Integrand size = 12, antiderivative size = 105 \[ \int \frac {1}{(c \csc (a+b x))^{7/2}} \, dx=-\frac {2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}-\frac {10 \cos (a+b x)}{21 b c^3 \sqrt {c \csc (a+b x)}}+\frac {10 \sqrt {c \csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{21 b c^4} \]
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Time = 0.06 (sec) , antiderivative size = 105, 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 = {3854, 3856, 2720} \[ \int \frac {1}{(c \csc (a+b x))^{7/2}} \, dx=\frac {10 \sqrt {\sin (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right ) \sqrt {c \csc (a+b x)}}{21 b c^4}-\frac {10 \cos (a+b x)}{21 b c^3 \sqrt {c \csc (a+b x)}}-\frac {2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}} \]
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Rule 2720
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}+\frac {5 \int \frac {1}{(c \csc (a+b x))^{3/2}} \, dx}{7 c^2} \\ & = -\frac {2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}-\frac {10 \cos (a+b x)}{21 b c^3 \sqrt {c \csc (a+b x)}}+\frac {5 \int \sqrt {c \csc (a+b x)} \, dx}{21 c^4} \\ & = -\frac {2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}-\frac {10 \cos (a+b x)}{21 b c^3 \sqrt {c \csc (a+b x)}}+\frac {\left (5 \sqrt {c \csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{21 c^4} \\ & = -\frac {2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}-\frac {10 \cos (a+b x)}{21 b c^3 \sqrt {c \csc (a+b x)}}+\frac {10 \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)}}{21 b c^4} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(c \csc (a+b x))^{7/2}} \, dx=-\frac {\sqrt {c \csc (a+b x)} \left (40 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sqrt {\sin (a+b x)}+26 \sin (2 (a+b x))-3 \sin (4 (a+b x))\right )}{84 b c^4} \]
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Result contains complex when optimal does not.
Time = 3.86 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.70
method | result | size |
default | \(\frac {\sin \left (x b +a \right )^{3} \left (5 i \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x b +a \right )+3 \sqrt {2}\, \cos \left (x b +a \right )^{3} \sin \left (x b +a \right )+5 i \sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (-i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}\, \sqrt {i \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x b +a \right )+\csc \left (x b +a \right )\right )}, \frac {\sqrt {2}}{2}\right )-8 \sqrt {2}\, \cos \left (x b +a \right ) \sin \left (x b +a \right )\right ) \sqrt {2}}{21 b \,c^{3} \sqrt {c \csc \left (x b +a \right )}\, \left (\cos \left (x b +a \right )-1\right )^{2} \left (1+\cos \left (x b +a \right )\right )^{2}}\) | \(283\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(c \csc (a+b x))^{7/2}} \, dx=\frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{3} - 8 \, \cos \left (b x + a\right )\right )} \sqrt {\frac {c}{\sin \left (b x + a\right )}} \sin \left (b x + a\right ) - 5 i \, \sqrt {2 i \, c} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 5 i \, \sqrt {-2 i \, c} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{21 \, b c^{4}} \]
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\[ \int \frac {1}{(c \csc (a+b x))^{7/2}} \, dx=\int \frac {1}{\left (c \csc {\left (a + b x \right )}\right )^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {1}{(c \csc (a+b x))^{7/2}} \, dx=\int { \frac {1}{\left (c \csc \left (b x + a\right )\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {1}{(c \csc (a+b x))^{7/2}} \, dx=\int { \frac {1}{\left (c \csc \left (b x + a\right )\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(c \csc (a+b x))^{7/2}} \, dx=\int \frac {1}{{\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^{7/2}} \,d x \]
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