Integrand size = 23, antiderivative size = 163 \[ \int \cot ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {11 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {2 a \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}+\frac {11 a \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 d} \]
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Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2797, 2725, 3123, 3059, 2851, 2852, 212} \[ \int \cot ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {11 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{8 d}-\frac {2 a \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}+\frac {11 a \cot (c+d x)}{8 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {a \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a \sin (c+d x)+a}} \]
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Rule 212
Rule 2725
Rule 2797
Rule 2851
Rule 2852
Rule 3059
Rule 3123
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {a+a \sin (c+d x)} \, dx+\int \csc ^4(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)}{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 d}+\frac {\int \csc ^3(c+d x) \left (\frac {a}{2}-\frac {9}{2} a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{3 a} \\ & = -\frac {2 a \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc (c+d x)}{12 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 d}-\frac {11}{8} \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {2 a \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}+\frac {11 a \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc (c+d x)}{12 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 d}-\frac {11}{16} \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {2 a \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}+\frac {11 a \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc (c+d x)}{12 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 d}+\frac {(11 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 {11 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {2 a \cos (c+d x)}{d \sqrt {a+a \sin (c+d x)}}+\frac {11 a \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc (c+d x)}{12 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 d} \\ \end{align*}
Time = 1.29 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.90 \[ \int \cot ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\csc ^{10}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (252 \cos \left (\frac {1}{2} (c+d x)\right )-250 \cos \left (\frac {3}{2} (c+d x)\right )-114 \cos \left (\frac {5}{2} (c+d x)\right )+48 \cos \left (\frac {7}{2} (c+d x)\right )-252 \sin \left (\frac {1}{2} (c+d x)\right )+99 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-99 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-250 \sin \left (\frac {3}{2} (c+d x)\right )+114 \sin \left (\frac {5}{2} (c+d x)\right )-33 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+33 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+48 \sin \left (\frac {7}{2} (c+d x)\right )\right )}{24 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^3} \]
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Time = 0.12 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.04
method | result | size |
default | \(\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (-48 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {7}{2}} \left (\sin ^{3}\left (d x +c \right )\right )+33 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {3}{2}}+33 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{4} \left (\sin ^{3}\left (d x +c \right )\right )-56 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {5}{2}}+15 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {7}{2}}\right )}{24 a^{\frac {7}{2}} \sin \left (d x +c \right )^{3} \cos \left (d x +c \right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, d}\) | \(170\) |
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Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (141) = 282\).
Time = 0.28 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.33 \[ \int \cot ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {33 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (48 \, \cos \left (d x + c\right )^{4} - 33 \, \cos \left (d x + c\right )^{3} - 139 \, \cos \left (d x + c\right )^{2} + {\left (48 \, \cos \left (d x + c\right )^{3} + 81 \, \cos \left (d x + c\right )^{2} - 58 \, \cos \left (d x + c\right ) - 83\right )} \sin \left (d x + c\right ) + 25 \, \cos \left (d x + c\right ) + 83\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{96 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \cot ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \cot ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{4} \,d x } \]
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Time = 0.36 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.29 \[ \int \cot ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {2} {\left (33 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 192 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {4 \, {\left (132 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 112 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}\right )} \sqrt {a}}{96 \, d} \]
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Timed out. \[ \int \cot ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{{\sin \left (c+d\,x\right )}^4} \,d x \]
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