Integrand size = 19, antiderivative size = 22 \[ \int \sqrt {d \cos (a+b x)} \sin (a+b x) \, dx=-\frac {2 (d \cos (a+b x))^{3/2}}{3 b d} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2645, 30} \[ \int \sqrt {d \cos (a+b x)} \sin (a+b x) \, dx=-\frac {2 (d \cos (a+b x))^{3/2}}{3 b d} \]
[In]
[Out]
Rule 30
Rule 2645
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \sqrt {x} \, dx,x,d \cos (a+b x)\right )}{b d} \\ & = -\frac {2 (d \cos (a+b x))^{3/2}}{3 b d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \sqrt {d \cos (a+b x)} \sin (a+b x) \, dx=-\frac {2 (d \cos (a+b x))^{3/2}}{3 b d} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(-\frac {2 \left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}}}{3 b d}\) | \(19\) |
default | \(-\frac {2 \left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}}}{3 b d}\) | \(19\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \sqrt {d \cos (a+b x)} \sin (a+b x) \, dx=-\frac {2 \, \sqrt {d \cos \left (b x + a\right )} \cos \left (b x + a\right )}{3 \, b} \]
[In]
[Out]
Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \sqrt {d \cos (a+b x)} \sin (a+b x) \, dx=\begin {cases} - \frac {2 \sqrt {d \cos {\left (a + b x \right )}} \cos {\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x \sqrt {d \cos {\left (a \right )}} \sin {\left (a \right )} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \sqrt {d \cos (a+b x)} \sin (a+b x) \, dx=-\frac {2 \, \left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}}{3 \, b d} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \sqrt {d \cos (a+b x)} \sin (a+b x) \, dx=-\frac {2 \, \sqrt {d \cos \left (b x + a\right )} \cos \left (b x + a\right )}{3 \, b} \]
[In]
[Out]
Time = 0.15 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \sqrt {d \cos (a+b x)} \sin (a+b x) \, dx=-\frac {2\,{\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}}{3\,b\,d} \]
[In]
[Out]