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 = 0.15, antiderivative size = 39, normalized size of antiderivative = 1.00, 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 191
Rule 434
Rule 453
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*}
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Mathematica [A] time = 0.08, 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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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {(-3+\cos (2 x)) \sec ^4(x)}{\sqrt {4-\cot ^2(x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.55, size = 33, normalized size = 0.85 \[ -\frac {{\left (\cos \relax (x)^{2} + 1\right )} \sqrt {\frac {5 \, \cos \relax (x)^{2} - 4}{\cos \relax (x)^{2} - 1}} \sin \relax (x)}{3 \, \cos \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 1.14, size = 340, normalized size = 8.72 \[ -\frac {2 \, {\left (15948 \, \sqrt {5} - 49185 i\right )} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right )}{-15296 i \, \sqrt {5} + 98560} + \frac {82 \, \sqrt {5} + \frac {939 \, \sqrt {5} {\left (4 \, \sqrt {5} - \sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 18 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 1}\right )}^{2}}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 9\right )}^{2}} + \frac {537 \, \sqrt {5} {\left (4 \, \sqrt {5} - \sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 18 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 1}\right )}^{4}}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 9\right )}^{4}} + \frac {975 \, {\left (4 \, \sqrt {5} - \sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 18 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 1}\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} - 9} + \frac {2255 \, {\left (4 \, \sqrt {5} - \sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 18 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 1}\right )}^{3}}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 9\right )}^{3}} + \frac {255 \, {\left (4 \, \sqrt {5} - \sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 18 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 1}\right )}^{5}}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 9\right )}^{5}}}{96 \, {\left (\frac {\sqrt {5} {\left (4 \, \sqrt {5} - \sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 18 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 1}\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} - 9} + \frac {{\left (4 \, \sqrt {5} - \sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 18 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 1}\right )}^{2}}{{\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 9\right )}^{2}} + 1\right )}^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.76, size = 64, normalized size = 1.64
method | result | size |
default | \(-\frac {\left (5 \left (\cos ^{2}\relax (x )\right )+2\right ) \sqrt {-\frac {5 \left (\cos ^{2}\relax (x )\right )-4}{\sin \relax (x )^{2}}}\, \sin \relax (x ) \sqrt {4}}{12 \cos \relax (x )^{3}}+\frac {\sqrt {4}\, \sin \relax (x ) \sqrt {-\frac {5 \left (\cos ^{2}\relax (x )\right )-4}{\sin \relax (x )^{2}}}}{4 \cos \relax (x )}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.48, size = 63, normalized size = 1.62 \[ -\frac {1}{48} \, {\left (-\frac {1}{\tan \relax (x)^{2}} + 4\right )}^{\frac {3}{2}} \tan \relax (x)^{3} + \frac {3}{16} \, \sqrt {-\frac {1}{\tan \relax (x)^{2}} + 4} \tan \relax (x) - \frac {8 \, \tan \relax (x)^{4} + 26 \, \tan \relax (x)^{2} - 7}{8 \, \sqrt {2 \, \tan \relax (x) + 1} \sqrt {2 \, \tan \relax (x) - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.76, size = 20, normalized size = 0.51 \[ -\frac {\mathrm {tan}\relax (x)\,\left ({\mathrm {tan}\relax (x)}^2+2\right )\,\sqrt {4-\frac {1}{{\mathrm {tan}\relax (x)}^2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (2 x \right )} - 3}{\sqrt {- \left (\cot {\relax (x )} - 2\right ) \left (\cot {\relax (x )} + 2\right )} \cos ^{4}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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