Optimal. Leaf size=27 \[ \text {Int}\left (\frac {x^m \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}},x\right ) \]
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Rubi [A] time = 0.10, 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 = {} \[ \int \frac {x^m \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^m \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\int \frac {x^m \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ \end {align*}
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Mathematica [A] time = 0.53, size = 0, normalized size = 0.00 \[ \int \frac {x^m \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{m} \operatorname {artanh}\left (a x\right )^{3}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1}, x\right ) \]
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 \[ \int \frac {x^{m} \operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.09, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \arctanh \left (a x \right )^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x^{m} \operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^m\,{\mathrm {atanh}\left (a\,x\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{m} \operatorname {atanh}^{3}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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