Integrand size = 25, antiderivative size = 297 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{a b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(7 a+8 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 f}-\frac {(3 a+4 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 b f}+\frac {(7 a+8 b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {4 (a+b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a^2 f \sqrt {a+b \sin ^2(e+f x)}} \]
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Time = 0.59 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3275, 479, 597, 538, 437, 435, 432, 430} \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {(7 a+8 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{3 a^3 f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}+\frac {(7 a+8 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 f}-\frac {4 (a+b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{3 a^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(3 a+4 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 b f}+\frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{a b f \sqrt {a+b \sin ^2(e+f x)}} \]
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Rule 430
Rule 432
Rule 435
Rule 437
Rule 479
Rule 538
Rule 597
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 {\left (1-x^2\right )^{3/2}}{x^4 \left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{a b f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-3 a-4 b+(2 a+3 b) x^2}{x^4 \sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{a b f} \\ & = \frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{a b f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(3 a+4 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 b f}+\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-b (7 a+8 b)+b (3 a+4 b) x^2}{x^2 \sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 b f} \\ & = \frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{a b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(7 a+8 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 f}-\frac {(3 a+4 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 b f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-a b (3 a+4 b)-b^2 (7 a+8 b) x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^3 b f} \\ & = \frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{a b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(7 a+8 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 f}-\frac {(3 a+4 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 b f}-\frac {\left (4 (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 a^2 f}+\frac {\left ((7 a+8 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 a^3 f} \\ & = \frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{a b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(7 a+8 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 f}-\frac {(3 a+4 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 b f}+\frac {\left ((7 a+8 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 a^3 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {\left (4 (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 a^2 f \sqrt {a+b \sin ^2(e+f x)}} \\ & = \frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{a b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(7 a+8 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 f}-\frac {(3 a+4 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^2 b f}+\frac {(7 a+8 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 a^3 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {4 (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 a^2 f \sqrt {a+b \sin ^2(e+f x)}} \\ \end{align*}
Time = 4.30 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.67 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {\frac {\left (8 a^2+37 a b+24 b^2-4 \left (4 a^2+11 a b+8 b^2\right ) \cos (2 (e+f x))+b (7 a+8 b) \cos (4 (e+f x))\right ) \cot (e+f x) \csc ^2(e+f x)}{2 \sqrt {2}}+2 a (7 a+8 b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )-8 a (a+b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )}{6 a^3 f \sqrt {2 a+b-b \cos (2 (e+f x))}} \]
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Time = 3.73 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.19
method | result | size |
default | \(-\frac {4 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} \left (\sin ^{3}\left (f x +e \right )\right )+4 b \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \left (\sin ^{3}\left (f x +e \right )\right )-7 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} \left (\sin ^{3}\left (f x +e \right )\right )-8 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b \left (\sin ^{3}\left (f x +e \right )\right )+7 a b \left (\sin ^{6}\left (f x +e \right )\right )+8 b^{2} \left (\sin ^{6}\left (f x +e \right )\right )+4 a^{2} \left (\sin ^{4}\left (f x +e \right )\right )-3 a b \left (\sin ^{4}\left (f x +e \right )\right )-8 b^{2} \left (\sin ^{4}\left (f x +e \right )\right )-5 a^{2} \left (\sin ^{2}\left (f x +e \right )\right )-4 a b \left (\sin ^{2}\left (f x +e \right )\right )+a^{2}}{3 a^{3} \sin \left (f x +e \right )^{3} \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(353\) |
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Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 1465, normalized size of antiderivative = 4.93 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\cot ^{4}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
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Timed out. \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\cot \left (f x + e\right )^{4}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (e+f\,x\right )}^4}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]
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