Optimal. Leaf size=254 \[ -\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{4 a^3}+\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{4 a^3}+\frac {i \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{4 a^3}-\frac {i \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{4 a^3}-\frac {\sin ^{-1}(a x)}{6 a^3}+\frac {\tanh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )}{4 a^3}+\frac {x \sqrt {1-a^2 x^2}}{12 a^2}+\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 a}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{12 a^3} \]
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Rubi [A] time = 0.85, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6014, 6016, 5994, 216, 5952, 4180, 2531, 2282, 6589, 321} \[ -\frac {i \tanh ^{-1}(a x) \text {PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )}{4 a^3}+\frac {i \tanh ^{-1}(a x) \text {PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )}{4 a^3}+\frac {i \text {PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )}{4 a^3}-\frac {i \text {PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )}{4 a^3}+\frac {x \sqrt {1-a^2 x^2}}{12 a^2}+\frac {1}{4} x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 a}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{8 a^2}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{12 a^3}-\frac {\sin ^{-1}(a x)}{6 a^3}+\frac {\tanh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )}{4 a^3} \]
Antiderivative was successfully verified.
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Rule 216
Rule 321
Rule 2282
Rule 2531
Rule 4180
Rule 5952
Rule 5994
Rule 6014
Rule 6016
Rule 6589
Rubi steps
\begin {align*} \int x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2 \, dx &=-\left (a^2 \int \frac {x^4 \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\right )+\int \frac {x^2 \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a^2}+\frac {1}{4} x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2-\frac {3}{4} \int \frac {x^2 \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx+\frac {\int \frac {\tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{2 a^2}+\frac {\int \frac {x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{a}-\frac {1}{2} a \int \frac {x^3 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 a}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2-\frac {1}{6} \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx+\frac {\operatorname {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\tanh ^{-1}(a x)\right )}{2 a^3}-\frac {3 \int \frac {\tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{8 a^2}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^2}-\frac {\int \frac {x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{3 a}-\frac {3 \int \frac {x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{4 a}\\ &=\frac {x \sqrt {1-a^2 x^2}}{12 a^2}+\frac {\sin ^{-1}(a x)}{a^3}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{12 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 a}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} x^3 \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^3}-\frac {i \operatorname {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}+\frac {i \operatorname {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}-\frac {3 \operatorname {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^3}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{12 a^2}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{3 a^2}-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{4 a^2}\\ &=\frac {x \sqrt {1-a^2 x^2}}{12 a^2}-\frac {\sin ^{-1}(a x)}{6 a^3}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{12 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 a}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} x^3 \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}{4 a^3}-\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {(3 i) \operatorname {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^3}-\frac {(3 i) \operatorname {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^3}+\frac {i \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}-\frac {i \operatorname {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=\frac {x \sqrt {1-a^2 x^2}}{12 a^2}-\frac {\sin ^{-1}(a x)}{6 a^3}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{12 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 a}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} x^3 \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}{4 a^3}-\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{4 a^3}+\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{4 a^3}-\frac {(3 i) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^3}+\frac {(3 i) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^3}+\frac {i \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {i \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{a^3}\\ &=\frac {x \sqrt {1-a^2 x^2}}{12 a^2}-\frac {\sin ^{-1}(a x)}{6 a^3}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{12 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 a}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} x^3 \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}{4 a^3}-\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{4 a^3}+\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{4 a^3}+\frac {i \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {i \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {(3 i) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{4 a^3}+\frac {(3 i) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{4 a^3}\\ &=\frac {x \sqrt {1-a^2 x^2}}{12 a^2}-\frac {\sin ^{-1}(a x)}{6 a^3}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{12 a^3}+\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 a}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} x^3 \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}{4 a^3}-\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{4 a^3}+\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{4 a^3}+\frac {i \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{4 a^3}-\frac {i \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{4 a^3}\\ \end {align*}
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Mathematica [A] time = 1.23, size = 228, normalized size = 0.90 \[ \frac {\sqrt {1-a^2 x^2} \left (-\frac {i \left (6 \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{-\tanh ^{-1}(a x)}\right )-6 \tanh ^{-1}(a x) \text {Li}_2\left (i e^{-\tanh ^{-1}(a x)}\right )+6 \text {Li}_3\left (-i e^{-\tanh ^{-1}(a x)}\right )-6 \text {Li}_3\left (i e^{-\tanh ^{-1}(a x)}\right )+3 \tanh ^{-1}(a x)^2 \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-3 \tanh ^{-1}(a x)^2 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )-8 i \tan ^{-1}\left (\tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right )\right )}{\sqrt {1-a^2 x^2}}-6 a x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2-4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+6 \tanh ^{-1}(a x)+a x \left (3 \tanh ^{-1}(a x)^2+2\right )\right )}{24 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-a^{2} x^{2} + 1} x^{2} \operatorname {artanh}\left (a x\right )^{2}, x\right ) \]
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 \[ \int \sqrt {-a^{2} x^{2} + 1} x^{2} \operatorname {artanh}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.50, size = 0, normalized size = 0.00 \[ \int x^{2} \arctanh \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, dx \]
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 \[ \int \sqrt {-a^{2} x^{2} + 1} x^{2} \operatorname {artanh}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\mathrm {atanh}\left (a\,x\right )}^2\,\sqrt {1-a^2\,x^2} \,d x \]
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 \[ \int x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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