Integrand size = 29, antiderivative size = 156 \[ \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {13 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 d}-\frac {2 a \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a+a \sin (c+d x)}}{2 d} \]
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Time = 0.29 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2960, 2830, 2725, 3123, 3059, 2852, 212} \[ \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {13 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{4 d}-\frac {2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {2 a \cos (c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {a \cot (c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a \sin (c+d x)+a}}{2 d} \]
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Rule 212
Rule 2725
Rule 2830
Rule 2852
Rule 2960
Rule 3059
Rule 3123
Rubi steps \begin{align*} \text {integral}& = \int \sin (c+d x) \sqrt {a+a \sin (c+d x)} \, dx+\int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \left (1-2 \sin ^2(c+d x)\right ) \, dx \\ & = -\frac {2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a+a \sin (c+d x)}}{2 d}+\frac {1}{3} \int \sqrt {a+a \sin (c+d x)} \, dx+\frac {\int \csc ^2(c+d x) \left (\frac {a}{2}-\frac {7}{2} a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{2 a} \\ & = -\frac {2 a \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a+a \sin (c+d x)}}{2 d}-\frac {13}{8} \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {2 a \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a+a \sin (c+d x)}}{2 d}+\frac {(13 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 )}{4 d} \\ & = \frac {13 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 d}-\frac {2 a \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}-\frac {\cot (c+d x) \csc (c+d x) \sqrt {a+a \sin (c+d x)}}{2 d} \\ \end{align*}
Time = 1.06 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.90 \[ \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\csc ^7\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (-26 \cos \left (\frac {1}{2} (c+d x)\right )-14 \cos \left (\frac {3}{2} (c+d x)\right )+12 \cos \left (\frac {5}{2} (c+d x)\right )+4 \cos \left (\frac {7}{2} (c+d x)\right )+39 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-39 \cos (2 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-39 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+39 \cos (2 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+26 \sin \left (\frac {1}{2} (c+d x)\right )-14 \sin \left (\frac {3}{2} (c+d x)\right )-12 \sin \left (\frac {5}{2} (c+d x)\right )+4 \sin \left (\frac {7}{2} (c+d x)\right )\right )}{12 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^2} \]
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Time = 0.14 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.14
method | result | size |
default | \(\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (8 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {a}-24 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {3}{2}} \left (\sin ^{2}\left (d x +c \right )\right )+39 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) \left (\sin ^{2}\left (d x +c \right )\right ) a^{2}+9 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} \sqrt {a}-15 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {3}{2}}\right )}{12 a^{\frac {3}{2}} \sin \left (d x +c \right )^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(178\) |
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Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (132) = 264\).
Time = 0.28 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.30 \[ \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {39 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) - 4 \, {\left (8 \, \cos \left (d x + c\right )^{4} + 16 \, \cos \left (d x + c\right )^{3} - 9 \, \cos \left (d x + c\right )^{2} + {\left (8 \, \cos \left (d x + c\right )^{3} - 8 \, \cos \left (d x + c\right )^{2} - 17 \, \cos \left (d x + c\right ) + 5\right )} \sin \left (d x + c\right ) - 22 \, \cos \left (d x + c\right ) - 5\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{48 \, {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right ) - d\right )}} \]
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Timed out. \[ \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{3} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.35 \[ \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\sqrt {2} {\left (64 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 39 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 96 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {12 \, {\left (6 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}\right )} \sqrt {a}}{48 \, d} \]
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Timed out. \[ \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{{\sin \left (c+d\,x\right )}^3} \,d x \]
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