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

Optimal. Leaf size=388 \[ -\frac{d e \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{2 b c^3}-\frac{d e \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b c^3}-\frac{e^2 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{8 b c^5}-\frac{3 e^2 \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c^5}-\frac{e^2 \sinh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c^5}+\frac{d e \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{2 b c^3}+\frac{d e \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b c^3}+\frac{e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{8 b c^5}+\frac{3 e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c^5}+\frac{e^2 \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c^5}-\frac{d^2 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{b c}+\frac{d^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{b c} \]

[Out]

-((d^2*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(b*c)) - (d*e*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh
[a/b])/(2*b*c^3) - (e^2*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(8*b*c^5) - (d*e*CoshIntegral[(3*(a +
b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/(2*b*c^3) - (3*e^2*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/
(16*b*c^5) - (e^2*CoshIntegral[(5*(a + b*ArcCosh[c*x]))/b]*Sinh[(5*a)/b])/(16*b*c^5) + (d^2*Cosh[a/b]*SinhInte
gral[(a + b*ArcCosh[c*x])/b])/(b*c) + (d*e*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(2*b*c^3) + (e^2*Co
sh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b])/(8*b*c^5) + (d*e*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*
x]))/b])/(2*b*c^3) + (3*e^2*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b*c^5) + (e^2*Cosh[(5*
a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b*c^5)

________________________________________________________________________________________

Rubi [A]  time = 0.792853, antiderivative size = 380, normalized size of antiderivative = 0.98, number of steps used = 27, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {5707, 5658, 3303, 3298, 3301, 5670, 5448} \[ -\frac{d e \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{2 b c^3}-\frac{d e \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{2 b c^3}-\frac{e^2 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{8 b c^5}-\frac{3 e^2 \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b c^5}-\frac{e^2 \sinh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b c^5}+\frac{d e \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{2 b c^3}+\frac{d e \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{2 b c^3}+\frac{e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{8 b c^5}+\frac{3 e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b c^5}+\frac{e^2 \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b c^5}-\frac{d^2 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{b c}+\frac{d^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*e*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b])/(2*b*c^3) - (e^2*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b])
/(8*b*c^5) - (d^2*CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(b*c) - (d*e*CoshIntegral[(3*a)/b + 3*ArcCos
h[c*x]]*Sinh[(3*a)/b])/(2*b*c^3) - (3*e^2*CoshIntegral[(3*a)/b + 3*ArcCosh[c*x]]*Sinh[(3*a)/b])/(16*b*c^5) - (
e^2*CoshIntegral[(5*a)/b + 5*ArcCosh[c*x]]*Sinh[(5*a)/b])/(16*b*c^5) + (d*e*Cosh[a/b]*SinhIntegral[a/b + ArcCo
sh[c*x]])/(2*b*c^3) + (e^2*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/(8*b*c^5) + (d*e*Cosh[(3*a)/b]*SinhInte
gral[(3*a)/b + 3*ArcCosh[c*x]])/(2*b*c^3) + (3*e^2*Cosh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcCosh[c*x]])/(16*b
*c^5) + (e^2*Cosh[(5*a)/b]*SinhIntegral[(5*a)/b + 5*ArcCosh[c*x]])/(16*b*c^5) + (d^2*Cosh[a/b]*SinhIntegral[(a
+ b*ArcCosh[c*x])/b])/(b*c)

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 5658

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Dist[(b*c)^(-1), Subst[Int[x^n*Sinh[a/b - x/b], x]
, x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 5670

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*
Cosh[x]^m*Sinh[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^2}{a+b \cosh ^{-1}(c x)} \, dx &=\int \left (\frac{d^2}{a+b \cosh ^{-1}(c x)}+\frac{2 d e x^2}{a+b \cosh ^{-1}(c x)}+\frac{e^2 x^4}{a+b \cosh ^{-1}(c x)}\right ) \, dx\\ &=d^2 \int \frac{1}{a+b \cosh ^{-1}(c x)} \, dx+(2 d e) \int \frac{x^2}{a+b \cosh ^{-1}(c x)} \, dx+e^2 \int \frac{x^4}{a+b \cosh ^{-1}(c x)} \, dx\\ &=-\frac{d^2 \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}-\frac{x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{b c}+\frac{(2 d e) \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^3}+\frac{e^2 \operatorname{Subst}\left (\int \frac{\cosh ^4(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^5}\\ &=\frac{(2 d e) \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{4 (a+b x)}+\frac{\sinh (3 x)}{4 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3}+\frac{e^2 \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{8 (a+b x)}+\frac{3 \sinh (3 x)}{16 (a+b x)}+\frac{\sinh (5 x)}{16 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^5}+\frac{\left (d^2 \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{b c}-\frac{\left (d^2 \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c x)\right )}{b c}\\ &=-\frac{d^2 \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right ) \sinh \left (\frac{a}{b}\right )}{b c}+\frac{d^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{b c}+\frac{(d e) \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3}+\frac{(d e) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3}+\frac{e^2 \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^5}+\frac{e^2 \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^5}+\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^5}\\ &=-\frac{d^2 \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right ) \sinh \left (\frac{a}{b}\right )}{b c}+\frac{d^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{b c}+\frac{\left (d e \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3}+\frac{\left (e^2 \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^5}+\frac{\left (d e \cosh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3}+\frac{\left (3 e^2 \cosh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^5}+\frac{\left (e^2 \cosh \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^5}-\frac{\left (d e \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3}-\frac{\left (e^2 \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^5}-\frac{\left (d e \sinh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^3}-\frac{\left (3 e^2 \sinh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^5}-\frac{\left (e^2 \sinh \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^5}\\ &=-\frac{d e \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac{a}{b}\right )}{2 b c^3}-\frac{e^2 \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac{a}{b}\right )}{8 b c^5}-\frac{d^2 \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right ) \sinh \left (\frac{a}{b}\right )}{b c}-\frac{d e \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right ) \sinh \left (\frac{3 a}{b}\right )}{2 b c^3}-\frac{3 e^2 \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right ) \sinh \left (\frac{3 a}{b}\right )}{16 b c^5}-\frac{e^2 \text{Chi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right ) \sinh \left (\frac{5 a}{b}\right )}{16 b c^5}+\frac{d e \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{2 b c^3}+\frac{e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{8 b c^5}+\frac{d e \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{2 b c^3}+\frac{3 e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b c^5}+\frac{e^2 \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b c^5}+\frac{d^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{b c}\\ \end{align*}

