Optimal. Leaf size=34 \[ \frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x \]
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Rubi [A] time = 0.0583988, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2757, 3770, 3767, 8} \[ \frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x \]
Antiderivative was successfully verified.
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Rule 2757
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^2 \sec ^2(c+d x) \, dx &=\int \left (a^2+2 a^2 \sec (c+d x)+a^2 \sec ^2(c+d x)\right ) \, dx\\ &=a^2 x+a^2 \int \sec ^2(c+d x) \, dx+\left (2 a^2\right ) \int \sec (c+d x) \, dx\\ &=a^2 x+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^2 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=a^2 x+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0122583, size = 28, normalized size = 0.82 \[ a^2 \left (\frac{\tan (c+d x)}{d}+\frac{2 \tanh ^{-1}(\sin (c+d x))}{d}+x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 50, normalized size = 1.5 \begin{align*}{a}^{2}x+2\,{\frac{{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{2}\tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{2}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07548, size = 66, normalized size = 1.94 \begin{align*} \frac{{\left (d x + c\right )} a^{2} + a^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + a^{2} \tan \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.65433, size = 193, normalized size = 5.68 \begin{align*} \frac{a^{2} d x \cos \left (d x + c\right ) + a^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - a^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + a^{2} \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.
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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 ^{2}{\left (c + d x \right )}\, dx + \int \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sec ^{2}{\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 = 1.33591, size = 107, normalized size = 3.15 \begin{align*} \frac{{\left (d x + c\right )} a^{2} + 2 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, 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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