Optimal. Leaf size=246 \[ -\frac {6}{a^3 \sqrt {1-a^2 x^2}}+\frac {6 x \tanh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {3 \tanh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^3}{a^2 \sqrt {1-a^2 x^2}}-\frac {2 \text {ArcTan}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3}{a^3}+\frac {3 i \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {3 i \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {6 i \tanh ^{-1}(a x) \text {PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {6 i \tanh ^{-1}(a x) \text {PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )}{a^3}+\frac {6 i \text {PolyLog}\left (4,-i e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {6 i \text {PolyLog}\left (4,i e^{\tanh ^{-1}(a x)}\right )}{a^3} \]
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Rubi [A]
time = 0.23, 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 = {6175, 6099,
4265, 2611, 6744, 2320, 6724, 6109, 6105} \begin {gather*} -\frac {2 \tanh ^{-1}(a x)^3 \text {ArcTan}\left (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 {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 {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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2611
Rule 4265
Rule 6099
Rule 6105
Rule 6109
Rule 6175
Rule 6724
Rule 6744
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 {\text {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) \text {Subst}\left (\int x^2 \log \left (1-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}-\frac {(3 i) \text {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) \text {Subst}\left (\int x \text {Li}_2\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}+\frac {(6 i) \text {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) \text {Subst}\left (\int \text {Li}_3\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}-\frac {(6 i) \text {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) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{a^3}-\frac {(6 i) \text {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] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(541\) vs. \(2(246)=492\).
time = 0.63, size = 541, normalized size = 2.20 \begin {gather*} \frac {7 i \pi ^4-\frac {384}{\sqrt {1-a^2 x^2}}-8 \pi ^3 \tanh ^{-1}(a x)+\frac {384 a x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}}+24 i \pi ^2 \tanh ^{-1}(a x)^2-\frac {192 \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}+32 \pi \tanh ^{-1}(a x)^3+\frac {64 a x \tanh ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}}-16 i \tanh ^{-1}(a x)^4-8 \pi ^3 \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 )+96 \pi \tanh ^{-1}(a x)^2 \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 )-48 i \pi ^2 \tanh ^{-1}(a x) \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 )+8 \pi ^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 )-8 \pi ^3 \log \left (\tan \left (\frac {1}{4} \left (\pi +2 i \tanh ^{-1}(a x)\right )\right )\right )-48 i \left (\pi -2 i \tanh ^{-1}(a x)\right )^2 \text {PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )+192 i \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )-48 i \pi ^2 \text {PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )-192 \pi \tanh ^{-1}(a x) \text {PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )-192 \pi \text {PolyLog}\left (3,-i e^{-\tanh ^{-1}(a x)}\right )+384 i \tanh ^{-1}(a x) \text {PolyLog}\left (3,-i e^{-\tanh ^{-1}(a x)}\right )-384 i \tanh ^{-1}(a x) \text {PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )+192 \pi \text {PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )+384 i \text {PolyLog}\left (4,-i e^{-\tanh ^{-1}(a x)}\right )+384 i \text {PolyLog}\left (4,-i e^{\tanh ^{-1}(a x)}\right )}{64 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, 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 \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 \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 \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {x^2\,{\mathrm {atanh}\left (a\,x\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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