3.194 \(\int \frac{(c e+d e x)^2}{(a+b \cosh ^{-1}(c+d x))^{7/2}} \, dx\)

Optimal. Leaf size=431 \[ \frac{\sqrt{\pi } e^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d}+\frac{3 \sqrt{3 \pi } e^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{5 b^{7/2} d}+\frac{\sqrt{\pi } e^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d}+\frac{3 \sqrt{3 \pi } e^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{5 b^{7/2} d}-\frac{4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{24 e^2 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^2}{5 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}+\frac{8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac{16 e^2 \sqrt{c+d x-1} \sqrt{c+d x+1}}{15 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{2 e^2 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^2}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \]

[Out]

(-2*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(5*b*d*(a + b*ArcCosh[c + d*x])^(5/2)) + (8*e^2*(c +
d*x))/(15*b^2*d*(a + b*ArcCosh[c + d*x])^(3/2)) - (4*e^2*(c + d*x)^3)/(5*b^2*d*(a + b*ArcCosh[c + d*x])^(3/2)
) + (16*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(15*b^3*d*Sqrt[a + b*ArcCosh[c + d*x]]) - (24*e^2*Sqrt[-1 +
c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(5*b^3*d*Sqrt[a + b*ArcCosh[c + d*x]]) + (e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqr
t[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(15*b^(7/2)*d) + (3*e^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*Arc
Cosh[c + d*x]])/Sqrt[b]])/(5*b^(7/2)*d) + (e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(15*b^(7/2
)*d*E^(a/b)) + (3*e^2*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(5*b^(7/2)*d*E^((3*a)/b
))

________________________________________________________________________________________

Rubi [A]  time = 1.47498, antiderivative size = 431, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {5866, 12, 5668, 5775, 5666, 3307, 2180, 2204, 2205, 5656, 5781} \[ \frac{\sqrt{\pi } e^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d}+\frac{3 \sqrt{3 \pi } e^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{5 b^{7/2} d}+\frac{\sqrt{\pi } e^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d}+\frac{3 \sqrt{3 \pi } e^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{5 b^{7/2} d}-\frac{4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{24 e^2 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^2}{5 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}+\frac{8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac{16 e^2 \sqrt{c+d x-1} \sqrt{c+d x+1}}{15 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{2 e^2 \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)^2}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x])^(7/2),x]

[Out]

(-2*e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(5*b*d*(a + b*ArcCosh[c + d*x])^(5/2)) + (8*e^2*(c +
d*x))/(15*b^2*d*(a + b*ArcCosh[c + d*x])^(3/2)) - (4*e^2*(c + d*x)^3)/(5*b^2*d*(a + b*ArcCosh[c + d*x])^(3/2)
) + (16*e^2*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(15*b^3*d*Sqrt[a + b*ArcCosh[c + d*x]]) - (24*e^2*Sqrt[-1 +
c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(5*b^3*d*Sqrt[a + b*ArcCosh[c + d*x]]) + (e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqr
t[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(15*b^(7/2)*d) + (3*e^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*Arc
Cosh[c + d*x]])/Sqrt[b]])/(5*b^(7/2)*d) + (e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(15*b^(7/2
)*d*E^(a/b)) + (3*e^2*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(5*b^(7/2)*d*E^((3*a)/b
))

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 5666

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[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a +
b*x)^(n + 1)*Cosh[x]^(m - 1)*(m - (m + 1)*Cosh[x]^2), x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 3307

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 5656

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

Rule 5781

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCos
h[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p
+ 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 0])

Rubi steps

\begin{align*} \int \frac{(c e+d e x)^2}{\left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^2 x^2}{\left (a+b \cosh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b \cosh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}-\frac{\left (4 e^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x} \sqrt{1+x} \left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d}+\frac{\left (6 e^2\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{-1+x} \sqrt{1+x} \left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d}\\ &=-\frac{2 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac{8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{\left (8 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d}+\frac{\left (12 e^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{5 b^2 d}\\ &=-\frac{2 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac{8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac{16 e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x}}{15 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{24 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{5 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{\left (16 e^2\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x} \sqrt{1+x} \sqrt{a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{15 b^3 d}-\frac{\left (24 e^2\right ) \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 \sqrt{a+b x}}-\frac{3 \cosh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}\\ &=-\frac{2 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac{8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac{16 e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x}}{15 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{24 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{5 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{\left (16 e^2\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac{\left (6 e^2\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}+\frac{\left (18 e^2\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}\\ &=-\frac{2 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac{8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac{16 e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x}}{15 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{24 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{5 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{\left (8 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}-\frac{\left (8 e^2\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}+\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}+\frac{\left (9 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}+\frac{\left (9 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}\\ &=-\frac{2 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac{8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac{16 e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x}}{15 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{24 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{5 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{\left (16 e^2\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{15 b^4 d}-\frac{\left (16 e^2\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{15 b^4 d}+\frac{\left (6 e^2\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{5 b^4 d}+\frac{\left (6 e^2\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{5 b^4 d}+\frac{\left (18 e^2\right ) \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{5 b^4 d}+\frac{\left (18 e^2\right ) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{5 b^4 d}\\ &=-\frac{2 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac{8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac{16 e^2 \sqrt{-1+c+d x} \sqrt{1+c+d x}}{15 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}-\frac{24 e^2 \sqrt{-1+c+d x} (c+d x)^2 \sqrt{1+c+d x}}{5 b^3 d \sqrt{a+b \cosh ^{-1}(c+d x)}}+\frac{e^2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d}+\frac{3 e^2 e^{\frac{3 a}{b}} \sqrt{3 \pi } \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{5 b^{7/2} d}+\frac{e^2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{15 b^{7/2} d}+\frac{3 e^2 e^{-\frac{3 a}{b}} \sqrt{3 \pi } \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{5 b^{7/2} d}\\ \end{align*}

