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 = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 43
Rule 2667
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*}
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Mathematica [A] time = 1.09, 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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fricas [B] time = 0.75, size = 136, normalized size = 3.02 \[ \frac {44 \, a^{8} \cos \left (d x + c\right )^{10} - 550 \, a^{8} \cos \left (d x + c\right )^{8} + 1760 \, a^{8} \cos \left (d x + c\right )^{6} - 1760 \, a^{8} \cos \left (d x + c\right )^{4} + {\left (5 \, a^{8} \cos \left (d x + c\right )^{10} - 190 \, a^{8} \cos \left (d x + c\right )^{8} + 1040 \, a^{8} \cos \left (d x + c\right )^{6} - 1568 \, a^{8} \cos \left (d x + c\right )^{4} + 256 \, a^{8} \cos \left (d x + c\right )^{2} + 512 \, a^{8}\right )} \sin \left (d x + c\right )}{55 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.57, size = 134, normalized size = 2.98 \[ -\frac {5 \, a^{8} \sin \left (d x + c\right )^{11} + 44 \, a^{8} \sin \left (d x + c\right )^{10} + 165 \, a^{8} \sin \left (d x + c\right )^{9} + 330 \, a^{8} \sin \left (d x + c\right )^{8} + 330 \, a^{8} \sin \left (d x + c\right )^{7} - 462 \, a^{8} \sin \left (d x + c\right )^{5} - 660 \, a^{8} \sin \left (d x + c\right )^{4} - 495 \, a^{8} \sin \left (d x + c\right )^{3} - 220 \, a^{8} \sin \left (d x + c\right )^{2} - 55 \, a^{8} \sin \left (d x + c\right )}{55 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 463, normalized size = 10.29 \[ \frac {a^{8} \left (-\frac {\left (\sin ^{7}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{11}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{99}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{99}-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{33}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{99}\right )+8 a^{8} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{10}-\frac {3 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{40}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{20}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{40}\right )+28 a^{8} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{9}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{63}-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{21}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{63}\right )+56 a^{8} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{8}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{12}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{24}\right )+70 a^{8} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{7}-\frac {3 \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{35}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{35}\right )+56 a^{8} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )+28 a^{8} \left (-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{5}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{15}\right )-2 \left (\cos ^{4}\left (d x +c \right )\right ) a^{8}+\frac {a^{8} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 134, normalized size = 2.98 \[ -\frac {5 \, a^{8} \sin \left (d x + c\right )^{11} + 44 \, a^{8} \sin \left (d x + c\right )^{10} + 165 \, a^{8} \sin \left (d x + c\right )^{9} + 330 \, a^{8} \sin \left (d x + c\right )^{8} + 330 \, a^{8} \sin \left (d x + c\right )^{7} - 462 \, a^{8} \sin \left (d x + c\right )^{5} - 660 \, a^{8} \sin \left (d x + c\right )^{4} - 495 \, a^{8} \sin \left (d x + c\right )^{3} - 220 \, a^{8} \sin \left (d x + c\right )^{2} - 55 \, a^{8} \sin \left (d x + c\right )}{55 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 132, normalized size = 2.93 \[ \frac {-\frac {a^8\,{\sin \left (c+d\,x\right )}^{11}}{11}-\frac {4\,a^8\,{\sin \left (c+d\,x\right )}^{10}}{5}-3\,a^8\,{\sin \left (c+d\,x\right )}^9-6\,a^8\,{\sin \left (c+d\,x\right )}^8-6\,a^8\,{\sin \left (c+d\,x\right )}^7+\frac {42\,a^8\,{\sin \left (c+d\,x\right )}^5}{5}+12\,a^8\,{\sin \left (c+d\,x\right )}^4+9\,a^8\,{\sin \left (c+d\,x\right )}^3+4\,a^8\,{\sin \left (c+d\,x\right )}^2+a^8\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 50.56, size = 422, normalized size = 9.38 \[ \begin {cases} \frac {2 a^{8} \sin ^{11}{\left (c + d x \right )}}{99 d} + \frac {a^{8} \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{9 d} + \frac {8 a^{8} \sin ^{9}{\left (c + d x \right )}}{9 d} + \frac {4 a^{8} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {4 a^{8} \sin ^{7}{\left (c + d x \right )}}{d} - \frac {2 a^{8} \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {14 a^{8} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {56 a^{8} \sin ^{5}{\left (c + d x \right )}}{15 d} - \frac {2 a^{8} \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {14 a^{8} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {28 a^{8} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {2 a^{8} \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} - \frac {28 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {14 a^{8} \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {a^{8} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac {a^{8} \cos ^{10}{\left (c + d x \right )}}{5 d} - \frac {7 a^{8} \cos ^{8}{\left (c + d x \right )}}{3 d} - \frac {14 a^{8} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {2 a^{8} \cos ^{4}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{8} \cos ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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