Integrand size = 10, antiderivative size = 67 \[ \int \frac {1}{\csc ^{\frac {3}{2}}(a+b x)} \, dx=-\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {2 \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{3 b} \]
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Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3854, 3856, 2720} \[ \int \frac {1}{\csc ^{\frac {3}{2}}(a+b x)} \, dx=\frac {2 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{3 b}-\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}} \]
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Rule 2720
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos (a+b x)}{3 b \sqrt {\csc (a+b x)}}+\frac {1}{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 (\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 {2 \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 = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\csc ^{\frac {3}{2}}(a+b x)} \, dx=-\frac {\sqrt {\csc (a+b x)} \left (2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sqrt {\sin (a+b x)}+\sin (2 (a+b x))\right )}{3 b} \]
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Time = 0.43 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.31
method | result | size |
default | \(\frac {\frac {\sqrt {\sin \left (x b +a \right )+1}\, \sqrt {-2 \sin \left (x b +a \right )+2}\, \sqrt {-\sin \left (x b +a \right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (x b +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{3}-\frac {2 \cos \left (x b +a \right )^{2} \sin \left (x b +a \right )}{3}}{\cos \left (x b +a \right ) \sqrt {\sin \left (x b +a \right )}\, b}\) | \(88\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\csc ^{\frac {3}{2}}(a+b x)} \, dx=-\frac {2 \, \cos \left (b x + a\right ) \sqrt {\sin \left (b x + a\right )} + i \, \sqrt {2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - i \, \sqrt {-2 i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{3 \, b} \]
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\[ \int \frac {1}{\csc ^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {1}{\csc ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {1}{\csc ^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {1}{\csc \left (b x + a\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{\csc ^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {1}{\csc \left (b x + a\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\csc ^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {1}{{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{3/2}} \,d x \]
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