Integrand size = 29, antiderivative size = 130 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{\sqrt {a} d}+\frac {32 \cos (c+d x)}{15 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{15 a d} \]
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Time = 0.43 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2960, 2857, 3047, 3102, 2830, 2728, 212, 3125, 3064, 2852} \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {2 \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}+\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{15 a d}+\frac {32 \cos (c+d x)}{15 d \sqrt {a \sin (c+d x)+a}} \]
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Rule 212
Rule 2728
Rule 2830
Rule 2852
Rule 2857
Rule 2960
Rule 3047
Rule 3064
Rule 3102
Rule 3125
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sin ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx+\int \frac {\csc (c+d x) \left (1-2 \sin ^2(c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = \frac {4 \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}-\frac {\int \frac {\sin (c+d x) (-4 a+a \sin (c+d x))}{\sqrt {a+a \sin (c+d x)}} \, dx}{5 a}+\frac {2 \int \frac {\csc (c+d x) \left (\frac {a}{2}+a \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{a} \\ & = \frac {4 \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}-\frac {\int \frac {-4 a \sin (c+d x)+a \sin ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{5 a}+\frac {\int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{a}+\int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = \frac {4 \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{15 a d}-\frac {2 \int \frac {\frac {a^2}{2}-7 a^2 \sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{15 a^2}-\frac {2 \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}-\frac {2 \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 )}{d} \\ & = -\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{\sqrt {a} d}+\frac {32 \cos (c+d x)}{15 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{15 a d}-\int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = -\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{\sqrt {a} d}+\frac {32 \cos (c+d x)}{15 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{15 a d}+\frac {2 \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 )}{d} \\ & = -\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{\sqrt {a} d}+\frac {32 \cos (c+d x)}{15 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^2(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{15 a d} \\ \end{align*}
Time = 0.63 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.30 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (60 \cos \left (\frac {1}{2} (c+d x)\right )+5 \cos \left (\frac {3}{2} (c+d x)\right )+3 \cos \left (\frac {5}{2} (c+d x)\right )-30 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+30 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-60 \sin \left (\frac {1}{2} (c+d x)\right )+5 \sin \left (\frac {3}{2} (c+d x)\right )-3 \sin \left (\frac {5}{2} (c+d x)\right )\right )}{30 d \sqrt {a (1+\sin (c+d x))}} \]
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Time = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (15 a^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}}{\sqrt {a}}\right )+3 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}}-5 a \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}}-15 a^{2} \sqrt {a -a \sin \left (d x +c \right )}\right )}{15 a^{3} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(123\) |
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Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (112) = 224\).
Time = 0.27 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.15 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {15 \, \sqrt {a} {\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (3 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} - {\left (3 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) + 13\right )} \sin \left (d x + c\right ) + 14 \, \cos \left (d x + c\right ) + 13\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{30 \, {\left (a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d\right )}} \]
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\[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {\cos ^{4}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}\, dx \]
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\[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \csc \left (d x + c\right )}{\sqrt {a \sin \left (d x + c\right ) + a}} \,d x } \]
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Time = 0.36 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {\frac {15 \, \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, \sqrt {2} {\left (12 \, a^{\frac {9}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 10 \, a^{\frac {9}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{\frac {9}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{5} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{15 \, d} \]
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Timed out. \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4}{\sin \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]
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