Integrand size = 31, antiderivative size = 205 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {9 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{128 \sqrt {a} d}-\frac {9 \cot (c+d x)}{128 d \sqrt {a+a \sin (c+d x)}}-\frac {3 \cot (c+d x) \csc (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}+\frac {29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt {a+a \sin (c+d x)}}+\frac {\cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}} \]
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Time = 0.85 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {2960, 2858, 3064, 2728, 212, 2852, 3123, 3063} \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {9 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{128 \sqrt {a} d}-\frac {9 \cot (c+d x)}{128 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}+\frac {\cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt {a \sin (c+d x)+a}}+\frac {29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt {a \sin (c+d x)+a}}-\frac {3 \cot (c+d x) \csc (c+d x)}{64 d \sqrt {a \sin (c+d x)+a}} \]
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Rule 212
Rule 2728
Rule 2852
Rule 2858
Rule 2960
Rule 3063
Rule 3064
Rule 3123
Rubi steps \begin{align*} \text {integral}& = \int \frac {\csc ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx+\int \frac {\csc ^6(c+d x) \left (1-2 \sin ^2(c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = -\frac {\cot (c+d x)}{d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {\int \frac {\csc ^5(c+d x) \left (-\frac {a}{2}-\frac {11}{2} a \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{5 a}-\frac {\int \frac {\csc (c+d x) (a-a \sin (c+d x))}{\sqrt {a+a \sin (c+d x)}} \, dx}{2 a} \\ & = -\frac {\cot (c+d x)}{d \sqrt {a+a \sin (c+d x)}}+\frac {\cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {\int \frac {\csc ^4(c+d x) \left (-\frac {87 a^2}{4}-\frac {7}{4} a^2 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{20 a^2}-\frac {\int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{2 a}+\int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = -\frac {\cot (c+d x)}{d \sqrt {a+a \sin (c+d x)}}+\frac {29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt {a+a \sin (c+d x)}}+\frac {\cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {\int \frac {\csc ^3(c+d x) \left (\frac {45 a^3}{8}-\frac {435}{8} a^3 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{60 a^3}+\frac {\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 {\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 {\cot (c+d x)}{d \sqrt {a+a \sin (c+d x)}}-\frac {3 \cot (c+d x) \csc (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}+\frac {29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt {a+a \sin (c+d x)}}+\frac {\cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {\int \frac {\csc ^2(c+d x) \left (-\frac {1785 a^4}{16}+\frac {135}{16} a^4 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{120 a^4} \\ & = \frac {\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 {9 \cot (c+d x)}{128 d \sqrt {a+a \sin (c+d x)}}-\frac {3 \cot (c+d x) \csc (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}+\frac {29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt {a+a \sin (c+d x)}}+\frac {\cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {\int \frac {\csc (c+d x) \left (\frac {2055 a^5}{32}-\frac {1785}{32} a^5 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{120 a^5} \\ & = \frac {\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 {9 \cot (c+d x)}{128 d \sqrt {a+a \sin (c+d x)}}-\frac {3 \cot (c+d x) \csc (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}+\frac {29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt {a+a \sin (c+d x)}}+\frac {\cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {137 \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{256 a}-\int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = \frac {\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 {9 \cot (c+d x)}{128 d \sqrt {a+a \sin (c+d x)}}-\frac {3 \cot (c+d x) \csc (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}+\frac {29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt {a+a \sin (c+d x)}}+\frac {\cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}-\frac {137 \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 )}{128 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 {9 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{128 \sqrt {a} d}-\frac {9 \cot (c+d x)}{128 d \sqrt {a+a \sin (c+d x)}}-\frac {3 \cot (c+d x) \csc (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}+\frac {29 \cot (c+d x) \csc ^2(c+d x)}{80 d \sqrt {a+a \sin (c+d x)}}+\frac {\cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}} \\ \end{align*}
Time = 1.81 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.00 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {\csc ^{15}\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (820 \cos \left (\frac {1}{2} (c+d x)\right )+1600 \cos \left (\frac {3}{2} (c+d x)\right )+1616 \cos \left (\frac {5}{2} (c+d x)\right )-30 \cos \left (\frac {7}{2} (c+d x)\right )+90 \cos \left (\frac {9}{2} (c+d x)\right )-820 \sin \left (\frac {1}{2} (c+d x)\right )+450 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-450 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+1600 \sin \left (\frac {3}{2} (c+d x)\right )-1616 \sin \left (\frac {5}{2} (c+d x)\right )-225 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+225 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-30 \sin \left (\frac {7}{2} (c+d x)\right )-90 \sin \left (\frac {9}{2} (c+d x)\right )+45 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-45 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))\right )}{640 d \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^5 \sqrt {a (1+\sin (c+d x))}} \]
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Time = 0.13 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (45 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {9}{2}} a^{\frac {5}{2}}-210 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {7}{2}}+45 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{7} \left (\sin ^{5}\left (d x +c \right )\right )+128 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {9}{2}}+210 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {11}{2}}-45 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {13}{2}}\right )}{640 a^{\frac {15}{2}} \sin \left (d x +c \right )^{5} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(180\) |
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Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (177) = 354\).
Time = 0.27 (sec) , antiderivative size = 472, normalized size of antiderivative = 2.30 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {45 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 1\right )} \sqrt {a} \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 (45 \, \cos \left (d x + c\right )^{5} + 15 \, \cos \left (d x + c\right )^{4} + 142 \, \cos \left (d x + c\right )^{3} + 186 \, \cos \left (d x + c\right )^{2} - {\left (45 \, \cos \left (d x + c\right )^{4} + 30 \, \cos \left (d x + c\right )^{3} + 172 \, \cos \left (d x + c\right )^{2} - 14 \, \cos \left (d x + c\right ) - 73\right )} \sin \left (d x + c\right ) - 59 \, \cos \left (d x + c\right ) - 73\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{2560 \, {\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d - {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a d\right )} \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{6}}{\sqrt {a \sin \left (d x + c\right ) + a}} \,d x } \]
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Time = 0.43 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.09 \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\frac {45 \, \log \left ({\left | \frac {1}{2} \, \sqrt {2} + \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {45 \, \log \left ({\left | -\frac {1}{2} \, \sqrt {2} + \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, \sqrt {2} {\left (720 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1680 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 512 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 420 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 45 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5} a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{1280 \, d} \]
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Timed out. \[ \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4}{{\sin \left (c+d\,x\right )}^6\,\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]
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