Optimal. Leaf size=125 \[ \frac{a \cos ^9(c+d x)}{9 d}-\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5 a x}{128} \]
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Rubi [A] time = 0.154914, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2838, 2568, 2635, 8, 2565, 14} \[ \frac{a \cos ^9(c+d x)}{9 d}-\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5 a x}{128} \]
Antiderivative was successfully verified.
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Rule 2838
Rule 2568
Rule 2635
Rule 8
Rule 2565
Rule 14
Rubi steps
\begin{align*} \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+a \int \cos ^6(c+d x) \sin ^3(c+d x) \, dx\\ &=-\frac{a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{8} a \int \cos ^6(c+d x) \, dx-\frac{a \operatorname{Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{48} (5 a) \int \cos ^4(c+d x) \, dx-\frac{a \operatorname{Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a \cos ^7(c+d x)}{7 d}+\frac{a \cos ^9(c+d x)}{9 d}+\frac{5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{64} (5 a) \int \cos ^2(c+d x) \, dx\\ &=-\frac{a \cos ^7(c+d x)}{7 d}+\frac{a \cos ^9(c+d x)}{9 d}+\frac{5 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac{5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{128} (5 a) \int 1 \, dx\\ &=\frac{5 a x}{128}-\frac{a \cos ^7(c+d x)}{7 d}+\frac{a \cos ^9(c+d x)}{9 d}+\frac{5 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac{5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{a \cos ^7(c+d x) \sin (c+d x)}{8 d}\\ \end{align*}
Mathematica [A] time = 0.275032, size = 91, normalized size = 0.73 \[ \frac{a (1008 \sin (2 (c+d x))-504 \sin (4 (c+d x))-336 \sin (6 (c+d x))-63 \sin (8 (c+d x))-1512 \cos (c+d x)-672 \cos (3 (c+d x))+108 \cos (7 (c+d x))+28 \cos (9 (c+d x))+2520 d x)}{64512 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 98, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( 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 ) +a \left ( -{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8}}+{\frac{\sin \left ( dx+c \right ) }{48} \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}{128}}+{\frac{5\,c}{128}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04192, size = 103, normalized size = 0.82 \begin{align*} \frac{1024 \,{\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a + 21 \,{\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}{64512 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.20244, size = 232, normalized size = 1.86 \begin{align*} \frac{896 \, a \cos \left (d x + c\right )^{9} - 1152 \, a \cos \left (d x + c\right )^{7} + 315 \, a d x - 21 \,{\left (48 \, a \cos \left (d x + c\right )^{7} - 8 \, a \cos \left (d x + c\right )^{5} - 10 \, a \cos \left (d x + c\right )^{3} - 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8064 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.5605, size = 248, normalized size = 1.98 \begin{align*} \begin{cases} \frac{5 a x \sin ^{8}{\left (c + d x \right )}}{128} + \frac{5 a x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac{15 a x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac{5 a x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac{5 a x \cos ^{8}{\left (c + d x \right )}}{128} + \frac{5 a \sin ^{7}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{128 d} + \frac{55 a \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac{73 a \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} - \frac{a \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac{5 a \sin{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac{2 a \cos ^{9}{\left (c + d x \right )}}{63 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right ) \sin ^{2}{\left (c \right )} \cos ^{6}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26139, size = 165, normalized size = 1.32 \begin{align*} \frac{5}{128} \, a x + \frac{a \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac{3 \, a \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac{a \cos \left (3 \, d x + 3 \, c\right )}{96 \, d} - \frac{3 \, a \cos \left (d x + c\right )}{128 \, d} - \frac{a \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac{a \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac{a \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac{a \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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