3.529 \(\int \frac{(a+b \cosh ^{-1}(c x))^2}{d+e x^2} \, dx\)

Optimal. Leaf size=763 \[ -\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \text{PolyLog}\left (3,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{PolyLog}\left (3,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \text{PolyLog}\left (3,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{PolyLog}\left (3,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{2 \sqrt{-d} \sqrt{e}} \]

[Out]

((a + b*ArcCosh[c*x])^2*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*Sqrt[-d]*Sqrt[
e]) - ((a + b*ArcCosh[c*x])^2*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*Sqrt[-d]
*Sqrt[e]) + ((a + b*ArcCosh[c*x])^2*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*Sq
rt[-d]*Sqrt[e]) - ((a + b*ArcCosh[c*x])^2*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])
/(2*Sqrt[-d]*Sqrt[e]) - (b*(a + b*ArcCosh[c*x])*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2
*d) - e]))])/(Sqrt[-d]*Sqrt[e]) + (b*(a + b*ArcCosh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sq
rt[-(c^2*d) - e])])/(Sqrt[-d]*Sqrt[e]) - (b*(a + b*ArcCosh[c*x])*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt
[-d] + Sqrt[-(c^2*d) - e]))])/(Sqrt[-d]*Sqrt[e]) + (b*(a + b*ArcCosh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])
/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(Sqrt[-d]*Sqrt[e]) + (b^2*PolyLog[3, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-
d] - Sqrt[-(c^2*d) - e]))])/(Sqrt[-d]*Sqrt[e]) - (b^2*PolyLog[3, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-
(c^2*d) - e])])/(Sqrt[-d]*Sqrt[e]) + (b^2*PolyLog[3, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) -
e]))])/(Sqrt[-d]*Sqrt[e]) - (b^2*PolyLog[3, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(Sqrt
[-d]*Sqrt[e])

________________________________________________________________________________________

Rubi [A]  time = 1.30796, antiderivative size = 763, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {5707, 5800, 5562, 2190, 2531, 2282, 6589} \[ -\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \text{PolyLog}\left (3,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{PolyLog}\left (3,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \text{PolyLog}\left (3,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{PolyLog}\left (3,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{2 \sqrt{-d} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])^2/(d + e*x^2),x]

[Out]

((a + b*ArcCosh[c*x])^2*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*Sqrt[-d]*Sqrt[
e]) - ((a + b*ArcCosh[c*x])^2*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*Sqrt[-d]
*Sqrt[e]) + ((a + b*ArcCosh[c*x])^2*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*Sq
rt[-d]*Sqrt[e]) - ((a + b*ArcCosh[c*x])^2*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])
/(2*Sqrt[-d]*Sqrt[e]) - (b*(a + b*ArcCosh[c*x])*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2
*d) - e]))])/(Sqrt[-d]*Sqrt[e]) + (b*(a + b*ArcCosh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sq
rt[-(c^2*d) - e])])/(Sqrt[-d]*Sqrt[e]) - (b*(a + b*ArcCosh[c*x])*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt
[-d] + Sqrt[-(c^2*d) - e]))])/(Sqrt[-d]*Sqrt[e]) + (b*(a + b*ArcCosh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])
/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(Sqrt[-d]*Sqrt[e]) + (b^2*PolyLog[3, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-
d] - Sqrt[-(c^2*d) - e]))])/(Sqrt[-d]*Sqrt[e]) - (b^2*PolyLog[3, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-
(c^2*d) - e])])/(Sqrt[-d]*Sqrt[e]) + (b^2*PolyLog[3, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) -
e]))])/(Sqrt[-d]*Sqrt[e]) - (b^2*PolyLog[3, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(Sqrt
[-d]*Sqrt[e])

Rule 5707

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p
] && (p > 0 || IGtQ[n, 0])

Rule 5800

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Subst[Int[((a + b*x)^n*Sinh[x
])/(c*d + e*Cosh[x]), x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 5562

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 - b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 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 \cosh ^{-1}(c x)\right )^2}{d+e x^2} \, dx &=\int \left (\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx\\ &=-\frac{\int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 \sqrt{-d}}-\frac{\int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 \sqrt{-d}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^2 \sinh (x)}{c \sqrt{-d}-\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d}}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^2 \sinh (x)}{c \sqrt{-d}+\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)^2}{c \sqrt{-d}-\sqrt{-c^2 d-e}-\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d}}-\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)^2}{c \sqrt{-d}+\sqrt{-c^2 d-e}-\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d}}-\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)^2}{c \sqrt{-d}-\sqrt{-c^2 d-e}+\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d}}-\frac{\operatorname{Subst}\left (\int \frac{e^x (a+b x)^2}{c \sqrt{-d}+\sqrt{-c^2 d-e}+\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt{-d}}\\ &=\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{b \operatorname{Subst}\left (\int (a+b x) \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \operatorname{Subst}\left (\int (a+b x) \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}-\frac{b \operatorname{Subst}\left (\int (a+b x) \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \operatorname{Subst}\left (\int (a+b x) \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}\\ &=\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \operatorname{Subst}\left (\int \text{Li}_2\left (\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \operatorname{Subst}\left (\int \text{Li}_2\left (\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-d} \sqrt{e}}\\ &=\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{\sqrt{e} x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{\sqrt{e} x}{-c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt{-d} \sqrt{e}}\\ &=\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \text{Li}_3\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{Li}_3\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \text{Li}_3\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{Li}_3\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{\sqrt{-d} \sqrt{e}}\\ \end{align*}

Mathematica [A]  time = 0.570427, size = 623, normalized size = 0.82 \[ \frac{2 b \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )-2 b \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-c \sqrt{-d}}\right )-2 b \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )+2 b \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )-2 b^2 \text{PolyLog}\left (3,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )+2 b^2 \text{PolyLog}\left (3,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-c \sqrt{-d}}\right )+2 b^2 \text{PolyLog}\left (3,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )-2 b^2 \text{PolyLog}\left (3,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )-\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )+\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-c \sqrt{-d}}+1\right )+\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )-\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{2 \sqrt{-d} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCosh[c*x])^2/(d + e*x^2),x]

[Out]

(-((a + b*ArcCosh[c*x])^2*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])]) + (a + b*ArcCos
h[c*x])^2*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])] + (a + b*ArcCosh[c*x])^2*Log[
1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])] - (a + b*ArcCosh[c*x])^2*Log[1 + (Sqrt[e]*E^Ar
cCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])] + 2*b*(a + b*ArcCosh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/
(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])] - 2*b*(a + b*ArcCosh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d
]) + Sqrt[-(c^2*d) - e])] - 2*b*(a + b*ArcCosh[c*x])*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[
-(c^2*d) - e]))] + 2*b*(a + b*ArcCosh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) -
e])] - 2*b^2*PolyLog[3, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])] + 2*b^2*PolyLog[3, (Sqrt[e
]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])] + 2*b^2*PolyLog[3, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[
-d] + Sqrt[-(c^2*d) - e]))] - 2*b^2*PolyLog[3, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2
*Sqrt[-d]*Sqrt[e])

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Maple [F]  time = 0.402, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}}{e{x}^{2}+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))^2/(e*x^2+d),x)

[Out]

int((a+b*arccosh(c*x))^2/(e*x^2+d),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)/(e*x^2 + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))**2/(e*x**2+d),x)

[Out]

Integral((a + b*acosh(c*x))**2/(d + e*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccosh(c*x) + a)^2/(e*x^2 + d), x)