Integrand size = 31, antiderivative size = 329 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {1587 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{16384 d}-\frac {1587 a^2 \cot (c+d x)}{16384 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc (c+d x)}{8192 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc ^2(c+d x)}{10240 d \sqrt {a+a \sin (c+d x)}}+\frac {8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt {a+a \sin (c+d x)}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d} \]
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Time = 0.86 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {2960, 2841, 21, 2851, 2852, 212, 3123, 3054, 3059} \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {1587 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{16384 d}-\frac {1587 a^2 \cot (c+d x)}{16384 d \sqrt {a \sin (c+d x)+a}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a \sin (c+d x)+a}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a \sin (c+d x)+a}}+\frac {8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt {a \sin (c+d x)+a}}-\frac {529 a^2 \cot (c+d x) \csc ^2(c+d x)}{10240 d \sqrt {a \sin (c+d x)+a}}-\frac {529 a^2 \cot (c+d x) \csc (c+d x)}{8192 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a \sin (c+d x)+a}}{112 d} \]
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Rule 21
Rule 212
Rule 2841
Rule 2851
Rule 2852
Rule 2960
Rule 3054
Rule 3059
Rule 3123
Rubi steps \begin{align*} \text {integral}& = \int \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx+\int \csc ^9(c+d x) (a+a \sin (c+d x))^{3/2} \left (1-2 \sin ^2(c+d x)\right ) \, dx \\ & = -\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {\int \csc ^8(c+d x) \left (\frac {3 a}{2}-\frac {21}{2} a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{8 a}-\frac {1}{4} a \int \frac {\csc ^4(c+d x) \left (-\frac {15 a}{2}-\frac {15}{2} a \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = -\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {\int \csc ^7(c+d x) \sqrt {a+a \sin (c+d x)} \left (-\frac {249 a^2}{4}-\frac {261}{4} a^2 \sin (c+d x)\right ) \, dx}{56 a}+\frac {1}{8} (15 a) \int \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {5 a^2 \cot (c+d x) \csc ^2(c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {1}{16} (25 a) \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {1}{896} (1957 a) \int \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {25 a^2 \cot (c+d x) \csc (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}-\frac {5 a^2 \cot (c+d x) \csc ^2(c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {1}{64} (75 a) \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {(17613 a) \int \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{8960} \\ & = -\frac {75 a^2 \cot (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}-\frac {25 a^2 \cot (c+d x) \csc (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}-\frac {5 a^2 \cot (c+d x) \csc ^2(c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}+\frac {8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt {a+a \sin (c+d x)}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {1}{128} (75 a) \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx-\frac {(17613 a) \int \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{10240} \\ & = -\frac {75 a^2 \cot (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}-\frac {25 a^2 \cot (c+d x) \csc (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc ^2(c+d x)}{10240 d \sqrt {a+a \sin (c+d x)}}+\frac {8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt {a+a \sin (c+d x)}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}-\frac {(5871 a) \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{4096}-\frac {\left (75 a^2\right ) \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 )}{64 d} \\ & = -\frac {75 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 d}-\frac {75 a^2 \cot (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc (c+d x)}{8192 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc ^2(c+d x)}{10240 d \sqrt {a+a \sin (c+d x)}}+\frac {8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt {a+a \sin (c+d x)}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}-\frac {(17613 a) \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{16384} \\ & = -\frac {75 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 d}-\frac {1587 a^2 \cot (c+d x)}{16384 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc (c+d x)}{8192 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc ^2(c+d x)}{10240 d \sqrt {a+a \sin (c+d x)}}+\frac {8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt {a+a \sin (c+d x)}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}-\frac {(17613 a) \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{32768} \\ & = -\frac {75 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 d}-\frac {1587 a^2 \cot (c+d x)}{16384 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc (c+d x)}{8192 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc ^2(c+d x)}{10240 d \sqrt {a+a \sin (c+d x)}}+\frac {8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt {a+a \sin (c+d x)}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {\left (17613 a^2\right ) \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 )}{16384 d} \\ & = -\frac {1587 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{16384 d}-\frac {1587 a^2 \cot (c+d x)}{16384 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc (c+d x)}{8192 d \sqrt {a+a \sin (c+d x)}}-\frac {529 a^2 \cot (c+d x) \csc ^2(c+d x)}{10240 d \sqrt {a+a \sin (c+d x)}}+\frac {8653 a^2 \cot (c+d x) \csc ^3(c+d x)}{35840 d \sqrt {a+a \sin (c+d x)}}+\frac {1957 a^2 \cot (c+d x) \csc ^4(c+d x)}{4480 d \sqrt {a+a \sin (c+d x)}}+\frac {83 a^2 \cot (c+d x) \csc ^5(c+d x)}{448 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)}}{112 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d} \\ \end{align*}
