Optimal. Leaf size=162 \[ -\frac {2 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{35 a^5}+\frac {2 \tanh ^{-1}(a x)^2}{35 a^5}-\frac {4 \tanh ^{-1}(a x)}{105 a^5}-\frac {4 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{35 a^5}+\frac {4 x}{105 a^4}+\frac {2 x^2 \tanh ^{-1}(a x)}{35 a^3}-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac {2 x^3}{315 a^2}-\frac {1}{21} a x^6 \tanh ^{-1}(a x)+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2+\frac {x^4 \tanh ^{-1}(a x)}{35 a}-\frac {x^5}{105} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.58, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6014, 5916, 5980, 302, 206, 321, 5984, 5918, 2402, 2315} \[ -\frac {2 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{35 a^5}-\frac {2 x^3}{315 a^2}-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac {2 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac {4 x}{105 a^4}+\frac {2 \tanh ^{-1}(a x)^2}{35 a^5}-\frac {4 \tanh ^{-1}(a x)}{105 a^5}-\frac {4 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{35 a^5}-\frac {1}{21} a x^6 \tanh ^{-1}(a x)+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2+\frac {x^4 \tanh ^{-1}(a x)}{35 a}-\frac {x^5}{105} \]
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 6014
Rubi steps
\begin {align*} \int x^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2 \, dx &=-\left (a^2 \int x^6 \tanh ^{-1}(a x)^2 \, dx\right )+\int x^4 \tanh ^{-1}(a x)^2 \, dx\\ &=\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {1}{7} a^2 x^7 \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 (2 a^3\right ) \int \frac {x^7 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {1}{7} a^2 x^7 \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} (2 a) \int x^5 \tanh ^{-1}(a x) \, dx+\frac {1}{7} (2 a) \int \frac {x^5 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx\\ &=\frac {x^4 \tanh ^{-1}(a x)}{10 a}-\frac {1}{21} a x^6 \tanh ^{-1}(a x)+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {1}{7} a^2 x^7 \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 {2 \int x^3 \tanh ^{-1}(a x) \, dx}{7 a}+\frac {2 \int \frac {x^3 \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{7 a}+\frac {1}{21} a^2 \int \frac {x^6}{1-a^2 x^2} \, dx\\ &=\frac {x^2 \tanh ^{-1}(a x)}{5 a^3}+\frac {x^4 \tanh ^{-1}(a x)}{35 a}-\frac {1}{21} a x^6 \tanh ^{-1}(a x)+\frac {\tanh ^{-1}(a x)^2}{5 a^5}+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)^2+\frac {1}{14} \int \frac {x^4}{1-a^2 x^2} \, dx-\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 {2 \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx}{5 a^4}-\frac {2 \int x \tanh ^{-1}(a x) \, dx}{7 a^3}+\frac {2 \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 {1}{21} 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 {53 x}{210 a^4}+\frac {11 x^3}{630 a^2}-\frac {x^5}{105}+\frac {2 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac {x^4 \tanh ^{-1}(a x)}{35 a}-\frac {1}{21} a x^6 \tanh ^{-1}(a x)+\frac {2 \tanh ^{-1}(a x)^2}{35 a^5}+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac {2 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{5 a^5}+\frac {1}{14} \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}{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 {\tanh ^{-1}(a x)}{1-a x} \, dx}{7 a^4}+\frac {2 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{5 a^4}+\frac {\int \frac {x^2}{1-a^2 x^2} \, dx}{7 a^2}\\ &=\frac {4 x}{105 a^4}-\frac {2 x^3}{315 a^2}-\frac {x^5}{105}-\frac {53 \tanh ^{-1}(a x)}{210 a^5}+\frac {2 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac {x^4 \tanh ^{-1}(a x)}{35 a}-\frac {1}{21} a x^6 \tanh ^{-1}(a x)+\frac {2 \tanh ^{-1}(a x)^2}{35 a^5}+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac {4 \tanh ^{-1}(a x) \log \left (\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 )}{5 a^5}+\frac {\int \frac {1}{1-a^2 x^2} \, dx}{14 a^4}+\frac {\int \frac {1}{1-a^2 x^2} \, dx}{7 a^4}-\frac {2 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{7 