Integrand size = 37, antiderivative size = 107 \[ \int \frac {a+b \sin ^2(e+f x)}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=\frac {2 (a+b) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {(2 a-b) \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{3 f g^2 \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}} \]
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Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {3281, 468, 335, 243, 342, 281, 238} \[ \int \frac {a+b \sin ^2(e+f x)}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=\frac {2 (a+b) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}-\frac {2 (2 a-b) \left (1-\csc ^2(e+f x)\right )^{3/4} (d \sin (e+f x))^{3/2} \operatorname {EllipticF}\left (\frac {1}{2} \csc ^{-1}(\sin (e+f x)),2\right )}{3 d^2 f g (g \cos (e+f x))^{3/2}} \]
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Rule 238
Rule 243
Rule 281
Rule 335
Rule 342
Rule 468
Rule 3281
Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^2(e+f x)^{3/4} \text {Subst}\left (\int \frac {a+b x^2}{\sqrt {d x} \left (1-x^2\right )^{7/4}} \, dx,x,\sin (e+f x)\right )}{f g (g \cos (e+f x))^{3/2}} \\ & = \frac {2 (a+b) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}-\frac {\left ((-2 a+b) \cos ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d x} \left (1-x^2\right )^{3/4}} \, dx,x,\sin (e+f x)\right )}{3 f g (g \cos (e+f x))^{3/2}} \\ & = \frac {2 (a+b) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}-\frac {\left (2 (-2 a+b) \cos ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {x^4}{d^2}\right )^{3/4}} \, dx,x,\sqrt {d \sin (e+f x)}\right )}{3 d f g (g \cos (e+f x))^{3/2}} \\ & = \frac {2 (a+b) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}-\frac {\left (2 (-2 a+b) \left (1-\csc ^2(e+f x)\right )^{3/4} (d \sin (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {d^2}{x^4}\right )^{3/4} x^3} \, dx,x,\sqrt {d \sin (e+f x)}\right )}{3 d f g (g \cos (e+f x))^{3/2}} \\ & = \frac {2 (a+b) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {\left (2 (-2 a+b) \left (1-\csc ^2(e+f x)\right )^{3/4} (d \sin (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {x}{\left (1-d^2 x^4\right )^{3/4}} \, dx,x,\frac {1}{\sqrt {d \sin (e+f x)}}\right )}{3 d f g (g \cos (e+f x))^{3/2}} \\ & = \frac {2 (a+b) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {\left ((-2 a+b) \left (1-\csc ^2(e+f x)\right )^{3/4} (d \sin (e+f x))^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-d^2 x^2\right )^{3/4}} \, dx,x,\frac {\csc (e+f x)}{d}\right )}{3 d f g (g \cos (e+f x))^{3/2}} \\ & = \frac {2 (a+b) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}-\frac {2 (2 a-b) \left (1-\csc ^2(e+f x)\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin (\csc (e+f x)),2\right ) (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \sin ^2(e+f x)}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=\frac {2 \cos ^2(e+f x)^{3/4} \left (5 a \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {7}{4},\frac {5}{4},\sin ^2(e+f x)\right ) \sin (e+f x)+b \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {7}{4},\frac {9}{4},\sin ^2(e+f x)\right ) \sin ^3(e+f x)\right )}{5 f g (g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(386\) vs. \(2(118)=236\).
Time = 7.07 (sec) , antiderivative size = 387, normalized size of antiderivative = 3.62
method | result | size |
default | \(\frac {\left (2 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) a \cos \left (f x +e \right )-\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) b \cos \left (f x +e \right )+2 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) a -\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) b +\tan \left (f x +e \right ) \sqrt {2}\, a +\tan \left (f x +e \right ) \sqrt {2}\, b \right ) \sqrt {2}}{3 g^{2} f \sqrt {g \cos \left (f x +e \right )}\, \sqrt {d \sin \left (f x +e \right )}}\) | \(387\) |
parts | \(\frac {a \left (2 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+2 \sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+\tan \left (f x +e \right ) \sqrt {2}\right ) \sqrt {2}}{3 f \sqrt {d \sin \left (f x +e \right )}\, \sqrt {g \cos \left (f x +e \right )}\, g^{2}}+\frac {b \left (-\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )-\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )+1}\, \sqrt {-\csc \left (f x +e \right )+\cot \left (f x +e \right )}\, F\left (\sqrt {-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1}, \frac {\sqrt {2}}{2}\right )+\tan \left (f x +e \right ) \sqrt {2}\right ) \sqrt {2}}{3 f \sqrt {d \sin \left (f x +e \right )}\, \sqrt {g \cos \left (f x +e \right )}\, g^{2}}\) | \(416\) |
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Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.17 \[ \int \frac {a+b \sin ^2(e+f x)}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=-\frac {\sqrt {i \, d g} {\left (2 \, a - b\right )} \cos \left (f x + e\right )^{2} F(\arcsin \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\,|\,-1) + \sqrt {-i \, d g} {\left (2 \, a - b\right )} \cos \left (f x + e\right )^{2} F(\arcsin \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\,|\,-1) - 2 \, \sqrt {g \cos \left (f x + e\right )} \sqrt {d \sin \left (f x + e\right )} {\left (a + b\right )}}{3 \, d f g^{3} \cos \left (f x + e\right )^{2}} \]
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Timed out. \[ \int \frac {a+b \sin ^2(e+f x)}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {a+b \sin ^2(e+f x)}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=\int { \frac {b \sin \left (f x + e\right )^{2} + a}{\left (g \cos \left (f x + e\right )\right )^{\frac {5}{2}} \sqrt {d \sin \left (f x + e\right )}} \,d x } \]
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\[ \int \frac {a+b \sin ^2(e+f x)}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=\int { \frac {b \sin \left (f x + e\right )^{2} + a}{\left (g \cos \left (f x + e\right )\right )^{\frac {5}{2}} \sqrt {d \sin \left (f x + e\right )}} \,d x } \]
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Timed out. \[ \int \frac {a+b \sin ^2(e+f x)}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx=\int \frac {b\,{\sin \left (e+f\,x\right )}^2+a}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{5/2}\,\sqrt {d\,\sin \left (e+f\,x\right )}} \,d x \]
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