Integrand size = 12, antiderivative size = 103 \[ \int (c \csc (a+b x))^{7/2} \, dx=-\frac {6 c^3 \cos (a+b x) \sqrt {c \csc (a+b x)}}{5 b}-\frac {2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b}-\frac {6 c^4 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{5 b \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \]
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Time = 0.06 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3853, 3856, 2719} \[ \int (c \csc (a+b x))^{7/2} \, dx=-\frac {6 c^4 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{5 b \sqrt {\sin (a+b x)} \sqrt {c \csc (a+b x)}}-\frac {6 c^3 \cos (a+b x) \sqrt {c \csc (a+b x)}}{5 b}-\frac {2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b} \]
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Rule 2719
Rule 3853
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b}+\frac {1}{5} \left (3 c^2\right ) \int (c \csc (a+b x))^{3/2} \, dx \\ & = -\frac {6 c^3 \cos (a+b x) \sqrt {c \csc (a+b x)}}{5 b}-\frac {2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b}-\frac {1}{5} \left (3 c^4\right ) \int \frac {1}{\sqrt {c \csc (a+b x)}} \, dx \\ & = -\frac {6 c^3 \cos (a+b x) \sqrt {c \csc (a+b x)}}{5 b}-\frac {2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b}-\frac {\left (3 c^4\right ) \int \sqrt {\sin (a+b x)} \, dx}{5 \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \\ & = -\frac {6 c^3 \cos (a+b x) \sqrt {c \csc (a+b x)}}{5 b}-\frac {2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b}-\frac {6 c^4 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{5 b \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.65 \[ \int (c \csc (a+b x))^{7/2} \, dx=\frac {(c \csc (a+b x))^{7/2} \left (24 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sin ^{\frac {7}{2}}(a+b x)-10 \sin (2 (a+b x))+3 \sin (4 (a+b x))\right )}{20 b} \]
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Result contains complex when optimal does not.
Time = 3.86 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.41
method | result | size |
default | \(-\frac {c^{3} \sqrt {c \csc \left (x b +a \right )}\, \left (\left (-6 \cos \left (x b +a \right )-6\right ) \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {-i \left (i+\cot \left (x b +a \right )-\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )+\left (3 \cos \left (x b +a \right )+3\right ) \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {-i \left (i+\cot \left (x b +a \right )-\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \left (3+\csc \left (x b +a \right ) \cot \left (x b +a \right )\right )\right ) \sqrt {2}}{5 b}\) | \(248\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.46 \[ \int (c \csc (a+b x))^{7/2} \, dx=-\frac {3 \, {\left (c^{3} \cos \left (b x + a\right )^{2} - c^{3}\right )} \sqrt {2 i \, c} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 3 \, {\left (c^{3} \cos \left (b x + a\right )^{2} - c^{3}\right )} \sqrt {-2 i \, c} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + 2 \, {\left (3 \, c^{3} \cos \left (b x + a\right )^{3} - 4 \, c^{3} \cos \left (b x + a\right )\right )} \sqrt {\frac {c}{\sin \left (b x + a\right )}}}{5 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )}} \]
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\[ \int (c \csc (a+b x))^{7/2} \, dx=\int \left (c \csc {\left (a + b x \right )}\right )^{\frac {7}{2}}\, dx \]
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\[ \int (c \csc (a+b x))^{7/2} \, dx=\int { \left (c \csc \left (b x + a\right )\right )^{\frac {7}{2}} \,d x } \]
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\[ \int (c \csc (a+b x))^{7/2} \, dx=\int { \left (c \csc \left (b x + a\right )\right )^{\frac {7}{2}} \,d x } \]
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Timed out. \[ \int (c \csc (a+b x))^{7/2} \, dx=\int {\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^{7/2} \,d x \]
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