Integrand size = 21, antiderivative size = 21 \[ \int \cot ^2(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx=-((a-b) x)-\frac {a \cot (e+f x)}{f} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3710, 8} \[ \int \cot ^2(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx=-(x (a-b))-\frac {a \cot (e+f x)}{f} \]
[In]
[Out]
Rule 8
Rule 3710
Rubi steps \begin{align*} \text {integral}& = -\frac {a \cot (e+f x)}{f}+\int (-a+b) \, dx \\ & = -((a-b) x)-\frac {a \cot (e+f x)}{f} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62 \[ \int \cot ^2(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx=b x-\frac {a \cot (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\tan ^2(e+f x)\right )}{f} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38
method | result | size |
risch | \(-a x +b x -\frac {2 i a}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}\) | \(29\) |
derivativedivides | \(\frac {-\frac {a}{\tan \left (f x +e \right )}+\left (-a +b \right ) \arctan \left (\tan \left (f x +e \right )\right )}{f}\) | \(30\) |
default | \(\frac {-\frac {a}{\tan \left (f x +e \right )}+\left (-a +b \right ) \arctan \left (\tan \left (f x +e \right )\right )}{f}\) | \(30\) |
norman | \(\frac {\left (-a +b \right ) x \tan \left (f x +e \right )-\frac {a}{f}}{\tan \left (f x +e \right )}\) | \(30\) |
parallelrisch | \(\frac {-\tan \left (f x +e \right ) f x \left (a -b \right )-a}{\tan \left (f x +e \right ) f}\) | \(32\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \cot ^2(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx=-\frac {{\left (a - b\right )} f x \tan \left (f x + e\right ) + a}{f \tan \left (f x + e\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (14) = 28\).
Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.00 \[ \int \cot ^2(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx=\begin {cases} \tilde {\infty } a x & \text {for}\: e = 0 \wedge f = 0 \\x \left (a + b \tan ^{2}{\left (e \right )}\right ) \cot ^{2}{\left (e \right )} & \text {for}\: f = 0 \\\tilde {\infty } a x & \text {for}\: e = - f x \\- a x - \frac {a}{f \tan {\left (e + f x \right )}} + b x & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \cot ^2(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx=-\frac {{\left (f x + e\right )} {\left (a - b\right )} + \frac {a}{\tan \left (f x + e\right )}}{f} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (21) = 42\).
Time = 0.52 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.05 \[ \int \cot ^2(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx=-\frac {2 \, {\left (f x + e\right )} {\left (a - b\right )} - a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + \frac {a}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}}{2 \, f} \]
[In]
[Out]
Time = 11.15 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \cot ^2(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx=-x\,\left (a-b\right )-\frac {a\,\mathrm {cot}\left (e+f\,x\right )}{f} \]
[In]
[Out]