Optimal. Leaf size=385 \[ -\frac{413312}{128625 a \sqrt{1-a^2 x^2}}-\frac{30256}{385875 a \left (1-a^2 x^2\right )^{3/2}}-\frac{2664}{214375 a \left (1-a^2 x^2\right )^{5/2}}-\frac{6}{2401 a \left (1-a^2 x^2\right )^{7/2}}+\frac{16 x \tanh ^{-1}(a x)^3}{35 \sqrt{1-a^2 x^2}}+\frac{8 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{6 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{x \tanh ^{-1}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac{48 \tanh ^{-1}(a x)^2}{35 a \sqrt{1-a^2 x^2}}-\frac{8 \tanh ^{-1}(a x)^2}{35 a \left (1-a^2 x^2\right )^{3/2}}-\frac{18 \tanh ^{-1}(a x)^2}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{3 \tanh ^{-1}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac{413312 x \tanh ^{-1}(a x)}{128625 \sqrt{1-a^2 x^2}}+\frac{30256 x \tanh ^{-1}(a x)}{128625 \left (1-a^2 x^2\right )^{3/2}}+\frac{2664 x \tanh ^{-1}(a x)}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac{6 x \tanh ^{-1}(a x)}{343 \left (1-a^2 x^2\right )^{7/2}} \]
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Rubi [A] time = 0.483868, antiderivative size = 385, 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, 5958, 5960} \[ -\frac{413312}{128625 a \sqrt{1-a^2 x^2}}-\frac{30256}{385875 a \left (1-a^2 x^2\right )^{3/2}}-\frac{2664}{214375 a \left (1-a^2 x^2\right )^{5/2}}-\frac{6}{2401 a \left (1-a^2 x^2\right )^{7/2}}+\frac{16 x \tanh ^{-1}(a x)^3}{35 \sqrt{1-a^2 x^2}}+\frac{8 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{6 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{x \tanh ^{-1}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}-\frac{48 \tanh ^{-1}(a x)^2}{35 a \sqrt{1-a^2 x^2}}-\frac{8 \tanh ^{-1}(a x)^2}{35 a \left (1-a^2 x^2\right )^{3/2}}-\frac{18 \tanh ^{-1}(a x)^2}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{3 \tanh ^{-1}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac{413312 x \tanh ^{-1}(a x)}{128625 \sqrt{1-a^2 x^2}}+\frac{30256 x \tanh ^{-1}(a x)}{128625 \left (1-a^2 x^2\right )^{3/2}}+\frac{2664 x \tanh ^{-1}(a x)}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac{6 x \tanh ^{-1}(a x)}{343 \left (1-a^2 x^2\right )^{7/2}} \]
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 )^{9/2}} \, dx &=-\frac{3 \tanh ^{-1}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac{x \tanh ^{-1}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{6}{49} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{9/2}} \, dx+\frac{6}{7} \int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{7/2}} \, dx\\ &=-\frac{6}{2401 a \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)}{343 \left (1-a^2 x^2\right )^{7/2}}-\frac{3 \tanh ^{-1}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac{18 \tanh ^{-1}(a x)^2}{175 a \left (1-a^2 x^2\right )^{5/2}}+\frac{x \tanh ^{-1}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{36}{343} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx+\frac{36}{175} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx+\frac{24}{35} \int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx\\ &=-\frac{6}{2401 a \left (1-a^2 x^2\right )^{7/2}}-\frac{2664}{214375 a \left (1-a^2 x^2\right )^{5/2}}+\frac{6 x \tanh ^{-1}(a x)}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac{2664 x \tanh ^{-1}(a x)}{42875 \left (1-a^2 x^2\right )^{5/2}}-\frac{3 \tanh ^{-1}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac{18 \tanh ^{-1}(a x)^2}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{8 \tanh ^{-1}(a x)^2}{35 a \left (1-a^2 x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{144 \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{1715}+\frac{144}{875} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac{16}{35} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac{16}{35} \int \frac{\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{6}{2401 a \left (1-a^2 x^2\right )^{7/2}}-\frac{2664}{214375 a \left (1-a^2 x^2\right )^{5/2}}-\frac{30256}{385875 a \left (1-a^2 x^2\right )^{3/2}}+\frac{6 x \tanh ^{-1}(a