3.19 \(\int \frac{(a+b \text{csch}^{-1}(c x))^2}{x} \, dx\)

Optimal. Leaf size=81 \[ -b \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{2} b^2 \text{PolyLog}\left (3,e^{2 \text{csch}^{-1}(c x)}\right )+\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{3 b}-\log \left (1-e^{2 \text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )^2 \]

[Out]

(a + b*ArcCsch[c*x])^3/(3*b) - (a + b*ArcCsch[c*x])^2*Log[1 - E^(2*ArcCsch[c*x])] - b*(a + b*ArcCsch[c*x])*Pol
yLog[2, E^(2*ArcCsch[c*x])] + (b^2*PolyLog[3, E^(2*ArcCsch[c*x])])/2

________________________________________________________________________________________

Rubi [A]  time = 0.137782, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6286, 3716, 2190, 2531, 2282, 6589} \[ -b \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{2} b^2 \text{PolyLog}\left (3,e^{2 \text{csch}^{-1}(c x)}\right )+\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{3 b}-\log \left (1-e^{2 \text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )^2 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsch[c*x])^2/x,x]

[Out]

(a + b*ArcCsch[c*x])^3/(3*b) - (a + b*ArcCsch[c*x])^2*Log[1 - E^(2*ArcCsch[c*x])] - b*(a + b*ArcCsch[c*x])*Pol
yLog[2, E^(2*ArcCsch[c*x])] + (b^2*PolyLog[3, E^(2*ArcCsch[c*x])])/2

Rule 6286

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b
*x)^n*Csch[x]^(m + 1)*Coth[x], x], x, ArcCsch[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] &
& (GtQ[n, 0] || LtQ[m, -1])

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{x} \, dx &=-\operatorname{Subst}\left (\int (a+b x)^2 \coth (x) \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{3 b}+2 \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)^2}{1-e^{2 x}} \, dx,x,\text{csch}^{-1}(c x)\right )\\ &=\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{3 b}-\left (a+b \text{csch}^{-1}(c x)\right )^2 \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )+(2 b) \operatorname{Subst}\left (\int (a+b x) \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 )^3}{3 b}-\left (a+b \text{csch}^{-1}(c x)\right )^2 \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )-b \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \text{csch}^{-1}(c x)}\right )+b^2 \operatorname{Subst}\left (\int \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 )^3}{3 b}-\left (a+b \text{csch}^{-1}(c x)\right )^2 \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )-b \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \text{csch}^{-1}(c x)}\right )+\frac{1}{2} b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 \text{csch}^{-1}(c x)}\right )\\ &=\frac{\left (a+b \text{csch}^{-1}(c x)\right )^3}{3 b}-\left (a+b \text{csch}^{-1}(c x)\right )^2 \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )-b \left (a+b \text{csch}^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \text{csch}^{-1}(c x)}\right )+\frac{1}{2} b^2 \text{Li}_3\left (e^{2 \text{csch}^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.131099, size = 115, normalized size = 1.42 \[ a b \left (\text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}(c x)}\right )-\text{csch}^{-1}(c x) \left (\text{csch}^{-1}(c x)+2 \log \left (1-e^{-2 \text{csch}^{-1}(c x)}\right )\right )\right )+b^2 \left (-\text{csch}^{-1}(c x) \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(c x)}\right )+\frac{1}{2} \text{PolyLog}\left (3,e^{2 \text{csch}^{-1}(c x)}\right )+\frac{1}{3} \text{csch}^{-1}(c x)^3-\text{csch}^{-1}(c x)^2 \log \left (1-e^{2 \text{csch}^{-1}(c x)}\right )\right )+a^2 \log (c x) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCsch[c*x])^2/x,x]

[Out]

a^2*Log[c*x] + a*b*(-(ArcCsch[c*x]*(ArcCsch[c*x] + 2*Log[1 - E^(-2*ArcCsch[c*x])])) + PolyLog[2, E^(-2*ArcCsch
[c*x])]) + b^2*(ArcCsch[c*x]^3/3 - ArcCsch[c*x]^2*Log[1 - E^(2*ArcCsch[c*x])] - ArcCsch[c*x]*PolyLog[2, E^(2*A
rcCsch[c*x])] + PolyLog[3, E^(2*ArcCsch[c*x])]/2)

