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

Optimal. Leaf size=543 \[ -\frac{b \sqrt{e} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 (-d)^{3/2}}+\frac{b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 (-d)^{3/2}}-\frac{b \sqrt{e} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 (-d)^{3/2}}+\frac{b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 (-d)^{3/2}}+\frac{\sqrt{e} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{2 (-d)^{3/2}}+\frac{\sqrt{e} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{2 (-d)^{3/2}}-\frac{a+b \cosh ^{-1}(c x)}{d x}+\frac{b c \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{d} \]

[Out]

-((a + b*ArcCosh[c*x])/(d*x)) + (b*c*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/d + (Sqrt[e]*(a + b*ArcCosh[c*x])*L
og[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*(-d)^(3/2)) - (Sqrt[e]*(a + b*ArcCosh[c
*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*(-d)^(3/2)) + (Sqrt[e]*(a + b*Arc
Cosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*(-d)^(3/2)) - (Sqrt[e]*(a +
b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*(-d)^(3/2)) - (b*Sqrt
[e]*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(2*(-d)^(3/2)) + (b*Sqrt[e]*Pol
yLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*(-d)^(3/2)) - (b*Sqrt[e]*PolyLog[2, -(
(Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(2*(-d)^(3/2)) + (b*Sqrt[e]*PolyLog[2, (Sqrt[e]*
E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*(-d)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.906472, antiderivative size = 543, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {5792, 5662, 92, 205, 5707, 5800, 5562, 2190, 2279, 2391} \[ -\frac{b \sqrt{e} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 (-d)^{3/2}}+\frac{b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 (-d)^{3/2}}-\frac{b \sqrt{e} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 (-d)^{3/2}}+\frac{b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 (-d)^{3/2}}+\frac{\sqrt{e} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{2 (-d)^{3/2}}+\frac{\sqrt{e} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{2 (-d)^{3/2}}-\frac{a+b \cosh ^{-1}(c x)}{d x}+\frac{b c \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*ArcCosh[c*x])/(d*x)) + (b*c*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/d + (Sqrt[e]*(a + b*ArcCosh[c*x])*L
og[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*(-d)^(3/2)) - (Sqrt[e]*(a + b*ArcCosh[c
*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*(-d)^(3/2)) + (Sqrt[e]*(a + b*Arc
Cosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*(-d)^(3/2)) - (Sqrt[e]*(a +
b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*(-d)^(3/2)) - (b*Sqrt
[e]*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(2*(-d)^(3/2)) + (b*Sqrt[e]*Pol
yLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*(-d)^(3/2)) - (b*Sqrt[e]*PolyLog[2, -(
(Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(2*(-d)^(3/2)) + (b*Sqrt[e]*PolyLog[2, (Sqrt[e]*
E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*(-d)^(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 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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

Mathematica [A]  time = 1.39716, size = 549, normalized size = 1.01 \[ \frac{1}{2} \left (\frac{b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{(-d)^{3/2}}+\frac{b d \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-c \sqrt{-d}}\right )}{(-d)^{5/2}}+\frac{b d \sqrt{e} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{(-d)^{5/2}}+\frac{b \sqrt{e} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{(-d)^{3/2}}+\frac{d \sqrt{e} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{(-d)^{5/2}}+\frac{\sqrt{e} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-c \sqrt{-d}}+1\right )}{(-d)^{3/2}}+\frac{\sqrt{e} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{(-d)^{3/2}}+\frac{d \sqrt{e} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{(-d)^{5/2}}-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{d x}+\frac{2 b c \sqrt{c^2 x^2-1} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{d \sqrt{c x-1} \sqrt{c x+1}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*(a + b*ArcCosh[c*x]))/(d*x) + (2*b*c*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(d*Sqrt[-1 + c*x]*Sqr
t[1 + c*x]) + (d*Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e
])])/(-d)^(5/2) + (Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d
) - e])])/(-d)^(3/2) + (Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2
*d) - e])])/(-d)^(3/2) + (d*Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-
(c^2*d) - e])])/(-d)^(5/2) + (b*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])]
)/(-d)^(3/2) + (b*d*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])])/(-d)^(5
/2) + (b*d*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(-d)^(5/2) + (b*
Sqrt[e]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(-d)^(3/2))/2

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Maple [C]  time = 0.774, size = 329, normalized size = 0.6 \begin{align*} -{\frac{a}{dx}}-{\frac{ae}{d}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{b{\rm arccosh} \left (cx\right )}{dx}}-{\frac{be}{8\,c{d}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( 4\,{c}^{2}d+2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{4\,{{\it \_R1}}^{2}{c}^{2}d+{{\it \_R1}}^{2}e+e}{{\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 ) }}+{\frac{be}{8\,c{d}^{2}}\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}}^{2}e+4\,{c}^{2}d+e}{{\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 ) }}+2\,{\frac{bc\arctan \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/d/x-a*e/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))-b*arccosh(c*x)/d/x-1/8*b/c/d^2*e*sum((4*_R1^2*c^2*d+_R1^2*e+e
)/_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))+1/8*b/c/d^2*e*sum((_R1^2*e+4*c^2*d+e)/_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))+2*c*b/d*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))

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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))/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 \operatorname{arcosh}\left (c x\right ) + a}{e x^{4} + d x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arccosh(c*x) + a)/(e*x^4 + d*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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