Optimal. Leaf size=26 \[ -\frac{\cot (c+d x) (a+b \sec (c+d x))}{d}-a x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0264657, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3882, 8} \[ -\frac{\cot (c+d x) (a+b \sec (c+d x))}{d}-a x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3882
Rule 8
Rubi steps
\begin{align*} \int \cot ^2(c+d x) (a+b \sec (c+d x)) \, dx &=-\frac{\cot (c+d x) (a+b \sec (c+d x))}{d}-\int a \, dx\\ &=-a x-\frac{\cot (c+d x) (a+b \sec (c+d x))}{d}\\ \end{align*}
Mathematica [C] time = 0.0203991, size = 43, normalized size = 1.65 \[ -\frac{a \cot (c+d x) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\tan ^2(c+d x)\right )}{d}-\frac{b \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.035, size = 35, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ( a \left ( -\cot \left ( dx+c \right ) -dx-c \right ) -{\frac{b}{\sin \left ( dx+c \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.47307, size = 42, normalized size = 1.62 \begin{align*} -\frac{{\left (d x + c + \frac{1}{\tan \left (d x + c\right )}\right )} a + \frac{b}{\sin \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.807459, size = 82, normalized size = 3.15 \begin{align*} -\frac{a d x \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) + b}{d \sin \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sec{\left (c + d x \right )}\right ) \cot ^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15339, size = 70, normalized size = 2.69 \begin{align*} -\frac{2 \,{\left (d x + c\right )} a - a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{a + b}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]