Optimal. Leaf size=105 \[ \frac {B (a \sin (c+d x)+a)^9}{9 a^6 d}+\frac {(A-5 B) (a \sin (c+d x)+a)^8}{8 a^5 d}-\frac {4 (A-2 B) (a \sin (c+d x)+a)^7}{7 a^4 d}+\frac {2 (A-B) (a \sin (c+d x)+a)^6}{3 a^3 d} \]
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Rubi [A] time = 0.15, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2836, 77} \[ \frac {(A-5 B) (a \sin (c+d x)+a)^8}{8 a^5 d}-\frac {4 (A-2 B) (a \sin (c+d x)+a)^7}{7 a^4 d}+\frac {2 (A-B) (a \sin (c+d x)+a)^6}{3 a^3 d}+\frac {B (a \sin (c+d x)+a)^9}{9 a^6 d} \]
Antiderivative was successfully verified.
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Rule 77
Rule 2836
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^2 (a+x)^5 \left (A+\frac {B x}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^2 (A-B) (a+x)^5-4 a (A-2 B) (a+x)^6+(A-5 B) (a+x)^7+\frac {B (a+x)^8}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {2 (A-B) (a+a \sin (c+d x))^6}{3 a^3 d}-\frac {4 (A-2 B) (a+a \sin (c+d x))^7}{7 a^4 d}+\frac {(A-5 B) (a+a \sin (c+d x))^8}{8 a^5 d}+\frac {B (a+a \sin (c+d x))^9}{9 a^6 d}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 70, normalized size = 0.67 \[ \frac {a^3 (\sin (c+d x)+1)^6 \left (21 (3 A-7 B) \sin ^2(c+d x)-6 (27 A-19 B) \sin (c+d x)+111 A+56 B \sin ^3(c+d x)-19 B\right )}{504 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 129, normalized size = 1.23 \[ \frac {63 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{8} - 336 \, {\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{6} + 8 \, {\left (7 \, B a^{3} \cos \left (d x + c\right )^{8} - {\left (27 \, A + 37 \, B\right )} a^{3} \cos \left (d x + c\right )^{6} + 6 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{4} + 8 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{2} + 16 \, {\left (3 \, A + B\right )} a^{3}\right )} \sin \left (d x + c\right )}{504 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.60, size = 230, normalized size = 2.19 \[ \frac {B a^{3} \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {{\left (A a^{3} + 3 \, B a^{3}\right )} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {{\left (5 \, A a^{3} - B a^{3}\right )} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {{\left (25 \, A a^{3} + 11 \, B a^{3}\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {{\left (33 \, A a^{3} + 19 \, B a^{3}\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} - \frac {{\left (12 \, A a^{3} + 11 \, B a^{3}\right )} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {{\left (A a^{3} + 2 \, B a^{3}\right )} \sin \left (5 \, d x + 5 \, c\right )}{64 \, d} + \frac {{\left (17 \, A a^{3} - 4 \, B a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {11 \, {\left (10 \, A a^{3} + 3 \, B a^{3}\right )} \sin \left (d x + c\right )}{128 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.49, size = 305, normalized size = 2.90 \[ \frac {a^{3} A \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )+B \,a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{9}-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{21}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{105}\right )+3 a^{3} A \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )+3 B \,a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )-\frac {a^{3} A \left (\cos ^{6}\left (d x +c \right )\right )}{2}+3 B \,a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )+\frac {a^{3} A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}-\frac {B \,a^{3} \left (\cos ^{6}\left (d x +c \right )\right )}{6}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 158, normalized size = 1.50 \[ \frac {56 \, B a^{3} \sin \left (d x + c\right )^{9} + 63 \, {\left (A + 3 \, B\right )} a^{3} \sin \left (d x + c\right )^{8} + 72 \, {\left (3 \, A + B\right )} a^{3} \sin \left (d x + c\right )^{7} + 84 \, {\left (A - 5 \, B\right )} a^{3} \sin \left (d x + c\right )^{6} - 504 \, {\left (A + B\right )} a^{3} \sin \left (d x + c\right )^{5} - 126 \, {\left (5 \, A - B\right )} a^{3} \sin \left (d x + c\right )^{4} + 168 \, {\left (A + 3 \, B\right )} a^{3} \sin \left (d x + c\right )^{3} + 252 \, {\left (3 \, A + B\right )} a^{3} \sin \left (d x + c\right )^{2} + 504 \, A a^{3} \sin \left (d x + c\right )}{504 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 156, normalized size = 1.49 \[ \frac {\frac {a^3\,{\sin \left (c+d\,x\right )}^2\,\left (3\,A+B\right )}{2}+\frac {a^3\,{\sin \left (c+d\,x\right )}^3\,\left (A+3\,B\right )}{3}+\frac {a^3\,{\sin \left (c+d\,x\right )}^7\,\left (3\,A+B\right )}{7}+\frac {a^3\,{\sin \left (c+d\,x\right )}^6\,\left (A-5\,B\right )}{6}+\frac {a^3\,{\sin \left (c+d\,x\right )}^8\,\left (A+3\,B\right )}{8}+\frac {B\,a^3\,{\sin \left (c+d\,x\right )}^9}{9}-\frac {a^3\,{\sin \left (c+d\,x\right )}^4\,\left (5\,A-B\right )}{4}+A\,a^3\,\sin \left (c+d\,x\right )-a^3\,{\sin \left (c+d\,x\right )}^5\,\left (A+B\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.66, size = 471, normalized size = 4.49 \[ \begin {cases} \frac {A a^{3} \sin ^{8}{\left (c + d x \right )}}{24 d} + \frac {8 A a^{3} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {A a^{3} \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{6 d} + \frac {4 A a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {8 A a^{3} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {A a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} + \frac {A a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {4 A a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {A a^{3} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {A a^{3} \cos ^{6}{\left (c + d x \right )}}{2 d} + \frac {8 B a^{3} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {B a^{3} \sin ^{8}{\left (c + d x \right )}}{8 d} + \frac {4 B a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {8 B a^{3} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {B a^{3} \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 d} + \frac {B a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {4 B a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {3 B a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} + \frac {B a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {B a^{3} \cos ^{6}{\left (c + d x \right )}}{6 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\relax (c )}\right ) \left (a \sin {\relax (c )} + a\right )^{3} \cos ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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