Optimal. Leaf size=23 \[ \text {Int}\left (x^m \left (a^2 c x^2+c\right ) \tan ^{-1}(a x)^2,x\right ) \]
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Rubi [A] time = 0.04, 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 = {} \[ \int x^m \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x^m \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx &=\int x^m \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx\\ \end {align*}
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Mathematica [A] time = 0.96, size = 0, normalized size = 0.00 \[ \int x^m \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{2} + c\right )} x^{m} \arctan \left (a x\right )^{2}, x\right ) \]
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 = 1.54, size = 0, normalized size = 0.00 \[ \int x^{m} \left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{2}\, dx \]
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 \[ \frac {7 \, {\left ({\left (a^{2} c m + a^{2} c\right )} x^{3} + {\left (c m + 3 \, c\right )} x\right )} x^{m} \arctan \left (a x\right )^{2} - \frac {3}{4} \, {\left ({\left (a^{2} c m + a^{2} c\right )} x^{3} + {\left (c m + 3 \, c\right )} x\right )} x^{m} \log \left (a^{2} x^{2} + 1\right )^{2} + {\left (m^{2} + 4 \, m + 3\right )} \int \frac {36 \, {\left ({\left (a^{4} c m^{2} + 4 \, a^{4} c m + 3 \, a^{4} c\right )} x^{4} + c m^{2} + 2 \, {\left (a^{2} c m^{2} + 4 \, a^{2} c m + 3 \, a^{2} c\right )} x^{2} + 4 \, c m + 3 \, c\right )} x^{m} \arctan \left (a x\right )^{2} + 3 \, {\left ({\left (a^{4} c m^{2} + 4 \, a^{4} c m + 3 \, a^{4} c\right )} x^{4} + c m^{2} + 2 \, {\left (a^{2} c m^{2} + 4 \, a^{2} c m + 3 \, a^{2} c\right )} x^{2} + 4 \, c m + 3 \, c\right )} x^{m} \log \left (a^{2} x^{2} + 1\right )^{2} - 56 \, {\left ({\left (a^{3} c m + a^{3} c\right )} x^{3} + {\left (a c m + 3 \, a c\right )} x\right )} x^{m} \arctan \left (a x\right ) + 12 \, {\left ({\left (a^{4} c m + a^{4} c\right )} x^{4} + {\left (a^{2} c m + 3 \, a^{2} c\right )} x^{2}\right )} x^{m} \log \left (a^{2} x^{2} + 1\right )}{4 \, {\left ({\left (a^{2} m^{2} + 4 \, a^{2} m + 3 \, a^{2}\right )} x^{2} + m^{2} + 4 \, m + 3\right )}}\,{d x}}{16 \, {\left (m^{2} + 4 \, m + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int x^m\,{\mathrm {atan}\left (a\,x\right )}^2\,\left (c\,a^2\,x^2+c\right ) \,d x \]
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 \[ c \left (\int x^{m} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{2} x^{2} x^{m} \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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