Integrand size = 19, antiderivative size = 67 \[ \int \cos ^2(a+b x) \sqrt {\csc (a+b x)} \, dx=\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {4 \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{3 b} \]
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Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2708, 3856, 2720} \[ \int \cos ^2(a+b x) \sqrt {\csc (a+b x)} \, dx=\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {4 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{3 b} \]
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Rule 2708
Rule 2720
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {2}{3} \int \sqrt {\csc (a+b x)} \, dx \\ & = \frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {1}{3} \left (2 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx \\ & = \frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {4 \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{3 b} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.79 \[ \int \cos ^2(a+b x) \sqrt {\csc (a+b x)} \, dx=\frac {\sqrt {\csc (a+b x)} \left (-4 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sqrt {\sin (a+b x)}+\sin (2 (a+b x))\right )}{3 b} \]
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Time = 1.17 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.31
method | result | size |
default | \(\frac {\frac {2 \sqrt {\sin \left (b x +a \right )+1}\, \sqrt {-2 \sin \left (b x +a \right )+2}\, \sqrt {-\sin \left (b x +a \right )}\, F\left (\sqrt {\sin \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{3}+\frac {2 \left (\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3}}{\cos \left (b x +a \right ) \sqrt {\sin \left (b x +a \right )}\, b}\) | \(88\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \cos ^2(a+b x) \sqrt {\csc (a+b x)} \, dx=\frac {2 \, {\left (\cos \left (b x + a\right ) \sqrt {\sin \left (b x + a\right )} - i \, \sqrt {2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, \sqrt {-2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}}{3 \, b} \]
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\[ \int \cos ^2(a+b x) \sqrt {\csc (a+b x)} \, dx=\int \cos ^{2}{\left (a + b x \right )} \sqrt {\csc {\left (a + b x \right )}}\, dx \]
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\[ \int \cos ^2(a+b x) \sqrt {\csc (a+b x)} \, dx=\int { \cos \left (b x + a\right )^{2} \sqrt {\csc \left (b x + a\right )} \,d x } \]
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\[ \int \cos ^2(a+b x) \sqrt {\csc (a+b x)} \, dx=\int { \cos \left (b x + a\right )^{2} \sqrt {\csc \left (b x + a\right )} \,d x } \]
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Timed out. \[ \int \cos ^2(a+b x) \sqrt {\csc (a+b x)} \, dx=\int {\cos \left (a+b\,x\right )}^2\,\sqrt {\frac {1}{\sin \left (a+b\,x\right )}} \,d x \]
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