Optimal. Leaf size=202 \[ -\frac {8 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{315 a^5}+\frac {8 \tanh ^{-1}(a x)^2}{315 a^5}-\frac {29 \tanh ^{-1}(a x)}{3780 a^5}-\frac {16 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{315 a^5}+\frac {1}{9} a^4 x^9 \tanh ^{-1}(a x)^2+\frac {29 x}{3780 a^4}+\frac {1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac {8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac {a^2 x^7}{252}-\frac {2}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac {67 x^3}{11340 a^2}-\frac {11}{189} a x^6 \tanh ^{-1}(a x)+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2+\frac {4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac {23 x^5}{3780} \]
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Rubi [A] time = 1.02, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 59, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {6012, 5916, 5980, 302, 206, 321, 5984, 5918, 2402, 2315} \[ -\frac {8 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{315 a^5}+\frac {a^2 x^7}{252}-\frac {67 x^3}{11340 a^2}+\frac {1}{9} a^4 x^9 \tanh ^{-1}(a x)^2+\frac {1}{36} a^3 x^8 \tanh ^{-1}(a x)-\frac {2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac {8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac {29 x}{3780 a^4}+\frac {8 \tanh ^{-1}(a x)^2}{315 a^5}-\frac {29 \tanh ^{-1}(a x)}{3780 a^5}-\frac {16 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{315 a^5}-\frac {11}{189} a x^6 \tanh ^{-1}(a x)+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2+\frac {4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac {23 x^5}{3780} \]
Antiderivative was successfully verified.
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Rule 206
Rule 302
Rule 321
Rule 2315
Rule 2402
Rule 5916
Rule 5918
Rule 5980
Rule 5984
Rule 6012
Rubi steps
\begin {align*} \int x^4 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2 \, dx &=\int \left (x^4 \tanh ^{-1}(a x)^2-2 a^2 x^6 \tanh ^{-1}(a x)^2+a^4 x^8 \tanh ^{-1}(a x)^2\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int x^6 \tanh ^{-1}(a x)^2 \, dx\right )+a^4 \int x^8 \tanh ^{-1}(a x)^2 \, dx+\int x^4 \tanh ^{-1}(a x)^2 \, dx\\ &=\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac {1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac {1}{5} (2 a) \int \frac {x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac {1}{7} \left (4 a^3\right ) \int \frac {x^7 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx-\frac {1}{9} \left (2 a^5\right ) \int \frac {x^9 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac {1}{9} a^4 x^9 \tanh ^{-1}(a x)^2+\frac {2 \int x^3 \tanh ^{-1}(a x) \, dx}{5 a}-\frac {2 \int \frac {x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a}-\frac {1}{7} (4 a) \int x^5 \tanh ^{-1}(a x) \, dx+\frac {1}{7} (4 a) \int \frac {x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac {1}{9} \left (2 a^3\right ) \int x^7 \tanh ^{-1}(a x) \, dx-\frac {1}{9} \left (2 a^3\right ) \int \frac {x^7 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {x^4 \tanh ^{-1}(a x)}{10 a}-\frac {2}{21} a x^6 \tanh ^{-1}(a x)+\frac {1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac {1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac {1}{10} \int \frac {x^4}{1-a^2 x^2} \, dx+\frac {2 \int x \tanh ^{-1}(a x) \, dx}{5 a^3}-\frac {2 \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a^3}-\frac {4 \int x^3 \tanh ^{-1}(a x) \, dx}{7 a}+\frac {4 \int \frac {x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{7 a}+\frac {1}{9} (2 a) \int x^5 \tanh ^{-1}(a x) \, dx-\frac {1}{9} (2 a) \int \frac {x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac {1}{21} \left (2 a^2\right ) \int \frac {x^6}{1-a^2 x^2} \, dx-\frac {1}{36} a^4 \int \frac {x^8}{1-a^2 x^2} \, dx\\ &=\frac {x^2 \tanh ^{-1}(a x)}{5 a^3}-\frac {3 x^4 \tanh ^{-1}(a x)}{70 a}-\frac {11}{189} a x^6 \tanh ^{-1}(a x)+\frac {1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac {\tanh ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac {1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac {1}{10} \int \left (-\frac {1}{a^4}-\frac {x^2}{a^2}+\frac {1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx+\frac {1}{7} \int \frac {x^4}{1-a^2 x^2} \, dx-\frac {2 \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx}{5 a^4}-\frac {4 \int x \tanh ^{-1}(a x) \, dx}{7 a^3}+\frac {4 \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{7 a^3}-\frac {\int \frac {x^2}{1-a^2 x^2} \, dx}{5 a^2}+\frac {2 \int x^3 \tanh ^{-1}(a x) \, dx}{9 a}-\frac {2 \int \frac {x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{9 a}-\frac {1}{27} a^2 \int \frac {x^6}{1-a^2 x^2} \, dx+\frac {1}{21} \left (2 a^2\right ) \int \left (-\frac {1}{a^6}-\frac {x^2}{a^4}-\frac {x^4}{a^2}+\frac {1}{a^6 \left (1-a^2 x^2\right )}\right ) \, dx-\frac {1}{36} a^4 \int \left (-\frac {1}{a^8}-\frac {x^2}{a^6}-\frac {x^4}{a^4}-\frac {x^6}{a^2}+\frac {1}{a^8 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=\frac {293 x}{1260 a^4}+\frac {41 x^3}{3780 a^2}-\frac {17 x^5}{1260}+\frac {a^2 x^7}{252}-\frac {3 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac {4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac {11}{189} a x^6 \tanh ^{-1}(a x)+\frac {1}{36} a^3 x^8 \tanh ^{-1}(a x)-\frac {3 \tanh ^{-1}(a x)^2}{35 a^5}+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac {1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac {2 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{5 a^5}-\frac {1}{18} \int \frac {x^4}{1-a^2 x^2} \, dx+\frac {1}{7} \int \left (-\frac {1}{a^4}-\frac {x^2}{a^2}+\frac {1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{36 a^4}+\frac {2 \int \frac {1}{1-a^2 x^2} \, dx}{21 a^4}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{10 a^4}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{5 a^4}+\frac {2 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^4}+\frac {4 \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx}{7 a^4}+\frac {2 \int x \tanh ^{-1}(a x) \, dx}{9 a^3}-\frac {2 \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{9 a^3}+\frac {2 \int \frac {x^2}{1-a^2 x^2} \, dx}{7 a^2}-\frac {1}{27} a^2 \int \left (-\frac {1}{a^6}-\frac {x^2}{a^4}-\frac {x^4}{a^2}+\frac {1}{a^6 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=-\frac {601 x}{3780 a^4}-\frac {277 x^3}{11340 a^2}-\frac {23 x^5}{3780}+\frac {a^2 x^7}{252}-\frac {293 \tanh ^{-1}(a x)}{1260 a^5}+\frac {8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac {4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac {11}{189} a x^6 \tanh ^{-1}(a x)+\frac {1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac {8 \tanh ^{-1}(a x)^2}{315 a^5}+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac {1}{9} a^4 x^9 \tanh ^{-1}(a x)^2+\frac {6 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{35 a^5}-\frac {1}{18} \int \left (-\frac {1}{a^4}-\frac {x^2}{a^2}+\frac {1}{a^4 \left (1-a^2 x^2\right )}\right ) \, dx-\frac {2 \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{5 a^5}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{27 a^4}+\frac {\int \frac {1}{1-a^2 x^2} \, dx}{7 a^4}-\frac {2 \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx}{9 a^4}+\frac {2 \int \frac {1}{1-a^2 x^2} \, dx}{7 a^4}-\frac {4 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{7 a^4}-\frac {\int \frac {x^2}{1-a^2 x^2} \, dx}{9 a^2}\\ &=\frac {29 x}{3780 a^4}-\frac {67 x^3}{11340 a^2}-\frac {23 x^5}{3780}+\frac {a^2 x^7}{252}+\frac {601 \tanh ^{-1}(a x)}{3780 a^5}+\frac {8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac {4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac {11}{189} a x^6 \tanh ^{-1}(a x)+\frac {1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac {8 \tanh ^{-1}(a x)^2}{315 a^5}+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac {1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac {16 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{315 a^5}-\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{5 a^5}+\frac {4 \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{7 a^5}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{18 a^4}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{9 a^4}+\frac {2 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{9 a^4}\\ &=\frac {29 x}{3780 a^4}-\frac {67 x^3}{11340 a^2}-\frac {23 x^5}{3780}+\frac {a^2 x^7}{252}-\frac {29 \tanh ^{-1}(a x)}{3780 a^5}+\frac {8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac {4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac {11}{189} a