Integrand size = 12, antiderivative size = 77 \[ \int \frac {1}{(c \csc (a+b x))^{5/2}} \, dx=-\frac {2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}}+\frac {6 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{5 b c^2 \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \]
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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, 2719} \[ \int \frac {1}{(c \csc (a+b x))^{5/2}} \, dx=\frac {6 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{5 b c^2 \sqrt {\sin (a+b x)} \sqrt {c \csc (a+b x)}}-\frac {2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}} \]
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Rule 2719
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}}+\frac {3 \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx}{5 c^2} \\ & = -\frac {2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}}+\frac {3 \int \sqrt {\sin (a+b x)} \, dx}{5 c^2 \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \\ & = -\frac {2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}}+\frac {6 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{5 b c^2 \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(c \csc (a+b x))^{5/2}} \, dx=\frac {-\frac {12 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right )}{\sqrt {\sin (a+b x)}}-2 \sin (2 (a+b x))}{10 b c^2 \sqrt {c \csc (a+b x)}} \]
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Result contains complex when optimal does not.
Time = 3.56 (sec) , antiderivative size = 445, normalized size of antiderivative = 5.78
method | result | size |
default | \(\frac {\csc \left (x b +a \right ) \left (3 \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x b +a \right )-6 \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x b +a \right )+\sqrt {2}\, \cos \left (x b +a \right )^{3}+3 \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )-6 \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )-4 \cos \left (x b +a \right ) \sqrt {2}+3 \sqrt {2}\right ) \sqrt {2}}{5 b \sqrt {c \csc \left (x b +a \right )}\, c^{2}}\) | \(445\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.25 \[ \int \frac {1}{(c \csc (a+b x))^{5/2}} \, dx=\frac {2 \, {\left (\cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )\right )} \sqrt {\frac {c}{\sin \left (b x + a\right )}} + 3 \, \sqrt {2 i \, c} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 3 \, \sqrt {-2 i \, c} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{5 \, b c^{3}} \]
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\[ \int \frac {1}{(c \csc (a+b x))^{5/2}} \, dx=\int \frac {1}{\left (c \csc {\left (a + b x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{(c \csc (a+b x))^{5/2}} \, dx=\int { \frac {1}{\left (c \csc \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{(c \csc (a+b x))^{5/2}} \, dx=\int { \frac {1}{\left (c \csc \left (b x + a\right )\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(c \csc (a+b x))^{5/2}} \, dx=\int \frac {1}{{\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^{5/2}} \,d x \]
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