Integrand size = 21, antiderivative size = 95 \[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=-\frac {5 b \csc (e+f x)}{6 f \sqrt {b \sec (e+f x)}}-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {5 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{6 f} \]
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Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2705, 3856, 2720} \[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}-\frac {5 b \csc (e+f x)}{6 f \sqrt {b \sec (e+f x)}}+\frac {5 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{6 f} \]
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Rule 2705
Rule 2720
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {5}{6} \int \csc ^2(e+f x) \sqrt {b \sec (e+f x)} \, dx \\ & = -\frac {5 b \csc (e+f x)}{6 f \sqrt {b \sec (e+f x)}}-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {5}{12} \int \sqrt {b \sec (e+f x)} \, dx \\ & = -\frac {5 b \csc (e+f x)}{6 f \sqrt {b \sec (e+f x)}}-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {1}{12} \left (5 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx \\ & = -\frac {5 b \csc (e+f x)}{6 f \sqrt {b \sec (e+f x)}}-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {5 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{6 f} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.66 \[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=\frac {\left (-\cot (e+f x) \left (5+2 \csc ^2(e+f x)\right )+5 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )\right ) \sqrt {b \sec (e+f x)}}{6 f} \]
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Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.96
method | result | size |
default | \(\frac {i \sqrt {b \sec \left (f x +e \right )}\, \left (-5 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right ) \left (\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-5 \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right ) \left (\sin ^{2}\left (f x +e \right )\right )+5 i \left (\cos ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right )-7 i \cot \left (f x +e \right )\right )}{6 f \left (\cos ^{2}\left (f x +e \right )-1\right )}\) | \(186\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.56 \[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=-\frac {5 \, \sqrt {2} {\left (i \, \cos \left (f x + e\right )^{2} - i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 5 \, \sqrt {2} {\left (-i \, \cos \left (f x + e\right )^{2} + i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, {\left (5 \, \cos \left (f x + e\right )^{3} - 7 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{12 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\right )} \]
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\[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int \sqrt {b \sec {\left (e + f x \right )}} \csc ^{4}{\left (e + f x \right )}\, dx \]
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\[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{4} \,d x } \]
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\[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{4} \,d x } \]
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Timed out. \[ \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx=\int \frac {\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}}{{\sin \left (e+f\,x\right )}^4} \,d x \]
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