Optimal. Leaf size=97 \[ -\frac {\text {Int}\left (\frac {1}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3},x\right )}{a^2}+\frac {\text {Chi}\left (\tanh ^{-1}(a x)\right )}{2 a^3}-\frac {x}{2 a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}-\frac {1}{2 a^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2} \]
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Rubi [A] time = 0.30, 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 = {} \[ \int \frac {x^2}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^2}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^3} \, dx &=\frac {\int \frac {1}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^3} \, dx}{a^2}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3} \, dx}{a^2}\\ &=-\frac {1}{2 a^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3} \, dx}{a^2}+\frac {\int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ &=-\frac {1}{2 a^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac {x}{2 a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\frac {\int \frac {1}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)} \, dx}{2 a^2}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3} \, dx}{a^2}\\ &=-\frac {1}{2 a^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac {x}{2 a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a^3}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3} \, dx}{a^2}\\ &=-\frac {1}{2 a^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac {x}{2 a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\frac {\text {Chi}\left (\tanh ^{-1}(a x)\right )}{2 a^3}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3} \, dx}{a^2}\\ \end {align*}
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Mathematica [A] time = 6.39, size = 0, normalized size = 0.00 \[ \int \frac {x^2}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )^{3}}, x\right ) \]
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 \[ \int \frac {x^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \arctanh \left (a x \right )^{3}}\, dx \]
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 \[ \int \frac {x^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{2}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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