a^4}\\ &=\frac {4 x}{105 a^4}-\frac {2 x^3}{315 a^2}-\frac {x^5}{105}-\frac {4 \tanh ^{-1}(a x)}{105 a^5}+\frac {2 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac {x^4 \tanh ^{-1}(a x)}{35 a}-\frac {1}{21} a x^6 \tanh ^{-1}(a x)+\frac {2 \tanh ^{-1}(a x)^2}{35 a^5}+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac {4 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{35 a^5}-\frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{5 a^5}+\frac {2 \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{7 a^5}\\ &=\frac {4 x}{105 a^4}-\frac {2 x^3}{315 a^2}-\frac {x^5}{105}-\frac {4 \tanh ^{-1}(a x)}{105 a^5}+\frac {2 x^2 \tanh ^{-1}(a x)}{35 a^3}+\frac {x^4 \tanh ^{-1}(a x)}{35 a}-\frac {1}{21} a x^6 \tanh ^{-1}(a x)+\frac {2 \tanh ^{-1}(a x)^2}{35 a^5}+\frac {1}{5} x^5 \tanh ^{-1}(a x)^2-\frac {1}{7} a^2 x^7 \tanh ^{-1}(a x)^2-\frac {4 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{35 a^5}-\frac {2 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{35 a^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.96, size = 113, normalized size = 0.70 \[ -\frac {3 a^5 x^5+2 a^3 x^3+9 \left (5 a^7 x^7-7 a^5 x^5+2\right ) \tanh ^{-1}(a x)^2+3 \tanh ^{-1}(a x) \left (5 a^6 x^6-3 a^4 x^4-6 a^2 x^2+12 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )+4\right )-18 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )-12 a x}{315 a^5} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} x^{6} - x^{4}\right )} \operatorname {artanh}\left (a x\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -{\left (a^{2} x^{2} - 1\right )} x^{4} \operatorname {artanh}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 225, normalized size = 1.39 \[ -\frac {a^{2} x^{7} \arctanh \left (a x \right )^{2}}{7}+\frac {x^{5} \arctanh \left (a x \right )^{2}}{5}-\frac {a \,x^{6} \arctanh \left (a x \right )}{21}+\frac {x^{4} \arctanh \left (a x \right )}{35 a}+\frac {2 x^{2} \arctanh \left (a x \right )}{35 a^{3}}+\frac {2 \arctanh \left (a x \right ) \ln \left (a x -1\right )}{35 a^{5}}+\frac {2 \arctanh \left (a x \right ) \ln \left (a x +1\right )}{35 a^{5}}+\frac {\ln \left (a x -1\right )^{2}}{70 a^{5}}-\frac {2 \dilog \left (\frac {1}{2}+\frac {a x}{2}\right )}{35 a^{5}}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{35 a^{5}}-\frac {\ln \left (a x +1\right )^{2}}{70 a^{5}}-\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {a x}{2}\right )}{35 a^{5}}+\frac {\ln \left (-\frac {a x}{2}+\frac {1}{2}\right ) \ln \left (a x +1\right )}{35 a^{5}}-\frac {x^{5}}{105}-\frac {2 x^{3}}{315 a^{2}}+\frac {4 x}{105 a^{4}}+\frac {2 \ln \left (a x -1\right )}{105 a^{5}}-\frac {2 \ln \left (a x +1\right )}{105 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 190, normalized size = 1.17 \[ -\frac {1}{630} \, a^{2} {\left (\frac {6 \, a^{5} x^{5} + 4 \, a^{3} x^{3} - 24 \, a x + 9 \, \log \left (a x + 1\right )^{2} - 18 \, \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 9 \, \log \left (a x - 1\right )^{2} - 12 \, \log \left (a x - 1\right )}{a^{7}} + \frac {36 \, {\left (\log \left (a x - 1\right ) \log \left (\frac {1}{2} \, a x + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, a x + \frac {1}{2}\right )\right )}}{a^{7}} + \frac {12 \, \log \left (a x + 1\right )}{a^{7}}\right )} - \frac {1}{105} \, a {\left (\frac {5 \, a^{4} x^{6} - 3 \, a^{2} x^{4} - 6 \, x^{2}}{a^{4}} - \frac {6 \, \log \left (a x + 1\right )}{a^{6}} - \frac {6 \, \log \left (a x - 1\right )}{a^{6}}\right )} \operatorname {artanh}\left (a x\right ) - \frac {1}{35} \, {\left (5 \, a^{2} x^{7} - 7 \, x^{5}\right )} \operatorname {artanh}\left (a x\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int x^4\,{\mathrm {atanh}\left (a\,x\right )}^2\,\left (a^2\,x^2-1\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- x^{4} \operatorname {atanh}^{2}{\left (a x \right )}\right )\, dx - \int a^{2} x^{6} \operatorname {atanh}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________