Integrand size = 29, antiderivative size = 159 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3 a^2 x}{64}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {3 a^2 \cos ^7(c+d x)}{7 d}-\frac {a^2 \cos ^9(c+d x)}{9 d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{32 d}-\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac {a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{4 d} \]
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Time = 0.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 2645, 14, 2648, 2715, 8, 276} \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \cos ^9(c+d x)}{9 d}+\frac {3 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{4 d}-\frac {a^2 \sin (c+d x) \cos ^5(c+d x)}{8 d}+\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{32 d}+\frac {3 a^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac {3 a^2 x}{64} \]
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Rule 8
Rule 14
Rule 276
Rule 2645
Rule 2648
Rule 2715
Rule 2952
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 \cos ^4(c+d x) \sin ^3(c+d x)+2 a^2 \cos ^4(c+d x) \sin ^4(c+d x)+a^2 \cos ^4(c+d x) \sin ^5(c+d x)\right ) \, dx \\ & = a^2 \int \cos ^4(c+d x) \sin ^3(c+d x) \, dx+a^2 \int \cos ^4(c+d x) \sin ^5(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx \\ & = -\frac {a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{4} \left (3 a^2\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac {a^2 \text {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {a^2 \text {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac {a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{8} a^2 \int \cos ^4(c+d x) \, dx-\frac {a^2 \text {Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {a^2 \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {3 a^2 \cos ^7(c+d x)}{7 d}-\frac {a^2 \cos ^9(c+d x)}{9 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{32 d}-\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac {a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{32} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {3 a^2 \cos ^7(c+d x)}{7 d}-\frac {a^2 \cos ^9(c+d x)}{9 d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{32 d}-\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac {a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{64} \left (3 a^2\right ) \int 1 \, dx \\ & = \frac {3 a^2 x}{64}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {3 a^2 \cos ^7(c+d x)}{7 d}-\frac {a^2 \cos ^9(c+d x)}{9 d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{32 d}-\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac {a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{4 d} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.54 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 (7560 c+7560 d x-11340 \cos (c+d x)-3360 \cos (3 (c+d x))+1008 \cos (5 (c+d x))+450 \cos (7 (c+d x))-70 \cos (9 (c+d x))-2520 \sin (4 (c+d x))+315 \sin (8 (c+d x)))}{161280 d} \]
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Time = 0.59 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.56
method | result | size |
parallelrisch | \(-\frac {a^{2} \left (-7560 d x +11340 \cos \left (d x +c \right )+2520 \sin \left (4 d x +4 c \right )-315 \sin \left (8 d x +8 c \right )-1008 \cos \left (5 d x +5 c \right )-450 \cos \left (7 d x +7 c \right )+3360 \cos \left (3 d x +3 c \right )+70 \cos \left (9 d x +9 c \right )+13312\right )}{161280 d}\) | \(89\) |
risch | \(\frac {3 a^{2} x}{64}-\frac {9 a^{2} \cos \left (d x +c \right )}{128 d}-\frac {a^{2} \cos \left (9 d x +9 c \right )}{2304 d}+\frac {a^{2} \sin \left (8 d x +8 c \right )}{512 d}+\frac {5 a^{2} \cos \left (7 d x +7 c \right )}{1792 d}+\frac {a^{2} \cos \left (5 d x +5 c \right )}{160 d}-\frac {a^{2} \sin \left (4 d x +4 c \right )}{64 d}-\frac {a^{2} \cos \left (3 d x +3 c \right )}{48 d}\) | \(124\) |
derivativedivides | \(\frac {a^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+2 a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )}{d}\) | \(162\) |
default | \(\frac {a^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+2 a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )}{d}\) | \(162\) |
norman | \(\frac {-\frac {52 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {27 a^{2} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {189 a^{2} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {63 a^{2} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {44 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {27 a^{2} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {27 a^{2} x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {3 a^{2} x \left (\tan ^{18}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}-\frac {4 a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {3 a^{2} x}{64}-\frac {52 a^{2}}{315 d}-\frac {155 a^{2} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {169 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {3 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d}+\frac {27 a^{2} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {63 a^{2} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {189 a^{2} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {3 a^{2} \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {12 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {13 a^{2} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {169 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {68 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {4 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {13 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {155 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {164 a^{2} \left (\tan ^{8}\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 )^{9}}\) | \(468\) |
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Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.70 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {2240 \, a^{2} \cos \left (d x + c\right )^{9} - 8640 \, a^{2} \cos \left (d x + c\right )^{7} + 8064 \, a^{2} \cos \left (d x + c\right )^{5} - 945 \, a^{2} d x - 315 \, {\left (16 \, a^{2} \cos \left (d x + c\right )^{7} - 24 \, a^{2} \cos \left (d x + c\right )^{5} + 2 \, a^{2} \cos \left (d x + c\right )^{3} + 3 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{20160 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (146) = 292\).
Time = 0.92 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.11 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\begin {cases} \frac {3 a^{2} x \sin ^{8}{\left (c + d x \right )}}{64} + \frac {3 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {9 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac {3 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{2} x \cos ^{8}{\left (c + d x \right )}}{64} + \frac {3 a^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{64 d} + \frac {11 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{64 d} - \frac {a^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {11 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{64 d} - \frac {4 a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {3 a^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac {8 a^{2} \cos ^{9}{\left (c + d x \right )}}{315 d} - \frac {2 a^{2} \cos ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{2} \sin ^{3}{\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.64 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {512 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{2} - 4608 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{2} - 315 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{161280 \, d} \]
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Time = 0.53 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.77 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3}{64} \, a^{2} x - \frac {a^{2} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {5 \, a^{2} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {a^{2} \cos \left (5 \, d x + 5 \, c\right )}{160 \, d} - \frac {a^{2} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {9 \, a^{2} \cos \left (d x + c\right )}{128 \, d} + \frac {a^{2} \sin \left (8 \, d x + 8 \, c\right )}{512 \, d} - \frac {a^{2} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} \]
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Time = 12.75 (sec) , antiderivative size = 437, normalized size of antiderivative = 2.75 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {3\,a^2\,x}{64}-\frac {\frac {3\,a^2\,\left (c+d\,x\right )}{64}+\frac {13\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{16}-\frac {155\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{16}+\frac {169\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{16}-\frac {169\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{16}+\frac {155\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{16}-\frac {13\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{16}-\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{32}-\frac {a^2\,\left (945\,c+945\,d\,x-3328\right )}{20160}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {27\,a^2\,\left (c+d\,x\right )}{64}-\frac {a^2\,\left (8505\,c+8505\,d\,x-29952\right )}{20160}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {27\,a^2\,\left (c+d\,x\right )}{16}-\frac {a^2\,\left (34020\,c+34020\,d\,x-39168\right )}{20160}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {27\,a^2\,\left (c+d\,x\right )}{16}-\frac {a^2\,\left (34020\,c+34020\,d\,x-80640\right )}{20160}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {63\,a^2\,\left (c+d\,x\right )}{16}-\frac {a^2\,\left (79380\,c+79380\,d\,x+16128\right )}{20160}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {63\,a^2\,\left (c+d\,x\right )}{16}-\frac {a^2\,\left (79380\,c+79380\,d\,x-295680\right )}{20160}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {189\,a^2\,\left (c+d\,x\right )}{32}-\frac {a^2\,\left (119070\,c+119070\,d\,x+241920\right )}{20160}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {189\,a^2\,\left (c+d\,x\right )}{32}-\frac {a^2\,\left (119070\,c+119070\,d\,x-661248\right )}{20160}\right )+\frac {3\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{32}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9} \]
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