Integrand size = 21, antiderivative size = 24 \[ \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=-\frac {2}{3 b d (a+b \sin (c+d x))^{3/2}} \]
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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 = {2747, 32} \[ \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=-\frac {2}{3 b d (a+b \sin (c+d x))^{3/2}} \]
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Rule 32
Rule 2747
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{(a+x)^{5/2}} \, dx,x,b \sin (c+d x)\right )}{b d} \\ & = -\frac {2}{3 b d (a+b \sin (c+d x))^{3/2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=-\frac {2}{3 b d (a+b \sin (c+d x))^{3/2}} \]
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Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(-\frac {2}{3 b d \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}\) | \(21\) |
default | \(-\frac {2}{3 b d \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}\) | \(21\) |
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Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (20) = 40\).
Time = 0.27 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.29 \[ \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {2 \, \sqrt {b \sin \left (d x + c\right ) + a}}{3 \, {\left (b^{3} d \cos \left (d x + c\right )^{2} - 2 \, a b^{2} d \sin \left (d x + c\right ) - {\left (a^{2} b + b^{3}\right )} d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (20) = 40\).
Time = 2.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.62 \[ \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\begin {cases} \frac {x \cos {\left (c \right )}}{a^{\frac {5}{2}}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sin {\left (c + d x \right )}}{a^{\frac {5}{2}} d} & \text {for}\: b = 0 \\\frac {x \cos {\left (c \right )}}{\left (a + b \sin {\left (c \right )}\right )^{\frac {5}{2}}} & \text {for}\: d = 0 \\- \frac {2}{3 a b d \sqrt {a + b \sin {\left (c + d x \right )}} + 3 b^{2} d \sqrt {a + b \sin {\left (c + d x \right )}} \sin {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
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none
Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=-\frac {2}{3 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} b d} \]
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\[ \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int { \frac {\cos \left (d x + c\right )}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 6.57 (sec) , antiderivative size = 157, normalized size of antiderivative = 6.54 \[ \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=-\frac {8\,\sqrt {a+b\,\sin \left (c+d\,x\right )}\,\left (2\,a^2+b^2-b^2\,\cos \left (2\,c+2\,d\,x\right )+4\,a\,b\,\sin \left (c+d\,x\right )\right )}{3\,b\,d\,\left (8\,a^4+3\,b^4+24\,a^2\,b^2-4\,b^4\,\cos \left (2\,c+2\,d\,x\right )+b^4\,\cos \left (4\,c+4\,d\,x\right )-8\,a\,b^3\,\sin \left (3\,c+3\,d\,x\right )-24\,a^2\,b^2\,\cos \left (2\,c+2\,d\,x\right )+24\,a\,b^3\,\sin \left (c+d\,x\right )+32\,a^3\,b\,\sin \left (c+d\,x\right )\right )} \]
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