Integrand size = 10, antiderivative size = 90 \[ \int \csc ^{\frac {7}{2}}(a+b x) \, dx=-\frac {6 \cos (a+b x) \sqrt {\csc (a+b x)}}{5 b}-\frac {2 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {6 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{5 b} \]
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Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3853, 3856, 2719} \[ \int \csc ^{\frac {7}{2}}(a+b x) \, dx=-\frac {2 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {6 \cos (a+b x) \sqrt {\csc (a+b x)}}{5 b}-\frac {6 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{5 b} \]
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Rule 2719
Rule 3853
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {3}{5} \int \csc ^{\frac {3}{2}}(a+b x) \, dx \\ & = -\frac {6 \cos (a+b x) \sqrt {\csc (a+b x)}}{5 b}-\frac {2 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {3}{5} \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx \\ & = -\frac {6 \cos (a+b x) \sqrt {\csc (a+b x)}}{5 b}-\frac {2 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {1}{5} \left (3 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \sqrt {\sin (a+b x)} \, dx \\ & = -\frac {6 \cos (a+b x) \sqrt {\csc (a+b x)}}{5 b}-\frac {2 \cos (a+b x) \csc ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {6 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{5 b} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.70 \[ \int \csc ^{\frac {7}{2}}(a+b x) \, dx=\frac {\csc ^{\frac {5}{2}}(a+b x) \left (-7 \cos (a+b x)+3 \cos (3 (a+b x))+12 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sin ^{\frac {5}{2}}(a+b x)\right )}{10 b} \]
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Time = 0.66 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.78
method | result | size |
default | \(\frac {6 \sqrt {\sin \left (x b +a \right )+1}\, \sqrt {-2 \sin \left (x b +a \right )+2}\, \sqrt {-\sin \left (x b +a \right )}\, \sin \left (x b +a \right )^{2} \operatorname {EllipticE}\left (\sqrt {\sin \left (x b +a \right )+1}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {\sin \left (x b +a \right )+1}\, \sqrt {-2 \sin \left (x b +a \right )+2}\, \sqrt {-\sin \left (x b +a \right )}\, \sin \left (x b +a \right )^{2} \operatorname {EllipticF}\left (\sqrt {\sin \left (x b +a \right )+1}, \frac {\sqrt {2}}{2}\right )+6 \sin \left (x b +a \right )^{4}-4 \sin \left (x b +a \right )^{2}-2}{5 \sin \left (x b +a \right )^{\frac {5}{2}} \cos \left (x b +a \right ) b}\) | \(160\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.33 \[ \int \csc ^{\frac {7}{2}}(a+b x) \, dx=-\frac {3 \, \sqrt {2 i} {\left (\cos \left (b x + a\right )^{2} - 1\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 3 \, \sqrt {-2 i} {\left (\cos \left (b x + a\right )^{2} - 1\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + \frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{3} - 4 \, \cos \left (b x + a\right )\right )}}{\sqrt {\sin \left (b x + a\right )}}}{5 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )}} \]
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\[ \int \csc ^{\frac {7}{2}}(a+b x) \, dx=\int \csc ^{\frac {7}{2}}{\left (a + b x \right )}\, dx \]
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\[ \int \csc ^{\frac {7}{2}}(a+b x) \, dx=\int { \csc \left (b x + a\right )^{\frac {7}{2}} \,d x } \]
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\[ \int \csc ^{\frac {7}{2}}(a+b x) \, dx=\int { \csc \left (b x + a\right )^{\frac {7}{2}} \,d x } \]
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Timed out. \[ \int \csc ^{\frac {7}{2}}(a+b x) \, dx=\int {\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{7/2} \,d x \]
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