Integrand size = 19, antiderivative size = 22 \[ \int \frac {\sin (a+b x)}{(d \cos (a+b x))^{5/2}} \, dx=\frac {2}{3 b d (d \cos (a+b x))^{3/2}} \]
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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} \[ \int \frac {\sin (a+b x)}{(d \cos (a+b x))^{5/2}} \, dx=\frac {2}{3 b d (d \cos (a+b x))^{3/2}} \]
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Rule 30
Rule 2645
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{x^{5/2}} \, dx,x,d \cos (a+b x)\right )}{b d} \\ & = \frac {2}{3 b d (d \cos (a+b x))^{3/2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (a+b x)}{(d \cos (a+b x))^{5/2}} \, dx=\frac {2}{3 b d (d \cos (a+b x))^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {2}{3 b d \left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}}}\) | \(19\) |
default | \(\frac {2}{3 b d \left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}}}\) | \(19\) |
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Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {\sin (a+b x)}{(d \cos (a+b x))^{5/2}} \, dx=\frac {2 \, \sqrt {d \cos \left (b x + a\right )}}{3 \, b d^{3} \cos \left (b x + a\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
Time = 3.46 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int \frac {\sin (a+b x)}{(d \cos (a+b x))^{5/2}} \, dx=\begin {cases} \frac {2 \cos {\left (a + b x \right )}}{3 b \left (d \cos {\left (a + b x \right )}\right )^{\frac {5}{2}}} & \text {for}\: b \neq 0 \\\frac {x \sin {\left (a \right )}}{\left (d \cos {\left (a \right )}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {\sin (a+b x)}{(d \cos (a+b x))^{5/2}} \, dx=\frac {2}{3 \, \left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}} b d} \]
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\[ \int \frac {\sin (a+b x)}{(d \cos (a+b x))^{5/2}} \, dx=\int { \frac {\sin \left (b x + a\right )}{\left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \]
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Time = 0.58 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.41 \[ \int \frac {\sin (a+b x)}{(d \cos (a+b x))^{5/2}} \, dx=\frac {8\,\left (\cos \left (2\,a+2\,b\,x\right )+1\right )\,\sqrt {d\,\cos \left (a+b\,x\right )}}{3\,b\,d^3\,\left (4\,\cos \left (2\,a+2\,b\,x\right )+\cos \left (4\,a+4\,b\,x\right )+3\right )} \]
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