Optimal. Leaf size=175 \[ \frac {\sqrt {1-a^2 x^2}}{3 a^2}+\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{3 a}-\frac {2 \text {ArcTan}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{3 a^2}-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}{3 a^2}-\frac {i \text {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{3 a^2}+\frac {i \text {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{3 a^2} \]
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Rubi [A]
time = 0.09, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6141, 6089,
6097} \begin {gather*} -\frac {2 \text {ArcTan}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)}{3 a^2}-\frac {i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{3 a^2}+\frac {i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{3 a^2}+\frac {\sqrt {1-a^2 x^2}}{3 a^2}-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}{3 a^2}+\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 6089
Rule 6097
Rule 6141
Rubi steps
\begin {align*} \int x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2 \, dx &=-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}{3 a^2}+\frac {2 \int \sqrt {1-a^2 x^2} \tanh ^{-1}(a x) \, dx}{3 a}\\ &=\frac {\sqrt {1-a^2 x^2}}{3 a^2}+\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{3 a}-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}{3 a^2}+\frac {\int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{3 a}\\ &=\frac {\sqrt {1-a^2 x^2}}{3 a^2}+\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{3 a}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{3 a^2}-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}{3 a^2}-\frac {i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{3 a^2}+\frac {i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{3 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 135, normalized size = 0.77 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (1+a x \tanh ^{-1}(a x)-\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2-\frac {i \left (\tanh ^{-1}(a x) \left (\log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-\log \left (1+i e^{-\tanh ^{-1}(a x)}\right )\right )+\text {PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-\text {PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )\right )}{\sqrt {1-a^2 x^2}}\right )}{3 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.70, size = 175, normalized size = 1.00
method | result | size |
default | \(\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (a^{2} x^{2} \arctanh \left (a x \right )^{2}+a x \arctanh \left (a x \right )-\arctanh \left (a x \right )^{2}+1\right )}{3 a^{2}}-\frac {i \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \arctanh \left (a x \right )}{3 a^{2}}+\frac {i \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \arctanh \left (a x \right )}{3 a^{2}}-\frac {i \dilog \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{3 a^{2}}+\frac {i \dilog \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{3 a^{2}}\) | \(175\) |
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 x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}^{2}{\left (a x \right )}\, 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 x\,{\mathrm {atanh}\left (a\,x\right )}^2\,\sqrt {1-a^2\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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