Optimal. Leaf size=110 \[ \frac{3}{2} b^2 \text{PolyLog}\left (3,e^{2 \text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )-\frac{3}{2} b \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )^2-\frac{3}{4} b^3 \text{PolyLog}\left (4,e^{2 \text{csch}^{-1}(c x)}\right )+\frac{\left (a+b \text{csch}^{-1}(c x)\right )^4}{4 b}-\log \left (1-e^{2 \text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )^3 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.153647, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6286, 3716, 2190, 2531, 6609, 2282, 6589} \[ \frac{3}{2} b^2 \text{PolyLog}\left (3,e^{2 \text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )-\frac{3}{2} b \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )^2-\frac{3}{4} b^3 \text{PolyLog}\left (4,e^{2 \text{csch}^{-1}(c x)}\right )+\frac{\left (a+b \text{csch}^{-1}(c x)\right )^4}{4 b}-\log \left (1-e^{2 \text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )^3 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6286
Rule 3716
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{x} \, dx &=-\operatorname{Subst}\left (\int (a+b x)^3 \coth (x) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=\frac{\left (a+b \text{csch}^{-1}(c x)\right )^4}{4 b}+2 \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)^3}{1-e^{2 x}} \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=\frac{\left (a+b \text{csch}^{-1}(c x)\right )^4}{4 b}-\left (a+b \text{csch}^{-1}(c x)\right )^3 \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )+(3 b) \operatorname{Subst}\left (\int (a+b x)^2 \log \left (1-e^{2 x}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=\frac{\left (a+b \text{csch}^{-1}(c x)\right )^4}{4 b}-\left (a+b \text{csch}^{-1}(c x)\right )^3 \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )-\frac{3}{2} b \left (a+b \text{csch}^{-1}(c x)\right )^2 \text{Li}_2\left (e^{2 \text{csch}^{-1}(c x)}\right )+\left (3 b^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{Li}_2\left (e^{2 x}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=\frac{\left (a+b \text{csch}^{-1}(c x)\right )^4}{4 b}-\left (a+b \text{csch}^{-1}(c x)\right )^3 \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )-\frac{3}{2} b \left (a+b \text{csch}^{-1}(c x)\right )^2 \text{Li}_2\left (e^{2 \text{csch}^{-1}(c x)}\right )+\frac{3}{2} b^2 \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_3\left (e^{2 \text{csch}^{-1}(c x)}\right )-\frac{1}{2} \left (3 b^3\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (e^{2 x}\right ) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=\frac{\left (a+b \text{csch}^{-1}(c x)\right )^4}{4 b}-\left (a+b \text{csch}^{-1}(c x)\right )^3 \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )-\frac{3}{2} b \left (a+b \text{csch}^{-1}(c x)\right )^2 \text{Li}_2\left (e^{2 \text{csch}^{-1}(c x)}\right )+\frac{3}{2} b^2 \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_3\left (e^{2 \text{csch}^{-1}(c x)}\right )-\frac{1}{4} \left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 \text{csch}^{-1}(c x)}\right )\\ &=\frac{\left (a+b \text{csch}^{-1}(c x)\right )^4}{4 b}-\left (a+b \text{csch}^{-1}(c x)\right )^3 \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )-\frac{3}{2} b \left (a+b \text{csch}^{-1}(c x)\right )^2 \text{Li}_2\left (e^{2 \text{csch}^{-1}(c x)}\right )+\frac{3}{2} b^2 \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_3\left (e^{2 \text{csch}^{-1}(c x)}\right )-\frac{3}{4} b^3 \text{Li}_4\left (e^{2 \text{csch}^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.215751, size = 198, normalized size = 1.8 \[ \frac{1}{4} \left (6 a^2 b \left (\text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}(c x)}\right )-\text{csch}^{-1}(c x) \left (\text{csch}^{-1}(c x)+2 \log \left (1-e^{-2 \text{csch}^{-1}(c x)}\right )\right )\right )+2 a b^2 \left (-6 \text{csch}^{-1}(c x) \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right )+3 \text{PolyLog}\left (3,e^{2 \text{csch}^{-1}(c x)}\right )+2 \text{csch}^{-1}(c x)^2 \left (\text{csch}^{-1}(c x)-3 \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )\right )\right )+b^3 \left (-6 \text{csch}^{-1}(c x)^2 \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right )+6 \text{csch}^{-1}(c x) \text{PolyLog}\left (3,e^{2 \text{csch}^{-1}(c x)}\right )-3 \text{PolyLog}\left (4,e^{2 \text{csch}^{-1}(c x)}\right )+\text{csch}^{-1}(c x)^4-4 \text{csch}^{-1}(c x)^3 \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )\right )+4 a^3 \log (c x)\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.2, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccsch} \left (cx\right ) \right ) ^{3}}{x}}\, dx \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 (\frac{b^{3} \operatorname{arcsch}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname{arcsch}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname{arcsch}\left (c x\right ) + a^{3}}{x}, 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 \frac{\left (a + b \operatorname{acsch}{\left (c x \right )}\right )^{3}}{x}\, 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 \frac{{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}^{3}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]