Optimal. Leaf size=336 \[ \frac {x \sqrt {1-a^2 x^2}}{18 a^4}+\frac {x^3 \sqrt {1-a^2 x^2}}{60 a^2}-\frac {19 \text {ArcSin}(a x)}{360 a^5}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{360 a^5}+\frac {11 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{180 a^3}+\frac {x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{16 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{24 a^2}+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {\text {ArcTan}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2}{8 a^5}-\frac {i \tanh ^{-1}(a x) \text {PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {i \tanh ^{-1}(a x) \text {PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {i \text {PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}-\frac {i \text {PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )}{8 a^5} \]
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Rubi [A]
time = 0.97, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps
used = 45, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6161, 6163,
327, 222, 6141, 6099, 4265, 2611, 2320, 6724} \begin {gather*} -\frac {19 \text {ArcSin}(a x)}{360 a^5}+\frac {\tanh ^{-1}(a x)^2 \text {ArcTan}\left (e^{\tanh ^{-1}(a x)}\right )}{8 a^5}-\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {i \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}-\frac {i \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a}+\frac {x^3 \sqrt {1-a^2 x^2}}{60 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{24 a^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{360 a^5}+\frac {x \sqrt {1-a^2 x^2}}{18 a^4}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{16 a^4}+\frac {11 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{180 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 327
Rule 2320
Rule 2611
Rule 4265
Rule 6099
Rule 6141
Rule 6161
Rule 6163
Rule 6724
Rubi steps
\begin {align*} \int x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2 \, dx &=-\left (a^2 \int \frac {x^6 \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\right )+\int \frac {x^4 \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{4 a^2}+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2-\frac {5}{6} \int \frac {x^4 \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx+\frac {3 \int \frac {x^2 \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{4 a^2}+\frac {\int \frac {x^3 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{2 a}-\frac {1}{3} a \int \frac {x^5 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 a^3}+\frac {x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a}-\frac {3 x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{24 a^2}+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2-\frac {1}{15} \int \frac {x^4}{\sqrt {1-a^2 x^2}} \, dx+\frac {3 \int \frac {\tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{8 a^4}+\frac {\int \frac {x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{3 a^3}+\frac {3 \int \frac {x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{4 a^3}+\frac {\int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{6 a^2}-\frac {5 \int \frac {x^2 \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{8 a^2}-\frac {4 \int \frac {x^3 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{15 a}-\frac {5 \int \frac {x^3 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{12 a}\\ &=-\frac {x \sqrt {1-a^2 x^2}}{12 a^4}+\frac {x^3 \sqrt {1-a^2 x^2}}{60 a^2}-\frac {13 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{12 a^5}+\frac {11 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{180 a^3}+\frac {x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{16 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{24 a^2}+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {3 \text {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^5}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{12 a^4}-\frac {5 \int \frac {\tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{16 a^4}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{3 a^4}+\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{4 a^4}-\frac {8 \int \frac {x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{45 a^3}-\frac {5 \int \frac {x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{18 a^3}-\frac {5 \int \frac {x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{8 a^3}-\frac {\int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{20 a^2}-\frac {4 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{45 a^2}-\frac {5 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{36 a^2}\\ &=\frac {x \sqrt {1-a^2 x^2}}{18 a^4}+\frac {x^3 \sqrt {1-a^2 x^2}}{60 a^2}+\frac {7 \sin ^{-1}(a x)}{6 a^5}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{360 a^5}+\frac {11 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{180 a^3}+\frac {x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{16 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{24 a^2}+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {3 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2}{4 a^5}-\frac {(3 i) \text {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^5}+\frac {(3 i) \text {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^5}-\frac {5 \text {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\tanh ^{-1}(a x)\right )}{16 a^5}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{40 a^4}-\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{45 a^4}-\frac {5 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{72 a^4}-\frac {8 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{45 a^4}-\frac {5 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{18 a^4}-\frac {5 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{8 a^4}\\ &=\frac {x \sqrt {1-a^2 x^2}}{18 a^4}+\frac {x^3 \sqrt {1-a^2 x^2}}{60 a^2}-\frac {19 \sin ^{-1}(a x)}{360 a^5}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{360 a^5}+\frac {11 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{180 a^3}+\frac {x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{16 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{24 a^2}+\frac {1}{6} x^5 \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}{8 a^5}-\frac {3 i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{4 a^5}+\frac {3 i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{4 a^5}+\frac {(5 i) \text {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^5}-\frac {(5 i) \text {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^5}+\frac {(3 i) \text {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^5}-\frac {(3 i) \text {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^5}\\ &=\frac {x \sqrt {1-a^2 x^2}}{18 a^4}+\frac {x^3 \sqrt {1-a^2 x^2}}{60 a^2}-\frac {19 \sin ^{-1}(a x)}{360 a^5}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{360 a^5}+\frac {11 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{180 a^3}+\frac {x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{16 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{24 a^2}+\frac {1}{6} x^5 \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}{8 a^5}-\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}-\frac {(5 i) \text {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^5}+\frac {(5 i) \text {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^5}+\frac {(3 i) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{4 a^5}-\frac {(3 i) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{4 a^5}\\ &=\frac {x \sqrt {1-a^2 x^2}}{18 a^4}+\frac {x^3 \sqrt {1-a^2 x^2}}{60 a^2}-\frac {19 \sin ^{-1}(a x)}{360 a^5}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{360 a^5}+\frac {11 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{180 a^3}+\frac {x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{16 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{24 a^2}+\frac {1}{6} x^5 \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}{8 a^5}-\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {3 i \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{4 a^5}-\frac {3 i \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{4 a^5}-\frac {(5 i) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {(5 i) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{8 a^5}\\ &=\frac {x \sqrt {1-a^2 x^2}}{18 a^4}+\frac {x^3 \sqrt {1-a^2 x^2}}{60 a^2}-\frac {19 \sin ^{-1}(a x)}{360 a^5}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{360 a^5}+\frac {11 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{180 a^3}+\frac {x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{16 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{24 a^2}+\frac {1}{6} x^5 \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}{8 a^5}-\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {i \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}-\frac {i \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}\\ \end {align*}
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Mathematica [A]
time = 1.52, size = 268, normalized size = 0.80 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (90 \tanh ^{-1}(a x)+140 \left (-1+a^2 x^2\right ) \tanh ^{-1}(a x)+48 \left (-1+a^2 x^2\right )^2 \tanh ^{-1}(a x)+120 a x \left (-1+a^2 x^2\right )^2 \tanh ^{-1}(a x)^2+6 a x \left (-1+a^2 x^2\right ) \left (2+35 \tanh ^{-1}(a x)^2\right )+a x \left (52+45 \tanh ^{-1}(a x)^2\right )-\frac {i \left (-76 i \text {ArcTan}\left (\tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right )+45 \tanh ^{-1}(a x)^2 \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-45 \tanh ^{-1}(a x)^2 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )+90 \tanh ^{-1}(a x) \text {PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-90 \tanh ^{-1}(a x) \text {PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )+90 \text {PolyLog}\left (3,-i e^{-\tanh ^{-1}(a x)}\right )-90 \text {PolyLog}\left (3,i e^{-\tanh ^{-1}(a x)}\right )\right )}{\sqrt {1-a^2 x^2}}\right )}{720 a^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.10, size = 0, normalized size = 0.00 \[\int x^{4} \arctanh \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, dx\]
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 \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 x^{4} \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]
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 x^4\,{\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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