Optimal. Leaf size=61 \[ \frac{(3 A+4 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{A \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{1}{8} x (3 A+4 C) \]
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Rubi [A] time = 0.0435039, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4045, 2635, 8} \[ \frac{(3 A+4 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{A \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{1}{8} x (3 A+4 C) \]
Antiderivative was successfully verified.
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Rule 4045
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^4(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{1}{4} (3 A+4 C) \int \cos ^2(c+d x) \, dx\\ &=\frac{(3 A+4 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{A \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{1}{8} (3 A+4 C) \int 1 \, dx\\ &=\frac{1}{8} (3 A+4 C) x+\frac{(3 A+4 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{A \cos ^3(c+d x) \sin (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.0865146, size = 45, normalized size = 0.74 \[ \frac{4 (3 A+4 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+A \sin (4 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 65, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( A \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +C \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40751, size = 99, normalized size = 1.62 \begin{align*} \frac{{\left (d x + c\right )}{\left (3 \, A + 4 \, C\right )} + \frac{{\left (3 \, A + 4 \, C\right )} \tan \left (d x + c\right )^{3} +{\left (5 \, A + 4 \, C\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.485769, size = 119, normalized size = 1.95 \begin{align*} \frac{{\left (3 \, A + 4 \, C\right )} d x +{\left (2 \, A \cos \left (d x + c\right )^{3} +{\left (3 \, A + 4 \, C\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, 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 = 1.16351, size = 99, normalized size = 1.62 \begin{align*} \frac{{\left (d x + c\right )}{\left (3 \, A + 4 \, C\right )} + \frac{3 \, A \tan \left (d x + c\right )^{3} + 4 \, C \tan \left (d x + c\right )^{3} + 5 \, A \tan \left (d x + c\right ) + 4 \, C \tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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