Integrand size = 21, antiderivative size = 93 \[ \int \frac {\csc ^3(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{4 b^{3/2} f}+\frac {\text {arctanh}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{4 b^{3/2} f}-\frac {\cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{2 b^3 f} \]
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Time = 0.05 (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, 304, 209, 212} \[ \int \frac {\csc ^3(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{4 b^{3/2} f}+\frac {\text {arctanh}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{4 b^{3/2} f}-\frac {\cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{2 b^3 f} \]
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Rule 209
Rule 212
Rule 296
Rule 304
Rule 335
Rule 2702
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\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) (b \sec (e+f x))^{3/2}}{2 b^3 f}-\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{-1+\frac {x^2}{b^2}} \, dx,x,b \sec (e+f x)\right )}{4 b^3 f} \\ & = -\frac {\cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{2 b^3 f}-\frac {\text {Subst}\left (\int \frac {x^2}{-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) (b \sec (e+f x))^{3/2}}{2 b^3 f}+\frac {\text {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{4 b f}-\frac {\text {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{4 b f} \\ & = -\frac {\arctan \left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{4 b^{3/2} f}+\frac {\text {arctanh}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{4 b^{3/2} f}-\frac {\cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{2 b^3 f} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.05 \[ \int \frac {\csc ^3(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\frac {-4 \csc ^2(e+f x)-2 \arctan \left (\sqrt {\sec (e+f x)}\right ) \sqrt {\sec (e+f x)}+\left (-\log \left (1-\sqrt {\sec (e+f x)}\right )+\log \left (1+\sqrt {\sec (e+f x)}\right )\right ) \sqrt {\sec (e+f x)}}{8 b f \sqrt {b \sec (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(73)=146\).
Time = 0.19 (sec) , antiderivative size = 281, normalized size of antiderivative = 3.02
method | result | size |
default | \(\frac {\ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right ) \cos \left (f x +e \right )-\cos \left (f x +e \right ) \arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\right )+4 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\ln \left (\frac {2 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )+1}\right )+\arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\right )}{8 f \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {b \sec \left (f x +e \right )}\, b \left (\cos ^{2}\left (f x +e \right )-1\right )}\) | \(281\) |
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Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (73) = 146\).
Time = 0.37 (sec) , antiderivative size = 364, normalized size of antiderivative = 3.91 \[ \int \frac {\csc ^3(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\left [-\frac {2 \, {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) + 1\right )}}{2 \, b}\right ) + {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {-b} \log \left (\frac {b \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt {-b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} - 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) - 8 \, \sqrt {\frac {b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{16 \, {\left (b^{2} f \cos \left (f x + e\right )^{2} - b^{2} f\right )}}, \frac {2 \, {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {b} \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) - 1\right )}}{2 \, \sqrt {b}}\right ) + {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {b} \log \left (\frac {b \cos \left (f x + e\right )^{2} + 4 \, {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} + 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1}\right ) + 8 \, \sqrt {\frac {b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{16 \, {\left (b^{2} f \cos \left (f x + e\right )^{2} - b^{2} f\right )}}\right ] \]
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\[ \int \frac {\csc ^3(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\int \frac {\csc ^{3}{\left (e + f x \right )}}{\left (b \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.14 \[ \int \frac {\csc ^3(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\frac {b {\left (\frac {4 \, \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {3}{2}}}{b^{4} - \frac {b^{4}}{\cos \left (f x + e\right )^{2}}} - \frac {2 \, \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (f x + e\right )}}}{\sqrt {b}}\right )}{b^{\frac {5}{2}}} - \frac {\log \left (-\frac {\sqrt {b} - \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{\sqrt {b} + \sqrt {\frac {b}{\cos \left (f x + e\right )}}}\right )}{b^{\frac {5}{2}}}\right )}}{8 \, f} \]
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Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00 \[ \int \frac {\csc ^3(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\frac {\frac {2 \, \sqrt {b \cos \left (f x + e\right )}}{b^{2} \cos \left (f x + e\right )^{2} - b^{2}} - \frac {\arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {-b}}\right )}{\sqrt {-b} b} + \frac {\arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {b}}\right )}{b^{\frac {3}{2}}}}{4 \, f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]
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Timed out. \[ \int \frac {\csc ^3(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^3\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
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