Optimal. Leaf size=34 \[ \frac{a^2 \sin (c+d x)}{d}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+2 a^2 x \]
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Rubi [A] time = 0.0575523, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2746, 2735, 3770} \[ \frac{a^2 \sin (c+d x)}{d}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+2 a^2 x \]
Antiderivative was successfully verified.
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Rule 2746
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^2 \sec (c+d x) \, dx &=\frac{a^2 \sin (c+d x)}{d}+\int \left (a^2+2 a^2 \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=2 a^2 x+\frac{a^2 \sin (c+d x)}{d}+a^2 \int \sec (c+d x) \, dx\\ &=2 a^2 x+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0115677, size = 47, normalized size = 1.38 \[ \frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \sin (c) \cos (d x)}{d}+\frac{a^2 \cos (c) \sin (d x)}{d}+2 a^2 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 51, normalized size = 1.5 \begin{align*} 2\,{a}^{2}x+{\frac{{a}^{2}\sin \left ( dx+c \right ) }{d}}+{\frac{{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{{a}^{2}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1559, size = 58, normalized size = 1.71 \begin{align*} \frac{2 \,{\left (d x + c\right )} a^{2} + a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + a^{2} \sin \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 = 1.64752, size = 131, normalized size = 3.85 \begin{align*} \frac{4 \, a^{2} d x + a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - a^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, a^{2} \sin \left (d x + c\right )}{2 \, d} \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*} a^{2} \left (\int 2 \cos{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int \cos ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int \sec{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.38499, size = 107, normalized size = 3.15 \begin{align*} \frac{2 \,{\left (d x + c\right )} a^{2} + a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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