Integrand size = 25, antiderivative size = 86 \[ \int \frac {\sqrt {b \tan (e+f x)}}{(a \sin (e+f x))^{5/2}} \, dx=-\frac {b}{a^2 f \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}}+\frac {\sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \tan (e+f x)}}{a^2 f \sqrt {a \sin (e+f x)}} \]
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Time = 0.12 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2679, 2681, 2720} \[ \int \frac {\sqrt {b \tan (e+f x)}}{(a \sin (e+f x))^{5/2}} \, dx=\frac {\sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \tan (e+f x)}}{a^2 f \sqrt {a \sin (e+f x)}}-\frac {b}{a^2 f \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}} \]
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Rule 2679
Rule 2681
Rule 2720
Rubi steps \begin{align*} \text {integral}& = -\frac {b}{a^2 f \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}}+\frac {\int \frac {\sqrt {b \tan (e+f x)}}{\sqrt {a \sin (e+f x)}} \, dx}{2 a^2} \\ & = -\frac {b}{a^2 f \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}}+\frac {\left (\sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx}{2 a^2 \sqrt {a \sin (e+f x)}} \\ & = -\frac {b}{a^2 f \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}}+\frac {\sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \tan (e+f x)}}{a^2 f \sqrt {a \sin (e+f x)}} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {b \tan (e+f x)}}{(a \sin (e+f x))^{5/2}} \, dx=\frac {b \left (-\sqrt [4]{\cos ^2(e+f x)}+\operatorname {EllipticF}\left (\frac {1}{2} \arcsin (\sin (e+f x)),2\right ) \sin (e+f x)\right )}{a^2 f \sqrt [4]{\cos ^2(e+f x)} \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)}} \]
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Result contains complex when optimal does not.
Time = 1.59 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.78
method | result | size |
default | \(\frac {\sqrt {b \tan \left (f x +e \right )}\, \left (i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right )+i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ), i\right )-\cot \left (f x +e \right )\right )}{f \,a^{2} \sqrt {\sin \left (f x +e \right ) a}}\) | \(153\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.71 \[ \int \frac {\sqrt {b \tan (e+f x)}}{(a \sin (e+f x))^{5/2}} \, dx=\frac {{\left (\sqrt {2} \cos \left (f x + e\right )^{2} - \sqrt {2}\right )} \sqrt {-a b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + {\left (\sqrt {2} \cos \left (f x + e\right )^{2} - \sqrt {2}\right )} \sqrt {-a b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{2 \, {\left (a^{3} f \cos \left (f x + e\right )^{2} - a^{3} f\right )}} \]
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Timed out. \[ \int \frac {\sqrt {b \tan (e+f x)}}{(a \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {b \tan (e+f x)}}{(a \sin (e+f x))^{5/2}} \, dx=\int { \frac {\sqrt {b \tan \left (f x + e\right )}}{\left (a \sin \left (f x + e\right )\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {b \tan (e+f x)}}{(a \sin (e+f x))^{5/2}} \, dx=\int { \frac {\sqrt {b \tan \left (f x + e\right )}}{\left (a \sin \left (f x + e\right )\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {b \tan (e+f x)}}{(a \sin (e+f x))^{5/2}} \, dx=\int \frac {\sqrt {b\,\mathrm {tan}\left (e+f\,x\right )}}{{\left (a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
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