Optimal. Leaf size=159 \[ -\frac{a^2 \cos ^9(c+d x)}{9 d}+\frac{3 a^2 \cos ^7(c+d x)}{7 d}-\frac{2 a^2 \cos ^5(c+d x)}{5 d}-\frac{a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{4 d}-\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{8 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{32 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac{3 a^2 x}{64} \]
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Rubi [A] time = 0.254925, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2873, 2565, 14, 2568, 2635, 8, 270} \[ -\frac{a^2 \cos ^9(c+d x)}{9 d}+\frac{3 a^2 \cos ^7(c+d x)}{7 d}-\frac{2 a^2 \cos ^5(c+d x)}{5 d}-\frac{a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{4 d}-\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{8 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{32 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac{3 a^2 x}{64} \]
Antiderivative was successfully verified.
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Rule 2873
Rule 2565
Rule 14
Rule 2568
Rule 2635
Rule 8
Rule 270
Rubi steps
\begin{align*} \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cos ^4(c+d x) \sin ^3(c+d x)+2 a^2 \cos ^4(c+d x) \sin ^4(c+d x)+a^2 \cos ^4(c+d x) \sin ^5(c+d x)\right ) \, dx\\ &=a^2 \int \cos ^4(c+d x) \sin ^3(c+d x) \, dx+a^2 \int \cos ^4(c+d x) \sin ^5(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx\\ &=-\frac{a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{4 d}+\frac{1}{4} \left (3 a^2\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac{a^2 \operatorname{Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac{a^2 \operatorname{Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac{a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{4 d}+\frac{1}{8} a^2 \int \cos ^4(c+d x) \, dx-\frac{a^2 \operatorname{Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac{a^2 \operatorname{Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{2 a^2 \cos ^5(c+d x)}{5 d}+\frac{3 a^2 \cos ^7(c+d x)}{7 d}-\frac{a^2 \cos ^9(c+d x)}{9 d}+\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{32 d}-\frac{a^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac{a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{4 d}+\frac{1}{32} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{2 a^2 \cos ^5(c+d x)}{5 d}+\frac{3 a^2 \cos ^7(c+d x)}{7 d}-\frac{a^2 \cos ^9(c+d x)}{9 d}+\frac{3 a^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{32 d}-\frac{a^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac{a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{4 d}+\frac{1}{64} \left (3 a^2\right ) \int 1 \, dx\\ &=\frac{3 a^2 x}{64}-\frac{2 a^2 \cos ^5(c+d x)}{5 d}+\frac{3 a^2 \cos ^7(c+d x)}{7 d}-\frac{a^2 \cos ^9(c+d x)}{9 d}+\frac{3 a^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{32 d}-\frac{a^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac{a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.697071, size = 86, normalized size = 0.54 \[ \frac{a^2 (-2520 \sin (4 (c+d x))+315 \sin (8 (c+d x))-11340 \cos (c+d x)-3360 \cos (3 (c+d x))+1008 \cos (5 (c+d x))+450 \cos (7 (c+d x))-70 \cos (9 (c+d x))+7560 c+7560 d x)}{161280 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 162, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{9}}-{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{63}}-{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{315}} \right ) +2\,{a}^{2} \left ( -1/8\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}-1/16\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{ \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) }{64}}+{\frac{3\,dx}{128}}+{\frac{3\,c}{128}} \right ) +{a}^{2} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{7}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09879, size = 136, normalized size = 0.86 \begin{align*} -\frac{512 \,{\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{2} - 4608 \,{\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{2} - 315 \,{\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{161280 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60285, size = 290, normalized size = 1.82 \begin{align*} -\frac{2240 \, a^{2} \cos \left (d x + c\right )^{9} - 8640 \, a^{2} \cos \left (d x + c\right )^{7} + 8064 \, a^{2} \cos \left (d x + c\right )^{5} - 945 \, a^{2} d x - 315 \,{\left (16 \, a^{2} \cos \left (d x + c\right )^{7} - 24 \, a^{2} \cos \left (d x + c\right )^{5} + 2 \, a^{2} \cos \left (d x + c\right )^{3} + 3 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{20160 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 22.0236, size = 335, normalized size = 2.11 \begin{align*} \begin{cases} \frac{3 a^{2} x \sin ^{8}{\left (c + d x \right )}}{64} + \frac{3 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac{9 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac{3 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{16} + \frac{3 a^{2} x \cos ^{8}{\left (c + d x \right )}}{64} + \frac{3 a^{2} \sin ^{7}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{64 d} + \frac{11 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{64 d} - \frac{a^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac{11 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{64 d} - \frac{4 a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac{a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac{3 a^{2} \sin{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac{8 a^{2} \cos ^{9}{\left (c + d x \right )}}{315 d} - \frac{2 a^{2} \cos ^{7}{\left (c + d x \right )}}{35 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{2} \sin ^{3}{\left (c \right )} \cos ^{4}{\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 = 1.42578, size = 166, normalized size = 1.04 \begin{align*} \frac{3}{64} \, a^{2} x - \frac{a^{2} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac{5 \, a^{2} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac{a^{2} \cos \left (5 \, d x + 5 \, c\right )}{160 \, d} - \frac{a^{2} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac{9 \, a^{2} \cos \left (d x + c\right )}{128 \, d} + \frac{a^{2} \sin \left (8 \, d x + 8 \, c\right )}{512 \, d} - \frac{a^{2} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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