Optimal. Leaf size=178 \[ \frac {1}{7} a^4 x^7 \tanh ^{-1}(a x)^2-\frac {8 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{105 a^3}+\frac {1}{21} a^3 x^6 \tanh ^{-1}(a x)+\frac {8 \tanh ^{-1}(a x)^2}{105 a^3}+\frac {\tanh ^{-1}(a x)}{210 a^3}-\frac {16 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{105 a^3}+\frac {a^2 x^5}{105}-\frac {2}{5} a^2 x^5 \tanh ^{-1}(a x)^2-\frac {x}{210 a^2}-\frac {9}{70} a x^4 \tanh ^{-1}(a x)+\frac {1}{3} x^3 \tanh ^{-1}(a x)^2+\frac {8 x^2 \tanh ^{-1}(a x)}{105 a}-\frac {17 x^3}{630} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.78, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 44, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {6012, 5916, 5980, 321, 206, 5984, 5918, 2402, 2315, 302} \[ -\frac {8 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{105 a^3}+\frac {a^2 x^5}{105}+\frac {1}{7} a^4 x^7 \tanh ^{-1}(a x)^2+\frac {1}{21} a^3 x^6 \tanh ^{-1}(a x)-\frac {2}{5} a^2 x^5 \tanh ^{-1}(a x)^2-\frac {x}{210 a^2}+\frac {8 \tanh ^{-1}(a x)^2}{105 a^3}+\frac {\tanh ^{-1}(a x)}{210 a^3}-\frac {16 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{105 a^3}-\frac {9}{70} a x^4 \tanh ^{-1}(a x)+\frac {1}{3} x^3 \tanh ^{-1}(a x)^2+\frac {8 x^2 \tanh ^{-1}(a x)}{105 a}-\frac {17 x^3}{630} \]
Antiderivative was successfully verified.
[In]
[Out]
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^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2 \, dx &=\int \left (x^2 \tanh ^{-1}(a x)^2-2 a^2 x^4 \tanh ^{-1}(a x)^2+a^4 x^6 \tanh ^{-1}(a x)^2\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int x^4 \tanh ^{-1}(a x)^2 \, dx\right )+a^4 \int x^6 \tanh ^{-1}(a x)^2 \, dx+\int x^2 \tanh ^{-1}(a x)^2 \, dx\\ &=\frac {1}{3} x^3 \tanh ^{-1}(a x)^2-\frac {2}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac {1}{7} a^4 x^7 \tanh ^{-1}(a x)^2-\frac {1}{3} (2 a) \int \frac {x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac {1}{5} \left (4 a^3\right ) \int \frac {x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx-\frac {1}{7} \left (2 a^5\right ) \int \frac {x^7 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {1}{3} x^3 \tanh ^{-1}(a x)^2-\frac {2}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac {1}{7} a^4 x^7 \tanh ^{-1}(a x)^2+\frac {2 \int x \tanh ^{-1}(a x) \, dx}{3 a}-\frac {2 \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a}-\frac {1}{5} (4 a) \int x^3 \tanh ^{-1}(a x) \, dx+\frac {1}{5} (4 a) \int \frac {x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac {1}{7} \left (2 a^3\right ) \int x^5 \tanh ^{-1}(a x) \, dx-\frac {1}{7} \left (2 a^3\right ) \int \frac {x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {x^2 \tanh ^{-1}(a x)}{3 a}-\frac {1}{5} a x^4 \tanh ^{-1}(a x)+\frac {1}{21} a^3 x^6 \tanh ^{-1}(a x)+\frac {\tanh ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \tanh ^{-1}(a x)^2-\frac {2}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac {1}{7} a^4 x^7 \tanh ^{-1}(a x)^2-\frac {1}{3} \int \frac {x^2}{1-a^2 x^2} \, dx-\frac {2 \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx}{3 a^2}-\frac {4 \int x \tanh ^{-1}(a x) \, dx}{5 a}+\frac {4 \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{5 a}+\frac {1}{7} (2 a) \int x^3 \tanh ^{-1}(a x) \, dx-\frac {1}{7} (2 a) \int \frac {x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx+\frac {1}{5} a^2 \int \frac {x^4}{1-a^2 x^2} \, dx-\frac {1}{21} a^4 \int \frac {x^6}{1-a^2 x^2} \, dx\\ &=\frac {x}{3 a^2}-\frac {x^2 \tanh ^{-1}(a x)}{15 a}-\frac {9}{70} a x^4 \tanh ^{-1}(a x)+\frac {1}{21} a^3 x^6 \tanh ^{-1}(a x)-\frac {\tanh ^{-1}(a x)^2}{15 a^3}+\frac {1}{3} x^3 \tanh ^{-1}(a x)^2-\frac {2}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac {1}{7} a^4 x^7 \tanh ^{-1}(a x)^2-\frac {2 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{3 a^3}+\frac {2}{5} \int \frac {x^2}{1-a^2 x^2} \, dx-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{3 a^2}+\frac {2 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{3 a^2}+\frac {4 \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx}{5 a^2}+\frac {2 \int x \tanh ^{-1}(a x) \, dx}{7 a}-\frac {2 \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{7 a}-\frac {1}{14} a^2 \int \frac {x^4}{1-a^2 x^2} \, dx+\frac {1}{5} a^2 \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}{21} a^4 \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 {23 x}{105 a^2}-\frac {16 x^3}{315}+\frac {a^2 x^5}{105}-\frac {\tanh ^{-1}(a x)}{3 a^3}+\frac {8 x^2 \tanh ^{-1}(a x)}{105 a}-\frac {9}{70} a x^4 \tanh ^{-1}(a x)+\frac {1}{21} a^3 x^6 \tanh ^{-1}(a x)+\frac {8 \tanh ^{-1}(a x)^2}{105 a^3}+\frac {1}{3} x^3 \tanh ^{-1}(a x)^2-\frac {2}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac {1}{7} a^4 x^7 \tanh ^{-1}(a x)^2+\frac {2 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{15 a^3}-\frac {1}{7} \int \frac {x^2}{1-a^2 x^2} \, dx-\frac {2 \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{3 a^3}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{21 a^2}+\frac {\int \frac {1}{1-a^2 x^2} \, dx}{5 a^2}-\frac {2 \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx}{7 a^2}+\frac {2 \int \frac {1}{1-a^2 x^2} \, dx}{5 a^2}-\frac {4 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^2}-\frac {1}{14} a^2 \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 {x}{210 a^2}-\frac {17 x^3}{630}+\frac {a^2 x^5}{105}+\frac {23 \tanh ^{-1}(a x)}{105 a^3}+\frac {8 x^2 \tanh ^{-1}(a x)}{105 a}-\frac {9}{70} a x^4 \tanh ^{-1}(a x)+\frac {1}{21} a^3 x^6 \tanh ^{-1}(a x)+\frac {8 \tanh ^{-1}(a x)^2}{105 a^3}+\frac {1}{3} x^3 \tanh ^{-1}(a x)^2-\frac {2}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac {1}{7} a^4 x^7 \tanh ^{-1}(a x)^2-\frac {16 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{105 a^3}-\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{3 a^3}+\frac {4 \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{5 a^3}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{14 a^2}-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{7 a^2}+\frac {2 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{7 a^2}\\ &=-\frac {x}{210 a^2}-\frac {17 x^3}{630}+\frac {a^2 x^5}{105}+\frac {\tanh ^{-1}(a x)}{210 a^3}+\frac {8 x^2 \tanh ^{-1}(a x)}{105 a}-\frac {9}{70} a x^4 \tanh ^{-1}(a x)+\frac {1}{21} a^3 x^6 \tanh ^{-1}(a x)+\frac {8 \tanh ^{-1}(a x)^2}{105 a^3}+\frac {1}{3} x^3 \tanh ^{-1}(a x)^2-\frac {2}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac {1}{7} a^4 x^7 \tanh ^{-1}(a x)^2-\frac {16 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{105 a^3}+\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{15 