Optimal. Leaf size=22 \[ \frac{(a \sin (c+d x)+a)^9}{9 a d} \]
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Rubi [A] time = 0.0243345, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2667, 32} \[ \frac{(a \sin (c+d x)+a)^9}{9 a d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \sin (c+d x))^8 \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^8 \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{(a+a \sin (c+d x))^9}{9 a d}\\ \end{align*}
Mathematica [B] time = 0.0894273, size = 147, normalized size = 6.68 \[ \frac{a^8 \sin ^9(c+d x)}{9 d}+\frac{a^8 \sin ^8(c+d x)}{d}+\frac{4 a^8 \sin ^7(c+d x)}{d}+\frac{28 a^8 \sin ^6(c+d x)}{3 d}+\frac{14 a^8 \sin ^5(c+d x)}{d}+\frac{14 a^8 \sin ^4(c+d x)}{d}+\frac{28 a^8 \sin ^3(c+d x)}{3 d}+\frac{4 a^8 \sin ^2(c+d x)}{d}+\frac{a^8 \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 21, normalized size = 1. \begin{align*}{\frac{ \left ( a+a\sin \left ( dx+c \right ) \right ) ^{9}}{9\,da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.93696, size = 27, normalized size = 1.23 \begin{align*} \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{9}}{9 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.84925, size = 304, normalized size = 13.82 \begin{align*} \frac{9 \, a^{8} \cos \left (d x + c\right )^{8} - 120 \, a^{8} \cos \left (d x + c\right )^{6} + 432 \, a^{8} \cos \left (d x + c\right )^{4} - 576 \, a^{8} \cos \left (d x + c\right )^{2} +{\left (a^{8} \cos \left (d x + c\right )^{8} - 40 \, a^{8} \cos \left (d x + c\right )^{6} + 240 \, a^{8} \cos \left (d x + c\right )^{4} - 448 \, a^{8} \cos \left (d x + c\right )^{2} + 256 \, a^{8}\right )} \sin \left (d x + c\right )}{9 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.8771, size = 148, normalized size = 6.73 \begin{align*} \begin{cases} \frac{a^{8} \sin ^{9}{\left (c + d x \right )}}{9 d} + \frac{a^{8} \sin ^{8}{\left (c + d x \right )}}{d} + \frac{4 a^{8} \sin ^{7}{\left (c + d x \right )}}{d} + \frac{28 a^{8} \sin ^{6}{\left (c + d x \right )}}{3 d} + \frac{14 a^{8} \sin ^{5}{\left (c + d x \right )}}{d} + \frac{14 a^{8} \sin ^{4}{\left (c + d x \right )}}{d} + \frac{28 a^{8} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{4 a^{8} \sin ^{2}{\left (c + d x \right )}}{d} + \frac{a^{8} \sin{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{8} \cos{\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 = 1.20482, size = 27, normalized size = 1.23 \begin{align*} \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{9}}{9 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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