3.186 \(\int \frac {(c e+d e x)^4}{(a+b \cosh ^{-1}(c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=444 \[ -\frac {\sqrt {\pi } e^4 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{12 b^{5/2} d}-\frac {3 \sqrt {3 \pi } e^4 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 b^{5/2} d}-\frac {5 \sqrt {5 \pi } e^4 e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{24 b^{5/2} d}+\frac {\sqrt {\pi } e^4 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{12 b^{5/2} d}+\frac {3 \sqrt {3 \pi } e^4 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 b^{5/2} d}+\frac {5 \sqrt {5 \pi } e^4 e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{24 b^{5/2} d}-\frac {20 e^4 (c+d x)^5}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {16 e^4 (c+d x)^3}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {2 e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \]

[Out]

-1/12*e^4*exp(a/b)*erf((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(5/2)/d+1/12*e^4*erfi((a+b*arccosh(d*x+c
))^(1/2)/b^(1/2))*Pi^(1/2)/b^(5/2)/d/exp(a/b)-3/8*e^4*exp(3*a/b)*erf(3^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2
))*3^(1/2)*Pi^(1/2)/b^(5/2)/d+3/8*e^4*erfi(3^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/b^(5/2
)/d/exp(3*a/b)-5/24*e^4*exp(5*a/b)*erf(5^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*5^(1/2)*Pi^(1/2)/b^(5/2)/d+
5/24*e^4*erfi(5^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*5^(1/2)*Pi^(1/2)/b^(5/2)/d/exp(5*a/b)-2/3*e^4*(d*x+c
)^4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c))^(3/2)+16/3*e^4*(d*x+c)^3/b^2/d/(a+b*arccosh(d*x+c
))^(1/2)-20/3*e^4*(d*x+c)^5/b^2/d/(a+b*arccosh(d*x+c))^(1/2)

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Rubi [A]  time = 1.77, antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5866, 12, 5668, 5775, 5670, 5448, 3308, 2180, 2204, 2205} \[ -\frac {\sqrt {\pi } e^4 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{12 b^{5/2} d}-\frac {3 \sqrt {3 \pi } e^4 e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 b^{5/2} d}-\frac {5 \sqrt {5 \pi } e^4 e^{\frac {5 a}{b}} \text {Erf}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{24 b^{5/2} d}+\frac {\sqrt {\pi } e^4 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{12 b^{5/2} d}+\frac {3 \sqrt {3 \pi } e^4 e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{8 b^{5/2} d}+\frac {5 \sqrt {5 \pi } e^4 e^{-\frac {5 a}{b}} \text {Erfi}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{24 b^{5/2} d}-\frac {20 e^4 (c+d x)^5}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {16 e^4 (c+d x)^3}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {2 e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^(5/2),x]

[Out]

(-2*e^4*Sqrt[-1 + c + d*x]*(c + d*x)^4*Sqrt[1 + c + d*x])/(3*b*d*(a + b*ArcCosh[c + d*x])^(3/2)) + (16*e^4*(c
+ d*x)^3)/(3*b^2*d*Sqrt[a + b*ArcCosh[c + d*x]]) - (20*e^4*(c + d*x)^5)/(3*b^2*d*Sqrt[a + b*ArcCosh[c + d*x]])
- (e^4*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]])/(12*b^(5/2)*d) - (3*e^4*E^((3*a)/b)*Sqrt[3
*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(8*b^(5/2)*d) - (5*e^4*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(S
qrt[5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(24*b^(5/2)*d) + (e^4*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]
]/Sqrt[b]])/(12*b^(5/2)*d*E^(a/b)) + (3*e^4*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(
8*b^(5/2)*d*E^((3*a)/b)) + (5*e^4*Sqrt[5*Pi]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(24*b^(5/2)
*d*E^((5*a)/b))

Rule 12

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

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 3308

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

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 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 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 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 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
]

