3.144 \(\int \frac{(c e+d e x)^2}{(a+b \cosh ^{-1}(c+d x))^3} \, dx\)

Optimal. Leaf size=252 \[ -\frac{e^2 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{8 b^3 d}-\frac{9 e^2 \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{8 b^3 d}+\frac{e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{8 b^3 d}+\frac{9 e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{8 b^3 d}-\frac{3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{e^2 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^2}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2} \]

[Out]

-(e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(2*b*d*(a + b*ArcCosh[c + d*x])^2) + (e^2*(c + d*x))/(
b^2*d*(a + b*ArcCosh[c + d*x])) - (3*e^2*(c + d*x)^3)/(2*b^2*d*(a + b*ArcCosh[c + d*x])) - (e^2*CoshIntegral[(
a + b*ArcCosh[c + d*x])/b]*Sinh[a/b])/(8*b^3*d) - (9*e^2*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]))/b]*Sinh[(3*
a)/b])/(8*b^3*d) + (e^2*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(8*b^3*d) + (9*e^2*Cosh[(3*a)/b]*S
inhIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(8*b^3*d)

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Rubi [A]  time = 0.799218, antiderivative size = 311, normalized size of antiderivative = 1.23, number of steps used = 18, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {5866, 12, 5668, 5775, 5670, 5448, 3303, 3298, 3301, 5658} \[ -\frac{9 e^2 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{8 b^3 d}+\frac{e^2 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{b^3 d}-\frac{9 e^2 \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{8 b^3 d}+\frac{9 e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{8 b^3 d}+\frac{9 e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{8 b^3 d}-\frac{e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{b^3 d}-\frac{3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{e^2 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^2}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x])^3,x]

[Out]

-(e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(2*b*d*(a + b*ArcCosh[c + d*x])^2) + (e^2*(c + d*x))/(
b^2*d*(a + b*ArcCosh[c + d*x])) - (3*e^2*(c + d*x)^3)/(2*b^2*d*(a + b*ArcCosh[c + d*x])) - (9*e^2*CoshIntegral
[a/b + ArcCosh[c + d*x]]*Sinh[a/b])/(8*b^3*d) + (e^2*CoshIntegral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b])/(b^3*
d) - (9*e^2*CoshIntegral[(3*a)/b + 3*ArcCosh[c + d*x]]*Sinh[(3*a)/b])/(8*b^3*d) + (9*e^2*Cosh[a/b]*SinhIntegra
l[a/b + ArcCosh[c + d*x]])/(8*b^3*d) + (9*e^2*Cosh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcCosh[c + d*x]])/(8*b^3
*d) - (e^2*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(b^3*d)

Rule 5866

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 5668

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(
a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(c*(m + 1))/(b*(n + 1)), Int[(x^(m + 1)*(a + b*ArcCosh
[c*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] + Dist[m/(b*c*(n + 1)), Int[(x^(m - 1)*(a + b*ArcCosh[c
*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 5775

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[((f*x)^m*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[(f
*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1
, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && GtQ[d1, 0] && LtQ[d2, 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]

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 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]

