Integrand size = 21, antiderivative size = 137 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {21 a^4 x}{16}-\frac {7 a^4 \cos ^3(c+d x)}{8 d}+\frac {21 a^4 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}-\frac {3 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{10 d}-\frac {21 \cos ^3(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{40 d} \]
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Time = 0.13 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2757, 2748, 2715, 8} \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {7 a^4 \cos ^3(c+d x)}{8 d}-\frac {21 \cos ^3(c+d x) \left (a^4 \sin (c+d x)+a^4\right )}{40 d}+\frac {21 a^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {21 a^4 x}{16}-\frac {3 \cos ^3(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^2}{10 d}-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d} \]
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Rule 8
Rule 2715
Rule 2748
Rule 2757
Rubi steps \begin{align*} \text {integral}& = -\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}+\frac {1}{2} (3 a) \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 \, dx \\ & = -\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}-\frac {3 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{10 d}+\frac {1}{10} \left (21 a^2\right ) \int \cos ^2(c+d x) (a+a \sin (c+d x))^2 \, dx \\ & = -\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}-\frac {3 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{10 d}-\frac {21 \cos ^3(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{40 d}+\frac {1}{8} \left (21 a^3\right ) \int \cos ^2(c+d x) (a+a \sin (c+d x)) \, dx \\ & = -\frac {7 a^4 \cos ^3(c+d x)}{8 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}-\frac {3 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{10 d}-\frac {21 \cos ^3(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{40 d}+\frac {1}{8} \left (21 a^4\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {7 a^4 \cos ^3(c+d x)}{8 d}+\frac {21 a^4 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}-\frac {3 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{10 d}-\frac {21 \cos ^3(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{40 d}+\frac {1}{16} \left (21 a^4\right ) \int 1 \, dx \\ & = \frac {21 a^4 x}{16}-\frac {7 a^4 \cos ^3(c+d x)}{8 d}+\frac {21 a^4 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}-\frac {3 \cos ^3(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{10 d}-\frac {21 \cos ^3(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{40 d} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.10 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {a^4 \cos ^3(c+d x) \left (630 \arcsin \left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}+\sqrt {1+\sin (c+d x)} \left (448-373 \sin (c+d x)-331 \sin ^2(c+d x)-94 \sin ^3(c+d x)+158 \sin ^4(c+d x)+152 \sin ^5(c+d x)+40 \sin ^6(c+d x)\right )\right )}{240 d (-1+\sin (c+d x))^2 (1+\sin (c+d x))^{3/2}} \]
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Time = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.55
method | result | size |
parallelrisch | \(\frac {\left (252 d x +\sin \left (6 d x +6 c \right )-288 \cos \left (d x +c \right )-80 \cos \left (3 d x +3 c \right )+\frac {48 \cos \left (5 d x +5 c \right )}{5}+45 \sin \left (2 d x +2 c \right )-39 \sin \left (4 d x +4 c \right )-\frac {1792}{5}\right ) a^{4}}{192 d}\) | \(76\) |
risch | \(\frac {21 a^{4} x}{16}-\frac {3 a^{4} \cos \left (d x +c \right )}{2 d}+\frac {a^{4} \sin \left (6 d x +6 c \right )}{192 d}+\frac {a^{4} \cos \left (5 d x +5 c \right )}{20 d}-\frac {13 a^{4} \sin \left (4 d x +4 c \right )}{64 d}-\frac {5 a^{4} \cos \left (3 d x +3 c \right )}{12 d}+\frac {15 a^{4} \sin \left (2 d x +2 c \right )}{64 d}\) | \(107\) |
derivativedivides | \(\frac {a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+4 a^{4} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+6 a^{4} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-\frac {4 a^{4} \left (\cos ^{3}\left (d x +c \right )\right )}{3}+a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(182\) |
default | \(\frac {a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+4 a^{4} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+6 a^{4} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-\frac {4 a^{4} \left (\cos ^{3}\left (d x +c \right )\right )}{3}+a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(182\) |
norman | \(\frac {\frac {21 a^{4} x}{16}-\frac {56 a^{4}}{15 d}-\frac {5 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {235 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {63 a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {63 a^{4} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {235 a^{4} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {5 a^{4} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {63 a^{4} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {315 a^{4} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {105 a^{4} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {315 a^{4} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {63 a^{4} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {21 a^{4} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {8 a^{4} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {40 a^{4} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {72 a^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {16 a^{4} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {112 a^{4} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\) | \(341\) |
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Time = 0.28 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.62 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {192 \, a^{4} \cos \left (d x + c\right )^{5} - 640 \, a^{4} \cos \left (d x + c\right )^{3} + 315 \, a^{4} d x + 5 \, {\left (8 \, a^{4} \cos \left (d x + c\right )^{5} - 86 \, a^{4} \cos \left (d x + c\right )^{3} + 63 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (128) = 256\).
Time = 0.37 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.78 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\begin {cases} \frac {a^{4} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 a^{4} x \sin ^{4}{\left (c + d x \right )}}{4} + \frac {3 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac {a^{4} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {a^{4} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{4} x \cos ^{4}{\left (c + d x \right )}}{4} + \frac {a^{4} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {a^{4} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {3 a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} - \frac {4 a^{4} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{4} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {3 a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} + \frac {a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} - \frac {8 a^{4} \cos ^{5}{\left (c + d x \right )}}{15 d} - \frac {4 a^{4} \cos ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{4} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.93 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=-\frac {1280 \, a^{4} \cos \left (d x + c\right )^{3} - 256 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{4} + 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} - 180 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{960 \, d} \]
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Time = 0.36 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.77 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {21}{16} \, a^{4} x + \frac {a^{4} \cos \left (5 \, d x + 5 \, c\right )}{20 \, d} - \frac {5 \, a^{4} \cos \left (3 \, d x + 3 \, c\right )}{12 \, d} - \frac {3 \, a^{4} \cos \left (d x + c\right )}{2 \, d} + \frac {a^{4} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {13 \, a^{4} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {15 \, a^{4} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
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Time = 11.83 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.55 \[ \int \cos ^2(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {21\,a^4\,x}{16}-\frac {\frac {63\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}-\frac {63\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}-\frac {235\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {235\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}-\frac {5\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {a^4\,\left (315\,c+315\,d\,x\right )}{240}-\frac {a^4\,\left (315\,c+315\,d\,x-896\right )}{240}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {a^4\,\left (315\,c+315\,d\,x\right )}{40}-\frac {a^4\,\left (1890\,c+1890\,d\,x-1920\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^4\,\left (315\,c+315\,d\,x\right )}{40}-\frac {a^4\,\left (1890\,c+1890\,d\,x-3456\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^4\,\left (315\,c+315\,d\,x\right )}{16}-\frac {a^4\,\left (4725\,c+4725\,d\,x-3840\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {a^4\,\left (315\,c+315\,d\,x\right )}{16}-\frac {a^4\,\left (4725\,c+4725\,d\,x-9600\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^4\,\left (315\,c+315\,d\,x\right )}{12}-\frac {a^4\,\left (6300\,c+6300\,d\,x-8960\right )}{240}\right )+\frac {5\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \]
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