {-\frac {b \left (\frac {a \tan \left (f x +e \right )}{2 \left (a +b \right ) \left (a +b +b \tan \left (f x +e \right )^{2}\right )}+\frac {\left (3 a +2 b \right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{2 \left (a +b \right ) \sqrt {\left (a +b \right ) b}}\right )}{a^{2}}+\frac {\arctan \left (\tan \left (f x +e \right )\right )}{a^{2}}}{f}\) | \(90\) |
risch | \(\frac {x}{a^{2}}-\frac {i b \left (a \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )}{a^{2} \left (a +b \right ) f \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+4 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )}+\frac {3 \sqrt {-\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-\left (a +b \right ) b}+a +2 b}{a}\right )}{4 \left (a +b \right )^{2} f a}+\frac {\sqrt {-\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-\left (a +b \right ) b}+a +2 b}{a}\right ) b}{2 \left (a +b \right )^{2} f \,a^{2}}-\frac {3 \sqrt {-\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-\left (a +b \right ) b}-a -2 b}{a}\right )}{4 \left (a +b \right )^{2} f a}-\frac {\sqrt {-\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-\left (a +b \right ) b}-a -2 b}{a}\right ) b}{2 \left (a +b \right )^{2} f \,a^{2}}\) | \(302\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (80) = 160\).
Time = 0.31 (sec) , antiderivative size = 435, normalized size of antiderivative = 4.73 \[ \int \frac {1}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\left [\frac {8 \, {\left (a^{2} + a b\right )} f x \cos \left (f x + e\right )^{2} - 4 \, a b \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 8 \, {\left (a b + b^{2}\right )} f x + {\left ({\left (3 \, a^{2} + 2 \, a b\right )} \cos \left (f x + e\right )^{2} + 3 \, a b + 2 \, b^{2}\right )} \sqrt {-\frac {b}{a + b}} \log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (a b + b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {b}{a + b}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right )}{8 \, {\left ({\left (a^{4} + a^{3} b\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{3} b + a^{2} b^{2}\right )} f\right )}}, \frac {4 \, {\left (a^{2} + a b\right )} f x \cos \left (f x + e\right )^{2} - 2 \, a b \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 4 \, {\left (a b + b^{2}\right )} f x + {\left ({\left (3 \, a^{2} + 2 \, a b\right )} \cos \left (f x + e\right )^{2} + 3 \, a b + 2 \, b^{2}\right )} \sqrt {\frac {b}{a + b}} \arctan \left (\frac {{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {b}{a + b}}}{2 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right )}{4 \, {\left ({\left (a^{4} + a^{3} b\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{3} b + a^{2} b^{2}\right )} f\right )}}\right ] \]
[In]
[Out]
\[ \int \frac {1}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\int \frac {1}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2}}\, dx \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=-\frac {\frac {b \tan \left (f x + e\right )}{a^{3} + 2 \, a^{2} b + a b^{2} + {\left (a^{2} b + a b^{2}\right )} \tan \left (f x + e\right )^{2}} + \frac {{\left (3 \, a b + 2 \, b^{2}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{{\left (a^{3} + a^{2} b\right )} \sqrt {{\left (a + b\right )} b}} - \frac {2 \, {\left (f x + e\right )}}{a^{2}}}{2 \, f} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.24 \[ \int \frac {1}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=-\frac {\frac {{\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right )\right )} {\left (3 \, a b + 2 \, b^{2}\right )}}{{\left (a^{3} + a^{2} b\right )} \sqrt {a b + b^{2}}} + \frac {b \tan \left (f x + e\right )}{{\left (b \tan \left (f x + e\right )^{2} + a + b\right )} {\left (a^{2} + a b\right )}} - \frac {2 \, {\left (f x + e\right )}}{a^{2}}}{2 \, f} \]
[In]
[Out]
Time = 20.41 (sec) , antiderivative size = 2056, normalized size of antiderivative = 22.35 \[ \int \frac {1}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\text {Too large to display} \]
[In]
[Out]