Optimal. Leaf size=126 \[ \frac {45 a^2 x}{128}-\frac {9 a^2 \cos ^7(c+d x)}{56 d}+\frac {45 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {15 a^2 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {3 a^2 \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{8 d} \]
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Rubi [A]
time = 0.07, antiderivative size = 126, 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 = {2757, 2748,
2715, 8} \begin {gather*} -\frac {9 a^2 \cos ^7(c+d x)}{56 d}-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}+\frac {3 a^2 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac {15 a^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {45 a^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {45 a^2 x}{128} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2748
Rule 2757
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 \, dx &=-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{8 d}+\frac {1}{8} (9 a) \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac {9 a^2 \cos ^7(c+d x)}{56 d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{8 d}+\frac {1}{8} \left (9 a^2\right ) \int \cos ^6(c+d x) \, dx\\ &=-\frac {9 a^2 \cos ^7(c+d x)}{56 d}+\frac {3 a^2 \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{8 d}+\frac {1}{16} \left (15 a^2\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac {9 a^2 \cos ^7(c+d x)}{56 d}+\frac {15 a^2 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {3 a^2 \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{8 d}+\frac {1}{64} \left (45 a^2\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {9 a^2 \cos ^7(c+d x)}{56 d}+\frac {45 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {15 a^2 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {3 a^2 \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{8 d}+\frac {1}{128} \left (45 a^2\right ) \int 1 \, dx\\ &=\frac {45 a^2 x}{128}-\frac {9 a^2 \cos ^7(c+d x)}{56 d}+\frac {45 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {15 a^2 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {3 a^2 \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{8 d}\\ \end {align*}
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Mathematica [A]
time = 0.98, size = 171, normalized size = 1.36 \begin {gather*} -\frac {a^2 \cos ^7(c+d x) \left (630 \sin ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}+\sqrt {1+\sin (c+d x)} \left (256-837 \sin (c+d x)-187 \sin ^2(c+d x)+978 \sin ^3(c+d x)+558 \sin ^4(c+d x)-600 \sin ^5(c+d x)-424 \sin ^6(c+d x)+144 \sin ^7(c+d x)+112 \sin ^8(c+d x)\right )\right )}{896 d (-1+\sin (c+d x))^4 (1+\sin (c+d x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 129, normalized size = 1.02
method | result | size |
risch | \(\frac {45 a^{2} x}{128}-\frac {5 a^{2} \cos \left (d x +c \right )}{32 d}-\frac {a^{2} \sin \left (8 d x +8 c \right )}{1024 d}-\frac {a^{2} \cos \left (7 d x +7 c \right )}{224 d}-\frac {a^{2} \cos \left (5 d x +5 c \right )}{32 d}+\frac {5 a^{2} \sin \left (4 d x +4 c \right )}{128 d}-\frac {3 a^{2} \cos \left (3 d x +3 c \right )}{32 d}+\frac {a^{2} \sin \left (2 d x +2 c \right )}{4 d}\) | \(124\) |
derivativedivides | \(\frac {a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right ) a^{2}}{7}+a^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(129\) |
default | \(\frac {a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right ) a^{2}}{7}+a^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(129\) |
norman | \(\frac {-\frac {83 a^{2} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {295 a^{2} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {315 a^{2} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {83 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {3 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {295 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {1575 a^{2} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {315 a^{2} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {815 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {4 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {45 a^{2} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {315 a^{2} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {4 a^{2}}{7 d}+\frac {315 a^{2} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {45 a^{2} x}{128}+\frac {45 a^{2} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {12 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d}-\frac {20 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {12 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {45 a^{2} x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}-\frac {20 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {3 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {815 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\) | \(451\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 115, normalized size = 0.91 \begin {gather*} -\frac {6144 \, a^{2} \cos \left (d x + c\right )^{7} - 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 )} a^{2} + 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}}{21504 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 85, normalized size = 0.67 \begin {gather*} -\frac {256 \, a^{2} \cos \left (d x + c\right )^{7} - 315 \, a^{2} d x + 7 \, {\left (16 \, a^{2} \cos \left (d x + c\right )^{7} - 24 \, a^{2} \cos \left (d x + c\right )^{5} - 30 \, a^{2} \cos \left (d x + c\right )^{3} - 45 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{896 \, 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.
\(2 (119) = 238\).
time = 0.99, size = 398, normalized size = 3.16 \begin {gather*} \begin {cases} \frac {5 a^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {5 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \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 ^{4}{\left (c + d x \right )}}{64} + \frac {15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {5 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \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 ^{8}{\left (c + d x \right )}}{128} + \frac {5 a^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 a^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {55 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {5 a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {73 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} + \frac {5 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {5 a^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} + \frac {11 a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {2 a^{2} \cos ^{7}{\left (c + d x \right )}}{7 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\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 = 5.47, size = 123, normalized size = 0.98 \begin {gather*} \frac {45}{128} \, a^{2} x - \frac {a^{2} \cos \left (7 \, d x + 7 \, c\right )}{224 \, d} - \frac {a^{2} \cos \left (5 \, d x + 5 \, c\right )}{32 \, d} - \frac {3 \, a^{2} \cos \left (3 \, d x + 3 \, c\right )}{32 \, d} - \frac {5 \, a^{2} \cos \left (d x + c\right )}{32 \, d} - \frac {a^{2} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {5 \, a^{2} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.92, size = 461, normalized size = 3.66 \begin {gather*} \frac {45\,a^2\,x}{128}-\frac {\frac {815\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{64}-\frac {815\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{64}-\frac {295\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64}+\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\frac {295\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}+\frac {83\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}+\frac {a^2\,\left (315\,c+315\,d\,x\right )}{896}-\frac {a^2\,\left (315\,c+315\,d\,x-512\right )}{896}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2\,\left (315\,c+315\,d\,x\right )}{112}-\frac {a^2\,\left (2520\,c+2520\,d\,x-512\right )}{896}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {a^2\,\left (315\,c+315\,d\,x\right )}{112}-\frac {a^2\,\left (2520\,c+2520\,d\,x-3584\right )}{896}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {a^2\,\left (315\,c+315\,d\,x\right )}{32}-\frac {a^2\,\left (8820\,c+8820\,d\,x-3584\right )}{896}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^2\,\left (315\,c+315\,d\,x\right )}{32}-\frac {a^2\,\left (8820\,c+8820\,d\,x-10752\right )}{896}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^2\,\left (315\,c+315\,d\,x\right )}{16}-\frac {a^2\,\left (17640\,c+17640\,d\,x-10752\right )}{896}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {a^2\,\left (315\,c+315\,d\,x\right )}{16}-\frac {a^2\,\left (17640\,c+17640\,d\,x-17920\right )}{896}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {5\,a^2\,\left (315\,c+315\,d\,x\right )}{64}-\frac {a^2\,\left (22050\,c+22050\,d\,x-17920\right )}{896}\right )-\frac {83\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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