Integrand size = 23, antiderivative size = 76 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{a^{3/2} d}+\frac {2 \cos (c+d x)}{a d \sqrt {a+a \sin (c+d x)}} \]
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Time = 0.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2758, 2728, 212} \[ \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {2 \cos (c+d x)}{a d \sqrt {a \sin (c+d x)+a}}-\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{a^{3/2} d} \]
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Rule 212
Rule 2728
Rule 2758
Rubi steps \begin{align*} \text {integral}& = \frac {2 \cos (c+d x)}{a d \sqrt {a+a \sin (c+d x)}}+\frac {2 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{a} \\ & = \frac {2 \cos (c+d x)}{a d \sqrt {a+a \sin (c+d x)}}-\frac {4 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{a d} \\ & = -\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{a^{3/2} d}+\frac {2 \cos (c+d x)}{a d \sqrt {a+a \sin (c+d x)}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.11 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {2 \cos ^3(c+d x) \left (-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right )+\sqrt {1-\sin (c+d x)}\right )}{d (1-\sin (c+d x))^{3/2} (a (1+\sin (c+d x)))^{3/2}} \]
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Time = 0.88 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.24
method | result | size |
default | \(\frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (-\sqrt {a}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )+\sqrt {a -a \sin \left (d x +c \right )}\right )}{a^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(94\) |
risch | \(\frac {\left ({\mathrm e}^{i \left (d x +c \right )}+3 i\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) \sqrt {2}\, {\mathrm e}^{-i \left (d x +c \right )}}{d a \sqrt {-a \left (i {\mathrm e}^{2 i \left (d x +c \right )}-i-2 \,{\mathrm e}^{i \left (d x +c \right )}\right ) {\mathrm e}^{-i \left (d x +c \right )}}}-\frac {4 i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) \left (\arctan \left (\frac {\sqrt {-i a \,{\mathrm e}^{i \left (d x +c \right )}}}{\sqrt {a}}\right ) a \sqrt {-i a \,{\mathrm e}^{i \left (d x +c \right )}}+a^{\frac {3}{2}}\right ) \sqrt {2}\, {\mathrm e}^{-i \left (d x +c \right )}}{d \,a^{\frac {5}{2}} \sqrt {-a \left (i {\mathrm e}^{2 i \left (d x +c \right )}-i-2 \,{\mathrm e}^{i \left (d x +c \right )}\right ) {\mathrm e}^{-i \left (d x +c \right )}}}\) | \(211\) |
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Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (65) = 130\).
Time = 0.28 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.58 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\frac {\sqrt {2} {\left (a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) - \frac {2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (d x + c\right ) + 2}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right )}{\sqrt {a}} + 2 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{a^{2} d \cos \left (d x + c\right ) + a^{2} d \sin \left (d x + c\right ) + a^{2} d} \]
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\[ \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {\cos ^{2}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.50 \[ \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\sqrt {2} \sqrt {a} {\left (\frac {\log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {\log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}\right )}}{d} \]
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Timed out. \[ \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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