Optimal. Leaf size=37 \[ \frac {2 \text {Int}\left (\frac {x}{\tan ^{-1}(a x)},x\right )}{a c}-\frac {x^2}{a c \tan ^{-1}(a x)} \]
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Rubi [A] time = 0.07, 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 = {} \[ \int \frac {x^2}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^2}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx &=-\frac {x^2}{a c \tan ^{-1}(a x)}+\frac {2 \int \frac {x}{\tan ^{-1}(a x)} \, dx}{a c}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2}}, x\right ) \]
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 = 0.84, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {-2 \, \mathit {sage}_{0} x \arctan \left (a x\right ) + x^{2}}{a c \arctan \left (a x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^2\,\left (c\,a^2\,x^2+c\right )} \,d x \]
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 \[ \frac {\int \frac {x^{2}}{a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )} + \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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