Optimal. Leaf size=58 \[ \frac{a A \sin (c+d x)}{d}+\frac{a A \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} a x (A+2 C)+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.126799, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {4075, 4047, 8, 4045, 3770} \[ \frac{a A \sin (c+d x)}{d}+\frac{a A \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} a x (A+2 C)+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 4075
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+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-a (A+2 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-2 a C \sec ^2(c+d x)\right ) \, dx+\frac{1}{2} (a (A+2 C)) \int 1 \, dx\\ &=\frac{1}{2} a (A+2 C) x+\frac{a A \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 C) x+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \sin (c+d x)}{d}+\frac{a A \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0808933, size = 52, normalized size = 0.9 \[ \frac{a \left (4 A \sin (c+d x)+A \sin (2 (c+d x))+2 A c+2 A d x+4 C \tanh ^{-1}(\sin (c+d x))+4 C d x\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.081, size = 77, normalized size = 1.3 \begin{align*}{\frac{Aa\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{aAx}{2}}+{\frac{Aac}{2\,d}}+aCx+{\frac{Cac}{d}}+{\frac{Aa\sin \left ( dx+c \right ) }{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 = 0.930492, size = 95, normalized size = 1.64 \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 )} 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 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.519841, size = 167, normalized size = 2.88 \begin{align*} \frac{{\left (A + 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 \, A 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 [A] time = 1.29866, size = 134, normalized size = 2.31 \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 \, 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} + 3 \, A 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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