Integrand size = 31, antiderivative size = 281 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {61 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{1024 d}-\frac {61 a \cot (c+d x)}{1024 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt {a+a \sin (c+d x)}}+\frac {579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d} \]
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Time = 0.68 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2960, 2851, 2852, 212, 3123, 3059} \[ \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {61 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{1024 d}-\frac {61 a \cot (c+d x)}{1024 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{7 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a \sin (c+d x)+a}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a \sin (c+d x)+a}}+\frac {579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a \sin (c+d x)+a}}-\frac {61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt {a \sin (c+d x)+a}}-\frac {61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt {a \sin (c+d x)+a}} \]
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Rule 212
Rule 2851
Rule 2852
Rule 2960
Rule 3059
Rule 3123
Rubi steps \begin{align*} \text {integral}& = \int \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx+\int \csc ^8(c+d x) \sqrt {a+a \sin (c+d x)} \left (1-2 \sin ^2(c+d x)\right ) \, dx \\ & = -\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}+\frac {5}{6} \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx+\frac {\int \csc ^7(c+d x) \left (\frac {a}{2}-\frac {17}{2} a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{7 a} \\ & = -\frac {5 a \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}+\frac {5}{8} \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {193}{168} \int \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {5 a \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {5 a \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}+\frac {5}{16} \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {579}{560} \int \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {5 a \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {5 a \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}+\frac {579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}-\frac {579}{640} \int \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {(5 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 )}{8 d} \\ & = -\frac {5 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {5 a \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {5 a \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt {a+a \sin (c+d x)}}+\frac {579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}-\frac {193}{256} \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {5 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {5 a \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt {a+a \sin (c+d x)}}+\frac {579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}-\frac {579 \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{1024} \\ & = -\frac {5 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {61 a \cot (c+d x)}{1024 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt {a+a \sin (c+d x)}}+\frac {579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}-\frac {579 \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{2048} \\ & = -\frac {5 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {61 a \cot (c+d x)}{1024 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt {a+a \sin (c+d x)}}+\frac {579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d}+\frac {(579 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 )}{1024 d} \\ & = -\frac {61 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{1024 d}-\frac {61 a \cot (c+d x)}{1024 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc (c+d x)}{1536 d \sqrt {a+a \sin (c+d x)}}-\frac {61 a \cot (c+d x) \csc ^2(c+d x)}{1920 d \sqrt {a+a \sin (c+d x)}}+\frac {579 a \cot (c+d x) \csc ^3(c+d x)}{2240 d \sqrt {a+a \sin (c+d x)}}+\frac {193 a \cot (c+d x) \csc ^4(c+d x)}{840 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{84 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{7 d} \\ \end{align*}
Time = 1.96 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.68 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {a (1+\sin (c+d x))} \left (-102480 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+102480 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\csc ^7(c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (-201298-244533 \cos (2 (c+d x))-52094 \cos (4 (c+d x))+6405 \cos (6 (c+d x))+49128 \sin (c+d x)-179636 \sin (3 (c+d x))-8540 \sin (5 (c+d x)))\right )}{3440640 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
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Time = 0.14 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.77
method | result | size |
default | \(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (6405 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {13}{2}} a^{\frac {7}{2}}-42700 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {11}{2}} a^{\frac {9}{2}}+120841 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {9}{2}} a^{\frac {11}{2}}+6405 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{10} \left (\sin ^{7}\left (d x +c \right )\right )-156672 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {13}{2}}+51191 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {15}{2}}+42700 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {17}{2}}-6405 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {19}{2}}\right )}{107520 a^{\frac {19}{2}} \sin \left (d x +c \right )^{7} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(216\) |
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Leaf count of result is larger than twice the leaf count of optimal. 567 vs. \(2 (245) = 490\).
Time = 0.29 (sec) , antiderivative size = 567, normalized size of antiderivative = 2.02 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {6405 \, {\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{7} + \cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{5} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{3} + 3 \, \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 (6405 \, \cos \left (d x + c\right )^{7} + 2135 \, \cos \left (d x + c\right )^{6} - 22631 \, \cos \left (d x + c\right )^{5} - 37613 \, \cos \left (d x + c\right )^{4} + 1343 \, \cos \left (d x + c\right )^{3} + 27477 \, \cos \left (d x + c\right )^{2} - {\left (6405 \, \cos \left (d x + c\right )^{6} + 4270 \, \cos \left (d x + c\right )^{5} - 18361 \, \cos \left (d x + c\right )^{4} + 19252 \, \cos \left (d x + c\right )^{3} + 20595 \, \cos \left (d x + c\right )^{2} - 6882 \, \cos \left (d x + c\right ) - 7359\right )} \sin \left (d x + c\right ) - 477 \, \cos \left (d x + c\right ) - 7359\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{430080 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{5} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{3} + 3 \, d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{8} \,d x } \]
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Time = 0.35 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.07 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\sqrt {2} {\left (6405 \, \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 ) + \frac {4 \, {\left (409920 \, \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 )^{13} - 1366400 \, \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 )^{11} + 1933456 \, \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 )^{9} - 1253376 \, \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} + 204764 \, \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} + 85400 \, \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} - 6405 \, \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 )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{7}}\right )} \sqrt {a}}{430080 \, d} \]
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Timed out. \[ \int \cot ^4(c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{{\sin \left (c+d\,x\right )}^8} \,d x \]
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