Optimal. Leaf size=136 \[ -\frac{1}{3} a^3 \text{Unintegrable}\left (\frac{1}{x \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right )+\frac{1}{6} a \text{Unintegrable}\left (\frac{1}{x^3 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right )+\frac{2 a^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{3 c x}-\frac{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{3 c x^3} \]
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Rubi [A] time = 0.424138, 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 = {} \[ \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^4 \sqrt{c+a^2 c x^2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^4 \sqrt{c+a^2 c x^2}} \, dx &=-\frac{\sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}{3 c x^3}+\frac{1}{6} a \int \frac{1}{x^3 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx-\frac{1}{3} \left (2 a^2\right ) \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{\sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}{3 c x^3}+\frac{2 a^2 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}}{3 c x}+\frac{1}{6} a \int \frac{1}{x^3 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx-\frac{1}{3} a^3 \int \frac{1}{x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx\\ \end{align*}
Mathematica [A] time = 19.2338, size = 0, normalized size = 0. \[ \int \frac{\sqrt{\tan ^{-1}(a x)}}{x^4 \sqrt{c+a^2 c x^2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 3.262, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}}\sqrt{\arctan \left ( ax \right ) }{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\sqrt{\arctan \left (a x\right )}}{\sqrt{a^{2} c x^{2} + c} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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