Integrand size = 21, antiderivative size = 131 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {1}{8} a \left (4 a^2+3 b^2\right ) x-\frac {b \left (27 a^2+8 b^2\right ) \cos ^3(c+d x)}{60 d}+\frac {a \left (4 a^2+3 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {7 a b \cos ^3(c+d x) (a+b \sin (c+d x))}{20 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d} \]
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Time = 0.14 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2771, 2941, 2748, 2715, 8} \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {b \left (27 a^2+8 b^2\right ) \cos ^3(c+d x)}{60 d}+\frac {a \left (4 a^2+3 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a x \left (4 a^2+3 b^2\right )-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}-\frac {7 a b \cos ^3(c+d x) (a+b \sin (c+d x))}{20 d} \]
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Rule 8
Rule 2715
Rule 2748
Rule 2771
Rule 2941
Rubi steps \begin{align*} \text {integral}& = -\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}+\frac {1}{5} \int \cos ^2(c+d x) (a+b \sin (c+d x)) \left (5 a^2+2 b^2+7 a b \sin (c+d x)\right ) \, dx \\ & = -\frac {7 a b \cos ^3(c+d x) (a+b \sin (c+d x))}{20 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}+\frac {1}{20} \int \cos ^2(c+d x) \left (5 a \left (4 a^2+3 b^2\right )+b \left (27 a^2+8 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = -\frac {b \left (27 a^2+8 b^2\right ) \cos ^3(c+d x)}{60 d}-\frac {7 a b \cos ^3(c+d x) (a+b \sin (c+d x))}{20 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}+\frac {1}{4} \left (a \left (4 a^2+3 b^2\right )\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {b \left (27 a^2+8 b^2\right ) \cos ^3(c+d x)}{60 d}+\frac {a \left (4 a^2+3 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {7 a b \cos ^3(c+d x) (a+b \sin (c+d x))}{20 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}+\frac {1}{8} \left (a \left (4 a^2+3 b^2\right )\right ) \int 1 \, dx \\ & = \frac {1}{8} a \left (4 a^2+3 b^2\right ) x-\frac {b \left (27 a^2+8 b^2\right ) \cos ^3(c+d x)}{60 d}+\frac {a \left (4 a^2+3 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}-\frac {7 a b \cos ^3(c+d x) (a+b \sin (c+d x))}{20 d}-\frac {b \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d} \\ \end{align*}
Time = 0.81 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {-60 b \left (6 a^2+b^2\right ) \cos (c+d x)-10 \left (12 a^2 b+b^3\right ) \cos (3 (c+d x))+6 b^3 \cos (5 (c+d x))+15 a \left (4 \left (4 a^2+3 b^2\right ) (c+d x)+8 a^2 \sin (2 (c+d x))-3 b^2 \sin (4 (c+d x))\right )}{480 d} \]
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Time = 0.76 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-a^{2} b \left (\cos ^{3}\left (d x +c \right )\right )+3 a \,b^{2} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+b^{3} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )}{d}\) | \(123\) |
default | \(\frac {a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-a^{2} b \left (\cos ^{3}\left (d x +c \right )\right )+3 a \,b^{2} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+b^{3} \left (-\frac {\left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )}{d}\) | \(123\) |
parallelrisch | \(\frac {240 a^{3} d x +180 a \,b^{2} d x +6 \cos \left (5 d x +5 c \right ) b^{3}-45 \sin \left (4 d x +4 c \right ) a \,b^{2}-120 \cos \left (3 d x +3 c \right ) a^{2} b -10 \cos \left (3 d x +3 c \right ) b^{3}+120 \sin \left (2 d x +2 c \right ) a^{3}-360 \cos \left (d x +c \right ) a^{2} b -60 \cos \left (d x +c \right ) b^{3}-480 a^{2} b -64 b^{3}}{480 d}\) | \(128\) |
risch | \(\frac {a^{3} x}{2}+\frac {3 a \,b^{2} x}{8}-\frac {3 b \cos \left (d x +c \right ) a^{2}}{4 d}-\frac {b^{3} \cos \left (d x +c \right )}{8 d}+\frac {b^{3} \cos \left (5 d x +5 c \right )}{80 d}-\frac {3 \sin \left (4 d x +4 c \right ) a \,b^{2}}{32 d}-\frac {b \cos \left (3 d x +3 c \right ) a^{2}}{4 d}-\frac {b^{3} \cos \left (3 d x +3 c \right )}{48 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3}}{4 d}\) | \(131\) |
norman | \(\frac {\left (\frac {1}{2} a^{3}+\frac {3}{8} a \,b^{2}\right ) x +\left (5 a^{3}+\frac {15}{4} a \,b^{2}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (5 a^{3}+\frac {15}{4} a \,b^{2}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {1}{2} a^{3}+\frac {3}{8} a \,b^{2}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{2} a^{3}+\frac {15}{8} a \,b^{2}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{2} a^{3}+\frac {15}{8} a \,b^{2}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {30 a^{2} b +4 b^{3}}{15 d}-\frac {6 a^{2} b \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (6 a^{2} b +2 b^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (12 a^{2} b -2 b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {\left (12 a^{2} b +4 b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a \left (4 a^{2}-3 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {a \left (4 a^{2}-3 b^{2}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {a \left (4 a^{2}+9 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {a \left (4 a^{2}+9 b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\) | \(388\) |
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Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.75 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {24 \, b^{3} \cos \left (d x + c\right )^{5} - 40 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (4 \, a^{3} + 3 \, a b^{2}\right )} d x - 15 \, {\left (6 \, a b^{2} \cos \left (d x + c\right )^{3} - {\left (4 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \]
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Time = 0.26 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.80 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\begin {cases} \frac {a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {a^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} - \frac {a^{2} b \cos ^{3}{\left (c + d x \right )}}{d} + \frac {3 a b^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 a b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 a b^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 a b^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {3 a b^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {2 b^{3} \cos ^{5}{\left (c + d x \right )}}{15 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{3} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.71 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {480 \, a^{2} b \cos \left (d x + c\right )^{3} - 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 45 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{2} - 32 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} b^{3}}{480 \, d} \]
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Time = 0.52 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.86 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {b^{3} \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {3 \, a b^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {a^{3} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {1}{8} \, {\left (4 \, a^{3} + 3 \, a b^{2}\right )} x - \frac {{\left (12 \, a^{2} b + b^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {{\left (6 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )}{8 \, d} \]
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Time = 6.09 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.72 \[ \int \cos ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,a^2+3\,b^2\right )}{4\,\left (a^3+\frac {3\,a\,b^2}{4}\right )}\right )\,\left (4\,a^2+3\,b^2\right )}{4\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a\,b^2}{4}-a^3\right )+2\,a^2\,b-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,a^3+\frac {9\,a\,b^2}{2}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {3\,a\,b^2}{4}-a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (2\,a^3+\frac {9\,a\,b^2}{2}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (4\,a^2\,b+\frac {4\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (8\,a^2\,b-\frac {4\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (12\,a^2\,b+4\,b^3\right )+\frac {4\,b^3}{15}+6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a\,\left (4\,a^2+3\,b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{4\,d} \]
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