Optimal. Leaf size=46 \[ \frac {x \tan ^{-1}(a x)^{n+1}}{a c (n+1)}-\frac {\text {Int}\left (\tan ^{-1}(a x)^{n+1},x\right )}{a c (n+1)} \]
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Rubi [A] time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}
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Mathematica [A] time = 0.95, size = 0, normalized size = 0.00 \[ \int \frac {x \tan ^{-1}(a x)^n}{c+a^2 c x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x \arctan \left (a x\right )^{n}}{a^{2} c x^{2} + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.80, size = 0, normalized size = 0.00 \[ \int \frac {x \arctan \left (a x \right )^{n}}{a^{2} c \,x^{2}+c}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^n}{c\,a^2\,x^2+c} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x \operatorname {atan}^{n}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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