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

Optimal. Leaf size=544 \[ -\frac{b \sqrt{-d} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^{3/2}}-\frac{b \sqrt{-d} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^{3/2}}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^{3/2}}-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{2 e^{3/2}}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^{3/2}}-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{2 e^{3/2}}+\frac{a x}{e}-\frac{b \sqrt{c x-1} \sqrt{c x+1}}{c e}+\frac{b x \cosh ^{-1}(c x)}{e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.901976, antiderivative size = 544, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5792, 5654, 74, 5707, 5800, 5562, 2190, 2279, 2391} \[ -\frac{b \sqrt{-d} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^{3/2}}-\frac{b \sqrt{-d} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^{3/2}}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^{3/2}}-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{2 e^{3/2}}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^{3/2}}-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{2 e^{3/2}}+\frac{a x}{e}-\frac{b \sqrt{c x-1} \sqrt{c x+1}}{c e}+\frac{b x \cosh ^{-1}(c x)}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5792

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

Rule 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

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 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

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

Mathematica [C]  time = 0.698351, size = 457, normalized size = 0.84 \[ \frac{i b \left (-c \sqrt{d} \left (-2 \text{PolyLog}\left (2,\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )-2 \text{PolyLog}\left (2,-\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-2 \left (\log \left (1+\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )+\log \left (1+\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )\right )\right )+c \sqrt{d} \left (-2 \text{PolyLog}\left (2,\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )-2 \text{PolyLog}\left (2,\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-2 \left (\log \left (1+\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )+\log \left (1-\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )\right )\right )+4 i \sqrt{e} \left (\sqrt{\frac{c x-1}{c x+1}} (c x+1)-c x \cosh ^{-1}(c x)\right )\right )-4 a c \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )+4 a c \sqrt{e} x}{4 c e^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*a*c*Sqrt[e]*x - 4*a*c*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + I*b*((4*I)*Sqrt[e]*(Sqrt[(-1 + c*x)/(1 + c*x)]*
(1 + c*x) - c*x*ArcCosh[c*x]) - c*Sqrt[d]*(ArcCosh[c*x]*(ArcCosh[c*x] - 2*(Log[1 + (I*Sqrt[e]*E^ArcCosh[c*x])/
(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])])) - 2*PolyL
og[2, (I*Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] - 2*PolyLog[2, ((-I)*Sqrt[e]*E^ArcCosh[c*x]
)/(c*Sqrt[d] + Sqrt[c^2*d + e])]) + c*Sqrt[d]*(ArcCosh[c*x]*(ArcCosh[c*x] - 2*(Log[1 + (I*Sqrt[e]*E^ArcCosh[c*
x])/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])])) -
2*PolyLog[2, (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] - Sqrt[c^2*d + e])] - 2*PolyLog[2, (I*Sqrt[e]*E^ArcCosh[c*x
])/(c*Sqrt[d] + Sqrt[c^2*d + e])])))/(4*c*e^(3/2))

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Maple [C]  time = 0.617, size = 284, normalized size = 0.5 \begin{align*}{\frac{ax}{e}}-{\frac{ad}{e}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{bx{\rm arccosh} \left (cx\right )}{e}}-{\frac{b}{ce}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{cbd}{2\,e}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( 4\,{c}^{2}d+2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{{\it \_R1}}{{{\it \_R1}}^{2}e+2\,{c}^{2}d+e} \left ({\rm arccosh} \left (cx\right )\ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) } \right ) \right ) }}+{\frac{cbd}{2\,e}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( 4\,{c}^{2}d+2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{1}{{\it \_R1}\, \left ({{\it \_R1}}^{2}e+2\,{c}^{2}d+e \right ) } \left ({\rm arccosh} \left (cx\right )\ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) } \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x/e-a*d/e/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+b*x*arccosh(c*x)/e-b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/e-1/2*c*b*d
/e*sum(_R1/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*ln((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x
-1)^(1/2)*(c*x+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))+1/2*c*b*d/e*sum(1/_R1/(_R1^2*e+2*c^2*d
+e)*(arccosh(c*x)*ln((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R
1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))

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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(x^2*(a+b*arccosh(c*x))/(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 x^{2} \operatorname{arcosh}\left (c x\right ) + a x^{2}}{e x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*(a + b*acosh(c*x))/(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 )} x^{2}}{e x^{2} + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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