Optimal. Leaf size=62 \[ \frac{a (A+2 (B+C)) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (A+B) \tan (c+d x)}{d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+a C x \]
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Rubi [A] time = 0.157749, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {3031, 3021, 2735, 3770} \[ \frac{a (A+2 (B+C)) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (A+B) \tan (c+d x)}{d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+a C x \]
Antiderivative was successfully verified.
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Rule 3031
Rule 3021
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=\frac{a A \sec (c+d x) \tan (c+d x)}{2 d}-\frac{1}{2} \int \left (-2 a (A+B)-a (A+2 (B+C)) \cos (c+d x)-2 a C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac{a (A+B) \tan (c+d x)}{d}+\frac{a A \sec (c+d x) \tan (c+d x)}{2 d}-\frac{1}{2} \int (-a (A+2 (B+C))-2 a C \cos (c+d x)) \sec (c+d x) \, dx\\ &=a C x+\frac{a (A+B) \tan (c+d x)}{d}+\frac{a A \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} (a (A+2 (B+C))) \int \sec (c+d x) \, dx\\ &=a C x+\frac{a (A+2 (B+C)) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (A+B) \tan (c+d x)}{d}+\frac{a A \sec (c+d x) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0344218, size = 92, normalized size = 1.48 \[ \frac{a A \tan (c+d x)}{d}+\frac{a A \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a B \tan (c+d x)}{d}+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}+a C x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 117, normalized size = 1.9 \begin{align*}{\frac{aA\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{aA\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{Ba\tan \left ( dx+c \right ) }{d}}+{\frac{aC\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{aA\tan \left ( dx+c \right ) }{d}}+{\frac{Ba\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+aCx+{\frac{Cac}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01262, size = 176, normalized size = 2.84 \begin{align*} \frac{4 \,{\left (d x + c\right )} C a - A a{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, 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{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, A a \tan \left (d x + c\right ) + 4 \, B a \tan \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03274, size = 292, normalized size = 4.71 \begin{align*} \frac{4 \, C a d x \cos \left (d x + c\right )^{2} +{\left (A + 2 \, B + 2 \, C\right )} a \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (A + 2 \, B + 2 \, C\right )} a \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (2 \,{\left (A + B\right )} a \cos \left (d x + c\right ) + A a\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \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 = 1.30916, size = 190, normalized size = 3.06 \begin{align*} \frac{2 \,{\left (d x + c\right )} C a +{\left (A a + 2 \, B a + 2 \, C a\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) -{\left (A a + 2 \, B a + 2 \, C a\right )} \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} + 2 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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