Integrand size = 21, antiderivative size = 137 \[ \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {9 \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{7/2}}-\frac {9 \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{7/2}}+\frac {9}{10 b d (d \cos (a+b x))^{5/2}}+\frac {9}{2 b d^3 \sqrt {d \cos (a+b x)}}-\frac {\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{5/2}} \]
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Time = 0.06 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2645, 296, 331, 335, 304, 209, 212} \[ \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {9 \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{7/2}}-\frac {9 \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{7/2}}+\frac {9}{2 b d^3 \sqrt {d \cos (a+b x)}}+\frac {9}{10 b d (d \cos (a+b x))^{5/2}}-\frac {\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{5/2}} \]
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Rule 209
Rule 212
Rule 296
Rule 304
Rule 331
Rule 335
Rule 2645
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{x^{7/2} \left (1-\frac {x^2}{d^2}\right )^2} \, dx,x,d \cos (a+b x)\right )}{b d} \\ & = -\frac {\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{5/2}}-\frac {9 \text {Subst}\left (\int \frac {1}{x^{7/2} \left (1-\frac {x^2}{d^2}\right )} \, dx,x,d \cos (a+b x)\right )}{4 b d} \\ & = \frac {9}{10 b d (d \cos (a+b x))^{5/2}}-\frac {\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{5/2}}-\frac {9 \text {Subst}\left (\int \frac {1}{x^{3/2} \left (1-\frac {x^2}{d^2}\right )} \, dx,x,d \cos (a+b x)\right )}{4 b d^3} \\ & = \frac {9}{10 b d (d \cos (a+b x))^{5/2}}+\frac {9}{2 b d^3 \sqrt {d \cos (a+b x)}}-\frac {\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{5/2}}-\frac {9 \text {Subst}\left (\int \frac {\sqrt {x}}{1-\frac {x^2}{d^2}} \, dx,x,d \cos (a+b x)\right )}{4 b d^5} \\ & = \frac {9}{10 b d (d \cos (a+b x))^{5/2}}+\frac {9}{2 b d^3 \sqrt {d \cos (a+b x)}}-\frac {\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{5/2}}-\frac {9 \text {Subst}\left (\int \frac {x^2}{1-\frac {x^4}{d^2}} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{2 b d^5} \\ & = \frac {9}{10 b d (d \cos (a+b x))^{5/2}}+\frac {9}{2 b d^3 \sqrt {d \cos (a+b x)}}-\frac {\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{5/2}}-\frac {9 \text {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{4 b d^3}+\frac {9 \text {Subst}\left (\int \frac {1}{d+x^2} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{4 b d^3} \\ & = \frac {9 \arctan \left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{7/2}}-\frac {9 \text {arctanh}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{7/2}}+\frac {9}{10 b d (d \cos (a+b x))^{5/2}}+\frac {9}{2 b d^3 \sqrt {d \cos (a+b x)}}-\frac {\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{5/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.53 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.74 \[ \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {45 \cot ^2(a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{4},\frac {5}{4},\csc ^2(a+b x)\right )+\left (-\cot ^2(a+b x)\right )^{3/4} \left (40-5 \cot ^2(a+b x)+4 \sec ^2(a+b x)\right )}{10 b d^3 \sqrt {d \cos (a+b x)} \left (-\cot ^2(a+b x)\right )^{3/4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1139\) vs. \(2(109)=218\).
Time = 0.18 (sec) , antiderivative size = 1140, normalized size of antiderivative = 8.32
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Time = 0.37 (sec) , antiderivative size = 438, normalized size of antiderivative = 3.20 \[ \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\left [\frac {90 \, {\left (\cos \left (b x + a\right )^{5} - \cos \left (b x + a\right )^{3}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )} \sqrt {-d} {\left (\cos \left (b x + a\right ) + 1\right )}}{2 \, d \cos \left (b x + a\right )}\right ) - 45 \, {\left (\cos \left (b x + a\right )^{5} - \cos \left (b x + a\right )^{3}\right )} \sqrt {-d} \log \left (\frac {d \cos \left (b x + a\right )^{2} - 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {-d} {\left (\cos \left (b x + a\right ) - 1\right )} - 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, {\left (45 \, \cos \left (b x + a\right )^{4} - 36 \, \cos \left (b x + a\right )^{2} - 4\right )} \sqrt {d \cos \left (b x + a\right )}}{80 \, {\left (b d^{4} \cos \left (b x + a\right )^{5} - b d^{4} \cos \left (b x + a\right )^{3}\right )}}, \frac {90 \, {\left (\cos \left (b x + a\right )^{5} - \cos \left (b x + a\right )^{3}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - 1\right )}}{2 \, \sqrt {d} \cos \left (b x + a\right )}\right ) + 45 \, {\left (\cos \left (b x + a\right )^{5} - \cos \left (b x + a\right )^{3}\right )} \sqrt {d} \log \left (\frac {d \cos \left (b x + a\right )^{2} - 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {d} {\left (\cos \left (b x + a\right ) + 1\right )} + 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, {\left (45 \, \cos \left (b x + a\right )^{4} - 36 \, \cos \left (b x + a\right )^{2} - 4\right )} \sqrt {d \cos \left (b x + a\right )}}{80 \, {\left (b d^{4} \cos \left (b x + a\right )^{5} - b d^{4} \cos \left (b x + a\right )^{3}\right )}}\right ] \]
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Timed out. \[ \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.98 \[ \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\frac {\frac {4 \, {\left (45 \, d^{4} \cos \left (b x + a\right )^{4} - 36 \, d^{4} \cos \left (b x + a\right )^{2} - 4 \, d^{4}\right )}}{\left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} d^{2} - \left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}} d^{4}} + \frac {90 \, \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )}}{\sqrt {d}}\right )}{d^{\frac {5}{2}}} + \frac {45 \, \log \left (\frac {\sqrt {d \cos \left (b x + a\right )} - \sqrt {d}}{\sqrt {d \cos \left (b x + a\right )} + \sqrt {d}}\right )}{d^{\frac {5}{2}}}}{40 \, b d} \]
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\[ \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\int { \frac {\csc \left (b x + a\right )^{3}}{\left (d \cos \left (b x + a\right )\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx=\int \frac {1}{{\sin \left (a+b\,x\right )}^3\,{\left (d\,\cos \left (a+b\,x\right )\right )}^{7/2}} \,d x \]
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