Optimal. Leaf size=14 \[ \frac{1}{3} \left (a \sec ^2(x)\right )^{3/2} \]
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Rubi [A] time = 0.0511205, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3657, 4124, 32} \[ \frac{1}{3} \left (a \sec ^2(x)\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4124
Rule 32
Rubi steps
\begin{align*} \int \tan (x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx &=\int \left (a \sec ^2(x)\right )^{3/2} \tan (x) \, dx\\ &=\frac{1}{2} a \operatorname{Subst}\left (\int \sqrt{a x} \, dx,x,\sec ^2(x)\right )\\ &=\frac{1}{3} \left (a \sec ^2(x)\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0149399, size = 14, normalized size = 1. \[ \frac{1}{3} \left (a \sec ^2(x)\right )^{3/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 13, normalized size = 0.9 \begin{align*}{\frac{1}{3} \left ( a+a \left ( \tan \left ( x \right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \tan \left (x\right )^{2} + a\right )}^{\frac{3}{2}} \tan \left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33693, size = 38, normalized size = 2.71 \begin{align*} \frac{1}{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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Sympy [A] time = 3.30181, size = 12, normalized size = 0.86 \begin{align*} \frac{\left (a \tan ^{2}{\left (x \right )} + a\right )^{\frac{3}{2}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05332, size = 16, normalized size = 1.14 \begin{align*} \frac{1}{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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