Integrand size = 25, antiderivative size = 87 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx=\frac {b x}{16}-\frac {a \cos ^5(c+d x)}{5 d}+\frac {b \cos (c+d x) \sin (c+d x)}{16 d}+\frac {b \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {b \cos ^5(c+d x) \sin (c+d x)}{6 d} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2917, 2645, 30, 2648, 2715, 8} \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \cos ^5(c+d x)}{5 d}-\frac {b \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {b \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {b \sin (c+d x) \cos (c+d x)}{16 d}+\frac {b x}{16} \]
[In]
[Out]
Rule 8
Rule 30
Rule 2645
Rule 2648
Rule 2715
Rule 2917
Rubi steps \begin{align*} \text {integral}& = a \int \cos ^4(c+d x) \sin (c+d x) \, dx+b \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx \\ & = -\frac {b \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{6} b \int \cos ^4(c+d x) \, dx-\frac {a \text {Subst}\left (\int x^4 \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a \cos ^5(c+d x)}{5 d}+\frac {b \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {b \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{8} b \int \cos ^2(c+d x) \, dx \\ & = -\frac {a \cos ^5(c+d x)}{5 d}+\frac {b \cos (c+d x) \sin (c+d x)}{16 d}+\frac {b \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {b \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{16} b \int 1 \, dx \\ & = \frac {b x}{16}-\frac {a \cos ^5(c+d x)}{5 d}+\frac {b \cos (c+d x) \sin (c+d x)}{16 d}+\frac {b \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {b \cos ^5(c+d x) \sin (c+d x)}{6 d} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.89 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx=-\frac {-60 b d x+120 a \cos (c+d x)+60 a \cos (3 (c+d x))+12 a \cos (5 (c+d x))-15 b \sin (2 (c+d x))+15 b \sin (4 (c+d x))+5 b \sin (6 (c+d x))}{960 d} \]
[In]
[Out]
Time = 0.48 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {-\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{5}+b \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )}{d}\) | \(68\) |
default | \(\frac {-\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{5}+b \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )}{d}\) | \(68\) |
parallelrisch | \(\frac {60 b x d -60 a \cos \left (3 d x +3 c \right )-120 a \cos \left (d x +c \right )-5 b \sin \left (6 d x +6 c \right )-12 a \cos \left (5 d x +5 c \right )-15 b \sin \left (4 d x +4 c \right )+15 b \sin \left (2 d x +2 c \right )-192 a}{960 d}\) | \(84\) |
risch | \(\frac {b x}{16}-\frac {a \cos \left (d x +c \right )}{8 d}-\frac {b \sin \left (6 d x +6 c \right )}{192 d}-\frac {a \cos \left (5 d x +5 c \right )}{80 d}-\frac {b \sin \left (4 d x +4 c \right )}{64 d}-\frac {a \cos \left (3 d x +3 c \right )}{16 d}+\frac {b \sin \left (2 d x +2 c \right )}{64 d}\) | \(93\) |
norman | \(\frac {\frac {b x}{16}-\frac {2 a}{5 d}-\frac {b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {47 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {13 b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {13 b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {47 b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {3 b x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {15 b x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {5 b x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {15 b x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {3 b x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {b x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {2 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {2 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\) | \(303\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.71 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx=-\frac {48 \, a \cos \left (d x + c\right )^{5} - 15 \, b d x + 5 \, {\left (8 \, b \cos \left (d x + c\right )^{5} - 2 \, b \cos \left (d x + c\right )^{3} - 3 \, b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (76) = 152\).
Time = 0.33 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.92 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx=\begin {cases} - \frac {a \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac {b x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 b x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 b x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {b x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {b \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {b \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {b \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right ) \sin {\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.60 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx=-\frac {192 \, a \cos \left (d x + c\right )^{5} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b}{960 \, d} \]
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.06 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx=\frac {1}{16} \, b x - \frac {a \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {a \cos \left (3 \, d x + 3 \, c\right )}{16 \, d} - \frac {a \cos \left (d x + c\right )}{8 \, d} - \frac {b \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {b \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {b \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
[In]
[Out]
Time = 14.66 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.08 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+b \sin (c+d x)) \, dx=\frac {b\,x}{16}-\frac {-\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {47\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {13\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {13\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {47\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {2\,a}{5}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \]
[In]
[Out]