Optimal. Leaf size=38 \[ \frac{B \sin (c+d x) \cos (c+d x)}{2 d}+\frac{B x}{2}+\frac{C \sin (c+d x)}{d} \]
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Rubi [A] time = 0.0422158, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {4047, 2635, 8, 12, 2637} \[ \frac{B \sin (c+d x) \cos (c+d x)}{2 d}+\frac{B x}{2}+\frac{C \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 4047
Rule 2635
Rule 8
Rule 12
Rule 2637
Rubi steps
\begin{align*} \int \cos ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=B \int \cos ^2(c+d x) \, dx+\int C \cos (c+d x) \, dx\\ &=\frac{B \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{2} B \int 1 \, dx+C \int \cos (c+d x) \, dx\\ &=\frac{B x}{2}+\frac{C \sin (c+d x)}{d}+\frac{B \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0622366, size = 35, normalized size = 0.92 \[ \frac{B (2 (c+d x)+\sin (2 (c+d x)))+4 C \sin (c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 38, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( B \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +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 = 0.926692, size = 46, normalized size = 1.21 \begin{align*} \frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B + 4 \, C \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 = 0.474546, size = 72, normalized size = 1.89 \begin{align*} \frac{B d x +{\left (B \cos \left (d x + c\right ) + 2 \, C\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 = 1.18198, size = 111, normalized size = 2.92 \begin{align*} \frac{{\left (d x + c\right )} B - \frac{2 \,{\left (B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, C \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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