Integrand size = 12, antiderivative size = 77 \[ \int \frac {1}{(c \csc (a+b x))^{3/2}} \, dx=-\frac {2 \cos (a+b x)}{3 b c \sqrt {c \csc (a+b x)}}+\frac {2 \sqrt {c \csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{3 b c^2} \]
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Time = 0.04 (sec) , antiderivative size = 77, 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 = {3854, 3856, 2720} \[ \int \frac {1}{(c \csc (a+b x))^{3/2}} \, dx=\frac {2 \sqrt {\sin (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right ) \sqrt {c \csc (a+b x)}}{3 b c^2}-\frac {2 \cos (a+b x)}{3 b c \sqrt {c \csc (a+b x)}} \]
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Rule 2720
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos (a+b x)}{3 b c \sqrt {c \csc (a+b x)}}+\frac {\int \sqrt {c \csc (a+b x)} \, dx}{3 c^2} \\ & = -\frac {2 \cos (a+b x)}{3 b c \sqrt {c \csc (a+b x)}}+\frac {\left (\sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{3 c^2} \\ & = -\frac {2 \cos (a+b x)}{3 b c \sqrt {c \csc (a+b x)}}+\frac {2 \sqrt {c \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 c^2} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(c \csc (a+b x))^{3/2}} \, dx=-\frac {\csc ^2(a+b x) \left (2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sqrt {\sin (a+b x)}+\sin (2 (a+b x))\right )}{3 b (c \csc (a+b x))^{3/2}} \]
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Result contains complex when optimal does not.
Time = 3.56 (sec) , antiderivative size = 261, normalized size of antiderivative = 3.39
method | result | size |
default | \(\frac {\sin \left (x b +a \right ) \left (-i \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x b +a \right )-i \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \cos \left (x b +a \right ) \sin \left (x b +a \right )\right ) \sqrt {2}}{3 b c \sqrt {c \csc \left (x b +a \right )}\, \left (\cos \left (x b +a \right )-1\right ) \left (1+\cos \left (x b +a \right )\right )}\) | \(261\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(c \csc (a+b x))^{3/2}} \, dx=-\frac {2 \, \sqrt {\frac {c}{\sin \left (b x + a\right )}} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + i \, \sqrt {2 i \, c} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - i \, \sqrt {-2 i \, c} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{3 \, b c^{2}} \]
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\[ \int \frac {1}{(c \csc (a+b x))^{3/2}} \, dx=\int \frac {1}{\left (c \csc {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{(c \csc (a+b x))^{3/2}} \, dx=\int { \frac {1}{\left (c \csc \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{(c \csc (a+b x))^{3/2}} \, dx=\int { \frac {1}{\left (c \csc \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(c \csc (a+b x))^{3/2}} \, dx=\int \frac {1}{{\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^{3/2}} \,d x \]
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