Optimal. Leaf size=191 \[ -\frac{40}{9 a \sqrt{1-a^2 x^2}}-\frac{2}{27 a \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(a x)^3}{3 \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \tanh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}-\frac{\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac{40 x \tanh ^{-1}(a x)}{9 \sqrt{1-a^2 x^2}}+\frac{2 x \tanh ^{-1}(a x)}{9 \left (1-a^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.164848, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {5964, 5962, 5958, 5960} \[ -\frac{40}{9 a \sqrt{1-a^2 x^2}}-\frac{2}{27 a \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(a x)^3}{3 \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \tanh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}-\frac{\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac{40 x \tanh ^{-1}(a x)}{9 \sqrt{1-a^2 x^2}}+\frac{2 x \tanh ^{-1}(a x)}{9 \left (1-a^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5964
Rule 5962
Rule 5958
Rule 5960
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=-\frac{\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac{2}{3} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac{2}{3} \int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{2}{27 a \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(a x)}{9 \left (1-a^2 x^2\right )^{3/2}}-\frac{\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \tanh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(a x)^3}{3 \sqrt{1-a^2 x^2}}+\frac{4}{9} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx+4 \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{2}{27 a \left (1-a^2 x^2\right )^{3/2}}-\frac{40}{9 a \sqrt{1-a^2 x^2}}+\frac{2 x \tanh ^{-1}(a x)}{9 \left (1-a^2 x^2\right )^{3/2}}+\frac{40 x \tanh ^{-1}(a x)}{9 \sqrt{1-a^2 x^2}}-\frac{\tanh ^{-1}(a x)^2}{3 a \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \tanh ^{-1}(a x)^2}{a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(a x)^3}{3 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0927984, size = 87, normalized size = 0.46 \[ \frac{120 a^2 x^2-9 a x \left (2 a^2 x^2-3\right ) \tanh ^{-1}(a x)^3+9 \left (6 a^2 x^2-7\right ) \tanh ^{-1}(a x)^2-6 a x \left (20 a^2 x^2-21\right ) \tanh ^{-1}(a x)-122}{27 a \left (1-a^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.178, size = 105, normalized size = 0.6 \begin{align*} -{\frac{18\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}{x}^{3}{a}^{3}+120\,{a}^{3}{x}^{3}{\it Artanh} \left ( ax \right ) -54\,{a}^{2}{x}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}-27\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}ax-120\,{a}^{2}{x}^{2}-126\,ax{\it Artanh} \left ( ax \right ) +63\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+122}{27\,a \left ({a}^{2}{x}^{2}-1 \right ) ^{2}}\sqrt{-{a}^{2}{x}^{2}+1}} \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{\operatorname{artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6996, size = 305, normalized size = 1.6 \begin{align*} \frac{{\left (960 \, a^{2} x^{2} - 9 \,{\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{3} + 18 \,{\left (6 \, a^{2} x^{2} - 7\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - 24 \,{\left (20 \, a^{3} x^{3} - 21 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) - 976\right )} \sqrt{-a^{2} x^{2} + 1}}{216 \,{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\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{\operatorname{atanh}^{3}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{5}{2}}}\, 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{\operatorname{artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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