Optimal. Leaf size=41 \[ \frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{2 a^3}-\frac {x^2}{a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \]
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Rubi [A] time = 0.29, antiderivative size = 60, normalized size of antiderivative = 1.46, number of steps used = 12, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6028, 5966, 6034, 5448, 12, 3298} \[ \frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{2 a^3}+\frac {1}{a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {1}{a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 5448
Rule 5966
Rule 6028
Rule 6034
Rubi steps
\begin {align*} \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx &=\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx}{a^2}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx}{a^2}\\ &=-\frac {1}{a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {1}{a^3 \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 {4 \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx}{a}\\ &=-\frac {1}{a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {1}{a^3 \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 {4 \operatorname {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {1}{a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {1}{a^3 \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 {4 \operatorname {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 x}+\frac {\sinh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {1}{a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {1}{a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac {1}{a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {1}{a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{2 a^3}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 56, normalized size = 1.37 \[ \frac {\left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x) \text {Shi}\left (4 \tanh ^{-1}(a x)\right )-2 a^2 x^2}{2 a^3 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 164, normalized size = 4.00 \[ -\frac {8 \, a^{2} x^{2} - {\left ({\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{4 \, {\left (a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )} \]
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 \[ \int -\frac {x^{2}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 38, normalized size = 0.93 \[ \frac {\frac {1}{8 \arctanh \left (a x \right )}-\frac {\cosh \left (4 \arctanh \left (a x \right )\right )}{8 \arctanh \left (a x \right )}+\frac {\Shi \left (4 \arctanh \left (a x \right )\right )}{2}}{a^{3}} \]
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 \[ -\frac {2 \, x^{2}}{{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right ) - {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-a x + 1\right )} + \int -\frac {4 \, {\left (a^{2} x^{3} + x\right )}}{{\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \log \left (a x + 1\right ) - {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \log \left (-a x + 1\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ -\int \frac {x^2}{{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^3} \,d x \]
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 \[ - \int \frac {x^{2}}{a^{6} x^{6} \operatorname {atanh}^{2}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}^{2}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}^{2}{\left (a x \right )} - \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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