Optimal. Leaf size=277 \[ \frac{413312 x}{385875 \sqrt{1-a^2 x^2}}+\frac{30256 x}{385875 \left (1-a^2 x^2\right )^{3/2}}+\frac{888 x}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac{2 x}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac{16 x \tanh ^{-1}(a x)^2}{35 \sqrt{1-a^2 x^2}}+\frac{8 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac{32 \tanh ^{-1}(a x)}{35 a \sqrt{1-a^2 x^2}}-\frac{16 \tanh ^{-1}(a x)}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac{12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}} \]
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Rubi [A] time = 0.228388, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {5964, 5962, 191, 192} \[ \frac{413312 x}{385875 \sqrt{1-a^2 x^2}}+\frac{30256 x}{385875 \left (1-a^2 x^2\right )^{3/2}}+\frac{888 x}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac{2 x}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac{16 x \tanh ^{-1}(a x)^2}{35 \sqrt{1-a^2 x^2}}+\frac{8 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac{32 \tanh ^{-1}(a x)}{35 a \sqrt{1-a^2 x^2}}-\frac{16 \tanh ^{-1}(a x)}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac{12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}} \]
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 )^{9/2}} \, dx &=-\frac{2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac{x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{2}{49} \int \frac{1}{\left (1-a^2 x^2\right )^{9/2}} \, dx+\frac{6}{7} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{7/2}} \, dx\\ &=\frac{2 x}{343 \left (1-a^2 x^2\right )^{7/2}}-\frac{2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac{12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}+\frac{x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{12}{343} \int \frac{1}{\left (1-a^2 x^2\right )^{7/2}} \, dx+\frac{12}{175} \int \frac{1}{\left (1-a^2 x^2\right )^{7/2}} \, dx+\frac{24}{35} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx\\ &=\frac{2 x}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac{888 x}{42875 \left (1-a^2 x^2\right )^{5/2}}-\frac{2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac{12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{16 \tanh ^{-1}(a x)}{105 a \left (1-a^2 x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{48 \int \frac{1}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{1715}+\frac{48}{875} \int \frac{1}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac{16}{105} \int \frac{1}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac{16}{35} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 x}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac{888 x}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac{30256 x}{385875 \left (1-a^2 x^2\right )^{3/2}}-\frac{2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac{12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{16 \tanh ^{-1}(a x)}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac{32 \tanh ^{-1}(a x)}{35 a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{16 x \tanh ^{-1}(a x)^2}{35 \sqrt{1-a^2 x^2}}+\frac{32 \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{1715}+\frac{32}{875} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac{32}{315} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac{32}{35} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 x}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac{888 x}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac{30256 x}{385875 \left (1-a^2 x^2\right )^{3/2}}+\frac{413312 x}{385875 \sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac{12 \tanh ^{-1}(a x)}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{16 \tanh ^{-1}(a x)}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac{32 \tanh ^{-1}(a x)}{35 a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)^2}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{16 x \tanh ^{-1}(a x)^2}{35 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.110049, size = 120, normalized size = 0.43 \[ \frac{2 a x \left (-206656 a^6 x^6+635096 a^4 x^4-654220 a^2 x^2+226905\right )-11025 a x \left (16 a^6 x^6-56 a^4 x^4+70 a^2 x^2-35\right ) \tanh ^{-1}(a x)^2+210 \left (1680 a^6 x^6-5320 a^4 x^4+5726 a^2 x^2-2161\right ) \tanh ^{-1}(a x)}{385875 a \left (1-a^2 x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.191, size = 152, normalized size = 0.6 \begin{align*} -{\frac{176400\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{7}{a}^{7}+413312\,{a}^{7}{x}^{7}-352800\,{\it Artanh} \left ( ax \right ){x}^{6}{a}^{6}-617400\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{5}{a}^{5}-1270192\,{x}^{5}{a}^{5}+1117200\,{a}^{4}{x}^{4}{\it Artanh} \left ( ax \right ) +771750\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{3}{a}^{3}+1308440\,{x}^{3}{a}^{3}-1202460\,{a}^{2}{x}^{2}{\it Artanh} \left ( ax \right ) -385875\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}xa-453810\,ax+453810\,{\it Artanh} \left ( ax \right ) }{385875\,a \left ({a}^{2}{x}^{2}-1 \right ) ^{4}}\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 = 1.79396, size = 1014, normalized size = 3.66 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70749, size = 428, normalized size = 1.55 \begin{align*} -\frac{{\left (1653248 \, a^{7} x^{7} - 5080768 \, a^{5} x^{5} + 5233760 \, a^{3} x^{3} + 11025 \,{\left (16 \, a^{7} x^{7} - 56 \, a^{5} x^{5} + 70 \, a^{3} x^{3} - 35 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - 1815240 \, a x - 420 \,{\left (1680 \, a^{6} x^{6} - 5320 \, a^{4} x^{4} + 5726 \, a^{2} x^{2} - 2161\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )\right )} \sqrt{-a^{2} x^{2} + 1}}{1543500 \,{\left (a^{9} x^{8} - 4 \, a^{7} x^{6} + 6 \, a^{5} x^{4} - 4 \, 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{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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