Optimal. Leaf size=30 \[ \frac{(A+C) \sin (c+d x)}{d}-\frac{A \sin ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.0473099, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {4044, 3013} \[ \frac{(A+C) \sin (c+d x)}{d}-\frac{A \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 4044
Rule 3013
Rubi steps
\begin{align*} \int \cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\int \cos (c+d x) \left (C+A \cos ^2(c+d x)\right ) \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \left (A+C-A x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac{(A+C) \sin (c+d x)}{d}-\frac{A \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0180238, size = 50, normalized size = 1.67 \[ -\frac{A \sin ^3(c+d x)}{3 d}+\frac{A \sin (c+d x)}{d}+\frac{C \sin (c) \cos (d x)}{d}+\frac{C \cos (c) \sin (d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 33, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{A \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+C\sin \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.921315, size = 36, normalized size = 1.2 \begin{align*} -\frac{A \sin \left (d x + c\right )^{3} - 3 \,{\left (A + C\right )} \sin \left (d x + c\right )}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.471226, size = 69, normalized size = 2.3 \begin{align*} \frac{{\left (A \cos \left (d x + c\right )^{2} + 2 \, A + 3 \, C\right )} \sin \left (d x + c\right )}{3 \, 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 = 1.12817, size = 46, normalized size = 1.53 \begin{align*} -\frac{A \sin \left (d x + c\right )^{3} - 3 \, A \sin \left (d x + c\right ) - 3 \, C \sin \left (d x + c\right )}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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