Optimal. Leaf size=54 \[ -\frac{B \sin ^3(c+d x)}{3 d}+\frac{B \sin (c+d x)}{d}+\frac{C \sin (c+d x) \cos (c+d x)}{2 d}+\frac{C x}{2} \]
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Rubi [A] time = 0.051129, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {4047, 2633, 12, 2635, 8} \[ -\frac{B \sin ^3(c+d x)}{3 d}+\frac{B \sin (c+d x)}{d}+\frac{C \sin (c+d x) \cos (c+d x)}{2 d}+\frac{C x}{2} \]
Antiderivative was successfully verified.
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Rule 4047
Rule 2633
Rule 12
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^4(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=B \int \cos ^3(c+d x) \, dx+\int C \cos ^2(c+d x) \, dx\\ &=C \int \cos ^2(c+d x) \, dx-\frac{B \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac{B \sin (c+d x)}{d}+\frac{C \cos (c+d x) \sin (c+d x)}{2 d}-\frac{B \sin ^3(c+d x)}{3 d}+\frac{1}{2} C \int 1 \, dx\\ &=\frac{C x}{2}+\frac{B \sin (c+d x)}{d}+\frac{C \cos (c+d x) \sin (c+d x)}{2 d}-\frac{B \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0624231, size = 57, normalized size = 1.06 \[ -\frac{B \sin ^3(c+d x)}{3 d}+\frac{B \sin (c+d x)}{d}+\frac{C (c+d x)}{2 d}+\frac{C \sin (2 (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 49, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({\frac{B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+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 = 0.936716, size = 62, normalized size = 1.15 \begin{align*} -\frac{4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B - 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.481587, size = 105, normalized size = 1.94 \begin{align*} \frac{3 \, C d x +{\left (2 \, B \cos \left (d x + c\right )^{2} + 3 \, C \cos \left (d x + c\right ) + 4 \, B\right )} \sin \left (d x + c\right )}{6 \, 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 = 1.22505, size = 132, normalized size = 2.44 \begin{align*} \frac{3 \,{\left (d x + c\right )} C + \frac{2 \,{\left (6 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 4 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, 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 )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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