Integrand size = 29, antiderivative size = 159 \[ \int \cos ^3(c+d x) \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}+\frac {8 a \cos (c+d x)}{15 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}+\frac {164 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{105 d}-\frac {12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d} \]
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Time = 0.36 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2960, 2849, 2838, 2830, 2725, 3125, 3060, 2852, 212} \[ \int \cos ^3(c+d x) \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}-\frac {12 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 a d}+\frac {164 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{105 d}+\frac {8 a \cos (c+d x)}{15 d \sqrt {a \sin (c+d x)+a}} \]
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Rule 212
Rule 2725
Rule 2830
Rule 2838
Rule 2849
Rule 2852
Rule 2960
Rule 3060
Rule 3125
Rubi steps \begin{align*} \text {integral}& = \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx+\int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \left (1-2 \sin ^2(c+d x)\right ) \, dx \\ & = -\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}+\frac {4 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}+\frac {6}{7} \int \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx+\frac {2 \int \csc (c+d x) \left (\frac {3 a}{2}-a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{3 a} \\ & = \frac {4 a \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}+\frac {4 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}-\frac {12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d}+\frac {12 \int \left (\frac {3 a}{2}-a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{35 a}+\int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = \frac {4 a \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}+\frac {164 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{105 d}-\frac {12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d}+\frac {2}{5} \int \sqrt {a+a \sin (c+d x)} \, dx-\frac {(2 a) \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 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}+\frac {8 a \cos (c+d x)}{15 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}+\frac {164 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{105 d}-\frac {12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d} \\ \end{align*}
Time = 0.93 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.23 \[ \int \cos ^3(c+d x) \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {a (1+\sin (c+d x))} \left (525 \cos \left (\frac {1}{2} (c+d x)\right )+175 \cos \left (\frac {3}{2} (c+d x)\right )+21 \cos \left (\frac {5}{2} (c+d x)\right )+15 \cos \left (\frac {7}{2} (c+d x)\right )-420 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+420 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-525 \sin \left (\frac {1}{2} (c+d x)\right )+175 \sin \left (\frac {3}{2} (c+d x)\right )-21 \sin \left (\frac {5}{2} (c+d x)\right )+15 \sin \left (\frac {7}{2} (c+d x)\right )\right )}{420 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
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Time = 0.12 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (105 a^{\frac {7}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}}{\sqrt {a}}\right )-15 \left (a -a \sin \left (d x +c \right )\right )^{\frac {7}{2}}+63 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} a -35 a^{2} \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}}-105 a^{3} \sqrt {a -a \sin \left (d x +c \right )}\right )}{105 a^{3} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(141\) |
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Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (137) = 274\).
Time = 0.29 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.85 \[ \int \cos ^3(c+d x) \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {105 \, \sqrt {a} {\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (15 \, \cos \left (d x + c\right )^{4} + 18 \, \cos \left (d x + c\right )^{3} + 34 \, \cos \left (d x + c\right )^{2} + {\left (15 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} + 31 \, \cos \left (d x + c\right ) - 43\right )} \sin \left (d x + c\right ) + 74 \, \cos \left (d x + c\right ) + 43\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{210 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
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\[ \int \cos ^3(c+d x) \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \cos ^{4}{\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx \]
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\[ \int \cos ^3(c+d x) \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right ) \,d x } \]
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Time = 0.36 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.19 \[ \int \cos ^3(c+d x) \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\sqrt {2} {\left (480 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1008 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 280 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 420 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{210 \, d} \]
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Timed out. \[ \int \cos ^3(c+d x) \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{\sin \left (c+d\,x\right )} \,d x \]
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