Integrand size = 25, antiderivative size = 208 \[ \int \cot ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=-\frac {\left (8 a^2-24 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 \sqrt {a} f}+\frac {\left (8 a^2-24 a b+3 b^2\right ) \sqrt {a+b \sin ^2(e+f x)}}{8 a f}+\frac {\left (8 a^2-24 a b+3 b^2\right ) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{24 a^2 f}+\frac {(8 a-b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{8 a^2 f}-\frac {\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{4 a f} \]
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Time = 0.24 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3273, 91, 79, 52, 65, 214} \[ \int \cot ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=-\frac {\left (8 a^2-24 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 \sqrt {a} f}+\frac {\left (8 a^2-24 a b+3 b^2\right ) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{24 a^2 f}+\frac {\left (8 a^2-24 a b+3 b^2\right ) \sqrt {a+b \sin ^2(e+f x)}}{8 a f}+\frac {(8 a-b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{8 a^2 f}-\frac {\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{4 a f} \]
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Rule 52
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 (a+b x)^{3/2}}{x^3} \, dx,x,\sin ^2(e+f x)\right )}{2 f} \\ & = -\frac {\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{4 a f}+\frac {\text {Subst}\left (\int \frac {\left (\frac {1}{2} (-8 a+b)+2 a x\right ) (a+b x)^{3/2}}{x^2} \, dx,x,\sin ^2(e+f x)\right )}{4 a f} \\ & = \frac {(8 a-b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{8 a^2 f}-\frac {\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{4 a f}+\frac {\left (2 a^2+\frac {3}{4} b (-8 a+b)\right ) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,\sin ^2(e+f x)\right )}{4 a^2 f} \\ & = \frac {\left (8 a^2-3 (8 a-b) b\right ) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{24 a^2 f}+\frac {(8 a-b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{8 a^2 f}-\frac {\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{4 a f}+\frac {\left (2 a^2+\frac {3}{4} b (-8 a+b)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\sin ^2(e+f x)\right )}{4 a f} \\ & = \frac {\left (8 a^2-3 (8 a-b) b\right ) \sqrt {a+b \sin ^2(e+f x)}}{8 a f}+\frac {\left (8 a^2-3 (8 a-b) b\right ) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{24 a^2 f}+\frac {(8 a-b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{8 a^2 f}-\frac {\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{4 a f}+\frac {\left (2 a^2+\frac {3}{4} b (-8 a+b)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{4 f} \\ & = \frac {\left (8 a^2-3 (8 a-b) b\right ) \sqrt {a+b \sin ^2(e+f x)}}{8 a f}+\frac {\left (8 a^2-3 (8 a-b) b\right ) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{24 a^2 f}+\frac {(8 a-b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{8 a^2 f}-\frac {\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{4 a f}+\frac {\left (2 a^2+\frac {3}{4} b (-8 a+b)\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 )}{2 b f} \\ & = -\frac {\left (8 a^2-3 (8 a-b) b\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 \sqrt {a} f}+\frac {\left (8 a^2-3 (8 a-b) b\right ) \sqrt {a+b \sin ^2(e+f x)}}{8 a f}+\frac {\left (8 a^2-3 (8 a-b) b\right ) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{24 a^2 f}+\frac {(8 a-b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{8 a^2 f}-\frac {\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{4 a f} \\ \end{align*}
Time = 0.84 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.59 \[ \int \cot ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {-3 \left (8 a^2-24 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )+\sqrt {a} \sqrt {a+b \sin ^2(e+f x)} \left (3 (8 a-5 b) \csc ^2(e+f x)-6 a \csc ^4(e+f x)+8 \left (4 a-6 b+b \sin ^2(e+f x)\right )\right )}{24 \sqrt {a} f} \]
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Time = 1.28 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.24
method | result | size |
default | \(\frac {\frac {b \left (\sin ^{2}\left (f x +e \right )\right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{3}+\frac {4 a \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{3}-\frac {a \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{4 \sin \left (f x +e \right )^{4}}-\frac {5 b \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{8 \sin \left (f x +e \right )^{2}}-\frac {3 b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{8 \sqrt {a}}-2 b \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}+\frac {a \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )^{2}}+3 \sqrt {a}\, b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )-a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{f}\) | \(257\) |
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Time = 1.35 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.12 \[ \int \cot ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\left [\frac {3 \, {\left ({\left (8 \, a^{2} - 24 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (8 \, a^{2} - 24 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, a^{2} - 24 \, a b + 3 \, b^{2}\right )} \sqrt {a} \log \left (\frac {2 \, {\left (b \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a} - 2 \, a - b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) - 2 \, {\left (8 \, a b \cos \left (f x + e\right )^{6} - 8 \, {\left (4 \, a^{2} - 3 \, a b\right )} \cos \left (f x + e\right )^{4} + {\left (88 \, a^{2} - 87 \, a b\right )} \cos \left (f x + e\right )^{2} - 50 \, a^{2} + 55 \, a b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{48 \, {\left (a f \cos \left (f x + e\right )^{4} - 2 \, a f \cos \left (f x + e\right )^{2} + a f\right )}}, \frac {3 \, {\left ({\left (8 \, a^{2} - 24 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (8 \, a^{2} - 24 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, a^{2} - 24 \, a b + 3 \, b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a}}{a}\right ) - {\left (8 \, a b \cos \left (f x + e\right )^{6} - 8 \, {\left (4 \, a^{2} - 3 \, a b\right )} \cos \left (f x + e\right )^{4} + {\left (88 \, a^{2} - 87 \, a b\right )} \cos \left (f x + e\right )^{2} - 50 \, a^{2} + 55 \, a b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{24 \, {\left (a f \cos \left (f x + e\right )^{4} - 2 \, a f \cos \left (f x + e\right )^{2} + a f\right )}}\right ] \]
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Timed out. \[ \int \cot ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.31 \[ \int \cot ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=-\frac {24 \, a^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right ) - 72 \, \sqrt {a} b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right ) + \frac {9 \, b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{\sqrt {a}} - 8 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} - 24 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} a + 72 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} b + \frac {24 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} b}{a} - \frac {3 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} b^{2}}{a^{2}} - \frac {9 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} b^{2}}{a} - \frac {24 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}{a \sin \left (f x + e\right )^{2}} + \frac {3 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}} b}{a^{2} \sin \left (f x + e\right )^{2}} + \frac {6 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}}{a \sin \left (f x + e\right )^{4}}}{24 \, f} \]
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Timed out. \[ \int \cot ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \cot ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^5\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]
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