Optimal. Leaf size=67 \[ \frac{(a \sin (c+d x)+a)^8}{8 a^5 d}-\frac{4 (a \sin (c+d x)+a)^7}{7 a^4 d}+\frac{2 (a \sin (c+d x)+a)^6}{3 a^3 d} \]
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Rubi [A] time = 0.0656316, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2667, 43} \[ \frac{(a \sin (c+d x)+a)^8}{8 a^5 d}-\frac{4 (a \sin (c+d x)+a)^7}{7 a^4 d}+\frac{2 (a \sin (c+d x)+a)^6}{3 a^3 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \cos ^5(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^2 (a+x)^5 \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (4 a^2 (a+x)^5-4 a (a+x)^6+(a+x)^7\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{2 (a+a \sin (c+d x))^6}{3 a^3 d}-\frac{4 (a+a \sin (c+d x))^7}{7 a^4 d}+\frac{(a+a \sin (c+d x))^8}{8 a^5 d}\\ \end{align*}
Mathematica [A] time = 0.0961315, size = 58, normalized size = 0.87 \[ -\frac{a^3 (\sin (c+d x)+1)^3 \left (21 \sin ^2(c+d x)-54 \sin (c+d x)+37\right ) \cos ^6(c+d x)}{168 d (\sin (c+d x)-1)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, size = 133, normalized size = 2. \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{8}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{24}} \right ) +3\,{a}^{3} \left ( -1/7\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}+1/35\, \left ( 8/3+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+4/3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{2}}+{\frac{{a}^{3}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.950401, size = 146, normalized size = 2.18 \begin{align*} \frac{21 \, a^{3} \sin \left (d x + c\right )^{8} + 72 \, a^{3} \sin \left (d x + c\right )^{7} + 28 \, a^{3} \sin \left (d x + c\right )^{6} - 168 \, a^{3} \sin \left (d x + c\right )^{5} - 210 \, a^{3} \sin \left (d x + c\right )^{4} + 56 \, a^{3} \sin \left (d x + c\right )^{3} + 252 \, a^{3} \sin \left (d x + c\right )^{2} + 168 \, a^{3} \sin \left (d x + c\right )}{168 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80867, size = 207, normalized size = 3.09 \begin{align*} \frac{21 \, a^{3} \cos \left (d x + c\right )^{8} - 112 \, a^{3} \cos \left (d x + c\right )^{6} - 8 \,{\left (9 \, a^{3} \cos \left (d x + c\right )^{6} - 6 \, a^{3} \cos \left (d x + c\right )^{4} - 8 \, a^{3} \cos \left (d x + c\right )^{2} - 16 \, a^{3}\right )} \sin \left (d x + c\right )}{168 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.7934, size = 270, normalized size = 4.03 \begin{align*} \begin{cases} \frac{a^{3} \sin ^{8}{\left (c + d x \right )}}{24 d} + \frac{8 a^{3} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac{a^{3} \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{6 d} + \frac{a^{3} \sin ^{6}{\left (c + d x \right )}}{2 d} + \frac{4 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac{8 a^{3} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} + \frac{3 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 d} + \frac{a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{4 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{3 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{2 d} + \frac{a^{3} \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{3} \cos ^{5}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20354, size = 181, normalized size = 2.7 \begin{align*} \frac{a^{3} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac{5 \, a^{3} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac{25 \, a^{3} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac{33 \, a^{3} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} - \frac{3 \, a^{3} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac{a^{3} \sin \left (5 \, d x + 5 \, c\right )}{64 \, d} + \frac{17 \, a^{3} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac{55 \, a^{3} \sin \left (d x + c\right )}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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