Optimal. Leaf size=32 \[ \frac{\left (a \sec ^2(x)\right )^{5/2}}{5 a}-\frac{1}{3} \left (a \sec ^2(x)\right )^{3/2} \]
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Rubi [A] time = 0.0999202, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {3657, 4124, 43} \[ \frac{\left (a \sec ^2(x)\right )^{5/2}}{5 a}-\frac{1}{3} \left (a \sec ^2(x)\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4124
Rule 43
Rubi steps
\begin{align*} \int \tan ^3(x) \left (a+a \tan ^2(x)\right )^{3/2} \, dx &=\int \left (a \sec ^2(x)\right )^{3/2} \tan ^3(x) \, dx\\ &=\frac{1}{2} a \operatorname{Subst}\left (\int (-1+x) \sqrt{a x} \, dx,x,\sec ^2(x)\right )\\ &=\frac{1}{2} a \operatorname{Subst}\left (\int \left (-\sqrt{a x}+\frac{(a x)^{3/2}}{a}\right ) \, dx,x,\sec ^2(x)\right )\\ &=-\frac{1}{3} \left (a \sec ^2(x)\right )^{3/2}+\frac{\left (a \sec ^2(x)\right )^{5/2}}{5 a}\\ \end{align*}
Mathematica [A] time = 0.0495771, size = 22, normalized size = 0.69 \[ \frac{1}{15} \left (3 \sec ^2(x)-5\right ) \left (a \sec ^2(x)\right )^{3/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 29, normalized size = 0.9 \begin{align*}{\frac{1}{5\,a} \left ( a+a \left ( \tan \left ( x \right ) \right ) ^{2} \right ) ^{{\frac{5}{2}}}}-{\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 [B] time = 1.93163, size = 755, normalized size = 23.59 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39101, size = 82, normalized size = 2.56 \begin{align*} \frac{1}{15} \,{\left (3 \, a \tan \left (x\right )^{4} + a \tan \left (x\right )^{2} - 2 \, a\right )} \sqrt{a \tan \left (x\right )^{2} + a} \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 \left (a \left (\tan ^{2}{\left (x \right )} + 1\right )\right )^{\frac{3}{2}} \tan ^{3}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13284, size = 42, normalized size = 1.31 \begin{align*} \frac{3 \,{\left (a \tan \left (x\right )^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (a \tan \left (x\right )^{2} + a\right )}^{\frac{3}{2}} a}{15 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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