Optimal. Leaf size=336 \[ -\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {i \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}-\frac {i \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}-\frac {19 \sin ^{-1}(a x)}{360 a^5}+\frac {\tanh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a}+\frac {x^3 \sqrt {1-a^2 x^2}}{60 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{24 a^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{360 a^5}+\frac {x \sqrt {1-a^2 x^2}}{18 a^4}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{16 a^4}+\frac {11 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{180 a^3} \]
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Rubi [A] time = 1.40, 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 = {6014, 6016, 321, 216, 5994, 5952, 4180, 2531, 2282, 6589} \[ -\frac {i \tanh ^{-1}(a x) \text {PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {i \tanh ^{-1}(a x) \text {PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {i \text {PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}-\frac {i \text {PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {x^3 \sqrt {1-a^2 x^2}}{60 a^2}+\frac {x \sqrt {1-a^2 x^2}}{18 a^4}+\frac {1}{6} x^5 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a}-\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{24 a^2}+\frac {11 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{180 a^3}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{16 a^4}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{360 a^5}-\frac {19 \sin ^{-1}(a x)}{360 a^5}+\frac {\tanh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right )}{8 a^5} \]
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^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 \operatorname {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) \operatorname {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^5}+\frac {(3 i) \operatorname {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^5}-\frac {5 \operatorname {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) \operatorname {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^5}-\frac {(5 i) \operatorname {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^5}+\frac {(3 i) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^5}-\frac {(3 i) \operatorname {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) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^5}+\frac {(5 i) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^5}+\frac {(3 i) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{4 a^5}-\frac {(3 i) \operatorname {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) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{8 a^5}+\frac {(5 i) \operatorname {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 = 1.69, size = 268, normalized size = 0.80 \[ \frac {\sqrt {1-a^2 x^2} \left (-\frac {i \left (90 \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{-\tanh ^{-1}(a x)}\right )-90 \tanh ^{-1}(a x) \text {Li}_2\left (i e^{-\tanh ^{-1}(a x)}\right )+90 \text {Li}_3\left (-i e^{-\tanh ^{-1}(a x)}\right )-90 \text {Li}_3\left (i e^{-\tanh ^{-1}(a x)}\right )+45 \tanh ^{-1}(a x)^2 \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-45 \tanh ^{-1}(a x)^2 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )-76 i \tan ^{-1}\left (\tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right )\right )}{\sqrt {1-a^2 x^2}}+120 a x \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^2+48 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)+140 \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)+6 a x \left (a^2 x^2-1\right ) \left (35 \tanh ^{-1}(a x)^2+2\right )+90 \tanh ^{-1}(a x)+a x \left (45 \tanh ^{-1}(a x)^2+52\right )\right )}{720 a^5} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-a^{2} x^{2} + 1} x^{4} \operatorname {artanh}\left (a x\right )^{2}, x\right ) \]
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 \[ \int \sqrt {-a^{2} x^{2} + 1} x^{4} \operatorname {artanh}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, 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 \[ \int \sqrt {-a^{2} x^{2} + 1} x^{4} \operatorname {artanh}\left (a x\right )^{2}\,{d x} \]
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 \[ \int x^4\,{\mathrm {atanh}\left (a\,x\right )}^2\,\sqrt {1-a^2\,x^2} \,d x \]
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 \[ \int x^{4} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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