Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{1}{x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)},x\right ) \]
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Rubi [A] time = 0.125433, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx &=\int \frac{1}{x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx\\ \end{align*}
Mathematica [A] time = 1.0901, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.456, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x\arctan \left ( ax \right ) } \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x \arctan \left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c}}{{\left (a^{4} c^{2} x^{5} + 2 \, a^{2} c^{2} x^{3} + c^{2} x\right )} \arctan \left (a x\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \operatorname{atan}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x \arctan \left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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