Optimal. Leaf size=146 \[ \frac {5}{128} \left (8 a^2+b^2\right ) x-\frac {9 a b \cos ^7(c+d x)}{56 d}+\frac {5 \left (8 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 \left (8 a^2+b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {\left (8 a^2+b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2771, 2748,
2715, 8} \begin {gather*} \frac {\left (8 a^2+b^2\right ) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {5 \left (8 a^2+b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {5 \left (8 a^2+b^2\right ) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {5}{128} x \left (8 a^2+b^2\right )-\frac {9 a b \cos ^7(c+d x)}{56 d}-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2748
Rule 2771
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx &=-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}+\frac {1}{8} \int \cos ^6(c+d x) \left (8 a^2+b^2+9 a b \sin (c+d x)\right ) \, dx\\ &=-\frac {9 a b \cos ^7(c+d x)}{56 d}-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}+\frac {1}{8} \left (8 a^2+b^2\right ) \int \cos ^6(c+d x) \, dx\\ &=-\frac {9 a b \cos ^7(c+d x)}{56 d}+\frac {\left (8 a^2+b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}+\frac {1}{48} \left (5 \left (8 a^2+b^2\right )\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac {9 a b \cos ^7(c+d x)}{56 d}+\frac {5 \left (8 a^2+b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {\left (8 a^2+b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}+\frac {1}{64} \left (5 \left (8 a^2+b^2\right )\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {9 a b \cos ^7(c+d x)}{56 d}+\frac {5 \left (8 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 \left (8 a^2+b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {\left (8 a^2+b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}+\frac {1}{128} \left (5 \left (8 a^2+b^2\right )\right ) \int 1 \, dx\\ &=\frac {5}{128} \left (8 a^2+b^2\right ) x-\frac {9 a b \cos ^7(c+d x)}{56 d}+\frac {5 \left (8 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 \left (8 a^2+b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {\left (8 a^2+b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 141, normalized size = 0.97 \begin {gather*} \frac {840 \left (8 a^2+b^2\right ) (c+d x)-3360 a b \cos (c+d x)-2016 a b \cos (3 (c+d x))-672 a b \cos (5 (c+d x))-96 a b \cos (7 (c+d x))+336 \left (15 a^2+b^2\right ) \sin (2 (c+d x))+168 \left (6 a^2-b^2\right ) \sin (4 (c+d x))+112 (a-b) (a+b) \sin (6 (c+d x))-21 b^2 \sin (8 (c+d x))}{21504 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 128, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {b^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {2 a b \left (\cos ^{7}\left (d x +c \right )\right )}{7}+a^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(128\) |
default | \(\frac {b^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {2 a b \left (\cos ^{7}\left (d x +c \right )\right )}{7}+a^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(128\) |
risch | \(\frac {5 a^{2} x}{16}+\frac {5 b^{2} x}{128}-\frac {5 a b \cos \left (d x +c \right )}{32 d}-\frac {b^{2} \sin \left (8 d x +8 c \right )}{1024 d}-\frac {a b \cos \left (7 d x +7 c \right )}{224 d}+\frac {a^{2} \sin \left (6 d x +6 c \right )}{192 d}-\frac {\sin \left (6 d x +6 c \right ) b^{2}}{192 d}-\frac {a b \cos \left (5 d x +5 c \right )}{32 d}+\frac {3 a^{2} \sin \left (4 d x +4 c \right )}{64 d}-\frac {\sin \left (4 d x +4 c \right ) b^{2}}{128 d}-\frac {3 a b \cos \left (3 d x +3 c \right )}{32 d}+\frac {15 a^{2} \sin \left (2 d x +2 c \right )}{64 d}+\frac {\sin \left (2 d x +2 c \right ) b^{2}}{64 d}\) | \(194\) |
norman | \(\frac {\left (\frac {5 a^{2}}{16}+\frac {5 b^{2}}{128}\right ) x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\left (488 a^{2}+397 b^{2}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {4 a b}{7 d}-\frac {12 a b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {20 a b \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d}-\frac {4 a b \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {12 a b \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {20 a b \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\left (\frac {5 a^{2}}{16}+\frac {5 b^{2}}{128}\right ) x -\frac {5 \left (136 a^{2}+353 b^{2}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {4 a b \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (488 a^{2}+397 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {\left (88 a^{2}-5 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {5 \left (136 a^{2}+353 b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\left (\frac {5 a^{2}}{2}+\frac {5 b^{2}}{16}\right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {35 a^{2}}{4}+\frac {35 b^{2}}{32}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {175 a^{2}}{8}+\frac {175 b^{2}}{64}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5 a^{2}}{2}+\frac {5 b^{2}}{16}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\left (88 a^{2}-5 b^{2}\right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\left (\frac {35 a^{2}}{2}+\frac {35 b^{2}}{16}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {35 a^{2}}{4}+\frac {35 b^{2}}{32}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {35 a^{2}}{2}+\frac {35 b^{2}}{16}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\left (904 a^{2}-895 b^{2}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {\left (904 a^{2}-895 b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\) | \(570\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 114, normalized size = 0.78 \begin {gather*} -\frac {6144 \, a b \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^{2} - 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^{2}}{21504 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 108, normalized size = 0.74 \begin {gather*} -\frac {768 \, a b \cos \left (d x + c\right )^{7} - 105 \, {\left (8 \, a^{2} + b^{2}\right )} d x + 7 \, {\left (48 \, b^{2} \cos \left (d x + c\right )^{7} - 8 \, {\left (8 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{5} - 10 \, {\left (8 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (8 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2688 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 398 vs.
\(2 (141) = 282\).
time = 0.93, size = 398, normalized size = 2.73 \begin {gather*} \begin {cases} \frac {5 a^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 a^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {2 a b \cos ^{7}{\left (c + d x \right )}}{7 d} + \frac {5 b^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {5 b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {15 b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {5 b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {5 b^{2} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {5 b^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {55 b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {73 b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} - \frac {5 b^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{2} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 7.27, size = 162, normalized size = 1.11 \begin {gather*} \frac {5}{128} \, {\left (8 \, a^{2} + b^{2}\right )} x - \frac {a b \cos \left (7 \, d x + 7 \, c\right )}{224 \, d} - \frac {a b \cos \left (5 \, d x + 5 \, c\right )}{32 \, d} - \frac {3 \, a b \cos \left (3 \, d x + 3 \, c\right )}{32 \, d} - \frac {5 \, a b \cos \left (d x + c\right )}{32 \, d} - \frac {b^{2} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {{\left (a^{2} - b^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (6 \, a^{2} - b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (15 \, a^{2} + b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.47, size = 178, normalized size = 1.22 \begin {gather*} \frac {5\,a^2\,x}{16}+\frac {5\,b^2\,x}{128}+\frac {5\,a^2\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{24\,d}+\frac {a^2\,{\cos \left (c+d\,x\right )}^5\,\sin \left (c+d\,x\right )}{6\,d}+\frac {5\,b^2\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{192\,d}+\frac {b^2\,{\cos \left (c+d\,x\right )}^5\,\sin \left (c+d\,x\right )}{48\,d}-\frac {b^2\,{\cos \left (c+d\,x\right )}^7\,\sin \left (c+d\,x\right )}{8\,d}-\frac {2\,a\,b\,{\cos \left (c+d\,x\right )}^7}{7\,d}+\frac {5\,a^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{16\,d}+\frac {5\,b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{128\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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