Integrand size = 10, antiderivative size = 63 \[ \int \csc ^{\frac {3}{2}}(a+b x) \, dx=-\frac {2 \cos (a+b x) \sqrt {\csc (a+b x)}}{b}-\frac {2 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{b} \]
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Time = 0.03 (sec) , antiderivative size = 63, 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 = {3853, 3856, 2719} \[ \int \csc ^{\frac {3}{2}}(a+b x) \, dx=-\frac {2 \cos (a+b x) \sqrt {\csc (a+b x)}}{b}-\frac {2 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b} \]
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Rule 2719
Rule 3853
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos (a+b x) \sqrt {\csc (a+b x)}}{b}-\int \frac {1}{\sqrt {\csc (a+b x)}} \, dx \\ & = -\frac {2 \cos (a+b x) \sqrt {\csc (a+b x)}}{b}-\left (\sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \sqrt {\sin (a+b x)} \, dx \\ & = -\frac {2 \cos (a+b x) \sqrt {\csc (a+b x)}}{b}-\frac {2 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{b} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.78 \[ \int \csc ^{\frac {3}{2}}(a+b x) \, dx=-\frac {2 \sqrt {\csc (a+b x)} \left (\cos (a+b x)-E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sqrt {\sin (a+b x)}\right )}{b} \]
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Time = 0.49 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.10
method | result | size |
default | \(\frac {2 \sqrt {\sin \left (x b +a \right )+1}\, \sqrt {-2 \sin \left (x b +a \right )+2}\, \sqrt {-\sin \left (x b +a \right )}\, \operatorname {EllipticE}\left (\sqrt {\sin \left (x b +a \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {\sin \left (x b +a \right )+1}\, \sqrt {-2 \sin \left (x b +a \right )+2}\, \sqrt {-\sin \left (x b +a \right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (x b +a \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \cos \left (x b +a \right )^{2}}{\cos \left (x b +a \right ) \sqrt {\sin \left (x b +a \right )}\, b}\) | \(132\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.14 \[ \int \csc ^{\frac {3}{2}}(a+b x) \, dx=-\frac {\sqrt {2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + \sqrt {-2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + \frac {2 \, \cos \left (b x + a\right )}{\sqrt {\sin \left (b x + a\right )}}}{b} \]
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\[ \int \csc ^{\frac {3}{2}}(a+b x) \, dx=\int \csc ^{\frac {3}{2}}{\left (a + b x \right )}\, dx \]
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\[ \int \csc ^{\frac {3}{2}}(a+b x) \, dx=\int { \csc \left (b x + a\right )^{\frac {3}{2}} \,d x } \]
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\[ \int \csc ^{\frac {3}{2}}(a+b x) \, dx=\int { \csc \left (b x + a\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \csc ^{\frac {3}{2}}(a+b x) \, dx=\int {\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{3/2} \,d x \]
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