Rubi steps

\begin{align*} \int \frac{(c e+d e x)^2}{\left (a+b \cosh ^{-1}(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^2 x^2}{\left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac{e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac{e^2 \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x} \sqrt{1+x} \left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{b d}+\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{-1+x} \sqrt{1+x} \left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{2 b d}\\ &=-\frac{e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{e^2 \operatorname{Subst}\left (\int \frac{1}{a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{b^2 d}+\frac{\left (9 e^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{2 b^2 d}\\ &=-\frac{e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{e^2 \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}-\frac{x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{b^3 d}+\frac{\left (9 e^2\right ) \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}\\ &=-\frac{e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{\left (9 e^2\right ) \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+d x)\right )}{2 b^2 d}-\frac{\left (e^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+d x)\right )}{b^3 d}+\frac{\left (e^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+d x)\right )}{b^3 d}\\ &=-\frac{e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{e^2 \text{Chi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right ) \sinh \left (\frac{a}{b}\right )}{b^3 d}-\frac{e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{b^3 d}+\frac{\left (9 e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}+\frac{\left (9 e^2\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}\\ &=-\frac{e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac{e^2 \text{Chi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right ) \sinh \left (\frac{a}{b}\right )}{b^3 d}-\frac{e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{b^3 d}+\frac{\left (9 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+d x)\right )}{8 b^2 d}+\frac{\left (9 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+d x)\right )}{8 b^2 d}-\frac{\left (9 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+d x)\right )}{8 b^2 d}-\frac{\left (9 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+d x)\right )}{8 b^2 d}\\ &=-\frac{e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac{e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac{9 e^2 \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right ) \sinh \left (\frac{a}{b}\right )}{8 b^3 d}+\frac{e^2 \text{Chi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right ) \sinh \left (\frac{a}{b}\right )}{b^3 d}-\frac{9 e^2 \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c+d x)\right ) \sinh \left (\frac{3 a}{b}\right )}{8 b^3 d}+\frac{9 e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )}{8 b^3 d}+\frac{9 e^2 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{8 b^3 d}-\frac{e^2 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )}{b^3 d}\\ \end{align*}

Mathematica [A]  time = 0.664219, size = 223, normalized size = 0.88 \[ \frac{e^2 \left (-\frac{4 b^2 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^2}{\left (a+b \cosh ^{-1}(c+d x)\right )^2}+9 \left (\sinh \left (\frac{a}{b}\right ) \left (-\text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )-\sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )+\cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )+\cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )+8 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )-8 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )+\frac{4 b \left (2 (c+d x)-3 (c+d x)^3\right )}{a+b \cosh ^{-1}(c+d x)}\right )}{8 b^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x])^3,x]

[Out]

(e^2*((-4*b^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(a + b*ArcCosh[c + d*x])^2 + (4*b*(2*(c + d*x)
- 3*(c + d*x)^3))/(a + b*ArcCosh[c + d*x]) + 8*CoshIntegral[a/b + ArcCosh[c + d*x]]*Sinh[a/b] - 8*Cosh[a/b]*S
inhIntegral[a/b + ArcCosh[c + d*x]] + 9*(-(CoshIntegral[a/b + ArcCosh[c + d*x]]*Sinh[a/b]) - CoshIntegral[3*(a
/b + ArcCosh[c + d*x])]*Sinh[(3*a)/b] + Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c + d*x]] + Cosh[(3*a)/b]*SinhInt
egral[3*(a/b + ArcCosh[c + d*x])])))/(8*b^3*d)

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Maple [B]  time = 0.116, size = 557, normalized size = 2.2 \begin{align*}{\frac{1}{d} \left ( -{\frac{{e}^{2} \left ( 3\,b{\rm arccosh} \left (dx+c\right )+3\,a-b \right ) }{16\,{b}^{2} \left ({b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}+2\,ab{\rm arccosh} \left (dx+c\right )+{a}^{2} \right ) } \left ( -4\, \left ( dx+c \right ) ^{2}\sqrt{dx+c-1}\sqrt{dx+c+1}+\sqrt{dx+c-1}\sqrt{dx+c+1}+4\, \left ( dx+c \right ) ^{3}-3\,dx-3\,c \right ) }+{\frac{9\,{e}^{2}}{16\,{b}^{3}}{{\rm e}^{3\,{\frac{a}{b}}}}{\it Ei} \left ( 1,3\,{\rm arccosh} \left (dx+c\right )+3\,{\frac{a}{b}} \right ) }-{\frac{{e}^{2} \left ( b{\rm arccosh} \left (dx+c\right )+a-b \right ) }{16\,{b}^{2} \left ({b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}+2\,ab{\rm arccosh} \left (dx+c\right )+{a}^{2} \right ) } \left ( -\sqrt{dx+c-1}\sqrt{dx+c+1}+dx+c \right ) }+{\frac{{e}^{2}}{16\,{b}^{3}}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\rm arccosh} \left (dx+c\right )+{\frac{a}{b}} \right ) }-{\frac{{e}^{2}}{16\,b \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{2}} \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) }-{\frac{{e}^{2}}{16\,{b}^{2} \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) } \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) }-{\frac{{e}^{2}}{16\,{b}^{3}}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arccosh} \left (dx+c\right )-{\frac{a}{b}} \right ) }-{\frac{{e}^{2}}{16\,b \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{2}} \left ( 4\, \left ( dx+c \right ) ^{3}-3\,dx-3\,c+4\, \left ( dx+c \right ) ^{2}\sqrt{dx+c-1}\sqrt{dx+c+1}-\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) }-{\frac{3\,{e}^{2}}{16\,{b}^{2} \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) } \left ( 4\, \left ( dx+c \right ) ^{3}-3\,dx-3\,c+4\, \left ( dx+c \right ) ^{2}\sqrt{dx+c-1}\sqrt{dx+c+1}-\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) }-{\frac{9\,{e}^{2}}{16\,{b}^{3}}{{\rm e}^{-3\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-3\,{\rm arccosh} \left (dx+c\right )-3\,{\frac{a}{b}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^3,x)

