Optimal. Leaf size=179 \[ -\frac {a^3 x}{\sqrt {1-a^2 x^2}}-\frac {a \sqrt {1-a^2 x^2}}{2 x}+\frac {a^2 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{2 x^2}-3 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+\frac {3}{2} a^2 \text {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {3}{2} a^2 \text {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \]
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Rubi [A]
time = 0.27, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6177, 6173,
270, 6165, 6141, 197} \begin {gather*} \frac {3}{2} a^2 \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\frac {3}{2} a^2 \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\frac {a \sqrt {1-a^2 x^2}}{2 x}+\frac {a^2 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{2 x^2}-3 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\frac {a^3 x}{\sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 270
Rule 6141
Rule 6165
Rule 6173
Rule 6177
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx &=a^2 \int \frac {\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )^{3/2}} \, dx+\int \frac {\tanh ^{-1}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{2 x^2}+\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx+\frac {1}{2} a^2 \int \frac {\tanh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \, dx+a^2 \int \frac {\tanh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \, dx+a^4 \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2}}{2 x}+\frac {a^2 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{2 x^2}-3 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+\frac {3}{2} a^2 \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {3}{2} a^2 \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-a^3 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {a^3 x}{\sqrt {1-a^2 x^2}}-\frac {a \sqrt {1-a^2 x^2}}{2 x}+\frac {a^2 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{2 x^2}-3 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+\frac {3}{2} a^2 \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {3}{2} a^2 \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )\\ \end {align*}
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Mathematica [A]
time = 1.10, size = 182, normalized size = 1.02 \begin {gather*} \frac {1}{8} a^2 \left (-\frac {8 a x}{\sqrt {1-a^2 x^2}}+\frac {8 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}-\frac {a x \text {csch}^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}-\tanh ^{-1}(a x) \text {csch}^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )+12 \tanh ^{-1}(a x) \log \left (1-e^{-\tanh ^{-1}(a x)}\right )-12 \tanh ^{-1}(a x) \log \left (1+e^{-\tanh ^{-1}(a x)}\right )+12 \text {PolyLog}\left (2,-e^{-\tanh ^{-1}(a x)}\right )-12 \text {PolyLog}\left (2,e^{-\tanh ^{-1}(a x)}\right )-\tanh ^{-1}(a x) \text {sech}^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )+2 \tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.38, size = 205, normalized size = 1.15
method | result | size |
default | \(-\frac {a^{2} \left (\arctanh \left (a x \right )-1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 \left (a x -1\right )}+\frac {\left (\arctanh \left (a x \right )+1\right ) a^{2} \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 a x +2}-\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +\arctanh \left (a x \right )\right )}{2 x^{2}}+\frac {3 a^{2} \arctanh \left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {3 a^{2} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {3 a^{2} \arctanh \left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {3 a^{2} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\) | \(205\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {atanh}{\left (a x \right )}}{x^{3} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {atanh}\left (a\,x\right )}{x^3\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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