Optimal. Leaf size=25 \[ \text {Int}\left (\frac {x^m}{\left (a^2 c x^2+c\right )^2 \tan ^{-1}(a x)^3},x\right ) \]
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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 = {} \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^m}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx &=\int \frac {x^m}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx\\ \end {align*}
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Mathematica [A] time = 1.00, size = 0, normalized size = 0.00 \[ \int \frac {x^m}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.57, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{3}}\, dx \]
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 \[ \frac {{\left (a^{4} c^{2} x^{3} + a^{2} c^{2} x\right )} \arctan \left (a x\right )^{2} \int \frac {{\left ({\left (a^{4} m^{2} - 3 \, a^{4} m + 2 \, a^{4}\right )} x^{4} + 2 \, {\left (a^{2} m^{2} - 2 \, a^{2} m - a^{2}\right )} x^{2} + m^{2} - m\right )} x^{m}}{{\left (a^{6} c^{2} x^{6} + 2 \, a^{4} c^{2} x^{4} + a^{2} c^{2} x^{2}\right )} \arctan \left (a x\right )}\,{d x} - a x x^{m} - {\left ({\left (a^{2} m - 2 \, a^{2}\right )} x^{2} + m\right )} x^{m} \arctan \left (a x\right )}{2 \, {\left (a^{4} c^{2} x^{3} + a^{2} c^{2} x\right )} \arctan \left (a x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^m}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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 \[ \frac {\int \frac {x^{m}}{a^{4} x^{4} \operatorname {atan}^{3}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )} + \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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