Integrand size = 27, antiderivative size = 103 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a x}{16}-\frac {a \cos ^5(c+d x)}{5 d}+\frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a \cos ^5(c+d x) \sin (c+d x)}{6 d} \]
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Time = 0.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2917, 2648, 2715, 8, 2645, 14} \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \cos ^5(c+d x)}{5 d}-\frac {a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {a \sin (c+d x) \cos (c+d x)}{16 d}+\frac {a x}{16} \]
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Rule 8
Rule 14
Rule 2645
Rule 2648
Rule 2715
Rule 2917
Rubi steps \begin{align*} \text {integral}& = a \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx+a \int \cos ^4(c+d x) \sin ^3(c+d x) \, dx \\ & = -\frac {a \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{6} a \int \cos ^4(c+d x) \, dx-\frac {a \text {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {a \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{8} a \int \cos ^2(c+d x) \, dx-\frac {a \text {Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a \cos ^5(c+d x)}{5 d}+\frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{16} a \int 1 \, dx \\ & = \frac {a x}{16}-\frac {a \cos ^5(c+d x)}{5 d}+\frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a \cos ^5(c+d x) \sin (c+d x)}{6 d} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.79 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a (420 d x-315 \cos (c+d x)-105 \cos (3 (c+d x))+21 \cos (5 (c+d x))+15 \cos (7 (c+d x))+105 \sin (2 (c+d x))-105 \sin (4 (c+d x))-35 \sin (6 (c+d x)))}{6720 d} \]
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Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.81
method | result | size |
parallelrisch | \(-\frac {\left (-4 d x +\sin \left (4 d x +4 c \right )+\frac {\sin \left (6 d x +6 c \right )}{3}+3 \cos \left (d x +c \right )+\cos \left (3 d x +3 c \right )-\frac {\cos \left (5 d x +5 c \right )}{5}-\frac {\cos \left (7 d x +7 c \right )}{7}-\sin \left (2 d x +2 c \right )+\frac {128}{35}\right ) a}{64 d}\) | \(83\) |
derivativedivides | \(\frac {a \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+a \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )}{d}\) | \(88\) |
default | \(\frac {a \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+a \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )}{d}\) | \(88\) |
risch | \(\frac {a x}{16}-\frac {3 a \cos \left (d x +c \right )}{64 d}+\frac {a \cos \left (7 d x +7 c \right )}{448 d}-\frac {a \sin \left (6 d x +6 c \right )}{192 d}+\frac {a \cos \left (5 d x +5 c \right )}{320 d}-\frac {a \sin \left (4 d x +4 c \right )}{64 d}-\frac {a \cos \left (3 d x +3 c \right )}{64 d}+\frac {a \sin \left (2 d x +2 c \right )}{64 d}\) | \(108\) |
norman | \(\frac {\frac {4 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a x}{16}-\frac {4 a}{35 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {11 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {31 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {31 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {11 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {7 a x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {21 a x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {21 a x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {7 a x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {a x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {4 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {8 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\) | \(318\) |
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Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.71 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {240 \, a \cos \left (d x + c\right )^{7} - 336 \, a \cos \left (d x + c\right )^{5} + 105 \, a d x - 35 \, {\left (8 \, a \cos \left (d x + c\right )^{5} - 2 \, a \cos \left (d x + c\right )^{3} - 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (90) = 180\).
Time = 0.46 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.86 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {a x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 a x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 a x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {a x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {a \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {a \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {2 a \cos ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{2}{\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.63 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {192 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a}{6720 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.04 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {1}{16} \, a x + \frac {a \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {a \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {a \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {3 \, a \cos \left (d x + c\right )}{64 \, d} - \frac {a \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {a \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
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Time = 13.85 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.83 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,x}{16}+\frac {\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{8}-\frac {11\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{6}+\left (\frac {a\,\left (2205\,c+2205\,d\,x-6720\right )}{1680}-\frac {21\,a\,\left (c+d\,x\right )}{16}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {31\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+\left (\frac {a\,\left (3675\,c+3675\,d\,x+6720\right )}{1680}-\frac {35\,a\,\left (c+d\,x\right )}{16}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\left (\frac {a\,\left (3675\,c+3675\,d\,x-13440\right )}{1680}-\frac {35\,a\,\left (c+d\,x\right )}{16}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {31\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{24}+\left (\frac {a\,\left (2205\,c+2205\,d\,x+2688\right )}{1680}-\frac {21\,a\,\left (c+d\,x\right )}{16}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {11\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6}+\left (\frac {a\,\left (735\,c+735\,d\,x-1344\right )}{1680}-\frac {7\,a\,\left (c+d\,x\right )}{16}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {a\,\left (105\,c+105\,d\,x-192\right )}{1680}-\frac {a\,\left (c+d\,x\right )}{16}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]
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