Optimal. Leaf size=62 \[ \frac{a (A+B) \sin (c+d x)}{d}+\frac{1}{2} a x (A+2 (B+C))+\frac{a A \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.149771, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.128, Rules used = {4074, 4047, 8, 4045, 3770} \[ \frac{a (A+B) \sin (c+d x)}{d}+\frac{1}{2} a x (A+2 (B+C))+\frac{a A \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 4074
Rule 4047
Rule 8
Rule 4045
Rule 3770
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{a A \cos (c+d x) \sin (c+d x)}{2 d}-\frac{1}{2} \int \cos (c+d x) \left (-2 a (A+B)-a (A+2 (B+C)) \sec (c+d x)-2 a C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a A \cos (c+d x) \sin (c+d x)}{2 d}-\frac{1}{2} \int \cos (c+d x) \left (-2 a (A+B)-2 a C \sec ^2(c+d x)\right ) \, dx+\frac{1}{2} (a (A+2 (B+C))) \int 1 \, dx\\ &=\frac{1}{2} a (A+2 (B+C)) x+\frac{a (A+B) \sin (c+d x)}{d}+\frac{a A \cos (c+d x) \sin (c+d x)}{2 d}+(a C) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} a (A+2 (B+C)) x+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a (A+B) \sin (c+d x)}{d}+\frac{a A \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.141427, size = 59, normalized size = 0.95 \[ \frac{a \left (4 (A+B) \sin (c+d x)+A \sin (2 (c+d x))+2 A c+2 A d x+4 B d x+4 C \tanh ^{-1}(\sin (c+d x))+4 C d x\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.09, size = 100, normalized size = 1.6 \begin{align*}{\frac{Aa\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{aAx}{2}}+{\frac{Aac}{2\,d}}+{\frac{Ba\sin \left ( dx+c \right ) }{d}}+aCx+{\frac{Cac}{d}}+{\frac{Aa\sin \left ( dx+c \right ) }{d}}+aBx+{\frac{Bac}{d}}+{\frac{aC\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.939819, size = 120, normalized size = 1.94 \begin{align*} \frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a + 4 \,{\left (d x + c\right )} B a + 4 \,{\left (d x + c\right )} C a + 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 \sin \left (d x + c\right ) + 4 \, B a \sin \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 = 0.519659, size = 184, normalized size = 2.97 \begin{align*} \frac{{\left (A + 2 \, B + 2 \, C\right )} a d x + C a \log \left (\sin \left (d x + c\right ) + 1\right ) - C a \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (A a \cos \left (d x + c\right ) + 2 \,{\left (A + B\right )} a\right )} \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 = 1.24857, size = 177, normalized size = 2.85 \begin{align*} \frac{2 \, C a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, C a \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 )}{\left (d x + c\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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