Optimal. Leaf size=45 \[ \frac{(a \sin (c+d x)+a)^{10}}{5 a^2 d}-\frac{(a \sin (c+d x)+a)^{11}}{11 a^3 d} \]
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Rubi [A] time = 0.0471371, antiderivative size = 45, 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} \[ \frac{(a \sin (c+d x)+a)^{10}}{5 a^2 d}-\frac{(a \sin (c+d x)+a)^{11}}{11 a^3 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+a \sin (c+d x))^8 \, dx &=\frac{\operatorname{Subst}\left (\int (a-x) (a+x)^9 \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (2 a (a+x)^9-(a+x)^{10}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{(a+a \sin (c+d x))^{10}}{5 a^2 d}-\frac{(a+a \sin (c+d x))^{11}}{11 a^3 d}\\ \end{align*}
Mathematica [A] time = 1.0431, size = 43, normalized size = 0.96 \[ -\frac{a^8 (5 \sin (c+d x)-6) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^{20}}{55 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 463, normalized size = 10.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.949323, size = 181, normalized size = 4.02 \begin{align*} -\frac{5 \, a^{8} \sin \left (d x + c\right )^{11} + 44 \, a^{8} \sin \left (d x + c\right )^{10} + 165 \, a^{8} \sin \left (d x + c\right )^{9} + 330 \, a^{8} \sin \left (d x + c\right )^{8} + 330 \, a^{8} \sin \left (d x + c\right )^{7} - 462 \, a^{8} \sin \left (d x + c\right )^{5} - 660 \, a^{8} \sin \left (d x + c\right )^{4} - 495 \, a^{8} \sin \left (d x + c\right )^{3} - 220 \, a^{8} \sin \left (d x + c\right )^{2} - 55 \, a^{8} \sin \left (d x + c\right )}{55 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.95231, size = 352, normalized size = 7.82 \begin{align*} \frac{44 \, a^{8} \cos \left (d x + c\right )^{10} - 550 \, a^{8} \cos \left (d x + c\right )^{8} + 1760 \, a^{8} \cos \left (d x + c\right )^{6} - 1760 \, a^{8} \cos \left (d x + c\right )^{4} +{\left (5 \, a^{8} \cos \left (d x + c\right )^{10} - 190 \, a^{8} \cos \left (d x + c\right )^{8} + 1040 \, a^{8} \cos \left (d x + c\right )^{6} - 1568 \, a^{8} \cos \left (d x + c\right )^{4} + 256 \, a^{8} \cos \left (d x + c\right )^{2} + 512 \, a^{8}\right )} \sin \left (d x + c\right )}{55 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 56.5975, size = 445, normalized size = 9.89 \begin{align*} \begin{cases} \frac{2 a^{8} \sin ^{11}{\left (c + d x \right )}}{99 d} + \frac{a^{8} \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{9 d} + \frac{8 a^{8} \sin ^{9}{\left (c + d x \right )}}{9 d} + \frac{4 a^{8} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{4 a^{8} \sin ^{7}{\left (c + d x \right )}}{d} - \frac{2 a^{8} \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{14 a^{8} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{56 a^{8} \sin ^{5}{\left (c + d x \right )}}{15 d} - \frac{2 a^{8} \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac{14 a^{8} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{2 a^{8} \sin ^{4}{\left (c + d x \right )}}{d} + \frac{28 a^{8} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{2 a^{8} \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac{a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} - \frac{28 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac{14 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{4 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{a^{8} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac{a^{8} \cos ^{10}{\left (c + d x \right )}}{5 d} - \frac{7 a^{8} \cos ^{8}{\left (c + d x \right )}}{3 d} - \frac{14 a^{8} \cos ^{6}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{8} \cos ^{3}{\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 = 1.2046, size = 181, normalized size = 4.02 \begin{align*} -\frac{5 \, a^{8} \sin \left (d x + c\right )^{11} + 44 \, a^{8} \sin \left (d x + c\right )^{10} + 165 \, a^{8} \sin \left (d x + c\right )^{9} + 330 \, a^{8} \sin \left (d x + c\right )^{8} + 330 \, a^{8} \sin \left (d x + c\right )^{7} - 462 \, a^{8} \sin \left (d x + c\right )^{5} - 660 \, a^{8} \sin \left (d x + c\right )^{4} - 495 \, a^{8} \sin \left (d x + c\right )^{3} - 220 \, a^{8} \sin \left (d x + c\right )^{2} - 55 \, a^{8} \sin \left (d x + c\right )}{55 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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