Optimal. Leaf size=111 \[ -\frac {a (6 A+B) \cos ^5(c+d x)}{30 d}+\frac {a (6 A+B) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {a (6 A+B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a x (6 A+B)-\frac {B \cos ^5(c+d x) (a \sin (c+d x)+a)}{6 d} \]
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Rubi [A] time = 0.11, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2860, 2669, 2635, 8} \[ -\frac {a (6 A+B) \cos ^5(c+d x)}{30 d}+\frac {a (6 A+B) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {a (6 A+B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a x (6 A+B)-\frac {B \cos ^5(c+d x) (a \sin (c+d x)+a)}{6 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2860
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))}{6 d}+\frac {1}{6} (6 A+B) \int \cos ^4(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac {a (6 A+B) \cos ^5(c+d x)}{30 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))}{6 d}+\frac {1}{6} (a (6 A+B)) \int \cos ^4(c+d x) \, dx\\ &=-\frac {a (6 A+B) \cos ^5(c+d x)}{30 d}+\frac {a (6 A+B) \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))}{6 d}+\frac {1}{8} (a (6 A+B)) \int \cos ^2(c+d x) \, dx\\ &=-\frac {a (6 A+B) \cos ^5(c+d x)}{30 d}+\frac {a (6 A+B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a (6 A+B) \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))}{6 d}+\frac {1}{16} (a (6 A+B)) \int 1 \, dx\\ &=\frac {1}{16} a (6 A+B) x-\frac {a (6 A+B) \cos ^5(c+d x)}{30 d}+\frac {a (6 A+B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a (6 A+B) \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 120, normalized size = 1.08 \[ -\frac {a (120 (A+B) \cos (c+d x)+60 (A+B) \cos (3 (c+d x))-240 A \sin (2 (c+d x))-30 A \sin (4 (c+d x))+12 A \cos (5 (c+d x))-360 A d x-15 B \sin (2 (c+d x))+15 B \sin (4 (c+d x))+5 B \sin (6 (c+d x))+12 B \cos (5 (c+d x))-60 B d x)}{960 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 81, normalized size = 0.73 \[ -\frac {48 \, {\left (A + B\right )} a \cos \left (d x + c\right )^{5} - 15 \, {\left (6 \, A + B\right )} a d x + 5 \, {\left (8 \, B a \cos \left (d x + c\right )^{5} - 2 \, {\left (6 \, A + B\right )} a \cos \left (d x + c\right )^{3} - 3 \, {\left (6 \, A + B\right )} a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 133, normalized size = 1.20 \[ \frac {1}{16} \, {\left (6 \, A a + B a\right )} x - \frac {B a \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {{\left (A a + B a\right )} \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {{\left (A a + B a\right )} \cos \left (3 \, d x + 3 \, c\right )}{16 \, d} - \frac {{\left (A a + B a\right )} \cos \left (d x + c\right )}{8 \, d} + \frac {{\left (2 \, A a - B a\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (16 \, A a + B a\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.49, size = 118, normalized size = 1.06 \[ \frac {a B \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {a A \left (\cos ^{5}\left (d x +c \right )\right )}{5}-\frac {a B \left (\cos ^{5}\left (d x +c \right )\right )}{5}+a A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 98, normalized size = 0.88 \[ -\frac {192 \, A a \cos \left (d x + c\right )^{5} + 192 \, B a \cos \left (d x + c\right )^{5} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} B a}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.51, size = 391, normalized size = 3.52 \[ \frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,A+B\right )}{8\,\left (\frac {3\,A\,a}{4}+\frac {B\,a}{8}\right )}\right )\,\left (6\,A+B\right )}{8\,d}-\frac {a\,\left (6\,A+B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d}-\frac {\left (\frac {5\,A\,a}{4}-\frac {B\,a}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (2\,A\,a+2\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\left (\frac {7\,A\,a}{4}+\frac {47\,B\,a}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (2\,A\,a+2\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\left (\frac {A\,a}{2}-\frac {13\,B\,a}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (4\,A\,a+4\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {13\,B\,a}{4}-\frac {A\,a}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (4\,A\,a+4\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-\frac {7\,A\,a}{4}-\frac {47\,B\,a}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {2\,A\,a}{5}+\frac {2\,B\,a}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\left (\frac {B\,a}{8}-\frac {5\,A\,a}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {2\,A\,a}{5}+\frac {2\,B\,a}{5}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.50, size = 306, normalized size = 2.76 \[ \begin {cases} \frac {3 A a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 A a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 A a \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {A a \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac {B a x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 B a x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 B a x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {B a x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {B a \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {B a \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {B a \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {B a \cos ^{5}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\relax (c )}\right ) \left (a \sin {\relax (c )} + a\right ) \cos ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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