Optimal. Leaf size=63 \[ \frac {2 x}{\sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {2 \tanh ^{-1}(a x)}{a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5962, 191} \[ \frac {2 x}{\sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {2 \tanh ^{-1}(a x)}{a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 5962
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=-\frac {2 \tanh ^{-1}(a x)}{a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}+2 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 x}{\sqrt {1-a^2 x^2}}-\frac {2 \tanh ^{-1}(a x)}{a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 38, normalized size = 0.60 \[ \frac {2 a x+a x \tanh ^{-1}(a x)^2-2 \tanh ^{-1}(a x)}{a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 69, normalized size = 1.10 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} {\left (a x \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 8 \, a x - 4 \, \log \left (-\frac {a x + 1}{a x - 1}\right )\right )}}{4 \, {\left (a^{3} x^{2} - a\right )}} \]
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 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 49, normalized size = 0.78 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \left (\arctanh \left (a x \right )^{2} a x +2 a x -2 \arctanh \left (a x \right )\right )}{a \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 57, normalized size = 0.90 \[ \frac {x \operatorname {artanh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {2 \, x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {2 \, \operatorname {artanh}\left (a x\right )}{\sqrt {-a^{2} x^{2} + 1} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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