Optimal. Leaf size=21 \[ \text {Int}\left ((d x)^m \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2,x\right ) \]
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Rubi [A] time = 0.02, 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 = {} \[ \int (d x)^m \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (d x)^m \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2 \, dx &=\int (d x)^m \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2 \, dx\\ \end {align*}
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Mathematica [A] time = 1.28, size = 0, normalized size = 0.00 \[ \int (d x)^m \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \arctan \left (c x^{2}\right )^{2} + 2 \, a b \arctan \left (c x^{2}\right ) + a^{2}\right )} \left (d x\right )^{m}, x\right ) \]
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 \[ \int {\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (a +b \arctan \left (c \,x^{2}\right )\right )^{2}\, dx \]
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 \[ \frac {\left (d x\right )^{m + 1} a^{2}}{d {\left (m + 1\right )}} + \frac {7 \, b^{2} d^{m} x x^{m} \arctan \left (c x^{2}\right )^{2} - \frac {3}{4} \, b^{2} d^{m} x x^{m} \log \left (c^{2} x^{4} + 1\right )^{2} + {\left (m + 1\right )} \int \frac {24 \, b^{2} c^{2} d^{m} x^{4} x^{m} \log \left (c^{2} x^{4} + 1\right ) + 36 \, {\left ({\left (b^{2} c^{2} d^{m} m + b^{2} c^{2} d^{m}\right )} x^{4} + b^{2} d^{m} m + b^{2} d^{m}\right )} x^{m} \arctan \left (c x^{2}\right )^{2} + 3 \, {\left ({\left (b^{2} c^{2} d^{m} m + b^{2} c^{2} d^{m}\right )} x^{4} + b^{2} d^{m} m + b^{2} d^{m}\right )} x^{m} \log \left (c^{2} x^{4} + 1\right )^{2} - 16 \, {\left (7 \, b^{2} c d^{m} x^{2} - 8 \, {\left (a b c^{2} d^{m} m + a b c^{2} d^{m}\right )} x^{4} - 8 \, a b d^{m} m - 8 \, a b d^{m}\right )} x^{m} \arctan \left (c x^{2}\right )}{4 \, {\left ({\left (c^{2} m + c^{2}\right )} x^{4} + m + 1\right )}}\,{d x}}{16 \, {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.05 \[ \int {\left (d\,x\right )}^m\,{\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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