Optimal. Leaf size=29 \[ \frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{4 a^2}+\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^2} \]
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Rubi [A] time = 0.09, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6034, 5448, 3298} \[ \frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{4 a^2}+\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^2} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 5448
Rule 6034
Rubi steps
\begin {align*} \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}\\ &=\frac {\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^2}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^2}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^2}\\ &=\frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{4 a^2}+\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^2}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 24, normalized size = 0.83 \[ \frac {2 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )+\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 102, normalized size = 3.52 \[ \frac {\operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) - \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + 2 \, \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) - 2 \, \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )}{16 \, a^{2}} \]
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 \[ \int -\frac {x}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 24, normalized size = 0.83 \[ \frac {\frac {\Shi \left (4 \arctanh \left (a x \right )\right )}{8}+\frac {\Shi \left (2 \arctanh \left (a x \right )\right )}{4}}{a^{2}} \]
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 \[ -\int \frac {x}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ -\int \frac {x}{\mathrm {atanh}\left (a\,x\right )\,{\left (a^2\,x^2-1\right )}^3} \,d x \]
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 \[ - \int \frac {x}{a^{6} x^{6} \operatorname {atanh}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}{\left (a x \right )} - \operatorname {atanh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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