Optimal. Leaf size=29 \[ \frac {2 \tan (x)}{3 \sqrt {\sec ^2(x)}}+\frac {\tan (x)}{3 \sec ^2(x)^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4122, 192, 191} \[ \frac {2 \tan (x)}{3 \sqrt {\sec ^2(x)}}+\frac {\tan (x)}{3 \sec ^2(x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\sec ^2(x)^{3/2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{5/2}} \, dx,x,\tan (x)\right )\\ &=\frac {\tan (x)}{3 \sec ^2(x)^{3/2}}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{3/2}} \, dx,x,\tan (x)\right )\\ &=\frac {\tan (x)}{3 \sec ^2(x)^{3/2}}+\frac {2 \tan (x)}{3 \sqrt {\sec ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 0.79 \[ \frac {(9 \sin (x)+\sin (3 x)) \sec (x)}{12 \sqrt {\sec ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 10, normalized size = 0.34 \[ -\frac {1}{3} \, {\left (\cos \relax (x)^{2} + 2\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 16, normalized size = 0.55 \[ -\frac {1}{3} \, \mathrm {sgn}\left (\cos \relax (x)\right ) \sin \relax (x)^{3} + \mathrm {sgn}\left (\cos \relax (x)\right ) \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 21, normalized size = 0.72 \[ \frac {\sin \relax (x ) \left (\cos ^{2}\relax (x )+2\right ) \left (\cos \left (2 x \right )+1\right ) \sqrt {2}}{12 \cos \relax (x )^{3} \sqrt {\frac {1}{\cos \left (2 x \right )+1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 25, normalized size = 0.86 \[ \frac {2 \, \tan \relax (x)}{3 \, \sqrt {\tan \relax (x)^{2} + 1}} + \frac {\tan \relax (x)}{3 \, {\left (\tan \relax (x)^{2} + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\left (\frac {1}{{\cos \relax (x)}^2}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.05, size = 27, normalized size = 0.93 \[ \frac {2 \tan ^{3}{\relax (x )}}{3 \left (\sec ^{2}{\relax (x )}\right )^{\frac {3}{2}}} + \frac {\tan {\relax (x )}}{\left (\sec ^{2}{\relax (x )}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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