Optimal. Leaf size=42 \[ \frac{a A \sin (c+d x)}{d}+a A x+\frac{a C \tan (c+d x)}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.100562, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {4077, 4047, 8, 4045, 3770} \[ \frac{a A \sin (c+d x)}{d}+a A x+\frac{a C \tan (c+d x)}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 4077
Rule 4047
Rule 8
Rule 4045
Rule 3770
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \sec (c+d x)) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{a C \tan (c+d x)}{d}+\int \cos (c+d x) \left (a A+a A \sec (c+d x)+a C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a C \tan (c+d x)}{d}+(a A) \int 1 \, dx+\int \cos (c+d x) \left (a A+a C \sec ^2(c+d x)\right ) \, dx\\ &=a A x+\frac{a A \sin (c+d x)}{d}+\frac{a C \tan (c+d x)}{d}+(a C) \int \sec (c+d x) \, dx\\ &=a A x+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \sin (c+d x)}{d}+\frac{a C \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.027497, size = 54, normalized size = 1.29 \[ \frac{a A \sin (c) \cos (d x)}{d}+\frac{a A \cos (c) \sin (d x)}{d}+a A x+\frac{a C \tan (c+d x)}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.071, size = 57, normalized size = 1.4 \begin{align*} aAx+{\frac{Aa\sin \left ( dx+c \right ) }{d}}+{\frac{Aac}{d}}+{\frac{aC\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{aC\tan \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.928543, size = 80, normalized size = 1.9 \begin{align*} \frac{2 \,{\left (d x + c\right )} A a + C a{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A a \sin \left (d x + c\right ) + 2 \, C a \tan \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.529703, size = 232, normalized size = 5.52 \begin{align*} \frac{2 \, A a d x \cos \left (d x + c\right ) + C a \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - C a \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (A a \cos \left (d x + c\right ) + C a\right )} \sin \left (d x + c\right )}{2 \, 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 \left (\int A \cos{\left (c + d x \right )}\, dx + \int A \cos{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int C \cos{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int C \cos{\left (c + d x \right )} \sec ^{3}{\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 = 1.27122, size = 161, normalized size = 3.83 \begin{align*} \frac{{\left (d x + c\right )} A a + C a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - C a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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