Optimal. Leaf size=81 \[ -\frac {a \cos ^6(c+d x)}{6 d}+\frac {a \cos ^8(c+d x)}{8 d}+\frac {a \sin ^3(c+d x)}{3 d}-\frac {2 a \sin ^5(c+d x)}{5 d}+\frac {a \sin ^7(c+d x)}{7 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2913, 2644,
276, 2645, 14} \begin {gather*} \frac {a \sin ^7(c+d x)}{7 d}-\frac {2 a \sin ^5(c+d x)}{5 d}+\frac {a \sin ^3(c+d x)}{3 d}+\frac {a \cos ^8(c+d x)}{8 d}-\frac {a \cos ^6(c+d x)}{6 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 276
Rule 2644
Rule 2645
Rule 2913
Rubi steps
\begin {align*} \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^5(c+d x) \sin ^2(c+d x) \, dx+a \int \cos ^5(c+d x) \sin ^3(c+d x) \, dx\\ &=-\frac {a \text {Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {a \text {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {a \text {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sin (c+d x)\right )}{d}-\frac {a \text {Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a \cos ^6(c+d x)}{6 d}+\frac {a \cos ^8(c+d x)}{8 d}+\frac {a \sin ^3(c+d x)}{3 d}-\frac {2 a \sin ^5(c+d x)}{5 d}+\frac {a \sin ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 87, normalized size = 1.07 \begin {gather*} -\frac {a (2520 \cos (2 (c+d x))+420 \cos (4 (c+d x))-280 \cos (6 (c+d x))-105 \cos (8 (c+d x))-8400 \sin (c+d x)+560 \sin (3 (c+d x))+1008 \sin (5 (c+d x))+240 \sin (7 (c+d x)))}{107520 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 84, normalized size = 1.04
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )+a \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )}{d}\) | \(84\) |
default | \(\frac {a \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )+a \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )}{d}\) | \(84\) |
risch | \(\frac {5 a \sin \left (d x +c \right )}{64 d}+\frac {a \cos \left (8 d x +8 c \right )}{1024 d}-\frac {a \sin \left (7 d x +7 c \right )}{448 d}+\frac {a \cos \left (6 d x +6 c \right )}{384 d}-\frac {3 a \sin \left (5 d x +5 c \right )}{320 d}-\frac {a \cos \left (4 d x +4 c \right )}{256 d}-\frac {a \sin \left (3 d x +3 c \right )}{192 d}-\frac {3 a \cos \left (2 d x +2 c \right )}{128 d}\) | \(119\) |
norman | \(\frac {\frac {8 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {8 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {688 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 d}+\frac {688 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 d}+\frac {8 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {8 a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {16 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {16 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {4 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {40 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\) | \(205\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 72, normalized size = 0.89 \begin {gather*} \frac {105 \, a \sin \left (d x + c\right )^{8} + 120 \, a \sin \left (d x + c\right )^{7} - 280 \, a \sin \left (d x + c\right )^{6} - 336 \, a \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} + 280 \, a \sin \left (d x + c\right )^{3}}{840 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 73, normalized size = 0.90 \begin {gather*} \frac {105 \, a \cos \left (d x + c\right )^{8} - 140 \, a \cos \left (d x + c\right )^{6} - 8 \, {\left (15 \, a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} - 8 \, a\right )} \sin \left (d x + c\right )}{840 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.44, size = 114, normalized size = 1.41 \begin {gather*} \begin {cases} \frac {8 a \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac {4 a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac {a \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{6 d} - \frac {a \cos ^{8}{\left (c + d x \right )}}{24 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{2}{\left (c \right )} \cos ^{5}{\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 = 0.50, size = 118, normalized size = 1.46 \begin {gather*} \frac {a \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {a \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {a \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {3 \, a \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} - \frac {a \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {3 \, a \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {a \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {5 \, a \sin \left (d x + c\right )}{64 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.69, size = 71, normalized size = 0.88 \begin {gather*} \frac {\frac {a\,{\sin \left (c+d\,x\right )}^8}{8}+\frac {a\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {a\,{\sin \left (c+d\,x\right )}^6}{3}-\frac {2\,a\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{4}+\frac {a\,{\sin \left (c+d\,x\right )}^3}{3}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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