Integrand size = 25, antiderivative size = 126 \[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=-\frac {\left (8 a^2+8 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{5/2} f}+\frac {(8 a+3 b) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac {\csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{4 a f} \]
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Time = 0.22 (sec) , antiderivative size = 126, 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, 91, 79, 65, 214} \[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\frac {(8 a+3 b) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac {\left (8 a^2+8 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{5/2} f}-\frac {\csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{4 a f} \]
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Rule 65
Rule 79
Rule 91
Rule 214
Rule 3273
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(1-x)^2}{x^3 \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{2 f} \\ & = -\frac {\csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{4 a f}+\frac {\text {Subst}\left (\int \frac {\frac {1}{2} (-8 a-3 b)+2 a x}{x^2 \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{4 a f} \\ & = \frac {(8 a+3 b) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac {\csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{4 a f}+\frac {\left (8 a^2+8 a b+3 b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{16 a^2 f} \\ & = \frac {(8 a+3 b) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac {\csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{4 a f}+\frac {\left (8 a^2+8 a b+3 b^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^2(e+f x)}\right )}{8 a^2 b f} \\ & = -\frac {\left (8 a^2+8 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{5/2} f}+\frac {(8 a+3 b) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 a^2 f}-\frac {\csc ^4(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{4 a f} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80 \[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\frac {-\left (\left (8 a^2+8 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )\right )+\sqrt {a} \csc ^2(e+f x) \left (8 a+3 b-2 a \csc ^2(e+f x)\right ) \sqrt {a+b \sin ^2(e+f x)}}{8 a^{5/2} f} \]
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Time = 1.12 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.63
method | result | size |
default | \(\frac {-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{\sqrt {a}}-\frac {\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{4 a \sin \left (f x +e \right )^{4}}+\frac {3 b \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{8 a^{2} \sin \left (f x +e \right )^{2}}-\frac {3 b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{8 a^{\frac {5}{2}}}+\frac {\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{a \sin \left (f x +e \right )^{2}}-\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{a^{\frac {3}{2}}}}{f}\) | \(205\) |
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Time = 0.39 (sec) , antiderivative size = 388, normalized size of antiderivative = 3.08 \[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\left [\frac {{\left ({\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \sqrt {a} \log \left (\frac {2 \, {\left (b \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a} - 2 \, a - b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) - 2 \, {\left ({\left (8 \, a^{2} + 3 \, a b\right )} \cos \left (f x + e\right )^{2} - 6 \, a^{2} - 3 \, a b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{16 \, {\left (a^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} f \cos \left (f x + e\right )^{2} + a^{3} f\right )}}, \frac {{\left ({\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a}}{a}\right ) - {\left ({\left (8 \, a^{2} + 3 \, a b\right )} \cos \left (f x + e\right )^{2} - 6 \, a^{2} - 3 \, a b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{8 \, {\left (a^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} f \cos \left (f x + e\right )^{2} + a^{3} f\right )}}\right ] \]
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\[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int \frac {\cot ^{5}{\left (e + f x \right )}}{\sqrt {a + b \sin ^{2}{\left (e + f x \right )}}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.25 \[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=-\frac {\frac {8 \, \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{\sqrt {a}} + \frac {8 \, b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {3}{2}}} + \frac {3 \, b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {5}{2}}} - \frac {8 \, \sqrt {b \sin \left (f x + e\right )^{2} + a}}{a \sin \left (f x + e\right )^{2}} - \frac {3 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} b}{a^{2} \sin \left (f x + e\right )^{2}} + \frac {2 \, \sqrt {b \sin \left (f x + e\right )^{2} + a}}{a \sin \left (f x + e\right )^{4}}}{8 \, f} \]
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\[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int { \frac {\cot \left (f x + e\right )^{5}}{\sqrt {b \sin \left (f x + e\right )^{2} + a}} \,d x } \]
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Timed out. \[ \int \frac {\cot ^5(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx=\int \frac {{\mathrm {cot}\left (e+f\,x\right )}^5}{\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a}} \,d x \]
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