Rubi steps

\begin {align*} \int \frac {(c e+d e x)^4}{\left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e^4 x^4}{\left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \operatorname {Subst}\left (\int \frac {x^4}{\left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {\left (8 e^4\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{3 b d}+\frac {\left (10 e^4\right ) \operatorname {Subst}\left (\int \frac {x^5}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac {2 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^4 (c+d x)^3}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {20 e^4 (c+d x)^5}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (16 e^4\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{b^2 d}+\frac {\left (100 e^4\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{3 b^2 d}\\ &=-\frac {2 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^4 (c+d x)^3}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {20 e^4 (c+d x)^5}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (16 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac {\left (100 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh ^4(x) \sinh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^2 d}\\ &=-\frac {2 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^4 (c+d x)^3}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {20 e^4 (c+d x)^5}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (16 e^4\right ) \operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{4 \sqrt {a+b x}}+\frac {\sinh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac {\left (100 e^4\right ) \operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{8 \sqrt {a+b x}}+\frac {3 \sinh (3 x)}{16 \sqrt {a+b x}}+\frac {\sinh (5 x)}{16 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^2 d}\\ &=-\frac {2 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^4 (c+d x)^3}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {20 e^4 (c+d x)^5}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int \frac {\sinh (5 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{12 b^2 d}-\frac {\left (4 e^4\right ) \operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac {\left (4 e^4\right ) \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{6 b^2 d}+\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{4 b^2 d}\\ &=-\frac {2 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^4 (c+d x)^3}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {20 e^4 (c+d x)^5}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{24 b^2 d}+\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{5 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{24 b^2 d}+\frac {\left (2 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac {\left (2 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac {\left (2 e^4\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac {\left (2 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}-\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{12 b^2 d}+\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{12 b^2 d}-\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}+\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}\\ &=-\frac {2 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^4 (c+d x)^3}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {20 e^4 (c+d x)^5}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int e^{\frac {5 a}{b}-\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{12 b^3 d}+\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int e^{-\frac {5 a}{b}+\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{12 b^3 d}+\frac {\left (4 e^4\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 )}{b^3 d}+\frac {\left (4 e^4\right ) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{b^3 d}-\frac {\left (4 e^4\right ) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{b^3 d}-\frac {\left (4 e^4\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 )}{b^3 d}-\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{6 b^3 d}+\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{6 b^3 d}-\frac {\left (25 e^4\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 )}{4 b^3 d}+\frac {\left (25 e^4\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 )}{4 b^3 d}\\ &=-\frac {2 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^4 (c+d x)^3}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {20 e^4 (c+d x)^5}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{12 b^{5/2} d}-\frac {3 e^4 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 )}{8 b^{5/2} d}-\frac {5 e^4 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{24 b^{5/2} d}+\frac {e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{12 b^{5/2} d}+\frac {3 e^4 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 )}{8 b^{5/2} d}+\frac {5 e^4 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{24 b^{5/2} d}\\ \end {align*}

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Mathematica [A]  time = 3.66, size = 615, normalized size = 1.39 \[ \frac {e^4 e^{-5 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )} \left (-10 \sqrt {5} b e^{5 \cosh ^{-1}(c+d x)} \left (-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {5 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )-18 \sqrt {3} b e^{\frac {2 a}{b}+5 \cosh ^{-1}(c+d x)} \left (-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+2 e^{4 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )} \left (2 e^{\frac {2 a}{b}+\cosh ^{-1}(c+d x)} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \Gamma \left (\frac {1}{2},\frac {a}{b}+\cosh ^{-1}(c+d x)\right )-2 \left (e^{a/b} \left (\left (e^{2 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )+b \sqrt {\frac {c+d x-1}{c+d x+1}} (c+d x+1) e^{\cosh ^{-1}(c+d x)}\right )+b e^{\cosh ^{-1}(c+d x)} \left (-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )\right )\right )+3 e^{\frac {5 a}{b}+2 \cosh ^{-1}(c+d x)} \left (6 \sqrt {3} e^{3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \Gamma \left (\frac {1}{2},\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )-6 a \left (e^{6 \cosh ^{-1}(c+d x)}+1\right )-6 b \cosh ^{-1}(c+d x)-b e^{6 \cosh ^{-1}(c+d x)} \left (6 \cosh ^{-1}(c+d x)+1\right )+b\right )+2 e^{\frac {5 a}{b}} \left (-5 \left (e^{10 \cosh ^{-1}(c+d x)}+1\right ) \left (a+b \cosh ^{-1}(c+d x)\right )+5 \sqrt {5} e^{5 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \Gamma \left (\frac {1}{2},\frac {5 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )-\frac {1}{2} b \left (e^{10 \cosh ^{-1}(c+d x)}-1\right )\right )\right )}{48 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c*e + d*e*x)^4/(a + b*ArcCosh[c + d*x])^(5/2),x]

