Optimal. Leaf size=92 \[ \frac{B \sin ^5(c+d x)}{5 d}-\frac{2 B \sin ^3(c+d x)}{3 d}+\frac{B \sin (c+d x)}{d}+\frac{C \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 C \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 C x}{8} \]
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Rubi [A] time = 0.0756441, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{B \sin ^5(c+d x)}{5 d}-\frac{2 B \sin ^3(c+d x)}{3 d}+\frac{B \sin (c+d x)}{d}+\frac{C \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 C \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 C x}{8} \]
Antiderivative was successfully verified.
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Rule 4047
Rule 2633
Rule 12
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^6(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=B \int \cos ^5(c+d x) \, dx+\int C \cos ^4(c+d x) \, dx\\ &=C \int \cos ^4(c+d x) \, dx-\frac{B \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac{B \sin (c+d x)}{d}+\frac{C \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{2 B \sin ^3(c+d x)}{3 d}+\frac{B \sin ^5(c+d x)}{5 d}+\frac{1}{4} (3 C) \int \cos ^2(c+d x) \, dx\\ &=\frac{B \sin (c+d x)}{d}+\frac{3 C \cos (c+d x) \sin (c+d x)}{8 d}+\frac{C \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{2 B \sin ^3(c+d x)}{3 d}+\frac{B \sin ^5(c+d x)}{5 d}+\frac{1}{8} (3 C) \int 1 \, dx\\ &=\frac{3 C x}{8}+\frac{B \sin (c+d x)}{d}+\frac{3 C \cos (c+d x) \sin (c+d x)}{8 d}+\frac{C \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{2 B \sin ^3(c+d x)}{3 d}+\frac{B \sin ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.10447, size = 89, normalized size = 0.97 \[ \frac{B \sin ^5(c+d x)}{5 d}-\frac{2 B \sin ^3(c+d x)}{3 d}+\frac{B \sin (c+d x)}{d}+\frac{3 C (c+d x)}{8 d}+\frac{C \sin (2 (c+d x))}{4 d}+\frac{C \sin (4 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.108, size = 70, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({\frac{B\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+C \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 ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.937335, size = 93, normalized size = 1.01 \begin{align*} \frac{32 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B + 15 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.494508, size = 173, normalized size = 1.88 \begin{align*} \frac{45 \, C d x +{\left (24 \, B \cos \left (d x + c\right )^{4} + 30 \, C \cos \left (d x + c\right )^{3} + 32 \, B \cos \left (d x + c\right )^{2} + 45 \, C \cos \left (d x + c\right ) + 64 \, B\right )} \sin \left (d x + c\right )}{120 \, 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 = 1.16142, size = 208, normalized size = 2.26 \begin{align*} \frac{45 \,{\left (d x + c\right )} C + \frac{2 \,{\left (120 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 75 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 160 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 30 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 464 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 160 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 30 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 120 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 75 \, 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 )}^{5}}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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