Optimal. Leaf size=231 \[ -\frac {11 a^3 (10 A+3 B) \cos ^7(c+d x)}{560 d}-\frac {11 (10 A+3 B) \cos ^7(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{720 d}+\frac {11 a^3 (10 A+3 B) \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac {11 a^3 (10 A+3 B) \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac {11 a^3 (10 A+3 B) \sin (c+d x) \cos (c+d x)}{256 d}+\frac {11}{256} a^3 x (10 A+3 B)-\frac {a (10 A+3 B) \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{90 d}-\frac {B \cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d} \]
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Rubi [A] time = 0.27, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2860, 2678, 2669, 2635, 8} \[ -\frac {11 a^3 (10 A+3 B) \cos ^7(c+d x)}{560 d}-\frac {11 (10 A+3 B) \cos ^7(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{720 d}+\frac {11 a^3 (10 A+3 B) \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac {11 a^3 (10 A+3 B) \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac {11 a^3 (10 A+3 B) \sin (c+d x) \cos (c+d x)}{256 d}+\frac {11}{256} a^3 x (10 A+3 B)-\frac {a (10 A+3 B) \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{90 d}-\frac {B \cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2678
Rule 2860
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx &=-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}+\frac {1}{10} (10 A+3 B) \int \cos ^6(c+d x) (a+a \sin (c+d x))^3 \, dx\\ &=-\frac {a (10 A+3 B) \cos ^7(c+d x) (a+a \sin (c+d x))^2}{90 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}+\frac {1}{90} (11 a (10 A+3 B)) \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=-\frac {a (10 A+3 B) \cos ^7(c+d x) (a+a \sin (c+d x))^2}{90 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 (10 A+3 B) \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{720 d}+\frac {1}{80} \left (11 a^2 (10 A+3 B)\right ) \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac {11 a^3 (10 A+3 B) \cos ^7(c+d x)}{560 d}-\frac {a (10 A+3 B) \cos ^7(c+d x) (a+a \sin (c+d x))^2}{90 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 (10 A+3 B) \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{720 d}+\frac {1}{80} \left (11 a^3 (10 A+3 B)\right ) \int \cos ^6(c+d x) \, dx\\ &=-\frac {11 a^3 (10 A+3 B) \cos ^7(c+d x)}{560 d}+\frac {11 a^3 (10 A+3 B) \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {a (10 A+3 B) \cos ^7(c+d x) (a+a \sin (c+d x))^2}{90 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 (10 A+3 B) \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{720 d}+\frac {1}{96} \left (11 a^3 (10 A+3 B)\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac {11 a^3 (10 A+3 B) \cos ^7(c+d x)}{560 d}+\frac {11 a^3 (10 A+3 B) \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {11 a^3 (10 A+3 B) \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {a (10 A+3 B) \cos ^7(c+d x) (a+a \sin (c+d x))^2}{90 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 (10 A+3 B) \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{720 d}+\frac {1}{128} \left (11 a^3 (10 A+3 B)\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {11 a^3 (10 A+3 B) \cos ^7(c+d x)}{560 d}+\frac {11 a^3 (10 A+3 B) \cos (c+d x) \sin (c+d x)}{256 d}+\frac {11 a^3 (10 A+3 B) \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {11 a^3 (10 A+3 B) \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {a (10 A+3 B) \cos ^7(c+d x) (a+a \sin (c+d x))^2}{90 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 (10 A+3 B) \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{720 d}+\frac {1}{256} \left (11 a^3 (10 A+3 B)\right ) \int 1 \, dx\\ &=\frac {11}{256} a^3 (10 A+3 B) x-\frac {11 a^3 (10 A+3 B) \cos ^7(c+d x)}{560 d}+\frac {11 a^3 (10 A+3 B) \cos (c+d x) \sin (c+d x)}{256 d}+\frac {11 a^3 (10 A+3 B) \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {11 a^3 (10 A+3 B) \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {a (10 A+3 B) \cos ^7(c+d x) (a+a \sin (c+d x))^2}{90 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 (10 A+3 B) \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{720 d}\\ \end {align*}
