Optimal. Leaf size=68 \[ \frac {2}{a^2 \sqrt {1-a^2 x^2}}+\frac {\tanh ^{-1}(a x)^2}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 x \tanh ^{-1}(a x)}{a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.10, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5994, 5958} \[ \frac {2}{a^2 \sqrt {1-a^2 x^2}}+\frac {\tanh ^{-1}(a x)^2}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 x \tanh ^{-1}(a x)}{a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5958
Rule 5994
Rubi steps
\begin {align*} \int \frac {x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\frac {\tanh ^{-1}(a x)^2}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a}\\ &=\frac {2}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 x \tanh ^{-1}(a x)}{a \sqrt {1-a^2 x^2}}+\frac {\tanh ^{-1}(a x)^2}{a^2 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 34, normalized size = 0.50 \[ \frac {\tanh ^{-1}(a x)^2-2 a x \tanh ^{-1}(a x)+2}{a^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 69, normalized size = 1.01 \[ \frac {\sqrt {-a^{2} x^{2} + 1} {\left (4 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) - \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 8\right )}}{4 \, {\left (a^{4} x^{2} - a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \operatorname {artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 82, normalized size = 1.21 \[ -\frac {\left (\arctanh \left (a x \right )^{2}-2 \arctanh \left (a x \right )+2\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 a^{2} \left (a x -1\right )}+\frac {\left (\arctanh \left (a x \right )^{2}+2 \arctanh \left (a x \right )+2\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 a^{2} \left (a x +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 62, normalized size = 0.91 \[ -\frac {2 \, x \operatorname {artanh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} a} + \frac {\operatorname {artanh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2}} + \frac {2}{\sqrt {-a^{2} x^{2} + 1} a^{2}} \]
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 \[ \int \frac {x\,{\mathrm {atanh}\left (a\,x\right )}^2}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \operatorname {atanh}^{2}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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