Optimal. Leaf size=77 \[ -\frac{\left (a^2-b^2\right ) (a+b \sin (c+d x))^9}{9 b^3 d}-\frac{(a+b \sin (c+d x))^{11}}{11 b^3 d}+\frac{a (a+b \sin (c+d x))^{10}}{5 b^3 d} \]
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Rubi [A] time = 0.150984, antiderivative size = 77, 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 = {2668, 697} \[ -\frac{\left (a^2-b^2\right ) (a+b \sin (c+d x))^9}{9 b^3 d}-\frac{(a+b \sin (c+d x))^{11}}{11 b^3 d}+\frac{a (a+b \sin (c+d x))^{10}}{5 b^3 d} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 697
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^8 \left (b^2-x^2\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\left (-a^2+b^2\right ) (a+x)^8+2 a (a+x)^9-(a+x)^{10}\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=-\frac{\left (a^2-b^2\right ) (a+b \sin (c+d x))^9}{9 b^3 d}+\frac{a (a+b \sin (c+d x))^{10}}{5 b^3 d}-\frac{(a+b \sin (c+d x))^{11}}{11 b^3 d}\\ \end{align*}
Mathematica [A] time = 0.884818, size = 56, normalized size = 0.73 \[ \frac{(a+b \sin (c+d x))^9 \left (-2 a^2+18 a b \sin (c+d x)+45 b^2 \cos (2 (c+d x))+65 b^2\right )}{990 b^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.085, size = 480, normalized size = 6.2 \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 = 0.971706, size = 315, normalized size = 4.09 \begin{align*} -\frac{45 \, b^{8} \sin \left (d x + c\right )^{11} + 396 \, a b^{7} \sin \left (d x + c\right )^{10} - 1980 \, a^{7} b \sin \left (d x + c\right )^{2} + 55 \,{\left (28 \, a^{2} b^{6} - b^{8}\right )} \sin \left (d x + c\right )^{9} - 495 \, a^{8} \sin \left (d x + c\right ) + 495 \,{\left (7 \, a^{3} b^{5} - a b^{7}\right )} \sin \left (d x + c\right )^{8} + 990 \,{\left (5 \, a^{4} b^{4} - 2 \, a^{2} b^{6}\right )} \sin \left (d x + c\right )^{7} + 4620 \,{\left (a^{5} b^{3} - a^{3} b^{5}\right )} \sin \left (d x + c\right )^{6} + 1386 \,{\left (2 \, a^{6} b^{2} - 5 \, a^{4} b^{4}\right )} \sin \left (d x + c\right )^{5} + 990 \,{\left (a^{7} b - 7 \, a^{5} b^{3}\right )} \sin \left (d x + c\right )^{4} + 165 \,{\left (a^{8} - 28 \, a^{6} b^{2}\right )} \sin \left (d x + c\right )^{3}}{495 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.02639, size = 740, normalized size = 9.61 \begin{align*} \frac{396 \, a b^{7} \cos \left (d x + c\right )^{10} - 495 \,{\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{8} + 660 \,{\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{6} - 990 \,{\left (a^{7} b + 7 \, a^{5} b^{3} + 7 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{4} +{\left (45 \, b^{8} \cos \left (d x + c\right )^{10} - 10 \,{\left (154 \, a^{2} b^{6} + 17 \, b^{8}\right )} \cos \left (d x + c\right )^{8} + 330 \, a^{8} + 1848 \, a^{6} b^{2} + 1980 \, a^{4} b^{4} + 440 \, a^{2} b^{6} + 10 \, b^{8} + 10 \,{\left (495 \, a^{4} b^{4} + 418 \, a^{2} b^{6} + 23 \, b^{8}\right )} \cos \left (d x + c\right )^{6} - 12 \,{\left (231 \, a^{6} b^{2} + 660 \, a^{4} b^{4} + 275 \, a^{2} b^{6} + 10 \, b^{8}\right )} \cos \left (d x + c\right )^{4} +{\left (165 \, a^{8} + 924 \, a^{6} b^{2} + 990 \, a^{4} b^{4} + 220 \, a^{2} b^{6} + 5 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{495 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 60.1598, size = 493, normalized size = 6.4 \begin{align*} \begin{cases} \frac{2 a^{8} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{a^{8} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{2 a^{7} b \sin ^{4}{\left (c + d x \right )}}{d} + \frac{4 a^{7} b \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{56 a^{6} b^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{28 a^{6} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} - \frac{14 a^{5} b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac{14 a^{5} b^{3} \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac{4 a^{4} b^{4} \sin ^{7}{\left (c + d x \right )}}{d} + \frac{14 a^{4} b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac{14 a^{3} b^{5} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac{28 a^{3} b^{5} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac{7 a^{3} b^{5} \cos ^{8}{\left (c + d x \right )}}{3 d} + \frac{8 a^{2} b^{6} \sin ^{9}{\left (c + d x \right )}}{9 d} + \frac{4 a^{2} b^{6} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac{2 a b^{7} \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac{2 a b^{7} \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac{a b^{7} \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} - \frac{a b^{7} \cos ^{10}{\left (c + d x \right )}}{5 d} + \frac{2 b^{8} \sin ^{11}{\left (c + d x \right )}}{99 d} + \frac{b^{8} \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{9 d} & \text{for}\: d \neq 0 \\x \left (a + b \sin{\left (c \right )}\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 = 1.16422, size = 367, normalized size = 4.77 \begin{align*} -\frac{45 \, b^{8} \sin \left (d x + c\right )^{11} + 396 \, a b^{7} \sin \left (d x + c\right )^{10} + 1540 \, a^{2} b^{6} \sin \left (d x + c\right )^{9} - 55 \, b^{8} \sin \left (d x + c\right )^{9} + 3465 \, a^{3} b^{5} \sin \left (d x + c\right )^{8} - 495 \, a b^{7} \sin \left (d x + c\right )^{8} + 4950 \, a^{4} b^{4} \sin \left (d x + c\right )^{7} - 1980 \, a^{2} b^{6} \sin \left (d x + c\right )^{7} + 4620 \, a^{5} b^{3} \sin \left (d x + c\right )^{6} - 4620 \, a^{3} b^{5} \sin \left (d x + c\right )^{6} + 2772 \, a^{6} b^{2} \sin \left (d x + c\right )^{5} - 6930 \, a^{4} b^{4} \sin \left (d x + c\right )^{5} + 990 \, a^{7} b \sin \left (d x + c\right )^{4} - 6930 \, a^{5} b^{3} \sin \left (d x + c\right )^{4} + 165 \, a^{8} \sin \left (d x + c\right )^{3} - 4620 \, a^{6} b^{2} \sin \left (d x + c\right )^{3} - 1980 \, a^{7} b \sin \left (d x + c\right )^{2} - 495 \, a^{8} \sin \left (d x + c\right )}{495 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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