Optimal. Leaf size=132 \[ \frac {(5 A+6 C) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {(5 A+6 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {A \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {1}{16} x (5 A+6 C)+\frac {B \sin ^5(c+d x)}{5 d}-\frac {2 B \sin ^3(c+d x)}{3 d}+\frac {B \sin (c+d x)}{d} \]
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Rubi [A] time = 0.10, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {4047, 2633, 4045, 2635, 8} \[ \frac {(5 A+6 C) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {(5 A+6 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {A \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {1}{16} x (5 A+6 C)+\frac {B \sin ^5(c+d x)}{5 d}-\frac {2 B \sin ^3(c+d x)}{3 d}+\frac {B \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 4045
Rule 4047
Rubi steps
\begin {align*} \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=B \int \cos ^5(c+d x) \, dx+\int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {A \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{6} (5 A+6 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 {(5 A+6 C) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {A \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 B \sin ^3(c+d x)}{3 d}+\frac {B \sin ^5(c+d x)}{5 d}+\frac {1}{8} (5 A+6 C) \int \cos ^2(c+d x) \, dx\\ &=\frac {B \sin (c+d x)}{d}+\frac {(5 A+6 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {(5 A+6 C) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {A \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 B \sin ^3(c+d x)}{3 d}+\frac {B \sin ^5(c+d x)}{5 d}+\frac {1}{16} (5 A+6 C) \int 1 \, dx\\ &=\frac {1}{16} (5 A+6 C) x+\frac {B \sin (c+d x)}{d}+\frac {(5 A+6 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {(5 A+6 C) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {A \cos ^5(c+d x) \sin (c+d x)}{6 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 = 0.35, size = 102, normalized size = 0.77 \[ \frac {5 ((45 A+48 C) \sin (2 (c+d x))+(9 A+6 C) \sin (4 (c+d x))+A \sin (6 (c+d x))+60 A c+60 A d x+72 c C+72 C d x)+192 B \sin ^5(c+d x)-640 B \sin ^3(c+d x)+960 B \sin (c+d x)}{960 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 93, normalized size = 0.70 \[ \frac {15 \, {\left (5 \, A + 6 \, C\right )} d x + {\left (40 \, A \cos \left (d x + c\right )^{5} + 48 \, B \cos \left (d x + c\right )^{4} + 10 \, {\left (5 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{3} + 64 \, B \cos \left (d x + c\right )^{2} + 15 \, {\left (5 \, A + 6 \, C\right )} \cos \left (d x + c\right ) + 128 \, B\right )} \sin \left (d x + c\right )}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 284, normalized size = 2.15 \[ \frac {15 \, {\left (d x + c\right )} {\left (5 \, A + 6 \, C\right )} - \frac {2 \, {\left (165 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 240 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 150 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 25 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 560 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 210 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 450 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1248 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 60 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 450 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1248 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 25 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 560 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 210 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 165 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 240 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 150 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.17, size = 115, normalized size = 0.87 \[ \frac {A \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 115, normalized size = 0.87 \[ -\frac {5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A - 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.67, size = 126, normalized size = 0.95 \[ \frac {5\,A\,x}{16}+\frac {3\,C\,x}{8}+\frac {15\,A\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {3\,A\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {A\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {5\,B\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {B\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {C\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {5\,B\,\sin \left (c+d\,x\right )}{8\,d} \]
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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