[Out]

1/d*(-1/16*(-4*(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+4*(d*x+c)^3-3*d*x-3*c
)*e^2*(3*b*arccosh(d*x+c)+3*a-b)/b^2/(b^2*arccosh(d*x+c)^2+2*a*b*arccosh(d*x+c)+a^2)+9/16*e^2/b^3*exp(3*a/b)*E
i(1,3*arccosh(d*x+c)+3*a/b)-1/16*(-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+d*x+c)*e^2*(b*arccosh(d*x+c)+a-b)/b^2/(b^2*
arccosh(d*x+c)^2+2*a*b*arccosh(d*x+c)+a^2)+1/16*e^2/b^3*exp(a/b)*Ei(1,arccosh(d*x+c)+a/b)-1/16*e^2/b*(d*x+c+(d
*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))^2-1/16*e^2/b^2*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+
b*arccosh(d*x+c))-1/16*e^2/b^3*exp(-a/b)*Ei(1,-arccosh(d*x+c)-a/b)-1/16*e^2/b*(4*(d*x+c)^3-3*d*x-3*c+4*(d*x+c)
^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))^2-3/16*e^2/b^2*(4*(d*
x+c)^3-3*d*x-3*c+4*(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x
+c))-9/16*e^2/b^3*exp(-3*a/b)*Ei(1,-3*arccosh(d*x+c)-3*a/b))

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^3,x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}{b^{3} \operatorname{arcosh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{arcosh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{arcosh}\left (d x + c\right ) + a^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^3,x, algorithm="fricas")

[Out]

integral((d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2)/(b^3*arccosh(d*x + c)^3 + 3*a*b^2*arccosh(d*x + c)^2 + 3*a^2*b*
arccosh(d*x + c) + a^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{2} \left (\int \frac{c^{2}}{a^{3} + 3 a^{2} b \operatorname{acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname{acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac{d^{2} x^{2}}{a^{3} + 3 a^{2} b \operatorname{acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname{acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac{2 c d x}{a^{3} + 3 a^{2} b \operatorname{acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname{acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname{acosh}^{3}{\left (c + d x \right )}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**2/(a+b*acosh(d*x+c))**3,x)

[Out]

e**2*(Integral(c**2/(a**3 + 3*a**2*b*acosh(c + d*x) + 3*a*b**2*acosh(c + d*x)**2 + b**3*acosh(c + d*x)**3), x)
+ Integral(d**2*x**2/(a**3 + 3*a**2*b*acosh(c + d*x) + 3*a*b**2*acosh(c + d*x)**2 + b**3*acosh(c + d*x)**3),
x) + Integral(2*c*d*x/(a**3 + 3*a**2*b*acosh(c + d*x) + 3*a*b**2*acosh(c + d*x)**2 + b**3*acosh(c + d*x)**3),
x))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^2/(b*arccosh(d*x + c) + a)^3, x)