Integrand size = 23, antiderivative size = 54 \[ \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {\sqrt {a+b \sin ^2(e+f x)}}{f} \]
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Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3273, 52, 65, 214} \[ \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {\sqrt {a+b \sin ^2(e+f x)}}{f}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{f} \]
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Rule 52
Rule 65
Rule 214
Rule 3273
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\sin ^2(e+f x)\right )}{2 f} \\ & = \frac {\sqrt {a+b \sin ^2(e+f x)}}{f}+\frac {a \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{2 f} \\ & = \frac {\sqrt {a+b \sin ^2(e+f x)}}{f}+\frac {a \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^2(e+f x)}\right )}{b f} \\ & = -\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {\sqrt {a+b \sin ^2(e+f x)}}{f} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.94 \[ \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )+\sqrt {a+b \sin ^2(e+f x)}}{f} \]
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Time = 0.82 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.07
method | result | size |
default | \(\frac {\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{f}\) | \(58\) |
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Time = 0.73 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.50 \[ \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\left [\frac {\sqrt {a} \log \left (\frac {2 \, {\left (b \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a} - 2 \, a - b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{2 \, f}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a}}{a}\right ) + \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{f}\right ] \]
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\[ \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int \sqrt {a + b \sin ^{2}{\left (e + f x \right )}} \cot {\left (e + f x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80 \[ \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=-\frac {\sqrt {a} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right ) - \sqrt {b \sin \left (f x + e\right )^{2} + a}}{f} \]
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\[ \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int { \sqrt {b \sin \left (f x + e\right )^{2} + a} \cot \left (f x + e\right ) \,d x } \]
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Timed out. \[ \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int \mathrm {cot}\left (e+f\,x\right )\,\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a} \,d x \]
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