Optimal. Leaf size=246 \[ \frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {6 i \tanh ^{-1}(a x) \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {6 i \tanh ^{-1}(a x) \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {6 i \text {Li}_4\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {6 i \text {Li}_4\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {2 \tanh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {x \tanh ^{-1}(a x)^3}{a^2 \sqrt {1-a^2 x^2}}+\frac {6 x \tanh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {6}{a^3 \sqrt {1-a^2 x^2}}-\frac {3 \tanh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.32, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6028, 5952, 4180, 2531, 6609, 2282, 6589, 5962, 5958} \[ \frac {3 i \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {3 i \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {6 i \tanh ^{-1}(a x) \text {PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {6 i \tanh ^{-1}(a x) \text {PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {6 i \text {PolyLog}\left (4,-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {6 i \text {PolyLog}\left (4,i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {6}{a^3 \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{a^2 \sqrt {1-a^2 x^2}}-\frac {3 \tanh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}+\frac {6 x \tanh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \tanh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )}{a^3} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4180
Rule 5952
Rule 5958
Rule 5962
Rule 6028
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {x^2 \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\frac {\int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a^2}-\frac {\int \frac {\tanh ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac {3 \tanh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{a^2 \sqrt {1-a^2 x^2}}-\frac {\operatorname {Subst}\left (\int x^3 \text {sech}(x) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}+\frac {6 \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a^2}\\ &=-\frac {6}{a^3 \sqrt {1-a^2 x^2}}+\frac {6 x \tanh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {3 \tanh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3}{a^3}+\frac {(3 i) \operatorname {Subst}\left (\int x^2 \log \left (1-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}-\frac {(3 i) \operatorname {Subst}\left (\int x^2 \log \left (1+i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {6}{a^3 \sqrt {1-a^2 x^2}}+\frac {6 x \tanh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {3 \tanh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3}{a^3}+\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {(6 i) \operatorname {Subst}\left (\int x \text {Li}_2\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}+\frac {(6 i) \operatorname {Subst}\left (\int x \text {Li}_2\left (i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {6}{a^3 \sqrt {1-a^2 x^2}}+\frac {6 x \tanh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {3 \tanh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3}{a^3}+\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {6 i \tanh ^{-1}(a x) \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {6 i \tanh ^{-1}(a x) \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {(6 i) \operatorname {Subst}\left (\int \text {Li}_3\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}-\frac {(6 i) \operatorname {Subst}\left (\int \text {Li}_3\left (i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {6}{a^3 \sqrt {1-a^2 x^2}}+\frac {6 x \tanh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {3 \tanh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3}{a^3}+\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {6 i \tanh ^{-1}(a x) \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {6 i \tanh ^{-1}(a x) \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {(6 i) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {(6 i) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{a^3}\\ &=-\frac {6}{a^3 \sqrt {1-a^2 x^2}}+\frac {6 x \tanh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {3 \tanh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3}{a^3}+\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {6 i \tanh ^{-1}(a x) \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {6 i \tanh ^{-1}(a x) \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {6 i \text {Li}_4\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {6 i \text {Li}_4\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}\\ \end {align*}
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Mathematica [B] time = 1.07, size = 541, normalized size = 2.20 \[ \frac {-\frac {384}{\sqrt {1-a^2 x^2}}+\frac {64 a x \tanh ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}}-\frac {192 \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}+\frac {384 a x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}+192 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )-192 \pi \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )+384 i \tanh ^{-1}(a x) \text {Li}_3\left (-i e^{-\tanh ^{-1}(a x)}\right )-384 i \tanh ^{-1}(a x) \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )-48 i \left (\pi -2 i \tanh ^{-1}(a x)\right )^2 \text {Li}_2\left (-i e^{-\tanh ^{-1}(a x)}\right )-48 i \pi ^2 \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )-192 \pi \text {Li}_3\left (-i e^{-\tanh ^{-1}(a x)}\right )+192 \pi \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )+384 i \text {Li}_4\left (-i e^{-\tanh ^{-1}(a x)}\right )+384 i \text {Li}_4\left (-i e^{\tanh ^{-1}(a x)}\right )-16 i \tanh ^{-1}(a x)^4+32 \pi \tanh ^{-1}(a x)^3+24 i \pi ^2 \tanh ^{-1}(a x)^2-8 \pi ^3 \tanh ^{-1}(a x)-64 i \tanh ^{-1}(a x)^3 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )+64 i \tanh ^{-1}(a x)^3 \log \left (1+i e^{\tanh ^{-1}(a x)}\right )+96 \pi \tanh ^{-1}(a x)^2 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )-96 \pi \tanh ^{-1}(a x)^2 \log \left (1-i e^{\tanh ^{-1}(a x)}\right )+48 i \pi ^2 \tanh ^{-1}(a x) \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )-48 i \pi ^2 \tanh ^{-1}(a x) \log \left (1-i e^{\tanh ^{-1}(a x)}\right )-8 \pi ^3 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )+8 \pi ^3 \log \left (1+i e^{\tanh ^{-1}(a x)}\right )-8 \pi ^3 \log \left (\tan \left (\frac {1}{4} \left (\pi +2 i \tanh ^{-1}(a x)\right )\right )\right )+7 i \pi ^4}{64 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2} \operatorname {artanh}\left (a x\right )^{3}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1}, x\right ) \]
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 \[ \int \frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \arctanh \left (a x \right )^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^2\,{\mathrm {atanh}\left (a\,x\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{2} \operatorname {atanh}^{3}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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