Integrand size = 17, antiderivative size = 32 \[ \int \sqrt {\csc (a+b x)} \sec (a+b x) \, dx=-\frac {\arctan \left (\sqrt {\csc (a+b x)}\right )}{b}+\frac {\text {arctanh}\left (\sqrt {\csc (a+b x)}\right )}{b} \]
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Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2701, 335, 304, 209, 212} \[ \int \sqrt {\csc (a+b x)} \sec (a+b x) \, dx=\frac {\text {arctanh}\left (\sqrt {\csc (a+b x)}\right )}{b}-\frac {\arctan \left (\sqrt {\csc (a+b x)}\right )}{b} \]
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Rule 209
Rule 212
Rule 304
Rule 335
Rule 2701
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{b} \\ & = -\frac {2 \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt {\csc (a+b x)}\right )}{b} \\ & = \frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\csc (a+b x)}\right )}{b}-\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\csc (a+b x)}\right )}{b} \\ & = -\frac {\arctan \left (\sqrt {\csc (a+b x)}\right )}{b}+\frac {\text {arctanh}\left (\sqrt {\csc (a+b x)}\right )}{b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.47 \[ \int \sqrt {\csc (a+b x)} \sec (a+b x) \, dx=\frac {\left (\arctan \left (\sqrt {\sin (a+b x)}\right )+\text {arctanh}\left (\sqrt {\sin (a+b x)}\right )\right ) \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}}{b} \]
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Time = 0.93 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {\operatorname {arctanh}\left (\sqrt {\sin }\left (b x +a \right )\right )+\arctan \left (\sqrt {\sin }\left (b x +a \right )\right )}{b}\) | \(24\) |
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Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (28) = 56\).
Time = 0.33 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.97 \[ \int \sqrt {\csc (a+b x)} \sec (a+b x) \, dx=\frac {2 \, \arctan \left (\frac {\sin \left (b x + a\right ) - 1}{2 \, \sqrt {\sin \left (b x + a\right )}}\right ) + \log \left (\frac {\cos \left (b x + a\right )^{2} + \frac {4 \, {\left (\cos \left (b x + a\right )^{2} - \sin \left (b x + a\right ) - 1\right )}}{\sqrt {\sin \left (b x + a\right )}} - 6 \, \sin \left (b x + a\right ) - 2}{\cos \left (b x + a\right )^{2} + 2 \, \sin \left (b x + a\right ) - 2}\right )}{4 \, b} \]
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\[ \int \sqrt {\csc (a+b x)} \sec (a+b x) \, dx=\int \sqrt {\csc {\left (a + b x \right )}} \sec {\left (a + b x \right )}\, dx \]
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none
Time = 0.45 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \sqrt {\csc (a+b x)} \sec (a+b x) \, dx=-\frac {2 \, \arctan \left (\frac {1}{\sqrt {\sin \left (b x + a\right )}}\right ) - \log \left (\frac {1}{\sqrt {\sin \left (b x + a\right )}} + 1\right ) + \log \left (\frac {1}{\sqrt {\sin \left (b x + a\right )}} - 1\right )}{2 \, b} \]
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Time = 0.31 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \sqrt {\csc (a+b x)} \sec (a+b x) \, dx=\frac {2 \, \arctan \left (\sqrt {\sin \left (b x + a\right )}\right ) + \log \left (\sqrt {\sin \left (b x + a\right )} + 1\right ) - \log \left ({\left | \sqrt {\sin \left (b x + a\right )} - 1 \right |}\right )}{2 \, b} \]
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Timed out. \[ \int \sqrt {\csc (a+b x)} \sec (a+b x) \, dx=\int \frac {\sqrt {\frac {1}{\sin \left (a+b\,x\right )}}}{\cos \left (a+b\,x\right )} \,d x \]
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