Time = 6.67 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.84 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {a \csc ^{25}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (3037258 \cos \left (\frac {1}{2} (c+d x)\right )+10394286 \cos \left (\frac {3}{2} (c+d x)\right )+3369650 \cos \left (\frac {5}{2} (c+d x)\right )+3171574 \cos \left (\frac {7}{2} (c+d x)\right )-2341070 \cos \left (\frac {9}{2} (c+d x)\right )+866502 \cos \left (\frac {11}{2} (c+d x)\right )-37030 \cos \left (\frac {13}{2} (c+d x)\right )-111090 \cos \left (\frac {15}{2} (c+d x)\right )+1944075 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-3110520 \cos (2 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+1555260 \cos (4 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-444360 \cos (6 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+55545 \cos (8 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-1944075 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+3110520 \cos (2 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-1555260 \cos (4 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+444360 \cos (6 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-55545 \cos (8 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-3037258 \sin \left (\frac {1}{2} (c+d x)\right )+10394286 \sin \left (\frac {3}{2} (c+d x)\right )-3369650 \sin \left (\frac {5}{2} (c+d x)\right )+3171574 \sin \left (\frac {7}{2} (c+d x)\right )+2341070 \sin \left (\frac {9}{2} (c+d x)\right )+866502 \sin \left (\frac {11}{2} (c+d x)\right )+37030 \sin \left (\frac {13}{2} (c+d x)\right )-111090 \sin \left (\frac {15}{2} (c+d x)\right )\right )}{573440 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^8} \]
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Time = 0.13 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.71
method | result | size |
default | \(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (-55545 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {15}{2}} a^{\frac {7}{2}}+425845 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {13}{2}} a^{\frac {9}{2}}-1418249 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {11}{2}} a^{\frac {11}{2}}+2509197 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {9}{2}} a^{\frac {13}{2}}-2176627 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {15}{2}}+416759 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {17}{2}}+425845 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {19}{2}}-55545 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {21}{2}}+55545 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{11} \left (\sin ^{8}\left (d x +c \right )\right )\right )}{573440 a^{\frac {19}{2}} \sin \left (d x +c \right )^{8} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(234\) |
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Leaf count of result is larger than twice the leaf count of optimal. 657 vs. \(2 (289) = 578\).
Time = 0.35 (sec) , antiderivative size = 657, normalized size of antiderivative = 2.00 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {55545 \, {\left (a \cos \left (d x + c\right )^{9} + a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{7} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{5} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{3} - 4 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{8} - 4 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + a\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 (55545 \, a \cos \left (d x + c\right )^{8} + 37030 \, a \cos \left (d x + c\right )^{7} - 214774 \, a \cos \left (d x + c\right )^{6} + 27358 \, a \cos \left (d x + c\right )^{5} + 199004 \, a \cos \left (d x + c\right )^{4} - 185006 \, a \cos \left (d x + c\right )^{3} - 153786 \, a \cos \left (d x + c\right )^{2} + 48938 \, a \cos \left (d x + c\right ) + {\left (55545 \, a \cos \left (d x + c\right )^{7} + 18515 \, a \cos \left (d x + c\right )^{6} - 196259 \, a \cos \left (d x + c\right )^{5} - 223617 \, a \cos \left (d x + c\right )^{4} - 24613 \, a \cos \left (d x + c\right )^{3} + 160393 \, a \cos \left (d x + c\right )^{2} + 6607 \, a \cos \left (d x + c\right ) - 42331 \, a\right )} \sin \left (d x + c\right ) + 42331 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{2293760 \, {\left (d \cos \left (d x + c\right )^{9} + d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{7} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{5} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{3} - 4 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) + {\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} + d\right )} \sin \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
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\[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{9} \,d x } \]
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Time = 0.36 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.03 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {\sqrt {2} {\left (55545 \, \sqrt {2} a \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 (7109760 \, a \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 )^{15} - 27254080 \, a \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} + 45383968 \, a \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} - 40147152 \, a \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} + 17413016 \, a \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} - 1667036 \, a \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} - 851690 \, a \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} + 55545 \, a \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 )}^{8}}\right )} \sqrt {a}}{2293760 \, d} \]
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Timed out. \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\sin \left (c+d\,x\right )}^9} \,d x \]
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