Optimal. Leaf size=31 \[ -\frac {\csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {\tan (x)}{\sqrt {a \sec ^2(x)}} \]
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Rubi [A] time = 0.09, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3657, 4125, 2590, 14} \[ -\frac {\csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {\tan (x)}{\sqrt {a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2590
Rule 3657
Rule 4125
Rubi steps
\begin {align*} \int \frac {\cot ^2(x)}{\sqrt {a+a \tan ^2(x)}} \, dx &=\int \frac {\cot ^2(x)}{\sqrt {a \sec ^2(x)}} \, dx\\ &=\frac {\sec (x) \int \cos (x) \cot ^2(x) \, dx}{\sqrt {a \sec ^2(x)}}\\ &=-\frac {\sec (x) \operatorname {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,-\sin (x)\right )}{\sqrt {a \sec ^2(x)}}\\ &=-\frac {\sec (x) \operatorname {Subst}\left (\int \left (-1+\frac {1}{x^2}\right ) \, dx,x,-\sin (x)\right )}{\sqrt {a \sec ^2(x)}}\\ &=-\frac {\csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}}-\frac {\tan (x)}{\sqrt {a \sec ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 22, normalized size = 0.71 \[ \frac {-\tan (x)-\csc (x) \sec (x)}{\sqrt {a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 33, normalized size = 1.06 \[ -\frac {\sqrt {a \tan \relax (x)^{2} + a} {\left (2 \, \tan \relax (x)^{2} + 1\right )}}{a \tan \relax (x)^{3} + a \tan \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 47, normalized size = 1.52 \[ -\frac {\tan \relax (x)}{\sqrt {a \tan \relax (x)^{2} + a}} + \frac {2 \, \sqrt {a}}{{\left (\sqrt {a} \tan \relax (x) - \sqrt {a \tan \relax (x)^{2} + a}\right )}^{2} - a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 24, normalized size = 0.77 \[ \frac {\cos ^{2}\relax (x )-2}{\sin \relax (x ) \cos \relax (x ) \sqrt {\frac {a}{\cos \relax (x )^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.48, size = 128, normalized size = 4.13 \[ \frac {{\left ({\left (\sin \left (3 \, x\right ) - \sin \relax (x)\right )} \cos \left (4 \, x\right ) - {\left (\cos \left (3 \, x\right ) - \cos \relax (x)\right )} \sin \left (4 \, x\right ) - {\left (6 \, \cos \left (2 \, x\right ) - 1\right )} \sin \left (3 \, x\right ) + 6 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) - 6 \, \cos \relax (x) \sin \left (2 \, x\right ) + 6 \, \cos \left (2 \, x\right ) \sin \relax (x) - \sin \relax (x)\right )} \sqrt {a}}{2 \, {\left (a \cos \left (3 \, x\right )^{2} - 2 \, a \cos \left (3 \, x\right ) \cos \relax (x) + a \cos \relax (x)^{2} + a \sin \left (3 \, x\right )^{2} - 2 \, a \sin \left (3 \, x\right ) \sin \relax (x) + a \sin \relax (x)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.82, size = 40, normalized size = 1.29 \[ \frac {\sqrt {2}\,\left (6\,\sin \left (2\,x\right )-2\,\sin \left (2\,x\right )\,\left (2\,{\cos \relax (x)}^2-1\right )\right )}{8\,\sqrt {a}\,\sqrt {2\,{\cos \relax (x)}^2}\,\left ({\cos \relax (x)}^2-1\right )} \]
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 \[ \int \frac {\cot ^{2}{\relax (x )}}{\sqrt {a \left (\tan ^{2}{\relax (x )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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