x^6 \tanh ^{-1}(a x)+\frac {1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac {8 \tanh ^{-1}(a x)^2}{315 a^5}+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac {1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac {16 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{315 a^5}+\frac {3 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{35 a^5}-\frac {2 \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{9 a^5}\\ &=\frac {29 x}{3780 a^4}-\frac {67 x^3}{11340 a^2}-\frac {23 x^5}{3780}+\frac {a^2 x^7}{252}-\frac {29 \tanh ^{-1}(a x)}{3780 a^5}+\frac {8 x^2 \tanh ^{-1}(a x)}{315 a^3}+\frac {4 x^4 \tanh ^{-1}(a x)}{315 a}-\frac {11}{189} a x^6 \tanh ^{-1}(a x)+\frac {1}{36} a^3 x^8 \tanh ^{-1}(a x)+\frac {8 \tanh ^{-1}(a x)^2}{315 a^5}+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {2}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac {1}{9} a^4 x^9 \tanh ^{-1}(a x)^2-\frac {16 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{315 a^5}-\frac {8 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{315 a^5}\\ \end {align*}
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Mathematica [A] time = 1.99, size = 138, normalized size = 0.68 \[ \frac {36 \left (35 a^9 x^9-90 a^7 x^7+63 a^5 x^5-8\right ) \tanh ^{-1}(a x)^2+a x \left (45 a^6 x^6-69 a^4 x^4-67 a^2 x^2+87\right )+3 \tanh ^{-1}(a x) \left (105 a^8 x^8-220 a^6 x^6+48 a^4 x^4+96 a^2 x^2-192 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )-29\right )+288 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )}{11340 a^5} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} x^{8} - 2 \, a^{2} x^{6} + x^{4}\right )} \operatorname {artanh}\left (a x\right )^{2}, x\right ) \]
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 \[ \int {\left (a^{2} x^{2} - 1\right )}^{2} x^{4} \operatorname {artanh}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 259, normalized size = 1.28 \[ \frac {a^{4} x^{9} \arctanh \left (a x \right )^{2}}{9}-\frac {2 a^{2} x^{7} \arctanh \left (a x \right )^{2}}{7}+\frac {x^{5} \arctanh \left (a x \right )^{2}}{5}+\frac {a^{3} x^{8} \arctanh \left (a x \right )}{36}-\frac {11 a \,x^{6} \arctanh \left (a x \right )}{189}+\frac {4 x^{4} \arctanh \left (a x \right )}{315 a}+\frac {8 x^{2} \arctanh \left (a x \right )}{315 a^{3}}+\frac {8 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{315 a^{5}}+\frac {8 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{315 a^{5}}+\frac {a^{2} x^{7}}{252}-\frac {23 x^{5}}{3780}-\frac {67 x^{3}}{11340 a^{2}}+\frac {29 x}{3780 a^{4}}+\frac {29 \ln \left (a x -1\right )}{7560 a^{5}}-\frac {29 \ln \left (a x +1\right )}{7560 a^{5}}+\frac {2 \ln \left (a x -1\right )^{2}}{315 a^{5}}-\frac {8 \dilog \left (\frac {1}{2}+\frac {a x}{2}\right )}{315 a^{5}}-\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{315 a^{5}}-\frac {2 \ln \left (a x +1\right )^{2}}{315 a^{5}}-\frac {4 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{315 a^{5}}+\frac {4 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{315 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 214, normalized size = 1.06 \[ \frac {1}{22680} \, a^{2} {\left (\frac {90 \, a^{7} x^{7} - 138 \, a^{5} x^{5} - 134 \, a^{3} x^{3} + 174 \, a x - 144 \, \log \left (a x + 1\right )^{2} + 288 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 144 \, \log \left (a x - 1\right )^{2} + 87 \, \log \left (a x - 1\right )}{a^{7}} - \frac {576 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{7}} - \frac {87 \, \log \left (a x + 1\right )}{a^{7}}\right )} + \frac {1}{3780} \, a {\left (\frac {105 \, a^{6} x^{8} - 220 \, a^{4} x^{6} + 48 \, a^{2} x^{4} + 96 \, x^{2}}{a^{4}} + \frac {96 \, \log \left (a x + 1\right )}{a^{6}} + \frac {96 \, \log \left (a x - 1\right )}{a^{6}}\right )} \operatorname {artanh}\left (a x\right ) + \frac {1}{315} \, {\left (35 \, a^{4} x^{9} - 90 \, a^{2} x^{7} + 63 \, x^{5}\right )} \operatorname {artanh}\left (a x\right )^{2} \]
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 \[ \int x^4\,{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^2 \,d x \]
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 \[ \int x^{4} \left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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