Integrand size = 25, antiderivative size = 223 \[ \int \cot ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {4 b \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}-\frac {\cot (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{f}-\frac {(7 a-b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {4 a (a+b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 f \sqrt {a+b \sin ^2(e+f x)}} \]
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Time = 0.30 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3275, 484, 542, 538, 437, 435, 432, 430} \[ \int \cot ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {4 a (a+b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{3 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(7 a-b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{3 f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}+\frac {4 b \sin (e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}-\frac {\cot (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{f} \]
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Rule 430
Rule 432
Rule 435
Rule 437
Rule 484
Rule 538
Rule 542
Rule 3275
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {1-x^2} \left (a+b x^2\right )^{3/2}}{x^2} \, dx,x,\sin (e+f x)\right )}{f} \\ & = -\frac {\cot (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{f}+\frac {\left (2 \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {\left (\frac {1}{2} (-a+3 b)-2 b x^2\right ) \sqrt {a+b x^2}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {4 b \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}-\frac {\cot (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{f}-\frac {\left (2 \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {\frac {1}{2} a (3 a-5 b)+\frac {1}{2} (7 a-b) b x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 f} \\ & = \frac {4 b \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}-\frac {\cot (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{f}-\frac {\left ((7 a-b) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 f}+\frac {\left (4 a (a+b) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 f} \\ & = \frac {4 b \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}-\frac {\cot (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{f}-\frac {\left ((7 a-b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {b x^2}{a}}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {\left (4 a (a+b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {b x^2}{a}}} \, dx,x,\sin (e+f x)\right )}{3 f \sqrt {a+b \sin ^2(e+f x)}} \\ & = \frac {4 b \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}-\frac {\cot (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{f}-\frac {(7 a-b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {4 a (a+b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 f \sqrt {a+b \sin ^2(e+f x)}} \\ \end{align*}
Time = 2.62 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.78 \[ \int \cot ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {\sqrt {2} \left (-24 a^2-8 a b+3 b^2+4 (2 a-b) b \cos (2 (e+f x))+b^2 \cos (4 (e+f x))\right ) \cot (e+f x)-8 a (7 a-b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )+32 a (a+b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )}{24 f \sqrt {2 a+b-b \cos (2 (e+f x))}} \]
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Time = 4.19 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {\left (\cos ^{6}\left (f x +e \right )\right ) b^{2}+\left (2 a b -2 b^{2}\right ) \left (\cos ^{4}\left (f x +e \right )\right )+\left (-3 a^{2}-2 a b +b^{2}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+\sin \left (f x +e \right ) \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, a \left (4 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a +4 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b -7 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a +E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b \right )}{3 \sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(204\) |
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\[ \int \cot ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cot \left (f x + e\right )^{2} \,d x } \]
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\[ \int \cot ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int \left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \cot ^{2}{\left (e + f x \right )}\, dx \]
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\[ \int \cot ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cot \left (f x + e\right )^{2} \,d x } \]
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Timed out. \[ \int \cot ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \cot ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^2\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]
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