________________________________________________________________________________________

Maple [F]  time = 0.201, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccsch} \left (cx\right ) \right ) ^{2}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsch(c*x))^2/x,x)

[Out]

int((a+b*arccsch(c*x))^2/x,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} b^{2} \log \left (x\right ) \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right )^{2} + a^{2} \log \left (x\right ) - \int -\frac{b^{2} \log \left (c\right )^{2} +{\left (b^{2} c^{2} \log \left (c\right )^{2} - 2 \, a b c^{2} \log \left (c\right )\right )} x^{2} - 2 \, a b \log \left (c\right ) +{\left (b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (x\right )^{2} + 2 \,{\left ({\left (b^{2} c^{2} \log \left (c\right ) - a b c^{2}\right )} x^{2} + b^{2} \log \left (c\right ) - a b\right )} \log \left (x\right ) - 2 \,{\left ({\left (b^{2} c^{2} \log \left (c\right ) - a b c^{2}\right )} x^{2} + b^{2} \log \left (c\right ) - a b +{\left (b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (x\right ) + \sqrt{c^{2} x^{2} + 1}{\left ({\left (b^{2} c^{2} \log \left (c\right ) - a b c^{2}\right )} x^{2} + b^{2} \log \left (c\right ) - a b +{\left (2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (x\right )\right )}\right )} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right ) + \sqrt{c^{2} x^{2} + 1}{\left (b^{2} \log \left (c\right )^{2} +{\left (b^{2} c^{2} \log \left (c\right )^{2} - 2 \, a b c^{2} \log \left (c\right )\right )} x^{2} - 2 \, a b \log \left (c\right ) +{\left (b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (x\right )^{2} + 2 \,{\left ({\left (b^{2} c^{2} \log \left (c\right ) - a b c^{2}\right )} x^{2} + b^{2} \log \left (c\right ) - a b\right )} \log \left (x\right )\right )}}{c^{2} x^{3} +{\left (c^{2} x^{3} + x\right )} \sqrt{c^{2} x^{2} + 1} + x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))^2/x,x, algorithm="maxima")

[Out]

b^2*log(x)*log(sqrt(c^2*x^2 + 1) + 1)^2 + a^2*log(x) - integrate(-(b^2*log(c)^2 + (b^2*c^2*log(c)^2 - 2*a*b*c^
2*log(c))*x^2 - 2*a*b*log(c) + (b^2*c^2*x^2 + b^2)*log(x)^2 + 2*((b^2*c^2*log(c) - a*b*c^2)*x^2 + b^2*log(c) -
a*b)*log(x) - 2*((b^2*c^2*log(c) - a*b*c^2)*x^2 + b^2*log(c) - a*b + (b^2*c^2*x^2 + b^2)*log(x) + sqrt(c^2*x^
2 + 1)*((b^2*c^2*log(c) - a*b*c^2)*x^2 + b^2*log(c) - a*b + (2*b^2*c^2*x^2 + b^2)*log(x)))*log(sqrt(c^2*x^2 +
1) + 1) + sqrt(c^2*x^2 + 1)*(b^2*log(c)^2 + (b^2*c^2*log(c)^2 - 2*a*b*c^2*log(c))*x^2 - 2*a*b*log(c) + (b^2*c^
2*x^2 + b^2)*log(x)^2 + 2*((b^2*c^2*log(c) - a*b*c^2)*x^2 + b^2*log(c) - a*b)*log(x)))/(c^2*x^3 + (c^2*x^3 + x
)*sqrt(c^2*x^2 + 1) + x), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \operatorname{arcsch}\left (c x\right )^{2} + 2 \, a b \operatorname{arcsch}\left (c x\right ) + a^{2}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))^2/x,x, algorithm="fricas")

[Out]

integral((b^2*arccsch(c*x)^2 + 2*a*b*arccsch(c*x) + a^2)/x, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acsch}{\left (c x \right )}\right )^{2}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsch(c*x))**2/x,x)

[Out]

Integral((a + b*acsch(c*x))**2/x, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}^{2}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))^2/x,x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)^2/x, x)