Integrand size = 12, antiderivative size = 75 \[ \int (c \csc (a+b x))^{5/2} \, dx=-\frac {2 c \cos (a+b x) (c \csc (a+b x))^{3/2}}{3 b}+\frac {2 c^2 \sqrt {c \csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{3 b} \]
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Time = 0.04 (sec) , antiderivative size = 75, 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, 2720} \[ \int (c \csc (a+b x))^{5/2} \, dx=\frac {2 c^2 \sqrt {\sin (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right ) \sqrt {c \csc (a+b x)}}{3 b}-\frac {2 c \cos (a+b x) (c \csc (a+b x))^{3/2}}{3 b} \]
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Rule 2720
Rule 3853
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c \cos (a+b x) (c \csc (a+b x))^{3/2}}{3 b}+\frac {1}{3} c^2 \int \sqrt {c \csc (a+b x)} \, dx \\ & = -\frac {2 c \cos (a+b x) (c \csc (a+b x))^{3/2}}{3 b}+\frac {1}{3} \left (c^2 \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx \\ & = -\frac {2 c \cos (a+b x) (c \csc (a+b x))^{3/2}}{3 b}+\frac {2 c^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} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.73 \[ \int (c \csc (a+b x))^{5/2} \, dx=-\frac {(c \csc (a+b x))^{5/2} \left (2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sin ^{\frac {5}{2}}(a+b x)+\sin (2 (a+b x))\right )}{3 b} \]
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Result contains complex when optimal does not.
Time = 3.64 (sec) , antiderivative size = 312, normalized size of antiderivative = 4.16
method | result | size |
default | \(\frac {\csc \left (x b +a \right )^{2} {\left (\frac {c \left (\csc \left (x b +a \right ) \left (1-\cos \left (x b +a \right )\right )^{2}+\sin \left (x b +a \right )\right )}{1-\cos \left (x b +a \right )}\right )}^{\frac {5}{2}} \left (1-\cos \left (x b +a \right )\right )^{2} \left (2 i \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {2}\, \sqrt {-i \left (i+\cot \left (x b +a \right )-\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right ) \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )+\csc \left (x b +a \right )^{4} \left (1-\cos \left (x b +a \right )\right )^{4}-1\right ) \sqrt {2}}{6 b \sqrt {\csc \left (x b +a \right )^{3} \left (1-\cos \left (x b +a \right )\right )^{3}+\csc \left (x b +a \right )-\cot \left (x b +a \right )}\, \sqrt {\csc \left (x b +a \right ) \left (1-\cos \left (x b +a \right )\right ) \left (\csc \left (x b +a \right )^{2} \left (1-\cos \left (x b +a \right )\right )^{2}+1\right )}\, \left (\csc \left (x b +a \right )^{2} \left (1-\cos \left (x b +a \right )\right )^{2}+1\right )^{2}}\) | \(312\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.40 \[ \int (c \csc (a+b x))^{5/2} \, dx=\frac {-i \, \sqrt {2 i \, c} c^{2} \sin \left (b x + a\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, \sqrt {-2 i \, c} c^{2} \sin \left (b x + a\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 2 \, c^{2} \sqrt {\frac {c}{\sin \left (b x + a\right )}} \cos \left (b x + a\right )}{3 \, b \sin \left (b x + a\right )} \]
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\[ \int (c \csc (a+b x))^{5/2} \, dx=\int \left (c \csc {\left (a + b x \right )}\right )^{\frac {5}{2}}\, dx \]
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\[ \int (c \csc (a+b x))^{5/2} \, dx=\int { \left (c \csc \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]
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\[ \int (c \csc (a+b x))^{5/2} \, dx=\int { \left (c \csc \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (c \csc (a+b x))^{5/2} \, dx=\int {\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^{5/2} \,d x \]
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