Optimal. Leaf size=22 \[ \frac{(a+b \sin (c+d x))^9}{9 b d} \]
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Rubi [A] time = 0.0263247, 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 = {2668, 32} \[ \frac{(a+b \sin (c+d x))^9}{9 b d} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 32
Rubi steps
\begin{align*} \int \cos (c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^8 \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{(a+b \sin (c+d x))^9}{9 b d}\\ \end{align*}
Mathematica [B] time = 0.359453, size = 137, normalized size = 6.23 \[ \frac{\sin (c+d x) \left (84 a^6 b^2 \sin ^2(c+d x)+126 a^5 b^3 \sin ^3(c+d x)+126 a^4 b^4 \sin ^4(c+d x)+84 a^3 b^5 \sin ^5(c+d x)+36 a^2 b^6 \sin ^6(c+d x)+36 a^7 b \sin (c+d x)+9 a^8+9 a b^7 \sin ^7(c+d x)+b^8 \sin ^8(c+d x)\right )}{9 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 21, normalized size = 1. \begin{align*}{\frac{ \left ( a+b\sin \left ( dx+c \right ) \right ) ^{9}}{9\,bd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.943892, size = 27, normalized size = 1.23 \begin{align*} \frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{9}}{9 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.10267, size = 582, normalized size = 26.45 \begin{align*} \frac{9 \, a b^{7} \cos \left (d x + c\right )^{8} - 12 \,{\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{6} + 18 \,{\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} - 36 \,{\left (a^{7} b + 7 \, a^{5} b^{3} + 7 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{2} +{\left (b^{8} \cos \left (d x + c\right )^{8} + 9 \, a^{8} + 84 \, a^{6} b^{2} + 126 \, a^{4} b^{4} + 36 \, a^{2} b^{6} + b^{8} - 4 \,{\left (9 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{6} + 6 \,{\left (21 \, a^{4} b^{4} + 18 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{4} - 4 \,{\left (21 \, a^{6} b^{2} + 63 \, a^{4} b^{4} + 27 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{2}\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 = 20.9601, size = 168, normalized size = 7.64 \begin{align*} \begin{cases} \frac{a^{8} \sin{\left (c + d x \right )}}{d} + \frac{4 a^{7} b \sin ^{2}{\left (c + d x \right )}}{d} + \frac{28 a^{6} b^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{14 a^{5} b^{3} \sin ^{4}{\left (c + d x \right )}}{d} + \frac{14 a^{4} b^{4} \sin ^{5}{\left (c + d x \right )}}{d} + \frac{28 a^{3} b^{5} \sin ^{6}{\left (c + d x \right )}}{3 d} + \frac{4 a^{2} b^{6} \sin ^{7}{\left (c + d x \right )}}{d} + \frac{a b^{7} \sin ^{8}{\left (c + d x \right )}}{d} + \frac{b^{8} \sin ^{9}{\left (c + d x \right )}}{9 d} & \text{for}\: d \neq 0 \\x \left (a + b \sin{\left (c \right )}\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 = 1.12865, size = 27, normalized size = 1.23 \begin{align*} \frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{9}}{9 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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