Optimal. Leaf size=289 \[ -\frac{4144}{1125 a \sqrt{1-a^2 x^2}}-\frac{272}{3375 a \left (1-a^2 x^2\right )^{3/2}}-\frac{6}{625 a \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)^3}{15 \sqrt{1-a^2 x^2}}+\frac{4 x \tanh ^{-1}(a x)^3}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac{8 \tanh ^{-1}(a x)^2}{5 a \sqrt{1-a^2 x^2}}-\frac{4 \tanh ^{-1}(a x)^2}{15 a \left (1-a^2 x^2\right )^{3/2}}-\frac{3 \tanh ^{-1}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac{4144 x \tanh ^{-1}(a x)}{1125 \sqrt{1-a^2 x^2}}+\frac{272 x \tanh ^{-1}(a x)}{1125 \left (1-a^2 x^2\right )^{3/2}}+\frac{6 x \tanh ^{-1}(a x)}{125 \left (1-a^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.304252, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {5964, 5962, 5958, 5960} \[ -\frac{4144}{1125 a \sqrt{1-a^2 x^2}}-\frac{272}{3375 a \left (1-a^2 x^2\right )^{3/2}}-\frac{6}{625 a \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)^3}{15 \sqrt{1-a^2 x^2}}+\frac{4 x \tanh ^{-1}(a x)^3}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac{8 \tanh ^{-1}(a x)^2}{5 a \sqrt{1-a^2 x^2}}-\frac{4 \tanh ^{-1}(a x)^2}{15 a \left (1-a^2 x^2\right )^{3/2}}-\frac{3 \tanh ^{-1}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac{4144 x \tanh ^{-1}(a x)}{1125 \sqrt{1-a^2 x^2}}+\frac{272 x \tanh ^{-1}(a x)}{1125 \left (1-a^2 x^2\right )^{3/2}}+\frac{6 x \tanh ^{-1}(a x)}{125 \left (1-a^2 x^2\right )^{5/2}} \]
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 )^{7/2}} \, dx &=-\frac{3 \tanh ^{-1}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac{x \tanh ^{-1}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac{6}{25} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx+\frac{4}{5} \int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx\\ &=-\frac{6}{625 a \left (1-a^2 x^2\right )^{5/2}}+\frac{6 x \tanh ^{-1}(a x)}{125 \left (1-a^2 x^2\right )^{5/2}}-\frac{3 \tanh ^{-1}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac{4 \tanh ^{-1}(a x)^2}{15 a \left (1-a^2 x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 x \tanh ^{-1}(a x)^3}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac{24}{125} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac{8}{15} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac{8}{15} \int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{6}{625 a \left (1-a^2 x^2\right )^{5/2}}-\frac{272}{3375 a \left (1-a^2 x^2\right )^{3/2}}+\frac{6 x \tanh ^{-1}(a x)}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac{272 x \tanh ^{-1}(a x)}{1125 \left (1-a^2 x^2\right )^{3/2}}-\frac{3 \tanh ^{-1}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac{4 \tanh ^{-1}(a x)^2}{15 a \left (1-a^2 x^2\right )^{3/2}}-\frac{8 \tanh ^{-1}(a x)^2}{5 a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 x \tanh ^{-1}(a x)^3}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac{8 x \tanh ^{-1}(a x)^3}{15 \sqrt{1-a^2 x^2}}+\frac{16}{125} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac{16}{45} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac{16}{5} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{6}{625 a \left (1-a^2 x^2\right )^{5/2}}-\frac{272}{3375 a \left (1-a^2 x^2\right )^{3/2}}-\frac{4144}{1125 a \sqrt{1-a^2 x^2}}+\frac{6 x \tanh ^{-1}(a x)}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac{272 x \tanh ^{-1}(a x)}{1125 \left (1-a^2 x^2\right )^{3/2}}+\frac{4144 x \tanh ^{-1}(a x)}{1125 \sqrt{1-a^2 x^2}}-\frac{3 \tanh ^{-1}(a x)^2}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac{4 \tanh ^{-1}(a x)^2}{15 a \left (1-a^2 x^2\right )^{3/2}}-\frac{8 \tanh ^{-1}(a x)^2}{5 a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 x \tanh ^{-1}(a x)^3}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac{8 x \tanh ^{-1}(a x)^3}{15 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.109655, size = 119, normalized size = 0.41 \[ \frac{-62160 a^4 x^4+125680 a^2 x^2+1125 a x \left (8 a^4 x^4-20 a^2 x^2+15\right ) \tanh ^{-1}(a x)^3+30 a x \left (2072 a^4 x^4-4280 a^2 x^2+2235\right ) \tanh ^{-1}(a x)-225 \left (120 a^4 x^4-260 a^2 x^2+149\right ) \tanh ^{-1}(a x)^2-63682}{16875 a \left (1-a^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.211, size = 153, normalized size = 0.5 \begin{align*} -{\frac{9000\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}{x}^{5}{a}^{5}+62160\,{\it Artanh} \left ( ax \right ){x}^{5}{a}^{5}-27000\,{a}^{4}{x}^{4} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}-22500\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}{x}^{3}{a}^{3}-62160\,{x}^{4}{a}^{4}-128400\,{a}^{3}{x}^{3}{\it Artanh} \left ( ax \right ) +58500\,{a}^{2}{x}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+16875\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}ax+125680\,{a}^{2}{x}^{2}+67050\,ax{\it Artanh} \left ( ax \right ) -33525\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}-63682}{16875\,a \left ({a}^{2}{x}^{2}-1 \right ) ^{3}}\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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71556, size = 433, normalized size = 1.5 \begin{align*} \frac{{\left (497280 \, a^{4} x^{4} - 1005440 \, a^{2} x^{2} - 1125 \,{\left (8 \, a^{5} x^{5} - 20 \, a^{3} x^{3} + 15 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{3} + 450 \,{\left (120 \, a^{4} x^{4} - 260 \, a^{2} x^{2} + 149\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - 120 \,{\left (2072 \, a^{5} x^{5} - 4280 \, a^{3} x^{3} + 2235 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) + 509456\right )} \sqrt{-a^{2} x^{2} + 1}}{135000 \,{\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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