Integrand size = 24, antiderivative size = 53 \[ \int \frac {\cot (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f}+\frac {1}{a f \sqrt {a \cos ^2(e+f x)}} \]
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Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3255, 3284, 53, 65, 212} \[ \int \frac {\cot (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {1}{a f \sqrt {a \cos ^2(e+f x)}}-\frac {\text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f} \]
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Rule 53
Rule 65
Rule 212
Rule 3255
Rule 3284
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot (e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}} \, dx \\ & = -\frac {\text {Subst}\left (\int \frac {1}{(1-x) (a x)^{3/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f} \\ & = \frac {1}{a f \sqrt {a \cos ^2(e+f x)}}-\frac {\text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a x}} \, dx,x,\cos ^2(e+f x)\right )}{2 a f} \\ & = \frac {1}{a f \sqrt {a \cos ^2(e+f x)}}-\frac {\text {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a \cos ^2(e+f x)}\right )}{a^2 f} \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f}+\frac {1}{a f \sqrt {a \cos ^2(e+f x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {\cot (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\cos ^2(e+f x)\right )}{a f \sqrt {a \cos ^2(e+f x)}} \]
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Time = 1.09 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.43
method | result | size |
default | \(-\frac {\ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}+2 a}{\sin \left (f x +e \right )}\right ) a^{2} \left (\cos ^{2}\left (f x +e \right )\right )-\sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, a^{\frac {3}{2}}}{a^{\frac {7}{2}} \cos \left (f x +e \right )^{2} f}\) | \(76\) |
risch | \(\frac {2}{a \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, f}-\frac {2 \ln \left ({\mathrm e}^{i f x}+{\mathrm e}^{-i e}\right ) \cos \left (f x +e \right )}{f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, a}+\frac {2 \ln \left ({\mathrm e}^{i f x}-{\mathrm e}^{-i e}\right ) \cos \left (f x +e \right )}{f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, a}\) | \(144\) |
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Time = 0.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.09 \[ \int \frac {\cot (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\sqrt {a \cos \left (f x + e\right )^{2}} {\left (\cos \left (f x + e\right ) \log \left (-\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1}\right ) - 2\right )}}{2 \, a^{2} f \cos \left (f x + e\right )^{2}} \]
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\[ \int \frac {\cot (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\cot {\left (e + f x \right )}}{\left (- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.38 \[ \int \frac {\cot (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\frac {\log \left (\frac {2 \, \sqrt {-a \sin \left (f x + e\right )^{2} + a} \sqrt {a}}{{\left | \sin \left (f x + e\right ) \right |}} + \frac {2 \, a}{{\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {3}{2}}} - \frac {1}{\sqrt {-a \sin \left (f x + e\right )^{2} + a} a}}{f} \]
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Exception generated. \[ \int \frac {\cot (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\cot (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\mathrm {cot}\left (e+f\,x\right )}{{\left (a-a\,{\sin \left (e+f\,x\right )}^2\right )}^{3/2}} \,d x \]
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