Optimal. Leaf size=38 \[ \frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{a^3}-\frac {x^2}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)} \]
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Rubi [A] time = 0.13, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6006, 6034, 5448, 12, 3298} \[ \frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{a^3}-\frac {x^2}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 5448
Rule 6006
Rule 6034
Rubi steps
\begin {align*} \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx &=-\frac {x^2}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx}{a}\\ &=-\frac {x^2}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {2 \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {x^2}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {2 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {x^2}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {x^2}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 36, normalized size = 0.95 \[ \frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{a^3}+\frac {x^2}{a \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.37, size = 111, normalized size = 2.92 \[ \frac {4 \, a^{2} x^{2} + {\left ({\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) - {\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{2 \, {\left (a^{5} x^{2} - a^{3}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )} \]
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 \[ \int \frac {x^{2}}{{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 36, normalized size = 0.95 \[ \frac {\frac {1}{2 \arctanh \left (a x \right )}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{2 \arctanh \left (a x \right )}+\Shi \left (2 \arctanh \left (a x \right )\right )}{a^{3}} \]
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 \[ \frac {2 \, x^{2}}{{\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right ) - {\left (a^{3} x^{2} - a\right )} \log \left (-a x + 1\right )} - 4 \, \int -\frac {x}{{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right ) - {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-a x + 1\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^2}{{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^2} \,d x \]
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 \[ \int \frac {x^{2}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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