Integrand size = 21, antiderivative size = 24 \[ \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2}{3 a d (a+a \sin (c+d x))^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2746, 32} \[ \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2}{3 a d (a \sin (c+d x)+a)^{3/2}} \]
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Rule 32
Rule 2746
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{(a+x)^{5/2}} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = -\frac {2}{3 a d (a+a \sin (c+d x))^{3/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2}{3 a d (a+a \sin (c+d x))^{3/2}} \]
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Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(-\frac {2}{3 a d \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}\) | \(21\) |
default | \(-\frac {2}{3 a d \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}\) | \(21\) |
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00 \[ \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a}}{3 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (20) = 40\).
Time = 2.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.71 \[ \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\begin {cases} - \frac {2}{3 a^{2} d \sqrt {a \sin {\left (c + d x \right )} + a} \sin {\left (c + d x \right )} + 3 a^{2} d \sqrt {a \sin {\left (c + d x \right )} + a}} & \text {for}\: d \neq 0 \\\frac {x \cos {\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2}{3 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a d} \]
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none
Time = 0.36 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {\sqrt {2}}{6 \, a^{\frac {5}{2}} d \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \]
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Time = 7.63 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.00 \[ \int \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {8\,{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,\sqrt {a+a\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}}{3\,a^3\,d\,{\left (-1+{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}\right )}^4} \]
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