Integrand size = 29, antiderivative size = 103 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3 a^2 x}{16}-\frac {a^2 \cos ^5(c+d x)}{10 d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac {\cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 a d} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2949, 2748, 2715, 8} \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \cos ^5(c+d x)}{10 d}+\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac {3 a^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {3 a^2 x}{16}-\frac {\cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 a d} \]
[In]
[Out]
Rule 8
Rule 2715
Rule 2748
Rule 2949
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 a d}+\frac {1}{2} a \int \cos ^4(c+d x) (a+a \sin (c+d x)) \, dx \\ & = -\frac {a^2 \cos ^5(c+d x)}{10 d}-\frac {\cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 a d}+\frac {1}{2} a^2 \int \cos ^4(c+d x) \, dx \\ & = -\frac {a^2 \cos ^5(c+d x)}{10 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac {\cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 a d}+\frac {1}{8} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {a^2 \cos ^5(c+d x)}{10 d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac {\cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 a d}+\frac {1}{16} \left (3 a^2\right ) \int 1 \, dx \\ & = \frac {3 a^2 x}{16}-\frac {a^2 \cos ^5(c+d x)}{10 d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac {\cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 a d} \\ \end{align*}
Time = 5.74 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.74 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 (180 c+180 d x-240 \cos (c+d x)-40 \cos (3 (c+d x))+24 \cos (5 (c+d x))-15 \sin (2 (c+d x))-45 \sin (4 (c+d x))+5 \sin (6 (c+d x)))}{960 d} \]
[In]
[Out]
Time = 0.23 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.74
method | result | size |
parallelrisch | \(\frac {\left (36 d x +\sin \left (6 d x +6 c \right )-48 \cos \left (d x +c \right )-8 \cos \left (3 d x +3 c \right )+\frac {24 \cos \left (5 d x +5 c \right )}{5}-3 \sin \left (2 d x +2 c \right )-9 \sin \left (4 d x +4 c \right )-\frac {256}{5}\right ) a^{2}}{192 d}\) | \(76\) |
risch | \(\frac {3 a^{2} x}{16}-\frac {a^{2} \cos \left (d x +c \right )}{4 d}+\frac {a^{2} \sin \left (6 d x +6 c \right )}{192 d}+\frac {a^{2} \cos \left (5 d x +5 c \right )}{40 d}-\frac {3 a^{2} \sin \left (4 d x +4 c \right )}{64 d}-\frac {a^{2} \cos \left (3 d x +3 c \right )}{24 d}-\frac {a^{2} \sin \left (2 d x +2 c \right )}{64 d}\) | \(107\) |
derivativedivides | \(\frac {a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+2 a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )}{d}\) | \(142\) |
default | \(\frac {a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+2 a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )}{d}\) | \(142\) |
norman | \(\frac {\frac {3 a^{2} x}{16}-\frac {8 a^{2}}{15 d}-\frac {3 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {13 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {25 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {25 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {13 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {3 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {9 a^{2} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {45 a^{2} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {15 a^{2} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {45 a^{2} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {9 a^{2} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {3 a^{2} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {16 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {16 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {8 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\) | \(303\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.83 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {96 \, a^{2} \cos \left (d x + c\right )^{5} - 160 \, a^{2} \cos \left (d x + c\right )^{3} + 45 \, a^{2} d x + 5 \, {\left (8 \, a^{2} \cos \left (d x + c\right )^{5} - 26 \, a^{2} \cos \left (d x + c\right )^{3} + 9 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (92) = 184\).
Time = 0.37 (sec) , antiderivative size = 309, normalized size of antiderivative = 3.00 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\begin {cases} \frac {a^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {a^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {2 a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {4 a^{2} \cos ^{5}{\left (c + d x \right )}}{15 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{2} \sin ^{2}{\left (c \right )} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.90 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {128 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{2} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} + 30 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{960 \, d} \]
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.03 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3}{16} \, a^{2} x + \frac {a^{2} \cos \left (5 \, d x + 5 \, c\right )}{40 \, d} - \frac {a^{2} \cos \left (3 \, d x + 3 \, c\right )}{24 \, d} - \frac {a^{2} \cos \left (d x + c\right )}{4 \, d} + \frac {a^{2} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {3 \, a^{2} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} - \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
[In]
[Out]
Time = 13.09 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.50 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3\,a^2\,x}{16}-\frac {\frac {3\,a^2\,\left (c+d\,x\right )}{16}-\frac {13\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}-\frac {25\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+\frac {25\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {13\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}-\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}-\frac {a^2\,\left (45\,c+45\,d\,x-128\right )}{240}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {9\,a^2\,\left (c+d\,x\right )}{8}-\frac {a^2\,\left (270\,c+270\,d\,x-768\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {15\,a^2\,\left (c+d\,x\right )}{4}-\frac {a^2\,\left (900\,c+900\,d\,x-1280\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {45\,a^2\,\left (c+d\,x\right )}{16}-\frac {a^2\,\left (675\,c+675\,d\,x-1920\right )}{240}\right )+\frac {3\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \]
[In]
[Out]