Mathematica [A]  time = 2.78614, size = 452, normalized size = 1.05 \[ \frac{e^2 \left (-2 e^{-\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \left (2 e^{\frac{a}{b}+\cosh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c+d x)\right )-2 a-2 b \cosh ^{-1}(c+d x)+b\right )-2 e^{-\frac{a}{b}} \left (a+b \cosh ^{-1}(c+d x)\right ) \left (2 b \left (-\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )+e^{\frac{a}{b}+\cosh ^{-1}(c+d x)} \left (2 a+2 b \cosh ^{-1}(c+d x)+b\right )\right )-3 \left (a+b \cosh ^{-1}(c+d x)\right ) \left (12 \sqrt{3} b e^{-\frac{3 a}{b}} \left (-\frac{a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+2 e^{-3 \cosh ^{-1}(c+d x)} \left (6 \sqrt{3} e^{3 \left (\frac{a}{b}+\cosh ^{-1}(c+d x)\right )} \sqrt{\frac{a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \text{Gamma}\left (\frac{1}{2},\frac{3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+6 a \left (e^{6 \cosh ^{-1}(c+d x)}-1\right )-6 b \cosh ^{-1}(c+d x)+b e^{6 \cosh ^{-1}(c+d x)} \left (6 \cosh ^{-1}(c+d x)+1\right )+b\right )\right )-6 b^2 \sqrt{\frac{c+d x-1}{c+d x+1}} (c+d x+1)-6 b^2 \sinh \left (3 \cosh ^{-1}(c+d x)\right )\right )}{60 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c*e + d*e*x)^2/(a + b*ArcCosh[c + d*x])^(7/2),x]

[Out]

(e^2*(-6*b^2*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x) - (2*(a + b*ArcCosh[c + d*x])*(-2*a + b - 2*b*Ar
cCosh[c + d*x] + 2*E^(a/b + ArcCosh[c + d*x])*Sqrt[a/b + ArcCosh[c + d*x]]*(a + b*ArcCosh[c + d*x])*Gamma[1/2,
a/b + ArcCosh[c + d*x]]))/E^ArcCosh[c + d*x] - (2*(a + b*ArcCosh[c + d*x])*(E^(a/b + ArcCosh[c + d*x])*(2*a +
b + 2*b*ArcCosh[c + d*x]) + 2*b*(-((a + b*ArcCosh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcCosh[c + d*x])/b
)]))/E^(a/b) - 3*(a + b*ArcCosh[c + d*x])*((12*Sqrt[3]*b*(-((a + b*ArcCosh[c + d*x])/b))^(3/2)*Gamma[1/2, (-3*
(a + b*ArcCosh[c + d*x]))/b])/E^((3*a)/b) + (2*(b + 6*a*(-1 + E^(6*ArcCosh[c + d*x])) - 6*b*ArcCosh[c + d*x] +
b*E^(6*ArcCosh[c + d*x])*(1 + 6*ArcCosh[c + d*x]) + 6*Sqrt[3]*E^(3*(a/b + ArcCosh[c + d*x]))*Sqrt[a/b + ArcCo
sh[c + d*x]]*(a + b*ArcCosh[c + d*x])*Gamma[1/2, (3*(a + b*ArcCosh[c + d*x]))/b]))/E^(3*ArcCosh[c + d*x])) - 6
*b^2*Sinh[3*ArcCosh[c + d*x]]))/(60*b^3*d*(a + b*ArcCosh[c + d*x])^(5/2))

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Maple [F]  time = 0.237, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dex+ce \right ) ^{2} \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{-{\frac{7}{2}}}}\, dx \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))^(7/2),x)

[Out]

int((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^(7/2),x)

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Maxima [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 )}^{\frac{7}{2}}}\,{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))^(7/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^2/(b*arccosh(d*x + c) + a)^(7/2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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))^(7/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [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*acosh(d*x+c))**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} 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))^(7/2),x, algorithm="giac")

[Out]

sage0*x