Integrand size = 25, antiderivative size = 110 \[ \int \cot ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {(2 a-b) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{2 \sqrt {a} f}-\frac {(2 a-b) \sqrt {a+b \sin ^2(e+f x)}}{2 a f}-\frac {\csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 a f} \]
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Time = 0.12 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3273, 79, 52, 65, 214} \[ \int \cot ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {(2 a-b) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{2 \sqrt {a} f}-\frac {(2 a-b) \sqrt {a+b \sin ^2(e+f x)}}{2 a f}-\frac {\csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 a f} \]
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Rule 52
Rule 65
Rule 79
Rule 214
Rule 3273
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(1-x) \sqrt {a+b x}}{x^2} \, dx,x,\sin ^2(e+f x)\right )}{2 f} \\ & = -\frac {\csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 a f}-\frac {(2 a-b) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\sin ^2(e+f x)\right )}{4 a f} \\ & = -\frac {(2 a-b) \sqrt {a+b \sin ^2(e+f x)}}{2 a f}-\frac {\csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 a f}-\frac {(2 a-b) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{4 f} \\ & = -\frac {(2 a-b) \sqrt {a+b \sin ^2(e+f x)}}{2 a f}-\frac {\csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 a f}-\frac {(2 a-b) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^2(e+f x)}\right )}{2 b f} \\ & = \frac {(2 a-b) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{2 \sqrt {a} f}-\frac {(2 a-b) \sqrt {a+b \sin ^2(e+f x)}}{2 a f}-\frac {\csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{2 a f} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.70 \[ \int \cot ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {(2 a-b) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )-\sqrt {a} \left (2+\csc ^2(e+f x)\right ) \sqrt {a+b \sin ^2(e+f x)}}{2 \sqrt {a} f} \]
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Time = 1.05 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.11
method | result | size |
default | \(\frac {-\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}-\frac {\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{2 \sin \left (f x +e \right )^{2}}-\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{2 \sqrt {a}}+\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{f}\) | \(122\) |
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Time = 0.94 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.17 \[ \int \cot ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\left [-\frac {{\left ({\left (2 \, a - b\right )} \cos \left (f x + e\right )^{2} - 2 \, a + b\right )} \sqrt {a} \log \left (\frac {2 \, {\left (b \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a} - 2 \, a - b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) + 2 \, {\left (2 \, a \cos \left (f x + e\right )^{2} - 3 \, a\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{4 \, {\left (a f \cos \left (f x + e\right )^{2} - a f\right )}}, -\frac {{\left ({\left (2 \, a - b\right )} \cos \left (f x + e\right )^{2} - 2 \, a + b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a}}{a}\right ) + {\left (2 \, a \cos \left (f x + e\right )^{2} - 3 \, a\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{2 \, {\left (a f \cos \left (f x + e\right )^{2} - a f\right )}}\right ] \]
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\[ \int \cot ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int \sqrt {a + b \sin ^{2}{\left (e + f x \right )}} \cot ^{3}{\left (e + f x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.03 \[ \int \cot ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {2 \, \sqrt {a} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right ) - \frac {b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{\sqrt {a}} - 2 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} + \frac {\sqrt {b \sin \left (f x + e\right )^{2} + a} b}{a} - \frac {{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}{a \sin \left (f x + e\right )^{2}}}{2 \, f} \]
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Timed out. \[ \int \cot ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \cot ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^3\,\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a} \,d x \]
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