Optimal. Leaf size=309 \[ -\frac {3 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{2 a^3 c}-\frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^3 c}+\frac {3 \text {Li}_4\left (1-\frac {2}{a x+1}\right )}{4 a^3 c}+\frac {3 \text {Li}_2\left (1-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2}{2 a^3 c}+\frac {3 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^3 c}+\frac {3 \text {Li}_3\left (1-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)}{2 a^3 c}-\frac {3 \tanh ^{-1}(a x)^3}{2 a^3 c}+\frac {3 \tanh ^{-1}(a x)^2}{2 a^3 c}-\frac {\log \left (\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^3}{a^3 c}+\frac {3 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^3 c}-\frac {3 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^3 c}-\frac {x \tanh ^{-1}(a x)^3}{a^2 c}+\frac {3 x \tanh ^{-1}(a x)^2}{2 a^2 c}+\frac {x^2 \tanh ^{-1}(a x)^3}{2 a c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.64, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {5930, 5916, 5980, 5910, 5984, 5918, 2402, 2315, 5948, 6058, 6610, 6056, 6060} \[ -\frac {3 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a^3 c}-\frac {3 \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^3 c}+\frac {3 \text {PolyLog}\left (4,1-\frac {2}{a x+1}\right )}{4 a^3 c}+\frac {3 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,1-\frac {2}{a x+1}\right )}{2 a^3 c}+\frac {3 \tanh ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \tanh ^{-1}(a x) \text {PolyLog}\left (3,1-\frac {2}{a x+1}\right )}{2 a^3 c}-\frac {x \tanh ^{-1}(a x)^3}{a^2 c}-\frac {3 \tanh ^{-1}(a x)^3}{2 a^3 c}+\frac {3 x \tanh ^{-1}(a x)^2}{2 a^2 c}+\frac {3 \tanh ^{-1}(a x)^2}{2 a^3 c}-\frac {\log \left (\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^3}{a^3 c}+\frac {3 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{a^3 c}-\frac {3 \log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{a^3 c}+\frac {x^2 \tanh ^{-1}(a x)^3}{2 a c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2315
Rule 2402
Rule 5910
Rule 5916
Rule 5918
Rule 5930
Rule 5948
Rule 5980
Rule 5984
Rule 6056
Rule 6058
Rule 6060
Rule 6610
Rubi steps
\begin {align*} \int \frac {x^2 \tanh ^{-1}(a x)^3}{c+a c x} \, dx &=-\frac {\int \frac {x \tanh ^{-1}(a x)^3}{c+a c x} \, dx}{a}+\frac {\int x \tanh ^{-1}(a x)^3 \, dx}{a c}\\ &=\frac {x^2 \tanh ^{-1}(a x)^3}{2 a c}+\frac {\int \frac {\tanh ^{-1}(a x)^3}{c+a c x} \, dx}{a^2}-\frac {3 \int \frac {x^2 \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{2 c}-\frac {\int \tanh ^{-1}(a x)^3 \, dx}{a^2 c}\\ &=-\frac {x \tanh ^{-1}(a x)^3}{a^2 c}+\frac {x^2 \tanh ^{-1}(a x)^3}{2 a c}-\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1+a x}\right )}{a^3 c}+\frac {3 \int \tanh ^{-1}(a x)^2 \, dx}{2 a^2 c}-\frac {3 \int \frac {\tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{2 a^2 c}+\frac {3 \int \frac {\tanh ^{-1}(a x)^2 \log \left (\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{a^2 c}+\frac {3 \int \frac {x \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{a c}\\ &=\frac {3 x \tanh ^{-1}(a x)^2}{2 a^2 c}-\frac {3 \tanh ^{-1}(a x)^3}{2 a^3 c}-\frac {x \tanh ^{-1}(a x)^3}{a^2 c}+\frac {x^2 \tanh ^{-1}(a x)^3}{2 a c}-\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1+a x}\right )}{a^3 c}+\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+a x}\right )}{2 a^3 c}+\frac {3 \int \frac {\tanh ^{-1}(a x)^2}{1-a x} \, dx}{a^2 c}-\frac {3 \int \frac {\tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{a^2 c}-\frac {3 \int \frac {x \tanh ^{-1}(a x)}{1-a^2 x^2} \, dx}{a c}\\ &=\frac {3 \tanh ^{-1}(a x)^2}{2 a^3 c}+\frac {3 x \tanh ^{-1}(a x)^2}{2 a^2 c}-\frac {3 \tanh ^{-1}(a x)^3}{2 a^3 c}-\frac {x \tanh ^{-1}(a x)^3}{a^2 c}+\frac {x^2 \tanh ^{-1}(a x)^3}{2 a c}+\frac {3 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3 c}-\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1+a x}\right )}{a^3 c}+\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+a x}\right )}{2 a^3 c}+\frac {3 \tanh ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+a x}\right )}{2 a^3 c}-\frac {3 \int \frac {\text {Li}_3\left (1-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx}{2 a^2 c}-\frac {3 \int \frac {\tanh ^{-1}(a x)}{1-a x} \, dx}{a^2 c}-\frac {6 \int \frac {\tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2 c}\\ &=\frac {3 \tanh ^{-1}(a x)^2}{2 a^3 c}+\frac {3 x \tanh ^{-1}(a x)^2}{2 a^2 c}-\frac {3 \tanh ^{-1}(a x)^3}{2 a^3 c}-\frac {x \tanh ^{-1}(a x)^3}{a^2 c}+\frac {x^2 \tanh ^{-1}(a x)^3}{2 a c}-\frac {3 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3 c}-\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1+a x}\right )}{a^3 c}+\frac {3 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+a x}\right )}{2 a^3 c}+\frac {3 \tanh ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+a x}\right )}{2 a^3 c}+\frac {3 \text {Li}_4\left (1-\frac {2}{1+a x}\right )}{4 a^3 c}+\frac {3 \int \frac {\log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2 c}-\frac {3 \int \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^2 c}\\ &=\frac {3 \tanh ^{-1}(a x)^2}{2 a^3 c}+\frac {3 x \tanh ^{-1}(a x)^2}{2 a^2 c}-\frac {3 \tanh ^{-1}(a x)^3}{2 a^3 c}-\frac {x \tanh ^{-1}(a x)^3}{a^2 c}+\frac {x^2 \tanh ^{-1}(a x)^3}{2 a c}-\frac {3 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3 c}-\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1+a x}\right )}{a^3 c}+\frac {3 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+a x}\right )}{2 a^3 c}-\frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^3 c}+\frac {3 \tanh ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+a x}\right )}{2 a^3 c}+\frac {3 \text {Li}_4\left (1-\frac {2}{1+a x}\right )}{4 a^3 c}-\frac {3 \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a x}\right )}{a^3 c}\\ &=\frac {3 \tanh ^{-1}(a x)^2}{2 a^3 c}+\frac {3 x \tanh ^{-1}(a x)^2}{2 a^2 c}-\frac {3 \tanh ^{-1}(a x)^3}{2 a^3 c}-\frac {x \tanh ^{-1}(a x)^3}{a^2 c}+\frac {x^2 \tanh ^{-1}(a x)^3}{2 a c}-\frac {3 \tanh ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a^3 c}-\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1+a x}\right )}{a^3 c}-\frac {3 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{2 a^3 c}+\frac {3 \tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a^3 c}+\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+a x}\right )}{2 a^3 c}-\frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^3 c}+\frac {3 \tanh ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+a x}\right )}{2 a^3 c}+\frac {3 \text {Li}_4\left (1-\frac {2}{1+a x}\right )}{4 a^3 c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.35, size = 172, normalized size = 0.56 \[ \frac {2 a^2 x^2 \tanh ^{-1}(a x)^3+6 \left (\tanh ^{-1}(a x)-1\right )^2 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(a x)}\right )+6 \left (\tanh ^{-1}(a x)-1\right ) \text {Li}_3\left (-e^{-2 \tanh ^{-1}(a x)}\right )+3 \text {Li}_4\left (-e^{-2 \tanh ^{-1}(a x)}\right )-4 a x \tanh ^{-1}(a x)^3+2 \tanh ^{-1}(a x)^3+6 a x \tanh ^{-1}(a x)^2-6 \tanh ^{-1}(a x)^2-4 \tanh ^{-1}(a x)^3 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )+12 \tanh ^{-1}(a x)^2 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )-12 \tanh ^{-1}(a x) \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )}{4 a^3 c} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{a c x + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \operatorname {artanh}\left (a x\right )^{3}}{a c x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 3.01, size = 400, normalized size = 1.29 \[ \frac {x^{2} \arctanh \left (a x \right )^{3}}{2 a c}-\frac {x \arctanh \left (a x \right )^{3}}{a^{2} c}+\frac {3 x \arctanh \left (a x \right )^{2}}{2 a^{2} c}-\frac {3 \arctanh \left (a x \right )^{3}}{2 a^{3} c}+\frac {3 \arctanh \left (a x \right )^{2}}{2 a^{3} c}+\frac {\arctanh \left (a x \right )^{4}}{2 a^{3} c}-\frac {\arctanh \left (a x \right )^{3} \ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3} c}-\frac {3 \arctanh \left (a x \right )^{2} \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{3} c}+\frac {3 \arctanh \left (a x \right ) \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{3} c}-\frac {3 \polylog \left (4, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{4 a^{3} c}-\frac {3 \arctanh \left (a x \right ) \ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3} c}-\frac {3 \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{3} c}+\frac {3 \arctanh \left (a x \right )^{2} \ln \left (1+\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3} c}+\frac {3 \arctanh \left (a x \right ) \polylog \left (2, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{a^{3} c}-\frac {3 \polylog \left (3, -\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}\right )}{2 a^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (a^{2} x^{2} - 2 \, a x + 2 \, \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{3}}{16 \, a^{3} c} + \frac {1}{8} \, \int \frac {2 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (a x + 1\right )^{3} - 6 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right ) + 3 \, {\left (a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 2 \, {\left (a^{3} x^{3} - a^{2} x^{2} + a x + 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{2}}{2 \, {\left (a^{4} c x^{2} - a^{2} c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,{\mathrm {atanh}\left (a\,x\right )}^3}{c+a\,c\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{2} \operatorname {atanh}^{3}{\left (a x \right )}}{a x + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________