Optimal. Leaf size=305 \[ -\frac{3 a^2 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )}{2 c}+\frac{3 a^2 \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )}{2 c}-\frac{3 a^2 \text{PolyLog}\left (4,\frac{2}{a x+1}-1\right )}{4 c}-\frac{3 a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )}{2 c}+\frac{3 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )}{c}-\frac{3 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )}{2 c}-\frac{a^2 \tanh ^{-1}(a x)^3}{2 c}+\frac{3 a^2 \tanh ^{-1}(a x)^2}{2 c}+\frac{a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^3}{c}-\frac{3 a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^2}{c}+\frac{3 a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)}{c}-\frac{\tanh ^{-1}(a x)^3}{2 c x^2}+\frac{a \tanh ^{-1}(a x)^3}{c x}-\frac{3 a \tanh ^{-1}(a x)^2}{2 c x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.748083, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {5934, 5916, 5982, 5988, 5932, 2447, 5948, 6056, 6610, 6060} \[ -\frac{3 a^2 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )}{2 c}+\frac{3 a^2 \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )}{2 c}-\frac{3 a^2 \text{PolyLog}\left (4,\frac{2}{a x+1}-1\right )}{4 c}-\frac{3 a^2 \tanh ^{-1}(a x)^2 \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )}{2 c}+\frac{3 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )}{c}-\frac{3 a^2 \tanh ^{-1}(a x) \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )}{2 c}-\frac{a^2 \tanh ^{-1}(a x)^3}{2 c}+\frac{3 a^2 \tanh ^{-1}(a x)^2}{2 c}+\frac{a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^3}{c}-\frac{3 a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^2}{c}+\frac{3 a^2 \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)}{c}-\frac{\tanh ^{-1}(a x)^3}{2 c x^2}+\frac{a \tanh ^{-1}(a x)^3}{c x}-\frac{3 a \tanh ^{-1}(a x)^2}{2 c x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5934
Rule 5916
Rule 5982
Rule 5988
Rule 5932
Rule 2447
Rule 5948
Rule 6056
Rule 6610
Rule 6060
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)^3}{x^3 (c+a c x)} \, dx &=-\left (a \int \frac{\tanh ^{-1}(a x)^3}{x^2 (c+a c x)} \, dx\right )+\frac{\int \frac{\tanh ^{-1}(a x)^3}{x^3} \, dx}{c}\\ &=-\frac{\tanh ^{-1}(a x)^3}{2 c x^2}+a^2 \int \frac{\tanh ^{-1}(a x)^3}{x (c+a c x)} \, dx-\frac{a \int \frac{\tanh ^{-1}(a x)^3}{x^2} \, dx}{c}+\frac{(3 a) \int \frac{\tanh ^{-1}(a x)^2}{x^2 \left (1-a^2 x^2\right )} \, dx}{2 c}\\ &=-\frac{\tanh ^{-1}(a x)^3}{2 c x^2}+\frac{a \tanh ^{-1}(a x)^3}{c x}+\frac{a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )}{c}+\frac{(3 a) \int \frac{\tanh ^{-1}(a x)^2}{x^2} \, dx}{2 c}-\frac{\left (3 a^2\right ) \int \frac{\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx}{c}+\frac{\left (3 a^3\right ) \int \frac{\tanh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{2 c}-\frac{\left (3 a^3\right ) \int \frac{\tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=-\frac{3 a \tanh ^{-1}(a x)^2}{2 c x}-\frac{a^2 \tanh ^{-1}(a x)^3}{2 c}-\frac{\tanh ^{-1}(a x)^3}{2 c x^2}+\frac{a \tanh ^{-1}(a x)^3}{c x}+\frac{a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )}{c}-\frac{3 a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{2 c}+\frac{\left (3 a^2\right ) \int \frac{\tanh ^{-1}(a x)}{x \left (1-a^2 x^2\right )} \, dx}{c}-\frac{\left (3 a^2\right ) \int \frac{\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx}{c}+\frac{\left (3 a^3\right ) \int \frac{\tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac{3 a^2 \tanh ^{-1}(a x)^2}{2 c}-\frac{3 a \tanh ^{-1}(a x)^2}{2 c x}-\frac{a^2 \tanh ^{-1}(a x)^3}{2 c}-\frac{\tanh ^{-1}(a x)^3}{2 c x^2}+\frac{a \tanh ^{-1}(a x)^3}{c x}-\frac{3 a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )}{c}+\frac{a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )}{c}-\frac{3 a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{2 c}-\frac{3 a^2 \tanh ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+a x}\right )}{2 c}+\frac{\left (3 a^2\right ) \int \frac{\tanh ^{-1}(a x)}{x (1+a x)} \, dx}{c}+\frac{\left (3 a^3\right ) \int \frac{\text{Li}_3\left (-1+\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx}{2 c}+\frac{\left (6 a^3\right ) \int \frac{\tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac{3 a^2 \tanh ^{-1}(a x)^2}{2 c}-\frac{3 a \tanh ^{-1}(a x)^2}{2 c x}-\frac{a^2 \tanh ^{-1}(a x)^3}{2 c}-\frac{\tanh ^{-1}(a x)^3}{2 c x^2}+\frac{a \tanh ^{-1}(a x)^3}{c x}+\frac{3 a^2 \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )}{c}-\frac{3 a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )}{c}+\frac{a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )}{c}+\frac{3 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{c}-\frac{3 a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{2 c}-\frac{3 a^2 \tanh ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+a x}\right )}{2 c}-\frac{3 a^2 \text{Li}_4\left (-1+\frac{2}{1+a x}\right )}{4 c}-\frac{\left (3 a^3\right ) \int \frac{\log \left (2-\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}-\frac{\left (3 a^3\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{1-a^2 x^2} \, dx}{c}\\ &=\frac{3 a^2 \tanh ^{-1}(a x)^2}{2 c}-\frac{3 a \tanh ^{-1}(a x)^2}{2 c x}-\frac{a^2 \tanh ^{-1}(a x)^3}{2 c}-\frac{\tanh ^{-1}(a x)^3}{2 c x^2}+\frac{a \tanh ^{-1}(a x)^3}{c x}+\frac{3 a^2 \tanh ^{-1}(a x) \log \left (2-\frac{2}{1+a x}\right )}{c}-\frac{3 a^2 \tanh ^{-1}(a x)^2 \log \left (2-\frac{2}{1+a x}\right )}{c}+\frac{a^2 \tanh ^{-1}(a x)^3 \log \left (2-\frac{2}{1+a x}\right )}{c}-\frac{3 a^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{2 c}+\frac{3 a^2 \tanh ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{c}-\frac{3 a^2 \tanh ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+a x}\right )}{2 c}+\frac{3 a^2 \text{Li}_3\left (-1+\frac{2}{1+a x}\right )}{2 c}-\frac{3 a^2 \tanh ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+a x}\right )}{2 c}-\frac{3 a^2 \text{Li}_4\left (-1+\frac{2}{1+a x}\right )}{4 c}\\ \end{align*}
Mathematica [C] time = 0.629177, size = 222, normalized size = 0.73 \[ \frac{a^2 \left (96 \left (\tanh ^{-1}(a x)-2\right ) \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )-96 \tanh ^{-1}(a x) \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )-96 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(a x)}\right )+96 \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )+48 \text{PolyLog}\left (4,e^{2 \tanh ^{-1}(a x)}\right )-\frac{32 \tanh ^{-1}(a x)^3}{a^2 x^2}-32 \tanh ^{-1}(a x)^4+\frac{64 \tanh ^{-1}(a x)^3}{a x}+96 \tanh ^{-1}(a x)^3-\frac{96 \tanh ^{-1}(a x)^2}{a x}+96 \tanh ^{-1}(a x)^2+64 \tanh ^{-1}(a x)^3 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-192 \tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )+192 \tanh ^{-1}(a x) \log \left (1-e^{-2 \tanh ^{-1}(a x)}\right )+\pi ^4-8 i \pi ^3\right )}{64 c} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.984, size = 664, normalized size = 2.2 \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*} \frac{{\left (2 \, a^{2} x^{2} \log \left (a x + 1\right ) - 2 \, a x + 1\right )} \log \left (-a x + 1\right )^{3}}{16 \, c x^{2}} - \frac{1}{8} \, \int -\frac{2 \,{\left (a x - 1\right )} \log \left (a x + 1\right )^{3} - 6 \,{\left (a x - 1\right )} \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right ) + 3 \,{\left (2 \, a^{3} x^{3} + a^{2} x^{2} - a x - 2 \,{\left (a^{4} x^{4} + a^{3} x^{3} - a x + 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{2}}{2 \,{\left (a^{2} c x^{5} - c x^{3}\right )}}\,{d x} \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 (\frac{\operatorname{artanh}\left (a x\right )^{3}}{a c x^{4} + c x^{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*} \frac{\int \frac{\operatorname{atanh}^{3}{\left (a x \right )}}{a x^{4} + x^{3}}\, dx}{c} \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 \frac{\operatorname{artanh}\left (a x\right )^{3}}{{\left (a c x + c\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]