[Out]

(e^4*(-10*Sqrt[5]*b*E^(5*ArcCosh[c + d*x])*(-((a + b*ArcCosh[c + d*x])/b))^(3/2)*Gamma[1/2, (-5*(a + b*ArcCosh
[c + d*x]))/b] - 18*Sqrt[3]*b*E^((2*a)/b + 5*ArcCosh[c + d*x])*(-((a + b*ArcCosh[c + d*x])/b))^(3/2)*Gamma[1/2
, (-3*(a + b*ArcCosh[c + d*x]))/b] + 2*E^(4*(a/b + ArcCosh[c + d*x]))*(2*E^((2*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]] - 2*(E^(a/b)*(b*E^ArcCosh[c
+ d*x]*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x) + (1 + E^(2*ArcCosh[c + d*x]))*(a + b*ArcCosh[c + d*x
])) + b*E^ArcCosh[c + d*x]*(-((a + b*ArcCosh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcCosh[c + d*x])/b)])) +
3*E^((5*a)/b + 2*ArcCosh[c + d*x])*(b - 6*a*(1 + E^(6*ArcCosh[c + d*x])) - 6*b*ArcCosh[c + d*x] - b*E^(6*ArcC
osh[c + d*x])*(1 + 6*ArcCosh[c + d*x]) + 6*Sqrt[3]*E^(3*(a/b + ArcCosh[c + d*x]))*Sqrt[a/b + ArcCosh[c + d*x]]
*(a + b*ArcCosh[c + d*x])*Gamma[1/2, (3*(a + b*ArcCosh[c + d*x]))/b]) + 2*E^((5*a)/b)*(-1/2*(b*(-1 + E^(10*Arc
Cosh[c + d*x]))) - 5*(1 + E^(10*ArcCosh[c + d*x]))*(a + b*ArcCosh[c + d*x]) + 5*Sqrt[5]*E^(5*(a/b + ArcCosh[c
+ d*x]))*Sqrt[a/b + ArcCosh[c + d*x]]*(a + b*ArcCosh[c + d*x])*Gamma[1/2, (5*(a + b*ArcCosh[c + d*x]))/b])))/(
48*b^2*d*E^(5*(a/b + ArcCosh[c + d*x]))*(a + b*ArcCosh[c + d*x])^(3/2))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^4/(b*arccosh(d*x + c) + a)^(5/2), x)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d e x +c e \right )^{4}}{\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(5/2),x)

[Out]

int((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(5/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^4/(a+b*arccosh(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^4/(b*arccosh(d*x + c) + a)^(5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^4/(a + b*acosh(c + d*x))^(5/2),x)

[Out]

int((c*e + d*e*x)^4/(a + b*acosh(c + d*x))^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{4} \left (\int \frac {c^{4}}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**4/(a+b*acosh(d*x+c))**(5/2),x)

[Out]

e**4*(Integral(c**4/(a**2*sqrt(a + b*acosh(c + d*x)) + 2*a*b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x) + b**2*
sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)**2), x) + Integral(d**4*x**4/(a**2*sqrt(a + b*acosh(c + d*x)) + 2*a*
b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x) + b**2*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)**2), x) + Integra
l(4*c*d**3*x**3/(a**2*sqrt(a + b*acosh(c + d*x)) + 2*a*b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x) + b**2*sqrt
(a + b*acosh(c + d*x))*acosh(c + d*x)**2), x) + Integral(6*c**2*d**2*x**2/(a**2*sqrt(a + b*acosh(c + d*x)) + 2
*a*b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x) + b**2*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)**2), x) + Inte
gral(4*c**3*d*x/(a**2*sqrt(a + b*acosh(c + d*x)) + 2*a*b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x) + b**2*sqrt
(a + b*acosh(c + d*x))*acosh(c + d*x)**2), x))

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