Optimal. Leaf size=305 \[ -\frac {3 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a^3}+\frac {3 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a^3}-\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{2 a^3}+\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{2 a^3}+\frac {3 i \tanh ^{-1}(a x) \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {3 i \tanh ^{-1}(a x) \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {3 i \text {Li}_4\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {3 i \text {Li}_4\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {\tanh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)}{a^3}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 a^2}-\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a^3} \]
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Rubi [A] time = 0.34, antiderivative size = 305, 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 = {6016, 5994, 5950, 5952, 4180, 2531, 6609, 2282, 6589} \[ -\frac {3 i \text {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a^3}+\frac {3 i \text {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a^3}-\frac {3 i \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )}{2 a^3}+\frac {3 i \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )}{2 a^3}+\frac {3 i \tanh ^{-1}(a x) \text {PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {3 i \tanh ^{-1}(a x) \text {PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {3 i \text {PolyLog}\left (4,-i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {3 i \text {PolyLog}\left (4,i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 a^2}-\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a^3}+\frac {\tanh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)}{a^3} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4180
Rule 5950
Rule 5952
Rule 5994
Rule 6016
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {x^2 \tanh ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx &=-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 a^2}+\frac {\int \frac {\tanh ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{2 a^2}+\frac {3 \int \frac {x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{2 a}\\ &=-\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a^3}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 a^2}+\frac {\operatorname {Subst}\left (\int x^3 \text {sech}(x) \, dx,x,\tanh ^{-1}(a x)\right )}{2 a^3}+\frac {3 \int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{a^3}-\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a^3}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 a^2}+\frac {\tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3}{a^3}-\frac {3 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a^3}+\frac {3 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {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 )}{2 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 )}{2 a^3}\\ &=-\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{a^3}-\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a^3}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 a^2}+\frac {\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 )}{2 a^3}+\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{2 a^3}-\frac {3 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a^3}+\frac {3 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a^3}+\frac {(3 i) \operatorname {Subst}\left (\int x \text {Li}_2\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}-\frac {(3 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 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{a^3}-\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a^3}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 a^2}+\frac {\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 )}{2 a^3}+\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{2 a^3}-\frac {3 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a^3}+\frac {3 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a^3}+\frac {3 i \tanh ^{-1}(a x) \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {3 i \tanh ^{-1}(a x) \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {(3 i) \operatorname {Subst}\left (\int \text {Li}_3\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}+\frac {(3 i) \operatorname {Subst}\left (\int \text {Li}_3\left (i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{a^3}-\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a^3}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 a^2}+\frac {\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 )}{2 a^3}+\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{2 a^3}-\frac {3 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a^3}+\frac {3 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a^3}+\frac {3 i \tanh ^{-1}(a x) \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {3 i \tanh ^{-1}(a x) \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {(3 i) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {(3 i) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{a^3}\\ &=-\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{a^3}-\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a^3}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3}{2 a^2}+\frac {\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 )}{2 a^3}+\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{2 a^3}-\frac {3 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a^3}+\frac {3 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a^3}+\frac {3 i \tanh ^{-1}(a x) \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {3 i \tanh ^{-1}(a x) \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {3 i \text {Li}_4\left (-i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {3 i \text {Li}_4\left (i e^{\tanh ^{-1}(a x)}\right )}{a^3}\\ \end {align*}
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Mathematica [A] time = 5.23, size = 570, normalized size = 1.87 \[ -\frac {i \left (-64 i a x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3-192 i \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+192 \tanh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )+192 i \pi \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )+384 \tanh ^{-1}(a x) \text {Li}_3\left (-i e^{-\tanh ^{-1}(a x)}\right )-384 \tanh ^{-1}(a x) \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )-48 \left (-4 i \pi \tanh ^{-1}(a x)-4 \left (\tanh ^{-1}(a x)^2+2\right )+\pi ^2\right ) \text {Li}_2\left (-i e^{-\tanh ^{-1}(a x)}\right )-384 \text {Li}_2\left (i e^{-\tanh ^{-1}(a x)}\right )-48 \pi ^2 \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )+192 i \pi \text {Li}_3\left (-i e^{-\tanh ^{-1}(a x)}\right )-192 i \pi \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )+384 \text {Li}_4\left (-i e^{-\tanh ^{-1}(a x)}\right )+384 \text {Li}_4\left (-i e^{\tanh ^{-1}(a x)}\right )-16 \tanh ^{-1}(a x)^4-32 i \pi \tanh ^{-1}(a x)^3+24 \pi ^2 \tanh ^{-1}(a x)^2+8 i \pi ^3 \tanh ^{-1}(a x)-64 \tanh ^{-1}(a x)^3 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )+64 \tanh ^{-1}(a x)^3 \log \left (1+i e^{\tanh ^{-1}(a x)}\right )-96 i \pi \tanh ^{-1}(a x)^2 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )+96 i \pi \tanh ^{-1}(a x)^2 \log \left (1-i e^{\tanh ^{-1}(a x)}\right )+384 \tanh ^{-1}(a x) \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )+48 \pi ^2 \tanh ^{-1}(a x) \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )-384 \tanh ^{-1}(a x) \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )-48 \pi ^2 \tanh ^{-1}(a x) \log \left (1-i e^{\tanh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )-8 i \pi ^3 \log \left (1+i e^{\tanh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (\tan \left (\frac {1}{4} \left (\pi +2 i \tanh ^{-1}(a x)\right )\right )\right )+7 \pi ^4\right )}{128 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} x^{2} \operatorname {artanh}\left (a x\right )^{3}}{a^{2} x^{2} - 1}, x\right ) \]
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 \[ \int \frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x^{2} \arctanh \left (a x \right )^{3}}{\sqrt {-a^{2} x^{2}+1}}\, dx \]
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 \[ \int \frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x^2\,{\mathrm {atanh}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{2} \operatorname {atanh}^{3}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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