Optimal. Leaf size=89 \[ \frac {(a \sin (c+d x)+a)^9}{9 a^6 d}-\frac {5 (a \sin (c+d x)+a)^8}{8 a^5 d}+\frac {8 (a \sin (c+d x)+a)^7}{7 a^4 d}-\frac {2 (a \sin (c+d x)+a)^6}{3 a^3 d} \]
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Rubi [A] time = 0.08, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2836, 12, 77} \[ \frac {(a \sin (c+d x)+a)^9}{9 a^6 d}-\frac {5 (a \sin (c+d x)+a)^8}{8 a^5 d}+\frac {8 (a \sin (c+d x)+a)^7}{7 a^4 d}-\frac {2 (a \sin (c+d x)+a)^6}{3 a^3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 77
Rule 2836
Rubi steps
\begin {align*} \int \cos ^5(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2 x (a+x)^5}{a} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int (a-x)^2 x (a+x)^5 \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-4 a^3 (a+x)^5+8 a^2 (a+x)^6-5 a (a+x)^7+(a+x)^8\right ) \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=-\frac {2 (a+a \sin (c+d x))^6}{3 a^3 d}+\frac {8 (a+a \sin (c+d x))^7}{7 a^4 d}-\frac {5 (a+a \sin (c+d x))^8}{8 a^5 d}+\frac {(a+a \sin (c+d x))^9}{9 a^6 d}\\ \end {align*}
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Mathematica [A] time = 0.68, size = 100, normalized size = 1.12 \[ \frac {a^3 (16632 \sin (c+d x)-1344 \sin (3 (c+d x))-2016 \sin (5 (c+d x))-396 \sin (7 (c+d x))+28 \sin (9 (c+d x))-9576 \cos (2 (c+d x))-2772 \cos (4 (c+d x))+168 \cos (6 (c+d x))+189 \cos (8 (c+d x))+4662)}{64512 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 98, normalized size = 1.10 \[ \frac {189 \, a^{3} \cos \left (d x + c\right )^{8} - 336 \, a^{3} \cos \left (d x + c\right )^{6} + 8 \, {\left (7 \, a^{3} \cos \left (d x + c\right )^{8} - 37 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} + 8 \, a^{3} \cos \left (d x + c\right )^{2} + 16 \, a^{3}\right )} \sin \left (d x + c\right )}{504 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 151, normalized size = 1.70 \[ \frac {3 \, a^{3} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {a^{3} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {11 \, a^{3} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {19 \, a^{3} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {a^{3} \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac {11 \, a^{3} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a^{3} \sin \left (5 \, d x + 5 \, c\right )}{32 \, d} - \frac {a^{3} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {33 \, a^{3} \sin \left (d x + c\right )}{128 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.26, size = 170, normalized size = 1.91 \[ \frac {a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{9}-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{21}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{105}\right )+3 a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {a^{3} \left (\cos ^{6}\left (d x +c \right )\right )}{6}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 110, normalized size = 1.24 \[ \frac {56 \, a^{3} \sin \left (d x + c\right )^{9} + 189 \, a^{3} \sin \left (d x + c\right )^{8} + 72 \, a^{3} \sin \left (d x + c\right )^{7} - 420 \, a^{3} \sin \left (d x + c\right )^{6} - 504 \, a^{3} \sin \left (d x + c\right )^{5} + 126 \, a^{3} \sin \left (d x + c\right )^{4} + 504 \, a^{3} \sin \left (d x + c\right )^{3} + 252 \, a^{3} \sin \left (d x + c\right )^{2}}{504 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 108, normalized size = 1.21 \[ \frac {\frac {a^3\,{\sin \left (c+d\,x\right )}^9}{9}+\frac {3\,a^3\,{\sin \left (c+d\,x\right )}^8}{8}+\frac {a^3\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {5\,a^3\,{\sin \left (c+d\,x\right )}^6}{6}-a^3\,{\sin \left (c+d\,x\right )}^5+\frac {a^3\,{\sin \left (c+d\,x\right )}^4}{4}+a^3\,{\sin \left (c+d\,x\right )}^3+\frac {a^3\,{\sin \left (c+d\,x\right )}^2}{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.30, size = 202, normalized size = 2.27 \[ \begin {cases} \frac {8 a^{3} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {4 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {8 a^{3} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {4 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{2 d} - \frac {a^{3} \cos ^{8}{\left (c + d x \right )}}{8 d} - \frac {a^{3} \cos ^{6}{\left (c + d x \right )}}{6 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{3} \sin {\relax (c )} \cos ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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