Integrand size = 16, antiderivative size = 223 \[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {b \cos (e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (2 a+b) \cos (e+f x) \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {2 (2 a+b) E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 (a+b)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {\operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a (a+b) f \sqrt {a+b \sin ^2(e+f x)}} \]
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Time = 0.29 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3263, 3252, 3251, 3257, 3256, 3262, 3261} \[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {2 b (2 a+b) \sin (e+f x) \cos (e+f x)}{3 a^2 f (a+b)^2 \sqrt {a+b \sin ^2(e+f x)}}+\frac {2 (2 a+b) \sqrt {a+b \sin ^2(e+f x)} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{3 a^2 f (a+b)^2 \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}+\frac {b \sin (e+f x) \cos (e+f x)}{3 a f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )}{3 a f (a+b) \sqrt {a+b \sin ^2(e+f x)}} \]
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Rule 3251
Rule 3252
Rule 3256
Rule 3257
Rule 3261
Rule 3262
Rule 3263
Rubi steps \begin{align*} \text {integral}& = \frac {b \cos (e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\int \frac {-3 a-2 b+b \sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx}{3 a (a+b)} \\ & = \frac {b \cos (e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (2 a+b) \cos (e+f x) \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\int \frac {-a (3 a+b)-2 b (2 a+b) \sin ^2(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx}{3 a^2 (a+b)^2} \\ & = \frac {b \cos (e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (2 a+b) \cos (e+f x) \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\int \frac {1}{\sqrt {a+b \sin ^2(e+f x)}} \, dx}{3 a (a+b)}+\frac {(2 (2 a+b)) \int \sqrt {a+b \sin ^2(e+f x)} \, dx}{3 a^2 (a+b)^2} \\ & = \frac {b \cos (e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (2 a+b) \cos (e+f x) \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\left (2 (2 a+b) \sqrt {a+b \sin ^2(e+f x)}\right ) \int \sqrt {1+\frac {b \sin ^2(e+f x)}{a}} \, dx}{3 a^2 (a+b)^2 \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {\sqrt {1+\frac {b \sin ^2(e+f x)}{a}} \int \frac {1}{\sqrt {1+\frac {b \sin ^2(e+f x)}{a}}} \, dx}{3 a (a+b) \sqrt {a+b \sin ^2(e+f x)}} \\ & = \frac {b \cos (e+f x) \sin (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (2 a+b) \cos (e+f x) \sin (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {2 (2 a+b) E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 (a+b)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {\operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right ) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a (a+b) f \sqrt {a+b \sin ^2(e+f x)}} \\ \end{align*}
Time = 1.34 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {2 a^2 (2 a+b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} E\left (e+f x\left |-\frac {b}{a}\right .\right )-a^2 (a+b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )-\sqrt {2} b \left (-5 a^2-5 a b-b^2+b (2 a+b) \cos (2 (e+f x))\right ) \sin (2 (e+f x))}{3 a^2 (a+b)^2 f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(546\) vs. \(2(245)=490\).
Time = 2.11 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.45
method | result | size |
default | \(-\frac {\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b \left (\sin ^{2}\left (f x +e \right )\right )+\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2} \left (\sin ^{2}\left (f x +e \right )\right )-4 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b \left (\sin ^{2}\left (f x +e \right )\right )-2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2} \left (\sin ^{2}\left (f x +e \right )\right )+4 a \,b^{2} \left (\sin ^{5}\left (f x +e \right )\right )+2 b^{3} \left (\sin ^{5}\left (f x +e \right )\right )+\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+a^{2} \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b -4 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}-2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b +5 a^{2} b \left (\sin ^{3}\left (f x +e \right )\right )-a \,b^{2} \left (\sin ^{3}\left (f x +e \right )\right )-2 b^{3} \left (\sin ^{3}\left (f x +e \right )\right )-5 a^{2} b \sin \left (f x +e \right )-3 a \,b^{2} \sin \left (f x +e \right )}{3 {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}} a^{2} \left (a +b \right )^{2} \cos \left (f x +e \right ) f}\) | \(547\) |
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Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 1531, normalized size of antiderivative = 6.87 \[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]
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