Optimal. Leaf size=94 \[ -\frac {\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}{5 x^5}-\frac {a \left (1-a^2 x^2\right )^{3/2}}{20 x^4}-\frac {3}{40} a^5 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\frac {3 a^3 \sqrt {1-a^2 x^2}}{40 x^2} \]
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Rubi [A] time = 0.10, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6008, 266, 47, 63, 208} \[ \frac {3 a^3 \sqrt {1-a^2 x^2}}{40 x^2}-\frac {a \left (1-a^2 x^2\right )^{3/2}}{20 x^4}-\frac {3}{40} a^5 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 6008
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{x^6} \, dx &=-\frac {\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}{5 x^5}+\frac {1}{5} a \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^5} \, dx\\ &=-\frac {\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}{5 x^5}+\frac {1}{10} a \operatorname {Subst}\left (\int \frac {\left (1-a^2 x\right )^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {a \left (1-a^2 x^2\right )^{3/2}}{20 x^4}-\frac {\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}{5 x^5}-\frac {1}{40} \left (3 a^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-a^2 x}}{x^2} \, dx,x,x^2\right )\\ &=\frac {3 a^3 \sqrt {1-a^2 x^2}}{40 x^2}-\frac {a \left (1-a^2 x^2\right )^{3/2}}{20 x^4}-\frac {\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}{5 x^5}+\frac {1}{80} \left (3 a^5\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac {3 a^3 \sqrt {1-a^2 x^2}}{40 x^2}-\frac {a \left (1-a^2 x^2\right )^{3/2}}{20 x^4}-\frac {\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}{5 x^5}-\frac {1}{40} \left (3 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )\\ &=\frac {3 a^3 \sqrt {1-a^2 x^2}}{40 x^2}-\frac {a \left (1-a^2 x^2\right )^{3/2}}{20 x^4}-\frac {\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}{5 x^5}-\frac {3}{40} a^5 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 104, normalized size = 1.11 \[ \frac {3}{40} a^5 \log (x)-\frac {\sqrt {1-a^2 x^2} \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)}{5 x^5}-\frac {3}{40} a^5 \log \left (\sqrt {1-a^2 x^2}+1\right )+\left (\frac {a^3}{8 x^2}-\frac {a}{20 x^4}\right ) \sqrt {1-a^2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 93, normalized size = 0.99 \[ \frac {3 \, a^{5} x^{5} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (5 \, a^{3} x^{3} - 2 \, a x - 4 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{40 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 116, normalized size = 1.23 \[ -\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (8 a^{4} x^{4} \arctanh \left (a x \right )-5 x^{3} a^{3}-16 a^{2} x^{2} \arctanh \left (a x \right )+2 a x +8 \arctanh \left (a x \right )\right )}{40 x^{5}}-\frac {3 a^{5} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{40}+\frac {3 a^{5} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )}{40} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 126, normalized size = 1.34 \[ \frac {1}{40} \, {\left ({\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{4} - 3 \, a^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + 3 \, \sqrt {-a^{2} x^{2} + 1} a^{4} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{2}}{x^{2}} - \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{x^{4}}\right )} a - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} \operatorname {artanh}\left (a x\right )}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )\,{\left (1-a^2\,x^2\right )}^{3/2}}{x^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}{\left (a x \right )}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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