Optimal. Leaf size=32 \[ \frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+a x (B+C)+\frac{a C \sin (c+d x)}{d} \]
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Rubi [A] time = 0.145278, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {3029, 2968, 3023, 2735, 3770} \[ \frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+a x (B+C)+\frac{a C \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2968
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=\int (a+a \cos (c+d x)) (B+C \cos (c+d x)) \sec (c+d x) \, dx\\ &=\int \left (a B+(a B+a C) \cos (c+d x)+a C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{a C \sin (c+d x)}{d}+\int (a B+a (B+C) \cos (c+d x)) \sec (c+d x) \, dx\\ &=a (B+C) x+\frac{a C \sin (c+d x)}{d}+(a B) \int \sec (c+d x) \, dx\\ &=a (B+C) x+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0245105, size = 46, normalized size = 1.44 \[ \frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+a B x+\frac{a C \sin (c) \cos (d x)}{d}+\frac{a C \cos (c) \sin (d x)}{d}+a C x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 56, normalized size = 1.8 \begin{align*} Bax+aCx+{\frac{Ba\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{Bac}{d}}+{\frac{aC\sin \left ( dx+c \right ) }{d}}+{\frac{Cac}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.2076, size = 78, normalized size = 2.44 \begin{align*} \frac{2 \,{\left (d x + c\right )} B a + 2 \,{\left (d x + c\right )} C a + B a{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, C a \sin \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71491, size = 139, normalized size = 4.34 \begin{align*} \frac{2 \,{\left (B + C\right )} a d x + B a \log \left (\sin \left (d x + c\right ) + 1\right ) - B a \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, C a \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.36079, size = 107, normalized size = 3.34 \begin{align*} \frac{B a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - B a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) +{\left (B a + C a\right )}{\left (d x + c\right )} + \frac{2 \, C a \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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