Integrand size = 21, antiderivative size = 93 \[ \int \frac {\csc ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {3 \arctan \left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{4 b^{5/2} f}+\frac {3 \text {arctanh}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{4 b^{5/2} f}-\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{2 b^3 f} \]
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Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2702, 296, 335, 218, 212, 209} \[ \int \frac {\csc ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {3 \arctan \left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{4 b^{5/2} f}+\frac {3 \text {arctanh}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{4 b^{5/2} f}-\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{2 b^3 f} \]
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Rule 209
Rule 212
Rule 218
Rule 296
Rule 335
Rule 2702
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+\frac {x^2}{b^2}\right )^2} \, dx,x,b \sec (e+f x)\right )}{b^3 f} \\ & = -\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{2 b^3 f}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-1+\frac {x^2}{b^2}\right )} \, dx,x,b \sec (e+f x)\right )}{4 b^3 f} \\ & = -\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{2 b^3 f}-\frac {3 \text {Subst}\left (\int \frac {1}{-1+\frac {x^4}{b^2}} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{2 b^3 f} \\ & = -\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{2 b^3 f}+\frac {3 \text {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{4 b^2 f}+\frac {3 \text {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{4 b^2 f} \\ & = \frac {3 \arctan \left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{4 b^{5/2} f}+\frac {3 \text {arctanh}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{4 b^{5/2} f}-\frac {\cot ^2(e+f x) \sqrt {b \sec (e+f x)}}{2 b^3 f} \\ \end{align*}
Time = 0.68 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.05 \[ \int \frac {\csc ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {\left (6 \arctan \left (\sqrt {\sec (e+f x)}\right )-3 \log \left (1-\sqrt {\sec (e+f x)}\right )+3 \log \left (1+\sqrt {\sec (e+f x)}\right )-\frac {4 \csc ^2(e+f x)}{\sec ^{\frac {3}{2}}(e+f x)}\right ) \sqrt {\sec (e+f x)}}{8 b^2 f \sqrt {b \sec (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(289\) vs. \(2(73)=146\).
Time = 0.19 (sec) , antiderivative size = 290, normalized size of antiderivative = 3.12
method | result | size |
default | \(\frac {4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-3 \cos \left (f x +e \right ) \arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\right )-3 \ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right ) \cos \left (f x +e \right )+3 \arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\right )+3 \ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right )}{8 f \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {b \sec \left (f x +e \right )}\, b^{2} \left (\cos ^{2}\left (f x +e \right )-1\right )}\) | \(290\) |
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Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (73) = 146\).
Time = 0.36 (sec) , antiderivative size = 370, normalized size of antiderivative = 3.98 \[ \int \frac {\csc ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\left [-\frac {6 \, {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) + 1\right )}}{2 \, b}\right ) - 8 \, \sqrt {\frac {b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2} + 3 \, {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {-b} \log \left (\frac {b \cos \left (f x + e\right )^{2} + 4 \, {\left (\cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt {-b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} - 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right )}{16 \, {\left (b^{3} f \cos \left (f x + e\right )^{2} - b^{3} f\right )}}, -\frac {6 \, {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {b} \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) - 1\right )}}{2 \, \sqrt {b}}\right ) - 8 \, \sqrt {\frac {b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2} - 3 \, {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {b} \log \left (\frac {b \cos \left (f x + e\right )^{2} + 4 \, {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} + 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1}\right )}{16 \, {\left (b^{3} f \cos \left (f x + e\right )^{2} - b^{3} f\right )}}\right ] \]
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\[ \int \frac {\csc ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\int \frac {\csc ^{3}{\left (e + f x \right )}}{\left (b \sec {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.14 \[ \int \frac {\csc ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {b {\left (\frac {4 \, \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{b^{4} - \frac {b^{4}}{\cos \left (f x + e\right )^{2}}} + \frac {6 \, \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (f x + e\right )}}}{\sqrt {b}}\right )}{b^{\frac {7}{2}}} - \frac {3 \, \log \left (-\frac {\sqrt {b} - \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{\sqrt {b} + \sqrt {\frac {b}{\cos \left (f x + e\right )}}}\right )}{b^{\frac {7}{2}}}\right )}}{8 \, f} \]
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Time = 0.30 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.09 \[ \int \frac {\csc ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\frac {\frac {2 \, \sqrt {b \cos \left (f x + e\right )} b \cos \left (f x + e\right )}{b^{2} \cos \left (f x + e\right )^{2} - b^{2}} - \frac {3 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {3 \, \arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {b}}\right )}{\sqrt {b}}}{4 \, b^{2} f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]
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Timed out. \[ \int \frac {\csc ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^3\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]
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