Optimal. Leaf size=19 \[ \frac{\cot (c+d x)}{a d}+\frac{x}{a} \]
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Rubi [A] time = 0.0214506, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4120, 3473, 8} \[ \frac{\cot (c+d x)}{a d}+\frac{x}{a} \]
Antiderivative was successfully verified.
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Rule 4120
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{a-a \sec ^2(c+d x)} \, dx &=-\frac{\int \cot ^2(c+d x) \, dx}{a}\\ &=\frac{\cot (c+d x)}{a d}+\frac{\int 1 \, dx}{a}\\ &=\frac{x}{a}+\frac{\cot (c+d x)}{a d}\\ \end{align*}
Mathematica [C] time = 0.0279704, size = 31, normalized size = 1.63 \[ \frac{\cot (c+d x) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\tan ^2(c+d x)\right )}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 31, normalized size = 1.6 \begin{align*}{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) }{ad}}+{\frac{1}{ad\tan \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52573, size = 35, normalized size = 1.84 \begin{align*} \frac{\frac{d x + c}{a} + \frac{1}{a \tan \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.463929, size = 73, normalized size = 3.84 \begin{align*} \frac{d x \sin \left (d x + c\right ) + \cos \left (d x + c\right )}{a d \sin \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{1}{\sec ^{2}{\left (c + d x \right )} - 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.29523, size = 61, normalized size = 3.21 \begin{align*} \frac{\frac{2 \,{\left (d x + c\right )}}{a} - \frac{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a} + \frac{1}{a \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.
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