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 = 0.268184, antiderivative size = 231, normalized size of antiderivative = 1., 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 2860
Rule 2678
Rule 2669
Rule 2635
Rule 8
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*}
Mathematica [A] time = 6.05706, 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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Maple [A] time = 0.072, size = 363, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ({a}^{3}A \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{9}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63}} \right ) +B{a}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{10}}-{\frac{3\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{80}}+{\frac{\sin \left ( dx+c \right ) }{160} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{3\,dx}{256}}+{\frac{3\,c}{256}} \right ) +3\,{a}^{3}A \left ( -1/8\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}+1/48\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+5/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) \sin \left ( dx+c \right ) +{\frac{5\,dx}{128}}+{\frac{5\,c}{128}} \right ) +3\,B{a}^{3} \left ( -1/9\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63}} \right ) -{\frac{3\,{a}^{3}A \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7}}+3\,B{a}^{3} \left ( -1/8\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}+1/48\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+5/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) \sin \left ( dx+c \right ) +{\frac{5\,dx}{128}}+{\frac{5\,c}{128}} \right ) +{a}^{3}A \left ({\frac{\sin \left ( dx+c \right ) }{6} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) -{\frac{B{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08704, size = 383, normalized size = 1.66 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1073, size = 410, normalized size = 1.77 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 46.5322, size = 1042, normalized size = 4.51 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.4006, size = 369, normalized size = 1.6 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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