Optimal. Leaf size=89 \[ -\frac {2}{3 a \sqrt {1-a^2 x^2}}-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)}{3 \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5960, 5958} \[ -\frac {2}{3 a \sqrt {1-a^2 x^2}}-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)}{3 \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5958
Rule 5960
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{3 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)}{3 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 49, normalized size = 0.55 \[ -\frac {\left (6 a^3 x^3-9 a x\right ) \tanh ^{-1}(a x)-6 a^2 x^2+7}{9 a \left (1-a^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 73, normalized size = 0.82 \[ \frac {{\left (12 \, a^{2} x^{2} - 3 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 14\right )} \sqrt {-a^{2} x^{2} + 1}}{18 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.71, size = 90, normalized size = 1.01 \[ -\frac {{\left (2 \, a^{2} x^{2} - 3\right )} \sqrt {-a^{2} x^{2} + 1} x \log \left (-\frac {a x + 1}{a x - 1}\right )}{6 \, {\left (a^{2} x^{2} - 1\right )}^{2}} - \frac {6 \, a^{2} x^{2} - 7}{9 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-a^{2} x^{2} + 1} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 59, normalized size = 0.66 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \left (6 a^{3} x^{3} \arctanh \left (a x \right )-6 a^{2} x^{2}-9 a x \arctanh \left (a x \right )+7\right )}{9 a \left (a^{2} x^{2}-1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 74, normalized size = 0.83 \[ -\frac {1}{9} \, a {\left (\frac {6}{\sqrt {-a^{2} x^{2} + 1} a^{2}} + \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}}\right )} + \frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\right )} \operatorname {artanh}\left (a x\right ) \]
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 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )}{{\left (1-a^2\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\operatorname {atanh}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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