Optimal. Leaf size=37 \[ -\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{\cot (c+d x)}{a^2 d}+\frac{x}{a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0299385, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4120, 3473, 8} \[ -\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{\cot (c+d x)}{a^2 d}+\frac{x}{a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4120
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{\left (a-a \sec ^2(c+d x)\right )^2} \, dx &=\frac{\int \cot ^4(c+d x) \, dx}{a^2}\\ &=-\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{\int \cot ^2(c+d x) \, dx}{a^2}\\ &=\frac{\cot (c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{\int 1 \, dx}{a^2}\\ &=\frac{x}{a^2}+\frac{\cot (c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}\\ \end{align*}
Mathematica [C] time = 0.0267814, size = 36, normalized size = 0.97 \[ -\frac{\cot ^3(c+d x) \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},-\tan ^2(c+d x)\right )}{3 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.05, size = 47, normalized size = 1.3 \begin{align*}{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{2}}}-{\frac{1}{3\,d{a}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}}+{\frac{1}{d{a}^{2}\tan \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.5722, size = 54, normalized size = 1.46 \begin{align*} \frac{\frac{3 \,{\left (d x + c\right )}}{a^{2}} + \frac{3 \, \tan \left (d x + c\right )^{2} - 1}{a^{2} \tan \left (d x + c\right )^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.472582, size = 177, normalized size = 4.78 \begin{align*} \frac{4 \, \cos \left (d x + c\right )^{3} + 3 \,{\left (d x \cos \left (d x + c\right )^{2} - d x\right )} \sin \left (d x + c\right ) - 3 \, \cos \left (d x + c\right )}{3 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \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*} \frac{\int \frac{1}{\sec ^{4}{\left (c + d x \right )} - 2 \sec ^{2}{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.20012, size = 108, normalized size = 2.92 \begin{align*} \frac{\frac{24 \,{\left (d x + c\right )}}{a^{2}} + \frac{15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1}{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}} + \frac{a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{6}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]