a^3}-\frac {2 \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{7 a^3}\\ &=-\frac {x}{210 a^2}-\frac {17 x^3}{630}+\frac {a^2 x^5}{105}+\frac {\tanh ^{-1}(a x)}{210 a^3}+\frac {8 x^2 \tanh ^{-1}(a x)}{105 a}-\frac {9}{70} a x^4 \tanh ^{-1}(a x)+\frac {1}{21} a^3 x^6 \tanh ^{-1}(a x)+\frac {8 \tanh ^{-1}(a x)^2}{105 a^3}+\frac {1}{3} x^3 \tanh ^{-1}(a x)^2-\frac {2}{5} a^2 x^5 \tanh ^{-1}(a x)^2+\frac {1}{7} a^4 x^7 \tanh ^{-1}(a x)^2-\frac {16 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{105 a^3}-\frac {8 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{105 a^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.26, size = 121, normalized size = 0.68 \[ \frac {a x \left (6 a^4 x^4-17 a^2 x^2-3\right )+6 \left (15 a^7 x^7-42 a^5 x^5+35 a^3 x^3-8\right ) \tanh ^{-1}(a x)^2+\tanh ^{-1}(a x) \left (30 a^6 x^6-81 a^4 x^4+48 a^2 x^2-96 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )+3\right )+48 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )}{630 a^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} x^{6} - 2 \, a^{2} x^{4} + x^{2}\right )} \operatorname {artanh}\left (a x\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a^{2} x^{2} - 1\right )}^{2} x^{2} \operatorname {artanh}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 239, normalized size = 1.34 \[ \frac {a^{4} x^{7} \arctanh \left (a x \right )^{2}}{7}-\frac {2 a^{2} x^{5} \arctanh \left (a x \right )^{2}}{5}+\frac {x^{3} \arctanh \left (a x \right )^{2}}{3}+\frac {a^{3} x^{6} \arctanh \left (a x \right )}{21}-\frac {9 a \,x^{4} \arctanh \left (a x \right )}{70}+\frac {8 x^{2} \arctanh \left (a x \right )}{105 a}+\frac {8 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{105 a^{3}}+\frac {8 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{105 a^{3}}+\frac {x^{5} a^{2}}{105}-\frac {17 x^{3}}{630}-\frac {x}{210 a^{2}}-\frac {\ln \left (a x -1\right )}{420 a^{3}}+\frac {\ln \left (a x +1\right )}{420 a^{3}}+\frac {2 \ln \left (a x -1\right )^{2}}{105 a^{3}}-\frac {8 \dilog \left (\frac {1}{2}+\frac {a x}{2}\right )}{105 a^{3}}-\frac {4 \ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{105 a^{3}}-\frac {2 \ln \left (a x +1\right )^{2}}{105 a^{3}}-\frac {4 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{105 a^{3}}+\frac {4 \ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{105 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 198, normalized size = 1.11 \[ \frac {1}{1260} \, a^{2} {\left (\frac {12 \, a^{5} x^{5} - 34 \, a^{3} x^{3} - 6 \, a x - 24 \, \log \left (a x + 1\right )^{2} + 48 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) + 24 \, \log \left (a x - 1\right )^{2} - 3 \, \log \left (a x - 1\right )}{a^{5}} - \frac {96 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{5}} + \frac {3 \, \log \left (a x + 1\right )}{a^{5}}\right )} + \frac {1}{210} \, a {\left (\frac {10 \, a^{4} x^{6} - 27 \, a^{2} x^{4} + 16 \, x^{2}}{a^{2}} + \frac {16 \, \log \left (a x + 1\right )}{a^{4}} + \frac {16 \, \log \left (a x - 1\right )}{a^{4}}\right )} \operatorname {artanh}\left (a x\right ) + \frac {1}{105} \, {\left (15 \, a^{4} x^{7} - 42 \, a^{2} x^{5} + 35 \, x^{3}\right )} \operatorname {artanh}\left (a x\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________