Optimal. Leaf size=27 \[ \frac {\text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^3}-\frac {\log \left (\tanh ^{-1}(a x)\right )}{8 a^3} \]
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Rubi [A] time = 0.11, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6034, 5448, 3301} \[ \frac {\text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^3}-\frac {\log \left (\tanh ^{-1}(a x)\right )}{8 a^3} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 5448
Rule 6034
Rubi steps
\begin {align*} \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{8 x}+\frac {\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {\log \left (\tanh ^{-1}(a x)\right )}{8 a^3}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^3}\\ &=\frac {\text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^3}-\frac {\log \left (\tanh ^{-1}(a x)\right )}{8 a^3}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 22, normalized size = 0.81 \[ \frac {\text {Chi}\left (4 \tanh ^{-1}(a x)\right )-\log \left (\tanh ^{-1}(a x)\right )}{8 a^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 88, normalized size = 3.26 \[ -\frac {2 \, \log \left (\log \left (-\frac {a x + 1}{a x - 1}\right )\right ) - \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) - \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right )}{16 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x^{2}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 24, normalized size = 0.89 \[ \frac {\Chi \left (4 \arctanh \left (a x \right )\right )}{8 a^{3}}-\frac {\ln \left (\arctanh \left (a x \right )\right )}{8 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {x^{2}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ -\int \frac {x^2}{\mathrm {atanh}\left (a\,x\right )\,{\left (a^2\,x^2-1\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x^{2}}{a^{6} x^{6} \operatorname {atanh}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}{\left (a x \right )} - \operatorname {atanh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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