Optimal. Leaf size=208 \[ \frac{4144 x}{3375 \sqrt{1-a^2 x^2}}+\frac{272 x}{3375 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)^2}{15 \sqrt{1-a^2 x^2}}+\frac{4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac{16 \tanh ^{-1}(a x)}{15 a \sqrt{1-a^2 x^2}}-\frac{8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.152059, antiderivative size = 208, 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, 191, 192} \[ \frac{4144 x}{3375 \sqrt{1-a^2 x^2}}+\frac{272 x}{3375 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)^2}{15 \sqrt{1-a^2 x^2}}+\frac{4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac{16 \tanh ^{-1}(a x)}{15 a \sqrt{1-a^2 x^2}}-\frac{8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 5964
Rule 5962
Rule 191
Rule 192
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{7/2}} \, dx &=-\frac{2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac{x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac{2}{25} \int \frac{1}{\left (1-a^2 x^2\right )^{7/2}} \, dx+\frac{4}{5} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx\\ &=\frac{2 x}{125 \left (1-a^2 x^2\right )^{5/2}}-\frac{2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac{8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac{8}{125} \int \frac{1}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac{8}{45} \int \frac{1}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac{8}{15} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 x}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac{272 x}{3375 \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac{8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac{16 \tanh ^{-1}(a x)}{15 a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac{8 x \tanh ^{-1}(a x)^2}{15 \sqrt{1-a^2 x^2}}+\frac{16}{375} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac{16}{135} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac{16}{15} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 x}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac{272 x}{3375 \left (1-a^2 x^2\right )^{3/2}}+\frac{4144 x}{3375 \sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac{8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac{16 \tanh ^{-1}(a x)}{15 a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac{4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac{8 x \tanh ^{-1}(a x)^2}{15 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0849286, size = 94, normalized size = 0.45 \[ \frac{4144 a^5 x^5-8560 a^3 x^3+225 a x \left (8 a^4 x^4-20 a^2 x^2+15\right ) \tanh ^{-1}(a x)^2-30 \left (120 a^4 x^4-260 a^2 x^2+149\right ) \tanh ^{-1}(a x)+4470 a x}{3375 a \left (1-a^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.178, size = 118, normalized size = 0.6 \begin{align*} -{\frac{1800\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{5}{a}^{5}+4144\,{x}^{5}{a}^{5}-3600\,{a}^{4}{x}^{4}{\it Artanh} \left ( ax \right ) -4500\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{3}{a}^{3}-8560\,{x}^{3}{a}^{3}+7800\,{a}^{2}{x}^{2}{\it Artanh} \left ( ax \right ) +3375\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}xa+4470\,ax-4470\,{\it Artanh} \left ( ax \right ) }{3375\,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 [B] time = 1.7301, size = 694, normalized size = 3.34 \begin{align*} \frac{1}{15} \,{\left (\frac{8 \, x}{\sqrt{-a^{2} x^{2} + 1}} + \frac{4 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{3 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}}}\right )} \operatorname{artanh}\left (a x\right )^{2} + \frac{1}{3375} \, a{\left (\frac{9 \,{\left (\frac{8 \, x}{\sqrt{-a^{2} x^{2} + 1}} + \frac{4 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{3}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2} x +{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a}\right )}}{a} + \frac{9 \,{\left (\frac{8 \, x}{\sqrt{-a^{2} x^{2} + 1}} + \frac{4 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}} - \frac{3}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2} x -{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a}\right )}}{a} + \frac{100 \,{\left (\frac{2 \, x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{1}{\sqrt{-a^{2} x^{2} + 1} a^{2} x + \sqrt{-a^{2} x^{2} + 1} a}\right )}}{a} + \frac{100 \,{\left (\frac{2 \, x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{1}{\sqrt{-a^{2} x^{2} + 1} a^{2} x - \sqrt{-a^{2} x^{2} + 1} a}\right )}}{a} - \frac{1800 \, \sqrt{-a^{2} x^{2} + 1}}{{\left (a^{2} x + a\right )} a} - \frac{1800 \, \sqrt{-a^{2} x^{2} + 1}}{{\left (a^{2} x - a\right )} a} - \frac{1800 \, \log \left (a x + 1\right )}{\sqrt{-a^{2} x^{2} + 1} a^{2}} + \frac{1800 \, \log \left (-a x + 1\right )}{\sqrt{-a^{2} x^{2} + 1} a^{2}} - \frac{300 \, \log \left (a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2}} + \frac{300 \, \log \left (-a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2}} - \frac{135 \, \log \left (a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} a^{2}} + \frac{135 \, \log \left (-a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72755, size = 329, normalized size = 1.58 \begin{align*} -\frac{{\left (16576 \, a^{5} x^{5} - 34240 \, a^{3} x^{3} + 225 \,{\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 )^{2} + 17880 \, a x - 60 \,{\left (120 \, a^{4} x^{4} - 260 \, a^{2} x^{2} + 149\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )\right )} \sqrt{-a^{2} x^{2} + 1}}{13500 \,{\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 )^{2}}{{\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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