Integrand size = 21, antiderivative size = 102 \[ \int \frac {\sin ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=-\frac {2 d^2 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac {10 \cos (e+f x)}{21 f \sqrt {d \csc (e+f x)}}+\frac {10 \sqrt {d \csc (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {\sin (e+f x)}}{21 d f} \]
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Time = 0.05 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 3854, 3856, 2720} \[ \int \frac {\sin ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=-\frac {2 d^2 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac {10 \cos (e+f x)}{21 f \sqrt {d \csc (e+f x)}}+\frac {10 \sqrt {\sin (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),2\right ) \sqrt {d \csc (e+f x)}}{21 d f} \]
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Rule 16
Rule 2720
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = d^3 \int \frac {1}{(d \csc (e+f x))^{7/2}} \, dx \\ & = -\frac {2 d^2 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}+\frac {1}{7} (5 d) \int \frac {1}{(d \csc (e+f x))^{3/2}} \, dx \\ & = -\frac {2 d^2 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac {10 \cos (e+f x)}{21 f \sqrt {d \csc (e+f x)}}+\frac {5 \int \sqrt {d \csc (e+f x)} \, dx}{21 d} \\ & = -\frac {2 d^2 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac {10 \cos (e+f x)}{21 f \sqrt {d \csc (e+f x)}}+\frac {\left (5 \sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\sin (e+f x)}} \, dx}{21 d} \\ & = -\frac {2 d^2 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac {10 \cos (e+f x)}{21 f \sqrt {d \csc (e+f x)}}+\frac {10 \sqrt {d \csc (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),2\right ) \sqrt {\sin (e+f x)}}{21 d f} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.69 \[ \int \frac {\sin ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=-\frac {\sqrt {d \csc (e+f x)} \left (40 \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),2\right ) \sqrt {\sin (e+f x)}+26 \sin (2 (e+f x))-3 \sin (4 (e+f x))\right )}{84 d f} \]
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Result contains complex when optimal does not.
Time = 1.15 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.41
method | result | size |
default | \(\frac {\sqrt {2}\, \left (5 i \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \cot \left (f x +e \right )+5 i \sqrt {-i \left (i+\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}\, \sqrt {i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {-i \left (i-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}\, \csc \left (f x +e \right )+3 \sqrt {2}\, \left (\cos ^{3}\left (f x +e \right )\right )-8 \sqrt {2}\, \cos \left (f x +e \right )\right )}{21 f \sqrt {d \csc \left (f x +e \right )}}\) | \(246\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.96 \[ \int \frac {\sin ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\frac {2 \, {\left (3 \, \cos \left (f x + e\right )^{3} - 8 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {d}{\sin \left (f x + e\right )}} \sin \left (f x + e\right ) - 5 i \, \sqrt {2 i \, d} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 5 i \, \sqrt {-2 i \, d} {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}{21 \, d f} \]
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Timed out. \[ \int \frac {\sin ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sin ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{3}}{\sqrt {d \csc \left (f x + e\right )}} \,d x } \]
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\[ \int \frac {\sin ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int { \frac {\sin \left (f x + e\right )^{3}}{\sqrt {d \csc \left (f x + e\right )}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^3(e+f x)}{\sqrt {d \csc (e+f x)}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^3}{\sqrt {\frac {d}{\sin \left (e+f\,x\right )}}} \,d x \]
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