x)}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac{2664 x \tanh ^{-1}(a x)}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac{30256 x \tanh ^{-1}(a x)}{128625 \left (1-a^2 x^2\right )^{3/2}}-\frac{3 \tanh ^{-1}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac{18 \tanh ^{-1}(a x)^2}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{8 \tanh ^{-1}(a x)^2}{35 a \left (1-a^2 x^2\right )^{3/2}}-\frac{48 \tanh ^{-1}(a x)^2}{35 a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{16 x \tanh ^{-1}(a x)^3}{35 \sqrt{1-a^2 x^2}}+\frac{96 \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{1715}+\frac{96}{875} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac{32}{105} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac{96}{35} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{6}{2401 a \left (1-a^2 x^2\right )^{7/2}}-\frac{2664}{214375 a \left (1-a^2 x^2\right )^{5/2}}-\frac{30256}{385875 a \left (1-a^2 x^2\right )^{3/2}}-\frac{413312}{128625 a \sqrt{1-a^2 x^2}}+\frac{6 x \tanh ^{-1}(a x)}{343 \left (1-a^2 x^2\right )^{7/2}}+\frac{2664 x \tanh ^{-1}(a x)}{42875 \left (1-a^2 x^2\right )^{5/2}}+\frac{30256 x \tanh ^{-1}(a x)}{128625 \left (1-a^2 x^2\right )^{3/2}}+\frac{413312 x \tanh ^{-1}(a x)}{128625 \sqrt{1-a^2 x^2}}-\frac{3 \tanh ^{-1}(a x)^2}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac{18 \tanh ^{-1}(a x)^2}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{8 \tanh ^{-1}(a x)^2}{35 a \left (1-a^2 x^2\right )^{3/2}}-\frac{48 \tanh ^{-1}(a x)^2}{35 a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^3}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)^3}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{16 x \tanh ^{-1}(a x)^3}{35 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.132237, size = 151, normalized size = 0.39 \[ \frac{43397760 a^6 x^6-131252240 a^4 x^4+132479032 a^2 x^2-385875 a x \left (16 a^6 x^6-56 a^4 x^4+70 a^2 x^2-35\right ) \tanh ^{-1}(a x)^3-210 a x \left (206656 a^6 x^6-635096 a^4 x^4+654220 a^2 x^2-226905\right ) \tanh ^{-1}(a x)+11025 \left (1680 a^6 x^6-5320 a^4 x^4+5726 a^2 x^2-2161\right ) \tanh ^{-1}(a x)^2-44658302}{13505625 a \left (1-a^2 x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.202, size = 201, normalized size = 0.5 \begin{align*} -{\frac{6174000\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}{x}^{7}{a}^{7}+43397760\,{\it Artanh} \left ( ax \right ){x}^{7}{a}^{7}-18522000\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}{x}^{6}{a}^{6}-21609000\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}{x}^{5}{a}^{5}-43397760\,{x}^{6}{a}^{6}-133370160\,{\it Artanh} \left ( ax \right ){x}^{5}{a}^{5}+58653000\,{a}^{4}{x}^{4} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+27011250\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}{x}^{3}{a}^{3}+131252240\,{x}^{4}{a}^{4}+137386200\,{a}^{3}{x}^{3}{\it Artanh} \left ( ax \right ) -63129150\,{a}^{2}{x}^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}-13505625\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}ax-132479032\,{a}^{2}{x}^{2}-47650050\,ax{\it Artanh} \left ( ax \right ) +23825025\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+44658302}{13505625\,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 [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{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71167, size = 574, normalized size = 1.49 \begin{align*} \frac{{\left (347182080 \, a^{6} x^{6} - 1050017920 \, a^{4} x^{4} + 1059832256 \, a^{2} x^{2} - 385875 \,{\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 )^{3} + 22050 \,{\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 )^{2} - 840 \,{\left (206656 \, a^{7} x^{7} - 635096 \, a^{5} x^{5} + 654220 \, a^{3} x^{3} - 226905 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) - 357266416\right )} \sqrt{-a^{2} x^{2} + 1}}{108045000 \,{\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 )^{3}}{{\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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