Optimal. Leaf size=65 \[ -\frac{x}{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}-\frac{1}{2 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}+\frac{\text{Chi}\left (\tanh ^{-1}(a x)\right )}{2 a} \]
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Rubi [A] time = 0.163092, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {5966, 6006, 5968, 3301} \[ -\frac{x}{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}-\frac{1}{2 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}+\frac{\text{Chi}\left (\tanh ^{-1}(a x)\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 5966
Rule 6006
Rule 5968
Rule 3301
Rubi steps
\begin{align*} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^3} \, dx &=-\frac{1}{2 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}+\frac{1}{2} a \int \frac{x}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac{1}{2 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac{x}{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}+\frac{1}{2} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)} \, dx\\ &=-\frac{1}{2 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac{x}{2 \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 )}{2 a}\\ &=-\frac{1}{2 a \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac{x}{2 \sqrt{1-a^2 x^2} \tanh ^{-1}(a x)}+\frac{\text{Chi}\left (\tanh ^{-1}(a x)\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0902135, size = 44, normalized size = 0.68 \[ \frac{\text{Chi}\left (\tanh ^{-1}(a x)\right )-\frac{a x \tanh ^{-1}(a x)+1}{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.228, size = 86, normalized size = 1.3 \begin{align*}{\frac{1}{2\,a \left ({\it Artanh} \left ( ax \right ) \right ) ^{2} \left ({a}^{2}{x}^{2}-1 \right ) } \left ( \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{\it Chi} \left ({\it Artanh} \left ( ax \right ) \right ){x}^{2}{a}^{2}+\sqrt{-{a}^{2}{x}^{2}+1}ax{\it Artanh} \left ( ax \right ) -{\it Chi} \left ({\it Artanh} \left ( ax \right ) \right ) \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+\sqrt{-{a}^{2}{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \operatorname{artanh}\left (a x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} x^{2} + 1}}{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname{artanh}\left (a x\right )^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}} \operatorname{atanh}^{3}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \operatorname{artanh}\left (a x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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