Integrand size = 15, antiderivative size = 15 \[ \int \cos (a+b x) \sin ^5(a+b x) \, dx=\frac {\sin ^6(a+b x)}{6 b} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2644, 30} \[ \int \cos (a+b x) \sin ^5(a+b x) \, dx=\frac {\sin ^6(a+b x)}{6 b} \]
[In]
[Out]
Rule 30
Rule 2644
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^5 \, dx,x,\sin (a+b x)\right )}{b} \\ & = \frac {\sin ^6(a+b x)}{6 b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \cos (a+b x) \sin ^5(a+b x) \, dx=\frac {\sin ^6(a+b x)}{6 b} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {\sin ^{6}\left (b x +a \right )}{6 b}\) | \(14\) |
default | \(\frac {\sin ^{6}\left (b x +a \right )}{6 b}\) | \(14\) |
norman | \(\frac {32 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b \left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{6}}\) | \(32\) |
parallelrisch | \(\frac {6 \cos \left (4 b x +4 a \right )+10-15 \cos \left (2 b x +2 a \right )-\cos \left (6 b x +6 a \right )}{192 b}\) | \(41\) |
risch | \(-\frac {\cos \left (6 b x +6 a \right )}{192 b}+\frac {\cos \left (4 b x +4 a \right )}{32 b}-\frac {5 \cos \left (2 b x +2 a \right )}{64 b}\) | \(44\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (13) = 26\).
Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.27 \[ \int \cos (a+b x) \sin ^5(a+b x) \, dx=-\frac {\cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2}}{6 \, b} \]
[In]
[Out]
Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33 \[ \int \cos (a+b x) \sin ^5(a+b x) \, dx=\begin {cases} \frac {\sin ^{6}{\left (a + b x \right )}}{6 b} & \text {for}\: b \neq 0 \\x \sin ^{5}{\left (a \right )} \cos {\left (a \right )} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \cos (a+b x) \sin ^5(a+b x) \, dx=\frac {\sin \left (b x + a\right )^{6}}{6 \, b} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \cos (a+b x) \sin ^5(a+b x) \, dx=\frac {\sin \left (b x + a\right )^{6}}{6 \, b} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \cos (a+b x) \sin ^5(a+b x) \, dx=\frac {{\sin \left (a+b\,x\right )}^6}{6\,b} \]
[In]
[Out]