Optimal. Leaf size=23 \[ \frac{\sin ^2(x) \tan (x)}{3 a \sqrt{a \sec ^2(x)}} \]
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Rubi [A] time = 0.101071, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {3657, 4125, 2564, 30} \[ \frac{\sin ^2(x) \tan (x)}{3 a \sqrt{a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4125
Rule 2564
Rule 30
Rubi steps
\begin{align*} \int \frac{\tan ^2(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx &=\int \frac{\tan ^2(x)}{\left (a \sec ^2(x)\right )^{3/2}} \, dx\\ &=\frac{\sec (x) \int \cos (x) \sin ^2(x) \, dx}{a \sqrt{a \sec ^2(x)}}\\ &=\frac{\sec (x) \operatorname{Subst}\left (\int x^2 \, dx,x,\sin (x)\right )}{a \sqrt{a \sec ^2(x)}}\\ &=\frac{\sin ^2(x) \tan (x)}{3 a \sqrt{a \sec ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0179405, size = 18, normalized size = 0.78 \[ \frac{\tan ^3(x)}{3 \left (a \sec ^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 56, normalized size = 2.4 \begin{align*}{\frac{\tan \left ( x \right ) }{a}{\frac{1}{\sqrt{a+a \left ( \tan \left ( x \right ) \right ) ^{2}}}}}-a \left ({\frac{\tan \left ( x \right ) }{3\,a} \left ( a+a \left ( \tan \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,\tan \left ( x \right ) }{3\,{a}^{2}}{\frac{1}{\sqrt{a+a \left ( \tan \left ( x \right ) \right ) ^{2}}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.92509, size = 19, normalized size = 0.83 \begin{align*} -\frac{\sin \left (3 \, x\right ) - 3 \, \sin \left (x\right )}{12 \, a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.34167, size = 99, normalized size = 4.3 \begin{align*} \frac{\sqrt{a \tan \left (x\right )^{2} + a} \tan \left (x\right )^{3}}{3 \,{\left (a^{2} \tan \left (x\right )^{4} + 2 \, a^{2} \tan \left (x\right )^{2} + a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{2}{\left (x \right )}}{\left (a \left (\tan ^{2}{\left (x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11288, size = 22, normalized size = 0.96 \begin{align*} \frac{\tan \left (x\right )^{3}}{3 \,{\left (a \tan \left (x\right )^{2} + a\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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