Optimal. Leaf size=33 \[ \frac{a^2 \tan (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+b^2 x \]
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Rubi [A] time = 0.0656476, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2789, 3770, 3012, 8} \[ \frac{a^2 \tan (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+b^2 x \]
Antiderivative was successfully verified.
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Rule 2789
Rule 3770
Rule 3012
Rule 8
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^2 \sec ^2(c+d x) \, dx &=(2 a b) \int \sec (c+d x) \, dx+\int \left (a^2+b^2 \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x)}{d}+b^2 \int 1 \, dx\\ &=b^2 x+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0845777, size = 32, normalized size = 0.97 \[ \frac{a^2 \tan (c+d x)+2 a b \tanh ^{-1}(\sin (c+d x))+b^2 d x}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 49, normalized size = 1.5 \begin{align*}{b}^{2}x+2\,{\frac{ab\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{2}\tan \left ( dx+c \right ) }{d}}+{\frac{{b}^{2}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975303, size = 65, normalized size = 1.97 \begin{align*} \frac{{\left (d x + c\right )} b^{2} + a b{\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 = 1.96573, size = 193, normalized size = 5.85 \begin{align*} \frac{b^{2} d x \cos \left (d x + c\right ) + a b \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - a b \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*} \int \left (a + b \cos{\left (c + d x \right )}\right )^{2} \sec ^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.57318, size = 104, normalized size = 3.15 \begin{align*} \frac{{\left (d x + c\right )} b^{2} + 2 \, a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, a b \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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