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 = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.00, 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 8
Rule 12
Rule 2633
Rule 2635
Rule 4047
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*}
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Mathematica [A] time = 0.12, 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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fricas [A] time = 0.43, size = 64, normalized size = 0.70 \[ \frac {45 \, C d x + {\left (24 \, B \cos \left (d x + c\right )^{4} + 30 \, C \cos \left (d x + c\right )^{3} + 32 \, B \cos \left (d x + c\right )^{2} + 45 \, C \cos \left (d x + c\right ) + 64 \, B\right )} \sin \left (d x + c\right )}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 154, normalized size = 1.67 \[ \frac {45 \, {\left (d x + c\right )} C + \frac {2 \, {\left (120 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 75 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 160 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 30 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 464 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 160 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 75 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.90, size = 70, normalized size = 0.76 \[ \frac {\frac {B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 69, normalized size = 0.75 \[ \frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.31, size = 113, normalized size = 1.23 \[ \frac {3\,C\,x}{8}+\frac {\left (2\,B-\frac {5\,C}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {8\,B}{3}-\frac {C}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {116\,B\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15}+\left (\frac {8\,B}{3}+\frac {C}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,B+\frac {5\,C}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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