Optimal. Leaf size=168 \[ -\frac {a \sqrt {1-a^2 x^2}}{2 x}+a^2 \text {ArcSin}(a x)-a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{2 x^2}+3 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {3}{2} a^2 \text {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+\frac {3}{2} a^2 \text {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \]
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Rubi [A]
time = 0.27, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6161, 6157,
6173, 270, 6165, 222} \begin {gather*} a^2 \text {ArcSin}(a x)-\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}-a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 270
Rule 6157
Rule 6161
Rule 6165
Rule 6173
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{x^3} \, dx &=-\left (a^2 \int \frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x} \, dx\right )+\int \frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x^3} \, dx\\ &=-a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x^2}+a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx-a^2 \int \frac {\tanh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \, dx+a^3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx-\int \frac {\tanh ^{-1}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2}}{x}+a^2 \sin ^{-1}(a x)-a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{2 x^2}+2 a^2 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-a^2 \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+a^2 \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\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\\ &=-\frac {a \sqrt {1-a^2 x^2}}{2 x}+a^2 \sin ^{-1}(a x)-a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\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 = 0.67, size = 158, normalized size = 0.94 \begin {gather*} \frac {1}{8} a^2 \left (16 \text {ArcTan}\left (\tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right )-8 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-2 \coth \left (\frac {1}{2} \tanh ^{-1}(a x)\right )-\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 = 1.47, size = 145, normalized size = 0.86
method | result | size |
default | \(-\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (2 a^{2} x^{2} \arctanh \left (a x \right )+a x +\arctanh \left (a x \right )\right )}{2 x^{2}}+2 a^{2} \arctan \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {3 a^{2} \dilog \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {3 a^{2} \arctanh \left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {3 a^{2} \dilog \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\) | \(145\) |
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 {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}{\left (a x \right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 )\,{\left (1-a^2\,x^2\right )}^{3/2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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