Optimal. Leaf size=102 \[ \frac {B (a \sin (c+d x)+a)^7}{7 a^6 d}+\frac {(A-5 B) (a \sin (c+d x)+a)^6}{6 a^5 d}-\frac {4 (A-2 B) (a \sin (c+d x)+a)^5}{5 a^4 d}+\frac {(A-B) (a \sin (c+d x)+a)^4}{a^3 d} \]
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Rubi [A] time = 0.11, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2836, 77} \[ \frac {(A-5 B) (a \sin (c+d x)+a)^6}{6 a^5 d}-\frac {4 (A-2 B) (a \sin (c+d x)+a)^5}{5 a^4 d}+\frac {(A-B) (a \sin (c+d x)+a)^4}{a^3 d}+\frac {B (a \sin (c+d x)+a)^7}{7 a^6 d} \]
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)) (A+B \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^2 (a+x)^3 \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)^3-4 a (A-2 B) (a+x)^4+(A-5 B) (a+x)^5+\frac {B (a+x)^6}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {(A-B) (a+a \sin (c+d x))^4}{a^3 d}-\frac {4 (A-2 B) (a+a \sin (c+d x))^5}{5 a^4 d}+\frac {(A-5 B) (a+a \sin (c+d x))^6}{6 a^5 d}+\frac {B (a+a \sin (c+d x))^7}{7 a^6 d}\\ \end {align*}
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Mathematica [A] time = 0.66, size = 130, normalized size = 1.27 \[ -\frac {a (525 (A+B) \cos (2 (c+d x))+210 (A+B) \cos (4 (c+d x))-4200 A \sin (c+d x)-700 A \sin (3 (c+d x))-84 A \sin (5 (c+d x))+35 A \cos (6 (c+d x))-525 B \sin (c+d x)+35 B \sin (3 (c+d x))+63 B \sin (5 (c+d x))+15 B \sin (7 (c+d x))+35 B \cos (6 (c+d x)))}{6720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 81, normalized size = 0.79 \[ -\frac {35 \, {\left (A + B\right )} a \cos \left (d x + c\right )^{6} + 2 \, {\left (15 \, B a \cos \left (d x + c\right )^{6} - 3 \, {\left (7 \, A + B\right )} a \cos \left (d x + c\right )^{4} - 4 \, {\left (7 \, A + B\right )} a \cos \left (d x + c\right )^{2} - 8 \, {\left (7 \, A + B\right )} a\right )} \sin \left (d x + c\right )}{210 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 145, normalized size = 1.42 \[ -\frac {B a \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {{\left (A a + B a\right )} \cos \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {{\left (A a + B a\right )} \cos \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac {5 \, {\left (A a + B a\right )} \cos \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (4 \, A a - 3 \, B a\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (20 \, A a - B a\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {5 \, {\left (8 \, A a + B a\right )} \sin \left (d x + c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 108, normalized size = 1.06 \[ \frac {a B \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {a A \left (\cos ^{6}\left (d x +c \right )\right )}{6}-\frac {a B \left (\cos ^{6}\left (d x +c \right )\right )}{6}+\frac {a A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 104, normalized size = 1.02 \[ \frac {30 \, B a \sin \left (d x + c\right )^{7} + 35 \, {\left (A + B\right )} a \sin \left (d x + c\right )^{6} + 42 \, {\left (A - 2 \, B\right )} a \sin \left (d x + c\right )^{5} - 105 \, {\left (A + B\right )} a \sin \left (d x + c\right )^{4} - 70 \, {\left (2 \, A - B\right )} a \sin \left (d x + c\right )^{3} + 105 \, {\left (A + B\right )} a \sin \left (d x + c\right )^{2} + 210 \, A a \sin \left (d x + c\right )}{210 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 102, normalized size = 1.00 \[ \frac {\frac {B\,a\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a\,\left (A+B\right )\,{\sin \left (c+d\,x\right )}^6}{6}+\frac {a\,\left (A-2\,B\right )\,{\sin \left (c+d\,x\right )}^5}{5}-\frac {a\,\left (A+B\right )\,{\sin \left (c+d\,x\right )}^4}{2}-\frac {a\,\left (2\,A-B\right )\,{\sin \left (c+d\,x\right )}^3}{3}+\frac {a\,\left (A+B\right )\,{\sin \left (c+d\,x\right )}^2}{2}+A\,a\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.33, size = 178, normalized size = 1.75 \[ \begin {cases} \frac {8 A a \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 A a \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {A a \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {A a \cos ^{6}{\left (c + d x \right )}}{6 d} + \frac {8 B a \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac {4 B a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac {B a \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} - \frac {B a \cos ^{6}{\left (c + d x \right )}}{6 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\relax (c )}\right ) \left (a \sin {\relax (c )} + a\right ) \cos ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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