Mathematica [A]  time = 0.50615, size = 254, normalized size = 0.65 \[ \frac{-2 \sinh \left (\frac{a}{b}\right ) \left (8 c^4 d^2+4 c^2 d e+e^2\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )-e \sinh \left (\frac{3 a}{b}\right ) \left (8 c^2 d+3 e\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+16 c^4 d^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )+8 c^2 d e \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )+8 c^2 d e \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-e^2 \sinh \left (\frac{5 a}{b}\right ) \text{Chi}\left (5 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+2 e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )+3 e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+e^2 \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (5 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )}{16 b c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(8*c^4*d^2 + 4*c^2*d*e + e^2)*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] - e*(8*c^2*d + 3*e)*CoshIntegral[
3*(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] - e^2*CoshIntegral[5*(a/b + ArcCosh[c*x])]*Sinh[(5*a)/b] + 16*c^4*d^2*Co
sh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 8*c^2*d*e*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 2*e^2*Cosh[a
/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 8*c^2*d*e*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 3*e^2*Co
sh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + e^2*Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])])/(16
*b*c^5)

________________________________________________________________________________________

Maple [A]  time = 0.123, size = 380, normalized size = 1. \begin{align*}{\frac{1}{c} \left ( -{\frac{{e}^{2}}{32\,{c}^{4}b}{{\rm e}^{-5\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-5\,{\rm arccosh} \left (cx\right )-5\,{\frac{a}{b}} \right ) }+{\frac{{e}^{2}}{32\,{c}^{4}b}{{\rm e}^{5\,{\frac{a}{b}}}}{\it Ei} \left ( 1,5\,{\rm arccosh} \left (cx\right )+5\,{\frac{a}{b}} \right ) }+{\frac{{d}^{2}}{2\,b}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\rm arccosh} \left (cx\right )+{\frac{a}{b}} \right ) }+{\frac{de}{4\,{c}^{2}b}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\rm arccosh} \left (cx\right )+{\frac{a}{b}} \right ) }+{\frac{{e}^{2}}{16\,{c}^{4}b}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\rm arccosh} \left (cx\right )+{\frac{a}{b}} \right ) }-{\frac{{d}^{2}}{2\,b}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arccosh} \left (cx\right )-{\frac{a}{b}} \right ) }-{\frac{de}{4\,{c}^{2}b}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arccosh} \left (cx\right )-{\frac{a}{b}} \right ) }-{\frac{{e}^{2}}{16\,{c}^{4}b}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arccosh} \left (cx\right )-{\frac{a}{b}} \right ) }+{\frac{de}{4\,{c}^{2}b}{{\rm e}^{3\,{\frac{a}{b}}}}{\it Ei} \left ( 1,3\,{\rm arccosh} \left (cx\right )+3\,{\frac{a}{b}} \right ) }+{\frac{3\,{e}^{2}}{32\,{c}^{4}b}{{\rm e}^{3\,{\frac{a}{b}}}}{\it Ei} \left ( 1,3\,{\rm arccosh} \left (cx\right )+3\,{\frac{a}{b}} \right ) }-{\frac{de}{4\,{c}^{2}b}{{\rm e}^{-3\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-3\,{\rm arccosh} \left (cx\right )-3\,{\frac{a}{b}} \right ) }-{\frac{3\,{e}^{2}}{32\,{c}^{4}b}{{\rm e}^{-3\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-3\,{\rm arccosh} \left (cx\right )-3\,{\frac{a}{b}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(-1/32/c^4*e^2/b*exp(-5*a/b)*Ei(1,-5*arccosh(c*x)-5*a/b)+1/32/c^4*e^2/b*exp(5*a/b)*Ei(1,5*arccosh(c*x)+5*a
/b)+1/2/b*exp(a/b)*Ei(1,arccosh(c*x)+a/b)*d^2+1/4/c^2/b*exp(a/b)*Ei(1,arccosh(c*x)+a/b)*d*e+1/16/c^4/b*exp(a/b
)*Ei(1,arccosh(c*x)+a/b)*e^2-1/2/b*exp(-a/b)*Ei(1,-arccosh(c*x)-a/b)*d^2-1/4/c^2/b*exp(-a/b)*Ei(1,-arccosh(c*x
)-a/b)*d*e-1/16/c^4/b*exp(-a/b)*Ei(1,-arccosh(c*x)-a/b)*e^2+1/4/c^2*e/b*exp(3*a/b)*Ei(1,3*arccosh(c*x)+3*a/b)*
d+3/32/c^4*e^2/b*exp(3*a/b)*Ei(1,3*arccosh(c*x)+3*a/b)-1/4/c^2*e/b*exp(-3*a/b)*Ei(1,-3*arccosh(c*x)-3*a/b)*d-3
/32/c^4*e^2/b*exp(-3*a/b)*Ei(1,-3*arccosh(c*x)-3*a/b))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}{b \operatorname{arcosh}\left (c x\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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