Integrand size = 25, antiderivative size = 49 \[ \int \cos ^3(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos ^4(c+d x)}{4 d}+\frac {a \sin ^3(c+d x)}{3 d}-\frac {a \sin ^5(c+d x)}{5 d} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2913, 2645, 30, 2644, 14} \[ \int \cos ^3(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \sin ^5(c+d x)}{5 d}+\frac {a \sin ^3(c+d x)}{3 d}-\frac {a \cos ^4(c+d x)}{4 d} \]
[In]
[Out]
Rule 14
Rule 30
Rule 2644
Rule 2645
Rule 2913
Rubi steps \begin{align*} \text {integral}& = a \int \cos ^3(c+d x) \sin (c+d x) \, dx+a \int \cos ^3(c+d x) \sin ^2(c+d x) \, dx \\ & = -\frac {a \text {Subst}\left (\int x^3 \, dx,x,\cos (c+d x)\right )}{d}+\frac {a \text {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {a \cos ^4(c+d x)}{4 d}+\frac {a \text {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {a \cos ^4(c+d x)}{4 d}+\frac {a \sin ^3(c+d x)}{3 d}-\frac {a \sin ^5(c+d x)}{5 d} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.18 \[ \int \cos ^3(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a (45+60 \cos (2 (c+d x))+15 \cos (4 (c+d x))-60 \sin (c+d x)+10 \sin (3 (c+d x))+6 \sin (5 (c+d x)))}{480 d} \]
[In]
[Out]
Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(-\frac {a \left (\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\) | \(48\) |
default | \(-\frac {a \left (\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\) | \(48\) |
parallelrisch | \(-\frac {a \left (10 \sin \left (3 d x +3 c \right )-60 \sin \left (d x +c \right )+6 \sin \left (5 d x +5 c \right )+15 \cos \left (4 d x +4 c \right )-75+60 \cos \left (2 d x +2 c \right )\right )}{480 d}\) | \(61\) |
risch | \(\frac {a \sin \left (d x +c \right )}{8 d}-\frac {a \sin \left (5 d x +5 c \right )}{80 d}-\frac {a \cos \left (4 d x +4 c \right )}{32 d}-\frac {a \sin \left (3 d x +3 c \right )}{48 d}-\frac {a \cos \left (2 d x +2 c \right )}{8 d}\) | \(74\) |
norman | \(\frac {\frac {2 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {16 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {8 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\) | \(137\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.04 \[ \int \cos ^3(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {15 \, a \cos \left (d x + c\right )^{4} + 4 \, {\left (3 \, a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2} - 2 \, a\right )} \sin \left (d x + c\right )}{60 \, d} \]
[In]
[Out]
Time = 0.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.35 \[ \int \cos ^3(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {2 a \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {a \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} - \frac {a \cos ^{4}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin {\left (c \right )} \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.02 \[ \int \cos ^3(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {12 \, a \sin \left (d x + c\right )^{5} + 15 \, a \sin \left (d x + c\right )^{4} - 20 \, a \sin \left (d x + c\right )^{3} - 30 \, a \sin \left (d x + c\right )^{2}}{60 \, d} \]
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.02 \[ \int \cos ^3(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {12 \, a \sin \left (d x + c\right )^{5} + 15 \, a \sin \left (d x + c\right )^{4} - 20 \, a \sin \left (d x + c\right )^{3} - 30 \, a \sin \left (d x + c\right )^{2}}{60 \, d} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \cos ^3(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^5}{5}-\frac {a\,{\sin \left (c+d\,x\right )}^4}{4}+\frac {a\,{\sin \left (c+d\,x\right )}^3}{3}+\frac {a\,{\sin \left (c+d\,x\right )}^2}{2}}{d} \]
[In]
[Out]