Optimal. Leaf size=248 \[ \frac{4 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{5 a}-\frac{8 \tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{5 a}-\frac{1-a^2 x^2}{20 a}-\frac{\log \left (1-a^2 x^2\right )}{2 a}+\frac{1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac{4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{5 a}-\frac{1}{10} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{8}{15} x \tanh ^{-1}(a x)^3+\frac{8 \tanh ^{-1}(a x)^3}{15 a}-x \tanh ^{-1}(a x)-\frac{8 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{5 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.251565, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {5944, 5910, 5984, 5918, 5948, 6058, 6610, 260, 5942} \[ \frac{4 \text{PolyLog}\left (3,1-\frac{2}{1-a x}\right )}{5 a}-\frac{8 \tanh ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1-a x}\right )}{5 a}-\frac{1-a^2 x^2}{20 a}-\frac{\log \left (1-a^2 x^2\right )}{2 a}+\frac{1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac{4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{5 a}-\frac{1}{10} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{8}{15} x \tanh ^{-1}(a x)^3+\frac{8 \tanh ^{-1}(a x)^3}{15 a}-x \tanh ^{-1}(a x)-\frac{8 \log \left (\frac{2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{5 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5944
Rule 5910
Rule 5984
Rule 5918
Rule 5948
Rule 6058
Rule 6610
Rule 260
Rule 5942
Rubi steps
\begin{align*} \int \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3 \, dx &=\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac{1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3-\frac{3}{10} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x) \, dx+\frac{4}{5} \int \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3 \, dx\\ &=-\frac{1-a^2 x^2}{20 a}-\frac{1}{10} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{5 a}+\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac{4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac{1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3-\frac{1}{5} \int \tanh ^{-1}(a x) \, dx+\frac{8}{15} \int \tanh ^{-1}(a x)^3 \, dx-\frac{4}{5} \int \tanh ^{-1}(a x) \, dx\\ &=-\frac{1-a^2 x^2}{20 a}-x \tanh ^{-1}(a x)-\frac{1}{10} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{5 a}+\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac{8}{15} x \tanh ^{-1}(a x)^3+\frac{4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac{1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3+\frac{1}{5} a \int \frac{x}{1-a^2 x^2} \, dx+\frac{1}{5} (4 a) \int \frac{x}{1-a^2 x^2} \, dx-\frac{1}{5} (8 a) \int \frac{x \tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=-\frac{1-a^2 x^2}{20 a}-x \tanh ^{-1}(a x)-\frac{1}{10} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{5 a}+\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac{8 \tanh ^{-1}(a x)^3}{15 a}+\frac{8}{15} x \tanh ^{-1}(a x)^3+\frac{4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac{1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3-\frac{\log \left (1-a^2 x^2\right )}{2 a}-\frac{8}{5} \int \frac{\tanh ^{-1}(a x)^2}{1-a x} \, dx\\ &=-\frac{1-a^2 x^2}{20 a}-x \tanh ^{-1}(a x)-\frac{1}{10} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{5 a}+\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac{8 \tanh ^{-1}(a x)^3}{15 a}+\frac{8}{15} x \tanh ^{-1}(a x)^3+\frac{4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac{1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3-\frac{8 \tanh ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{5 a}-\frac{\log \left (1-a^2 x^2\right )}{2 a}+\frac{16}{5} \int \frac{\tanh ^{-1}(a x) \log \left (\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{1-a^2 x^2}{20 a}-x \tanh ^{-1}(a x)-\frac{1}{10} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{5 a}+\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac{8 \tanh ^{-1}(a x)^3}{15 a}+\frac{8}{15} x \tanh ^{-1}(a x)^3+\frac{4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac{1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3-\frac{8 \tanh ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{5 a}-\frac{\log \left (1-a^2 x^2\right )}{2 a}-\frac{8 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{5 a}+\frac{8}{5} \int \frac{\text{Li}_2\left (1-\frac{2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac{1-a^2 x^2}{20 a}-x \tanh ^{-1}(a x)-\frac{1}{10} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)+\frac{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}{5 a}+\frac{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}{20 a}+\frac{8 \tanh ^{-1}(a x)^3}{15 a}+\frac{8}{15} x \tanh ^{-1}(a x)^3+\frac{4}{15} x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3+\frac{1}{5} x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3-\frac{8 \tanh ^{-1}(a x)^2 \log \left (\frac{2}{1-a x}\right )}{5 a}-\frac{\log \left (1-a^2 x^2\right )}{2 a}-\frac{8 \tanh ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1-a x}\right )}{5 a}+\frac{4 \text{Li}_3\left (1-\frac{2}{1-a x}\right )}{5 a}\\ \end{align*}
Mathematica [A] time = 0.602464, size = 183, normalized size = 0.74 \[ \frac{96 \tanh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+48 \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(a x)}\right )+3 a^2 x^2-30 \log \left (1-a^2 x^2\right )+12 a^5 x^5 \tanh ^{-1}(a x)^3+9 a^4 x^4 \tanh ^{-1}(a x)^2-40 a^3 x^3 \tanh ^{-1}(a x)^3+6 a^3 x^3 \tanh ^{-1}(a x)-42 a^2 x^2 \tanh ^{-1}(a x)^2+60 a x \tanh ^{-1}(a x)^3-66 a x \tanh ^{-1}(a x)-32 \tanh ^{-1}(a x)^3+33 \tanh ^{-1}(a x)^2-96 \tanh ^{-1}(a x)^2 \log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )-3}{60 a} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 1.285, size = 883, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname{artanh}\left (a x\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname{atanh}^{3}{\left (a x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname{artanh}\left (a x\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]