Integrand size = 23, antiderivative size = 144 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 a^{3/2} d}-\frac {\cot (c+d x)}{8 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 a^2 d} \]
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Time = 0.40 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2796, 2851, 2852, 212, 3123, 3059} \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{8 a^{3/2} d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 a^2 d}-\frac {\cot (c+d x)}{8 a d \sqrt {a \sin (c+d x)+a}}+\frac {11 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt {a \sin (c+d x)+a}} \]
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Rule 212
Rule 2796
Rule 2851
Rule 2852
Rule 3059
Rule 3123
Rubi steps \begin{align*} \text {integral}& = \frac {\int \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \left (1+\sin ^2(c+d x)\right ) \, dx}{a^2}-\frac {2 \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{a^2} \\ & = \frac {\cot (c+d x) \csc (c+d x)}{a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 a^2 d}+\frac {\int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \left (\frac {a}{2}+\frac {9}{2} a \sin (c+d x)\right ) \, dx}{3 a^3}-\frac {3 \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{2 a^2} \\ & = \frac {3 \cot (c+d x)}{2 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 a^2 d}-\frac {3 \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{4 a^2}+\frac {13 \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{8 a^2} \\ & = -\frac {\cot (c+d x)}{8 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 a^2 d}+\frac {13 \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{16 a^2}+\frac {3 \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 )}{2 a d} \\ & = \frac {3 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{2 a^{3/2} d}-\frac {\cot (c+d x)}{8 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 a^2 d}-\frac {13 \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 a d} \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 a^{3/2} d}-\frac {\cot (c+d x)}{8 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 a^2 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(294\) vs. \(2(144)=288\).
Time = 0.92 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.04 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\csc ^9\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (-132 \cos \left (\frac {1}{2} (c+d x)\right )+62 \cos \left (\frac {3}{2} (c+d x)\right )+6 \cos \left (\frac {5}{2} (c+d x)\right )+132 \sin \left (\frac {1}{2} (c+d x)\right )-9 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+9 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+62 \sin \left (\frac {3}{2} (c+d x)\right )-6 \sin \left (\frac {5}{2} (c+d x)\right )+3 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-3 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))\right )}{24 d \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^3 (a (1+\sin (c+d x)))^{3/2}} \]
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Time = 0.12 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (3 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {3}{2}}+3 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{4} \left (\sin ^{3}\left (d x +c \right )\right )+8 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {5}{2}}-3 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {7}{2}}\right )}{24 a^{\frac {11}{2}} \sin \left (d x +c \right )^{3} \cos \left (d x +c \right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, d}\) | \(144\) |
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Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (124) = 248\).
Time = 0.28 (sec) , antiderivative size = 383, normalized size of antiderivative = 2.66 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {3 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (3 \, \cos \left (d x + c\right )^{3} + 17 \, \cos \left (d x + c\right )^{2} - {\left (3 \, \cos \left (d x + c\right )^{2} - 14 \, \cos \left (d x + c\right ) - 25\right )} \sin \left (d x + c\right ) - 11 \, \cos \left (d x + c\right ) - 25\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{96 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d - {\left (a^{2} d \cos \left (d x + c\right )^{3} + a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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Time = 0.33 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.17 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {\sqrt {2} \sqrt {a} {\left (\frac {3 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {4 \, {\left (12 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 16 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}\right )}}{96 \, d} \]
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Timed out. \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4}{{\sin \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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