Integrand size = 19, antiderivative size = 22 \[ \int (d \cos (a+b x))^{3/2} \sin (a+b x) \, dx=-\frac {2 (d \cos (a+b x))^{5/2}}{5 b d} \]
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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} \[ \int (d \cos (a+b x))^{3/2} \sin (a+b x) \, dx=-\frac {2 (d \cos (a+b x))^{5/2}}{5 b d} \]
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Rule 30
Rule 2645
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int x^{3/2} \, dx,x,d \cos (a+b x)\right )}{b d} \\ & = -\frac {2 (d \cos (a+b x))^{5/2}}{5 b d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int (d \cos (a+b x))^{3/2} \sin (a+b x) \, dx=-\frac {2 (d \cos (a+b x))^{5/2}}{5 b d} \]
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Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(-\frac {2 \left (d \cos \left (b x +a \right )\right )^{\frac {5}{2}}}{5 b d}\) | \(19\) |
default | \(-\frac {2 \left (d \cos \left (b x +a \right )\right )^{\frac {5}{2}}}{5 b d}\) | \(19\) |
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Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int (d \cos (a+b x))^{3/2} \sin (a+b x) \, dx=-\frac {2 \, \sqrt {d \cos \left (b x + a\right )} d \cos \left (b x + a\right )^{2}}{5 \, b} \]
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Time = 2.89 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int (d \cos (a+b x))^{3/2} \sin (a+b x) \, dx=\begin {cases} - \frac {2 \left (d \cos {\left (a + b x \right )}\right )^{\frac {3}{2}} \cos {\left (a + b x \right )}}{5 b} & \text {for}\: b \neq 0 \\x \left (d \cos {\left (a \right )}\right )^{\frac {3}{2}} \sin {\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int (d \cos (a+b x))^{3/2} \sin (a+b x) \, dx=-\frac {2 \, \left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}}}{5 \, b d} \]
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\[ \int (d \cos (a+b x))^{3/2} \sin (a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}} \sin \left (b x + a\right ) \,d x } \]
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Time = 0.15 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int (d \cos (a+b x))^{3/2} \sin (a+b x) \, dx=-\frac {2\,{\left (d\,\cos \left (a+b\,x\right )\right )}^{5/2}}{5\,b\,d} \]
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