Integrand size = 12, antiderivative size = 71 \[ \int (c \csc (a+b x))^{3/2} \, dx=-\frac {2 c \cos (a+b x) \sqrt {c \csc (a+b x)}}{b}-\frac {2 c^2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{b \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \]
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Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3853, 3856, 2719} \[ \int (c \csc (a+b x))^{3/2} \, dx=-\frac {2 c^2 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {\sin (a+b x)} \sqrt {c \csc (a+b x)}}-\frac {2 c \cos (a+b x) \sqrt {c \csc (a+b x)}}{b} \]
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Rule 2719
Rule 3853
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c \cos (a+b x) \sqrt {c \csc (a+b x)}}{b}-c^2 \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx \\ & = -\frac {2 c \cos (a+b x) \sqrt {c \csc (a+b x)}}{b}-\frac {c^2 \int \sqrt {\sin (a+b x)} \, dx}{\sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \\ & = -\frac {2 c \cos (a+b x) \sqrt {c \csc (a+b x)}}{b}-\frac {2 c^2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{b \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76 \[ \int (c \csc (a+b x))^{3/2} \, dx=\frac {(c \csc (a+b x))^{3/2} \left (2 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sin ^{\frac {3}{2}}(a+b x)-\sin (2 (a+b x))\right )}{b} \]
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Result contains complex when optimal does not.
Time = 3.31 (sec) , antiderivative size = 413, normalized size of antiderivative = 5.82
method | result | size |
default | \(\frac {c \sqrt {c \csc \left (x b +a \right )}\, \left (2 \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x b +a \right )-\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x b +a \right )+2 \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\right ) \sqrt {2}}{b}\) | \(413\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.17 \[ \int (c \csc (a+b x))^{3/2} \, dx=-\frac {2 \, c \sqrt {\frac {c}{\sin \left (b x + a\right )}} \cos \left (b x + a\right ) + \sqrt {2 i \, c} c {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + \sqrt {-2 i \, c} c {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{b} \]
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\[ \int (c \csc (a+b x))^{3/2} \, dx=\int \left (c \csc {\left (a + b x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int (c \csc (a+b x))^{3/2} \, dx=\int { \left (c \csc \left (b x + a\right )\right )^{\frac {3}{2}} \,d x } \]
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\[ \int (c \csc (a+b x))^{3/2} \, dx=\int { \left (c \csc \left (b x + a\right )\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (c \csc (a+b x))^{3/2} \, dx=\int {\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^{3/2} \,d x \]
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