Optimal. Leaf size=158 \[ -\frac {\text {ArcSin}(a x)}{a}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {\text {ArcTan}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2}{a}-\frac {i \tanh ^{-1}(a x) \text {PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac {i \tanh ^{-1}(a x) \text {PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac {i \text {PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )}{a}-\frac {i \text {PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )}{a} \]
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Rubi [A]
time = 0.10, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6091, 6099,
4265, 2611, 2320, 6724, 222} \begin {gather*} \frac {1}{2} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a}-\frac {\text {ArcSin}(a x)}{a}+\frac {\tanh ^{-1}(a x)^2 \text {ArcTan}\left (e^{\tanh ^{-1}(a x)}\right )}{a}-\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac {i \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a}-\frac {i \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 2320
Rule 2611
Rule 4265
Rule 6091
Rule 6099
Rule 6724
Rubi steps
\begin {align*} \int \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2 \, dx &=\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {1}{2} \int \frac {\tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx-\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sin ^{-1}(a x)}{a}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {\text {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}\\ &=-\frac {\sin ^{-1}(a x)}{a}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {\tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2}{a}-\frac {i \text {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac {i \text {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {\sin ^{-1}(a x)}{a}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {\tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2}{a}-\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac {i \text {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}-\frac {i \text {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {\sin ^{-1}(a x)}{a}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {\tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2}{a}-\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac {i \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{a}-\frac {i \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{a}\\ &=-\frac {\sin ^{-1}(a x)}{a}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {\tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2}{a}-\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac {i \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a}-\frac {i \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.51, size = 187, normalized size = 1.18 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (2 \tanh ^{-1}(a x)+a x \tanh ^{-1}(a x)^2-\frac {i \left (-4 i \text {ArcTan}\left (\tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right )+\tanh ^{-1}(a x)^2 \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-\tanh ^{-1}(a x)^2 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )+2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-2 \tanh ^{-1}(a x) \text {PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )+2 \text {PolyLog}\left (3,-i e^{-\tanh ^{-1}(a x)}\right )-2 \text {PolyLog}\left (3,i e^{-\tanh ^{-1}(a x)}\right )\right )}{\sqrt {1-a^2 x^2}}\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 3.17, size = 0, normalized size = 0.00 \[\int \sqrt {-a^{2} x^{2}+1}\, \arctanh \left (a x \right )^{2}\, dx\]
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 \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 {\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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