Optimal. Leaf size=39 \[ -\frac{1}{3} \tan ^3(x) \sqrt{4-\cot ^2(x)}-\frac{2}{3} \tan (x) \sqrt{4-\cot ^2(x)} \]
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Rubi [A] time = 0.149407, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {12, 434, 453, 191} \[ -\frac{1}{3} \tan ^3(x) \sqrt{4-\cot ^2(x)}-\frac{2}{3} \tan (x) \sqrt{4-\cot ^2(x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 434
Rule 453
Rule 191
Rubi steps
\begin{align*} \int \frac{(-3+\cos (2 x)) \sec ^4(x)}{\sqrt{4-\cot ^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{2 \left (-1-2 x^2\right )}{\sqrt{4-\frac{1}{x^2}}} \, dx,x,\tan (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{-1-2 x^2}{\sqrt{4-\frac{1}{x^2}}} \, dx,x,\tan (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{\left (-2-\frac{1}{x^2}\right ) x^2}{\sqrt{4-\frac{1}{x^2}}} \, dx,x,\tan (x)\right )\\ &=-\frac{1}{3} \sqrt{4-\cot ^2(x)} \tan ^3(x)-\frac{8}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{4-\frac{1}{x^2}}} \, dx,x,\tan (x)\right )\\ &=-\frac{2}{3} \sqrt{4-\cot ^2(x)} \tan (x)-\frac{1}{3} \sqrt{4-\cot ^2(x)} \tan ^3(x)\\ \end{align*}
Mathematica [A] time = 0.0784498, size = 36, normalized size = 0.92 \[ \frac{(\cos (2 x)+3) (5 \cos (2 x)-3) \csc (x) \sec ^3(x)}{12 \sqrt{4-\cot ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.457, size = 61, normalized size = 1.6 \begin{align*} -{\frac{ \left ( 5\, \left ( \cos \left ( x \right ) \right ) ^{2}+2 \right ) \sin \left ( x \right ) \sqrt{4}}{12\, \left ( \cos \left ( x \right ) \right ) ^{3}}\sqrt{-{\frac{-4+5\, \left ( \cos \left ( x \right ) \right ) ^{2}}{ \left ( \sin \left ( x \right ) \right ) ^{2}}}}}+{\frac{\sin \left ( x \right ) }{2\,\cos \left ( x \right ) }\sqrt{-{\frac{-4+5\, \left ( \cos \left ( x \right ) \right ) ^{2}}{ \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.975043, size = 85, normalized size = 2.18 \begin{align*} -\frac{1}{48} \,{\left (-\frac{1}{\tan \left (x\right )^{2}} + 4\right )}^{\frac{3}{2}} \tan \left (x\right )^{3} + \frac{3}{16} \, \sqrt{-\frac{1}{\tan \left (x\right )^{2}} + 4} \tan \left (x\right ) - \frac{8 \, \tan \left (x\right )^{4} + 26 \, \tan \left (x\right )^{2} - 7}{8 \, \sqrt{2 \, \tan \left (x\right ) + 1} \sqrt{2 \, \tan \left (x\right ) - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.44099, size = 101, normalized size = 2.59 \begin{align*} -\frac{{\left (\cos \left (x\right )^{2} + 1\right )} \sqrt{\frac{5 \, \cos \left (x\right )^{2} - 4}{\cos \left (x\right )^{2} - 1}} \sin \left (x\right )}{3 \, \cos \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (2 \, x\right ) - 3}{\sqrt{-\cot \left (x\right )^{2} + 4} \cos \left (x\right )^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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