Integrand size = 25, antiderivative size = 348 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {8 (a+2 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^4 f}-\frac {(3 a+8 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 b f}+\frac {8 (a+2 b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^4 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {(5 a+8 b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a^3 f \sqrt {a+b \sin ^2(e+f x)}} \]
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Time = 0.59 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3275, 479, 593, 597, 538, 437, 435, 432, 430} \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {8 (a+2 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{3 a^4 f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}+\frac {8 (a+2 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^4 f}-\frac {(5 a+8 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{3 a^3 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(3 a+8 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 b f}+\frac {2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
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Rule 430
Rule 432
Rule 435
Rule 437
Rule 479
Rule 538
Rule 593
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 )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-3 (a+2 b)+(2 a+5 b) x^2}{x^4 \sqrt {1-x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 a b f} \\ & = \frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 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+b) (3 a+8 b)-6 (a+b) (a+3 b) x^2}{x^4 \sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 b (a+b) f} \\ & = \frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(3 a+8 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 b f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {24 b (a+b) (a+2 b)-3 b (a+b) (3 a+8 b) x^2}{x^2 \sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{9 a^3 b (a+b) f} \\ & = \frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {8 (a+2 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^4 f}-\frac {(3 a+8 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 b f}+\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {3 a b (a+b) (3 a+8 b)+24 b^2 (a+b) (a+2 b) x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{9 a^4 b (a+b) f} \\ & = \frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {8 (a+2 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^4 f}-\frac {(3 a+8 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 b f}+\frac {\left (8 (a+2 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^4 f}-\frac {\left ((5 a+8 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^3 f} \\ & = \frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {8 (a+2 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^4 f}-\frac {(3 a+8 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 b f}+\frac {\left (8 (a+2 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^4 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {\left ((5 a+8 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^3 f \sqrt {a+b \sin ^2(e+f x)}} \\ & = \frac {(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt {a+b \sin ^2(e+f x)}}+\frac {8 (a+2 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^4 f}-\frac {(3 a+8 b) \cot (e+f x) \csc ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 b f}+\frac {8 (a+2 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^4 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {(5 a+8 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^3 f \sqrt {a+b \sin ^2(e+f x)}} \\ \end{align*}
Time = 3.71 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.65 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {2 a^2 b \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \left (8 (a+2 b) E\left (e+f x\left |-\frac {b}{a}\right .\right )-(5 a+8 b) \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )\right )+\sqrt {2} b \left (4 (a+2 b) (2 a+b-b \cos (2 (e+f x)))^2 \cot (e+f x)-a (2 a+b-b \cos (2 (e+f x)))^2 \cot (e+f x) \csc ^2(e+f x)+2 a b (a+b) \sin (2 (e+f x))+4 b (a+2 b) (2 a+b-b \cos (2 (e+f x))) \sin (2 (e+f x))\right )}{6 a^4 b f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \]
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Time = 4.47 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.82
method | result | size |
default | \(-\frac {5 \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} b \left (\sin ^{5}\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}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2} \left (\sin ^{5}\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^{2} b \left (\sin ^{5}\left (f x +e \right )\right )-16 \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^{2} \left (\sin ^{5}\left (f x +e \right )\right )+8 \left (\sin ^{8}\left (f x +e \right )\right ) a \,b^{2}+16 \left (\sin ^{8}\left (f x +e \right )\right ) b^{3}+5 \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^{3} \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}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b \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^{3} \left (\sin ^{3}\left (f x +e \right )\right )-16 \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} b \left (\sin ^{3}\left (f x +e \right )\right )+13 \left (\sin ^{6}\left (f x +e \right )\right ) a^{2} b +16 \left (\sin ^{6}\left (f x +e \right )\right ) a \,b^{2}-16 b^{3} \left (\sin ^{6}\left (f x +e \right )\right )+4 \left (\sin ^{4}\left (f x +e \right )\right ) a^{3}-7 \left (\sin ^{4}\left (f x +e \right )\right ) a^{2} b -24 \left (\sin ^{4}\left (f x +e \right )\right ) a \,b^{2}-5 \left (\sin ^{2}\left (f x +e \right )\right ) a^{3}-6 \left (\sin ^{2}\left (f x +e \right )\right ) a^{2} b +a^{3}}{3 \sin \left (f x +e \right )^{3} a^{4} {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}} \cos \left (f x +e \right ) f}\) | \(633\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 1971, normalized size of antiderivative = 5.66 \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cot ^{4}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
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Timed out. \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cot \left (f x + e\right )^{4}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {cot}\left (e+f\,x\right )}^4}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]
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