Integrand size = 23, antiderivative size = 83 \[ \int \frac {\cot (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{a^{5/2} f}+\frac {1}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {1}{a^2 f \sqrt {a+b \sin ^2(e+f x)}} \]
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Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3273, 53, 65, 214} \[ \int \frac {\cot (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{a^{5/2} f}+\frac {1}{a^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {1}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
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Rule 53
Rule 65
Rule 214
Rule 3273
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x (a+b x)^{5/2}} \, dx,x,\sin ^2(e+f x)\right )}{2 f} \\ & = \frac {1}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\sin ^2(e+f x)\right )}{2 a f} \\ & = \frac {1}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {1}{a^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{2 a^2 f} \\ & = \frac {1}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {1}{a^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^2(e+f x)}\right )}{a^2 b f} \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{a^{5/2} f}+\frac {1}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {1}{a^2 f \sqrt {a+b \sin ^2(e+f x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.59 \[ \int \frac {\cot (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},1+\frac {b \sin ^2(e+f x)}{a}\right )}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(259\) vs. \(2(71)=142\).
Time = 1.64 (sec) , antiderivative size = 260, normalized size of antiderivative = 3.13
method | result | size |
default | \(\frac {-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{a^{\frac {5}{2}}}-\frac {7 \sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}}{12 a^{2} \sqrt {-a b}\, \left (\sin \left (f x +e \right )+\frac {\sqrt {-a b}}{b}\right )}+\frac {7 \sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}}{12 a^{2} \sqrt {-a b}\, \left (\sin \left (f x +e \right )-\frac {\sqrt {-a b}}{b}\right )}-\frac {\sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}}{12 a^{2} b \left (\sin \left (f x +e \right )-\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {-b \left (\cos ^{2}\left (f x +e \right )\right )+\frac {a b +b^{2}}{b}}}{12 a^{2} b \left (\sin \left (f x +e \right )+\frac {\sqrt {-a b}}{b}\right )^{2}}}{f}\) | \(260\) |
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Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (71) = 142\).
Time = 0.35 (sec) , antiderivative size = 382, normalized size of antiderivative = 4.60 \[ \int \frac {\cot (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (b^{2} \cos \left (f x + e\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a} \log \left (\frac {2 \, {\left (b \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a} - 2 \, a - b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) - 2 \, {\left (3 \, a b \cos \left (f x + e\right )^{2} - 4 \, a^{2} - 3 \, a b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{6 \, {\left (a^{3} b^{2} f \cos \left (f x + e\right )^{4} - 2 \, {\left (a^{4} b + a^{3} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} f\right )}}, \frac {3 \, {\left (b^{2} \cos \left (f x + e\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a}}{a}\right ) - {\left (3 \, a b \cos \left (f x + e\right )^{2} - 4 \, a^{2} - 3 \, a b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{3 \, {\left (a^{3} b^{2} f \cos \left (f x + e\right )^{4} - 2 \, {\left (a^{4} b + a^{3} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} f\right )}}\right ] \]
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\[ \int \frac {\cot (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cot {\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
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none
Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.80 \[ \int \frac {\cot (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\frac {3 \, \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {5}{2}}} - \frac {3}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a^{2}} - \frac {1}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a}}{3 \, f} \]
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\[ \int \frac {\cot (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cot \left (f x + e\right )}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cot (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\mathrm {cot}\left (e+f\,x\right )}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]
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