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Mathematica [A] time = 6.05, size = 344, normalized size = 1.49 \[ -\frac {32 \sqrt {2} a^2 (10 a A+3 a B) \left (\frac {1}{2} (\sin (c+d x)-1)+1\right )^{13/2} \left (\frac {385 \left (\frac {\sqrt {2} \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}}{\sqrt {\frac {1}{2} (\sin (c+d x)-1)+1}}-\frac {2}{15} (1-\sin (c+d x))^3-\frac {1}{3} (1-\sin (c+d x))^2+\sin (c+d x)-1\right )}{8192 \left (\frac {1}{2} (\sin (c+d x)-1)+1\right )^6 (1-\sin (c+d x))^4}+\frac {7}{18} \left (\frac {1}{\frac {1}{2} (\sin (c+d x)-1)+1}+\frac {11}{16 \left (\frac {1}{2} (\sin (c+d x)-1)+1\right )^2}+\frac {99}{224 \left (\frac {1}{2} (\sin (c+d x)-1)+1\right )^3}+\frac {33}{128 \left (\frac {1}{2} (\sin (c+d x)-1)+1\right )^4}+\frac {33}{256 \left (\frac {1}{2} (\sin (c+d x)-1)+1\right )^5}+\frac {99}{2048 \left (\frac {1}{2} (\sin (c+d x)-1)+1\right )^6}\right )\right ) \cos ^7(c+d x)}{35 d (\sin (c+d x)+1)^{7/2}}-\frac {B \cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 155, normalized size = 0.67 \[ \frac {8960 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{9} - 46080 \, {\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{7} + 3465 \, {\left (10 \, A + 3 \, B\right )} a^{3} d x + 21 \, {\left (384 \, B a^{3} \cos \left (d x + c\right )^{9} - 48 \, {\left (30 \, A + 41 \, B\right )} a^{3} \cos \left (d x + c\right )^{7} + 88 \, {\left (10 \, A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{5} + 110 \, {\left (10 \, A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 165 \, {\left (10 \, A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.74, size = 273, normalized size = 1.18 \[ \frac {B a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {11}{256} \, {\left (10 \, A a^{3} + 3 \, B a^{3}\right )} x + \frac {{\left (A a^{3} + 3 \, B a^{3}\right )} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac {{\left (9 \, A a^{3} - 5 \, B a^{3}\right )} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {{\left (3 \, A a^{3} + B a^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac {{\left (29 \, A a^{3} + 15 \, B a^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {{\left (33 \, A a^{3} + 19 \, B a^{3}\right )} \cos \left (d x + c\right )}{128 \, d} - \frac {{\left (6 \, A a^{3} + 5 \, B a^{3}\right )} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac {{\left (32 \, A a^{3} + 51 \, B a^{3}\right )} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} + \frac {{\left (6 \, A a^{3} - 7 \, B a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {{\left (144 \, A a^{3} + 25 \, B a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 363, normalized size = 1.57 \[ \frac {a^{3} A \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+B \,a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} A \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+3 B \,a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )-\frac {3 a^{3} A \left (\cos ^{7}\left (d x +c \right )\right )}{7}+3 B \,a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+a^{3} A \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-\frac {B \,a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 284, normalized size = 1.23 \[ -\frac {276480 \, A a^{3} \cos \left (d x + c\right )^{7} + 92160 \, B a^{3} \cos \left (d x + c\right )^{7} - 10240 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} A a^{3} - 630 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} A a^{3} + 3360 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 30720 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} B a^{3} - 63 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} B a^{3} - 630 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} B a^{3}}{645120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.83, size = 711, normalized size = 3.08 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 27.75, size = 1042, normalized size = 4.51 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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