Optimal. Leaf size=16 \[ a x-\frac{a \tan (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0126255, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3767, 8} \[ a x-\frac{a \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \left (a-a \sec ^2(c+d x)\right ) \, dx &=a x-a \int \sec ^2(c+d x) \, dx\\ &=a x+\frac{a \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=a x-\frac{a \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0062625, size = 26, normalized size = 1.62 \[ -a \left (\frac{\tan (c+d x)}{d}-\frac{\tan ^{-1}(\tan (c+d x))}{d}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.014, size = 17, normalized size = 1.1 \begin{align*} ax-{\frac{a\tan \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.05065, size = 22, normalized size = 1.38 \begin{align*} a x - \frac{a \tan \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.465543, size = 76, normalized size = 4.75 \begin{align*} \frac{a d x \cos \left (d x + c\right ) - a \sin \left (d x + c\right )}{d \cos \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*} - a \left (\int \left (-1\right )\, dx + \int \sec ^{2}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.31491, size = 22, normalized size = 1.38 \begin{align*} a x - \frac{a \tan \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]