Optimal. Leaf size=179 \[ -\frac {1}{2} a \sqrt {1-a^2 x^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {1}{2} a^2 x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+3 a \text {ArcTan}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)-a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\frac {3}{2} i a \text {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {3}{2} i a \text {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right ) \]
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Rubi [A]
time = 0.21, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6161, 6155,
272, 65, 214, 6097, 6089} \begin {gather*} -\frac {1}{2} a \sqrt {1-a^2 x^2}-\frac {1}{2} a^2 x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}+3 a \text {ArcTan}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)+\frac {3}{2} i a \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )-\frac {3}{2} i a \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 6089
Rule 6097
Rule 6155
Rule 6161
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{x^2} \, dx &=-\left (a^2 \int \sqrt {1-a^2 x^2} \tanh ^{-1}(a x) \, dx\right )+\int \frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x^2} \, dx\\ &=-\frac {1}{2} a \sqrt {1-a^2 x^2}-\frac {1}{2} a^2 x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {1}{2} a^2 \int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx-a^2 \int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx+\int \frac {\tanh ^{-1}(a x)}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {1}{2} a \sqrt {1-a^2 x^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {1}{2} a^2 x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+3 a \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)+\frac {3}{2} i a \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {3}{2} i a \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )+a \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {1}{2} a \sqrt {1-a^2 x^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {1}{2} a^2 x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+3 a \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)+\frac {3}{2} i a \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {3}{2} i a \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )+\frac {1}{2} a \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {1}{2} a \sqrt {1-a^2 x^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {1}{2} a^2 x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+3 a \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)+\frac {3}{2} i a \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {3}{2} i a \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {\text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a}\\ &=-\frac {1}{2} a \sqrt {1-a^2 x^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {1}{2} a^2 x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+3 a \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)-a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\frac {3}{2} i a \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {3}{2} i a \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 168, normalized size = 0.94 \begin {gather*} \frac {1}{2} \left (-a \sqrt {1-a^2 x^2}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-a^2 x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+3 i a \tanh ^{-1}(a x) \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-3 i a \tanh ^{-1}(a x) \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )+2 a \log \left (\tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right )+3 i a \text {PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-3 i a \text {PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.43, size = 205, normalized size = 1.15
method | result | size |
default | \(-\frac {\left (a^{2} x^{2} \arctanh \left (a x \right )+a x +2 \arctanh \left (a x \right )\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 x}-a \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+a \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )+\frac {3 i a \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \arctanh \left (a x \right )}{2}-\frac {3 i a \dilog \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {3 i a \dilog \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {3 i a \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \arctanh \left (a x \right )}{2}\) | \(205\) |
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 {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}{\left (a x \right )}}{x^{2}}\, 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^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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