Optimal. Leaf size=45 \[ \frac{x \tan ^{-1}(a x)^{n+1}}{a c (n+1)}-\frac{\text{Unintegrable}\left (\tan ^{-1}(a x)^{n+1},x\right )}{a c (n+1)} \]
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Rubi [A] time = 0.084999, 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 = {} \[ \int \frac{x \tan ^{-1}(a x)^n}{c+a^2 c x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{x \tan ^{-1}(a x)^n}{c+a^2 c x^2} \, dx &=\frac{x \tan ^{-1}(a x)^{1+n}}{a c (1+n)}-\frac{\int \tan ^{-1}(a x)^{1+n} \, dx}{a c (1+n)}\\ \end{align*}
Mathematica [A] time = 0.89217, size = 0, normalized size = 0. \[ \int \frac{x \tan ^{-1}(a x)^n}{c+a^2 c x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.602, size = 0, normalized size = 0. \begin{align*} \int{\frac{x \left ( \arctan \left ( ax \right ) \right ) ^{n}}{{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 [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x \arctan \left (a x\right )^{n}}{a^{2} c x^{2} + c}, 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*} \frac{\int \frac{x \operatorname{atan}^{n}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \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{x \arctan \left (a x\right )^{n}}{a^{2} c x^{2} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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