Optimal. Leaf size=138 \[ -\frac {a (8 A+B) \cos ^7(c+d x)}{56 d}+\frac {a (8 A+B) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {5 a (8 A+B) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {5 a (8 A+B) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {5}{128} a x (8 A+B)-\frac {B \cos ^7(c+d x) (a \sin (c+d x)+a)}{8 d} \]
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Rubi [A] time = 0.14, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2860, 2669, 2635, 8} \[ -\frac {a (8 A+B) \cos ^7(c+d x)}{56 d}+\frac {a (8 A+B) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {5 a (8 A+B) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {5 a (8 A+B) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {5}{128} a x (8 A+B)-\frac {B \cos ^7(c+d x) (a \sin (c+d x)+a)}{8 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2860
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))}{8 d}+\frac {1}{8} (8 A+B) \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac {a (8 A+B) \cos ^7(c+d x)}{56 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))}{8 d}+\frac {1}{8} (a (8 A+B)) \int \cos ^6(c+d x) \, dx\\ &=-\frac {a (8 A+B) \cos ^7(c+d x)}{56 d}+\frac {a (8 A+B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))}{8 d}+\frac {1}{48} (5 a (8 A+B)) \int \cos ^4(c+d x) \, dx\\ &=-\frac {a (8 A+B) \cos ^7(c+d x)}{56 d}+\frac {5 a (8 A+B) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a (8 A+B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))}{8 d}+\frac {1}{64} (5 a (8 A+B)) \int \cos ^2(c+d x) \, dx\\ &=-\frac {a (8 A+B) \cos ^7(c+d x)}{56 d}+\frac {5 a (8 A+B) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 a (8 A+B) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a (8 A+B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))}{8 d}+\frac {1}{128} (5 a (8 A+B)) \int 1 \, dx\\ &=\frac {5}{128} a (8 A+B) x-\frac {a (8 A+B) \cos ^7(c+d x)}{56 d}+\frac {5 a (8 A+B) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 a (8 A+B) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a (8 A+B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.84, size = 164, normalized size = 1.19 \[ -\frac {a (1680 (A+B) \cos (c+d x)+1008 (A+B) \cos (3 (c+d x))-5040 A \sin (2 (c+d x))-1008 A \sin (4 (c+d x))-112 A \sin (6 (c+d x))+336 A \cos (5 (c+d x))+48 A \cos (7 (c+d x))-6720 A d x-336 B \sin (2 (c+d x))+168 B \sin (4 (c+d x))+112 B \sin (6 (c+d x))+21 B \sin (8 (c+d x))+336 B \cos (5 (c+d x))+48 B \cos (7 (c+d x))-840 B d x)}{21504 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 97, normalized size = 0.70 \[ -\frac {384 \, {\left (A + B\right )} a \cos \left (d x + c\right )^{7} - 105 \, {\left (8 \, A + B\right )} a d x + 7 \, {\left (48 \, B a \cos \left (d x + c\right )^{7} - 8 \, {\left (8 \, A + B\right )} a \cos \left (d x + c\right )^{5} - 10 \, {\left (8 \, A + B\right )} a \cos \left (d x + c\right )^{3} - 15 \, {\left (8 \, A + B\right )} a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2688 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 176, normalized size = 1.28 \[ \frac {5}{128} \, {\left (8 \, A a + B a\right )} x - \frac {B a \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {{\left (A a + B a\right )} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {{\left (A a + B a\right )} \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac {3 \, {\left (A a + B a\right )} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {5 \, {\left (A a + B a\right )} \cos \left (d x + c\right )}{64 \, d} + \frac {{\left (A a - B a\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (6 \, A a - B a\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (15 \, A a + B a\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 138, normalized size = 1.00 \[ \frac {a B \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {a A \left (\cos ^{7}\left (d x +c \right )\right )}{7}-\frac {a B \left (\cos ^{7}\left (d x +c \right )\right )}{7}+a A \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 124, normalized size = 0.90 \[ -\frac {3072 \, A a \cos \left (d x + c\right )^{7} + 3072 \, B a \cos \left (d x + c\right )^{7} + 112 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 7 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} B a}{21504 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.72, size = 504, normalized size = 3.65 \[ \frac {5\,a\,\mathrm {atan}\left (\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,A+B\right )}{64\,\left (\frac {5\,A\,a}{8}+\frac {5\,B\,a}{64}\right )}\right )\,\left (8\,A+B\right )}{64\,d}-\frac {5\,a\,\left (8\,A+B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{64\,d}-\frac {\left (\frac {11\,A\,a}{8}-\frac {5\,B\,a}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+\left (2\,A\,a+2\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+\left (\frac {61\,A\,a}{24}+\frac {397\,B\,a}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (2\,A\,a+2\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\left (\frac {113\,A\,a}{24}-\frac {895\,B\,a}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (10\,A\,a+10\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\left (\frac {85\,A\,a}{24}+\frac {1765\,B\,a}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (10\,A\,a+10\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\left (-\frac {85\,A\,a}{24}-\frac {1765\,B\,a}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (6\,A\,a+6\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {895\,B\,a}{192}-\frac {113\,A\,a}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (6\,A\,a+6\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-\frac {61\,A\,a}{24}-\frac {397\,B\,a}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {2\,A\,a}{7}+\frac {2\,B\,a}{7}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\left (\frac {5\,B\,a}{64}-\frac {11\,A\,a}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {2\,A\,a}{7}+\frac {2\,B\,a}{7}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,{\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 = 9.70, size = 416, normalized size = 3.01 \[ \begin {cases} \frac {5 A a x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 A a x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 A a x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 A a x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 A a \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 A a \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 A a \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {A a \cos ^{7}{\left (c + d x \right )}}{7 d} + \frac {5 B a x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {5 B a x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {15 B a x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {5 B a x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {5 B a x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {5 B a \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {55 B a \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {73 B a \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} - \frac {5 B a \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {B a \cos ^{7}{\left (c + d x \right )}}{7 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\relax (c )}\right ) \left (a \sin {\relax (c )} + a\right ) \cos ^{6}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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