Optimal. Leaf size=26 \[ -\frac {\cot (c+d x) (a \sec (c+d x)+a)}{d}-a x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, 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} \[ -\frac {\cot (c+d x) (a \sec (c+d x)+a)}{d}-a x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3882
Rubi steps
\begin {align*} \int \cot ^2(c+d x) (a+a \sec (c+d x)) \, dx &=-\frac {\cot (c+d x) (a+a \sec (c+d x))}{d}-\int a \, dx\\ &=-a x-\frac {\cot (c+d x) (a+a \sec (c+d x))}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.03, size = 43, normalized size = 1.65 \[ -\frac {a \cot (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(c+d x)\right )}{d}-\frac {a \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.94, size = 33, normalized size = 1.27 \[ -\frac {a d x \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) + a}{d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.23, size = 26, normalized size = 1.00 \[ -\frac {{\left (d x + c\right )} a + \frac {a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.48, size = 35, normalized size = 1.35 \[ \frac {a \left (-\cot \left (d x +c \right )-d x -c \right )-\frac {a}{\sin \left (d x +c \right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.22, size = 31, normalized size = 1.19 \[ -\frac {{\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} a + \frac {a}{\sin \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.07, size = 19, normalized size = 0.73 \[ -\frac {a\,\left (\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \cot ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \cot ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________