Optimal. Leaf size=171 \[ \frac {2 a^2 x}{\sqrt {1-a^2 x^2}}+\frac {a^2 x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {2 a \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}+2 a \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-2 a \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-4 a \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \]
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Rubi [A] time = 0.31, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6030, 6008, 6018, 5962, 191} \[ 2 a \text {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-2 a \text {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\frac {2 a^2 x}{\sqrt {1-a^2 x^2}}+\frac {a^2 x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {2 a \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}-4 a \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 191
Rule 5962
Rule 6008
Rule 6018
Rule 6030
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx &=a^2 \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\int \frac {\tanh ^{-1}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 a \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}+\frac {a^2 x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}+(2 a) \int \frac {\tanh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \, dx+\left (2 a^2\right ) \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 a^2 x}{\sqrt {1-a^2 x^2}}-\frac {2 a \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}+\frac {a^2 x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}-4 a \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+2 a \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 a \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )\\ \end {align*}
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Mathematica [A] time = 1.15, size = 215, normalized size = 1.26 \[ \frac {a \left (4 \sqrt {1-a^2 x^2} \text {Li}_2\left (-e^{-\tanh ^{-1}(a x)}\right )-4 \sqrt {1-a^2 x^2} \text {Li}_2\left (e^{-\tanh ^{-1}(a x)}\right )+4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x) \log \left (1-e^{-\tanh ^{-1}(a x)}\right )-4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x) \log \left (e^{-\tanh ^{-1}(a x)}+1\right )-\frac {2 \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)^2 \sinh ^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )}{a x}+4 a x+2 a x \tanh ^{-1}(a x)^2-4 \tanh ^{-1}(a x)-\frac {1}{2} a x \tanh ^{-1}(a x)^2 \text {csch}^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right )}{2 \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )^{2}}{a^{4} x^{6} - 2 \, a^{2} x^{4} + x^{2}}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 207, normalized size = 1.21 \[ -\frac {a \left (\arctanh \left (a x \right )^{2}-2 \arctanh \left (a x \right )+2\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 \left (a x -1\right )}-\frac {\left (\arctanh \left (a x \right )^{2}+2 \arctanh \left (a x \right )+2\right ) a \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 \left (a x +1\right )}-\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \arctanh \left (a x \right )^{2}}{x}-2 a \arctanh \left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-2 a \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 a \arctanh \left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+2 a \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
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 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{x^2\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{x^{2} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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