Optimal. Leaf size=36 \[ \frac {\text {Chi}\left (\tanh ^{-1}(a x)\right )}{a^2}-\frac {x}{a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)} \]
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Rubi [A] time = 0.12, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6006, 5968, 3301} \[ \frac {\text {Chi}\left (\tanh ^{-1}(a x)\right )}{a^2}-\frac {x}{a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 5968
Rule 6006
Rubi steps
\begin {align*} \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx &=-\frac {x}{a \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}{a}\\ &=-\frac {x}{a \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 )}{a^2}\\ &=-\frac {x}{a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\frac {\text {Chi}\left (\tanh ^{-1}(a x)\right )}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 34, normalized size = 0.94 \[ \frac {\text {Chi}\left (\tanh ^{-1}(a x)\right )-\frac {a x}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}}{a^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} x}{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.35, size = 90, normalized size = 2.50 \[ \frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 a^{2} \left (a x -1\right ) \arctanh \left (a x \right )}-\frac {\Ei \left (1, -\arctanh \left (a x \right )\right )}{2 a^{2}}+\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 a^{2} \left (a x +1\right ) \arctanh \left (a x \right )}-\frac {\Ei \left (1, \arctanh \left (a x \right )\right )}{2 a^{2}} \]
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 \[ \int \frac {x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )^{2}}\,{d x} \]
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 \[ \int \frac {x}{{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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