Optimal. Leaf size=97 \[ -\frac{a \sin ^{11}(c+d x)}{11 d}+\frac{a \sin ^9(c+d x)}{3 d}-\frac{3 a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^5(c+d x)}{5 d}+\frac{a \cos ^{10}(c+d x)}{10 d}-\frac{a \cos ^8(c+d x)}{8 d} \]
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Rubi [A] time = 0.131623, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2834, 2565, 14, 2564, 270} \[ -\frac{a \sin ^{11}(c+d x)}{11 d}+\frac{a \sin ^9(c+d x)}{3 d}-\frac{3 a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^5(c+d x)}{5 d}+\frac{a \cos ^{10}(c+d x)}{10 d}-\frac{a \cos ^8(c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 2834
Rule 2565
Rule 14
Rule 2564
Rule 270
Rubi steps
\begin{align*} \int \cos ^7(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^7(c+d x) \sin ^3(c+d x) \, dx+a \int \cos ^7(c+d x) \sin ^4(c+d x) \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int x^7 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac{a \operatorname{Subst}\left (\int x^4 \left (1-x^2\right )^3 \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{a \operatorname{Subst}\left (\int \left (x^7-x^9\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac{a \operatorname{Subst}\left (\int \left (x^4-3 x^6+3 x^8-x^{10}\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{a \cos ^8(c+d x)}{8 d}+\frac{a \cos ^{10}(c+d x)}{10 d}+\frac{a \sin ^5(c+d x)}{5 d}-\frac{3 a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^9(c+d x)}{3 d}-\frac{a \sin ^{11}(c+d x)}{11 d}\\ \end{align*}
Mathematica [A] time = 0.605329, size = 117, normalized size = 1.21 \[ \frac{a (16170 \sin (c+d x)-2310 \sin (3 (c+d x))-2541 \sin (5 (c+d x))-165 \sin (7 (c+d x))+385 \sin (9 (c+d x))+105 \sin (11 (c+d x))-16170 \cos (2 (c+d x))-4620 \cos (4 (c+d x))+1155 \cos (6 (c+d x))+1155 \cos (8 (c+d x))+231 \cos (10 (c+d x)))}{1182720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 112, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{11}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{33}}+{\frac{\sin \left ( dx+c \right ) }{231} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) } \right ) +a \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{10}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{40}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02365, size = 127, normalized size = 1.31 \begin{align*} -\frac{840 \, a \sin \left (d x + c\right )^{11} + 924 \, a \sin \left (d x + c\right )^{10} - 3080 \, a \sin \left (d x + c\right )^{9} - 3465 \, a \sin \left (d x + c\right )^{8} + 3960 \, a \sin \left (d x + c\right )^{7} + 4620 \, a \sin \left (d x + c\right )^{6} - 1848 \, a \sin \left (d x + c\right )^{5} - 2310 \, a \sin \left (d x + c\right )^{4}}{9240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60166, size = 259, normalized size = 2.67 \begin{align*} \frac{924 \, a \cos \left (d x + c\right )^{10} - 1155 \, a \cos \left (d x + c\right )^{8} + 8 \,{\left (105 \, a \cos \left (d x + c\right )^{10} - 140 \, a \cos \left (d x + c\right )^{8} + 5 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} + 8 \, a \cos \left (d x + c\right )^{2} + 16 \, a\right )} \sin \left (d x + c\right )}{9240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 59.8253, size = 138, normalized size = 1.42 \begin{align*} \begin{cases} \frac{16 a \sin ^{11}{\left (c + d x \right )}}{1155 d} + \frac{8 a \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{105 d} + \frac{6 a \sin ^{7}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{35 d} + \frac{a \sin ^{5}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{5 d} - \frac{a \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{8 d} - \frac{a \cos ^{10}{\left (c + d x \right )}}{40 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right ) \sin ^{3}{\left (c \right )} \cos ^{7}{\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 = 1.25992, size = 220, normalized size = 2.27 \begin{align*} \frac{a \cos \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac{a \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac{a \cos \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac{a \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac{7 \, a \cos \left (2 \, d x + 2 \, c\right )}{512 \, d} + \frac{a \sin \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac{a \sin \left (9 \, d x + 9 \, c\right )}{3072 \, d} - \frac{a \sin \left (7 \, d x + 7 \, c\right )}{7168 \, d} - \frac{11 \, a \sin \left (5 \, d x + 5 \, c\right )}{5120 \, d} - \frac{a \sin \left (3 \, d x + 3 \, c\right )}{512 \, d} + \frac{7 \, a \sin \left (d x + c\right )}{512 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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