Optimal. Leaf size=60 \[ -\frac {\csc (x) \sec (x)}{a \sqrt {a \sec ^2(x)}}-\frac {2 \tan (x)}{a \sqrt {a \sec ^2(x)}}+\frac {\sin ^2(x) \tan (x)}{3 a \sqrt {a \sec ^2(x)}} \]
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Rubi [A] time = 0.11, antiderivative size = 60, 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, 270} \[ -\frac {\csc (x) \sec (x)}{a \sqrt {a \sec ^2(x)}}-\frac {2 \tan (x)}{a \sqrt {a \sec ^2(x)}}+\frac {\sin ^2(x) \tan (x)}{3 a \sqrt {a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2590
Rule 3657
Rule 4125
Rubi steps
\begin {align*} \int \frac {\cot ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx &=\int \frac {\cot ^2(x)}{\left (a \sec ^2(x)\right )^{3/2}} \, dx\\ &=\frac {\sec (x) \int \cos ^3(x) \cot ^2(x) \, dx}{a \sqrt {a \sec ^2(x)}}\\ &=-\frac {\sec (x) \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^2} \, dx,x,-\sin (x)\right )}{a \sqrt {a \sec ^2(x)}}\\ &=-\frac {\sec (x) \operatorname {Subst}\left (\int \left (-2+\frac {1}{x^2}+x^2\right ) \, dx,x,-\sin (x)\right )}{a \sqrt {a \sec ^2(x)}}\\ &=-\frac {\csc (x) \sec (x)}{a \sqrt {a \sec ^2(x)}}-\frac {2 \tan (x)}{a \sqrt {a \sec ^2(x)}}+\frac {\sin ^2(x) \tan (x)}{3 a \sqrt {a \sec ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 31, normalized size = 0.52 \[ \frac {\sec ^3(x) \left (\sin ^3(x)-6 \sin (x)-3 \csc (x)\right )}{3 \left (a \sec ^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 52, normalized size = 0.87 \[ -\frac {{\left (8 \, \tan \relax (x)^{4} + 12 \, \tan \relax (x)^{2} + 3\right )} \sqrt {a \tan \relax (x)^{2} + a}}{3 \, {\left (a^{2} \tan \relax (x)^{5} + 2 \, a^{2} \tan \relax (x)^{3} + a^{2} \tan \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.52, size = 55, normalized size = 0.92 \[ -\frac {{\left (5 \, \tan \relax (x)^{2} + 6\right )} \tan \relax (x)}{3 \, {\left (a \tan \relax (x)^{2} + a\right )}^{\frac {3}{2}}} + \frac {2}{{\left ({\left (\sqrt {a} \tan \relax (x) - \sqrt {a \tan \relax (x)^{2} + a}\right )}^{2} - a\right )} \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.53, size = 31, normalized size = 0.52 \[ \frac {\cos ^{4}\relax (x )+4 \left (\cos ^{2}\relax (x )\right )-8}{3 \sin \relax (x ) \cos \relax (x )^{3} \left (\frac {a}{\cos \relax (x )^{2}}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.34, size = 225, normalized size = 3.75 \[ \frac {{\left ({\left (\sin \left (5 \, x\right ) - \sin \left (3 \, x\right )\right )} \cos \left (8 \, x\right ) + 20 \, {\left (\sin \left (5 \, x\right ) - \sin \left (3 \, x\right )\right )} \cos \left (6 \, x\right ) + 10 \, {\left (9 \, \sin \left (4 \, x\right ) - 2 \, \sin \left (2 \, x\right )\right )} \cos \left (5 \, x\right ) - {\left (\cos \left (5 \, x\right ) - \cos \left (3 \, x\right )\right )} \sin \left (8 \, x\right ) - 20 \, {\left (\cos \left (5 \, x\right ) - \cos \left (3 \, x\right )\right )} \sin \left (6 \, x\right ) - {\left (90 \, \cos \left (4 \, x\right ) - 20 \, \cos \left (2 \, x\right ) - 1\right )} \sin \left (5 \, x\right ) - 90 \, \cos \left (3 \, x\right ) \sin \left (4 \, x\right ) - {\left (20 \, \cos \left (2 \, x\right ) + 1\right )} \sin \left (3 \, x\right ) + 90 \, \cos \left (4 \, x\right ) \sin \left (3 \, x\right ) + 20 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right )\right )} \sqrt {a}}{24 \, {\left (a^{2} \cos \left (5 \, x\right )^{2} - 2 \, a^{2} \cos \left (5 \, x\right ) \cos \left (3 \, x\right ) + a^{2} \cos \left (3 \, x\right )^{2} + a^{2} \sin \left (5 \, x\right )^{2} - 2 \, a^{2} \sin \left (5 \, x\right ) \sin \left (3 \, x\right ) + a^{2} \sin \left (3 \, x\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {cot}\relax (x)}^2}{{\left (a\,{\mathrm {tan}\relax (x)}^2+a\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\cot ^{2}{\relax (x )}}{\left (a \left (\tan ^{2}{\relax (x )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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