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

Optimal. Leaf size=374 \[ -\frac {3 \sqrt {\pi } b^{3/2} e^3 e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2048 d}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^3 e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}+\frac {3 \sqrt {\pi } b^{3/2} e^3 e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2048 d}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^3 e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {3 b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}-\frac {9 b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}}{64 d}-\frac {3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d} \]

[Out]

-3/32*e^3*(a+b*arccosh(d*x+c))^(3/2)/d+1/4*e^3*(d*x+c)^4*(a+b*arccosh(d*x+c))^(3/2)/d-3/256*b^(3/2)*e^3*exp(2*
a/b)*erf(2^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d+3/256*b^(3/2)*e^3*erfi(2^(1/2)*(a+b*ar
ccosh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d/exp(2*a/b)-3/2048*b^(3/2)*e^3*exp(4*a/b)*erf(2*(a+b*arccosh(d*
x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d+3/2048*b^(3/2)*e^3*erfi(2*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d/exp(4
*a/b)-9/64*b*e^3*(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/d-3/32*b*e^3*(d*x+c)^3*(d*
x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/d

________________________________________________________________________________________

Rubi [A]  time = 1.41, antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5866, 12, 5664, 5759, 5676, 5670, 5448, 3308, 2180, 2204, 2205} \[ -\frac {3 \sqrt {\pi } b^{3/2} e^3 e^{\frac {4 a}{b}} \text {Erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2048 d}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^3 e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}+\frac {3 \sqrt {\pi } b^{3/2} e^3 e^{-\frac {4 a}{b}} \text {Erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2048 d}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^3 e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {3 b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}-\frac {9 b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}}{64 d}-\frac {3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*b*e^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(64*d) - (3*b*e^3*Sqrt[
-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(32*d) - (3*e^3*(a + b*ArcCosh[c + d
*x])^(3/2))/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcCosh[c + d*x])^(3/2))/(4*d) - (3*b^(3/2)*e^3*E^((4*a)/b)*Sqrt[
Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(2048*d) - (3*b^(3/2)*e^3*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[
2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(128*d) + (3*b^(3/2)*e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c + d*
x]])/Sqrt[b]])/(2048*d*E^((4*a)/b)) + (3*b^(3/2)*e^3*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sq
rt[b]])/(128*d*E^((2*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 5664

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcCosh[c*x])^n)/
(m + 1), x] - Dist[(b*c*n)/(m + 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] && GtQ[n, 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 5676

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

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

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 (c e+d e x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int e^3 x^3 \left (a+b \cosh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \operatorname {Subst}\left (\int x^3 \left (a+b \cosh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {\left (3 b e^3\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {a+b \cosh ^{-1}(x)}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{8 d}\\ &=-\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {\left (9 b e^3\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a+b \cosh ^{-1}(x)}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{32 d}+\frac {\left (3 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{64 d}\\ &=-\frac {9 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{64 d}-\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {\left (9 b e^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b \cosh ^{-1}(x)}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{64 d}+\frac {\left (3 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{64 d}+\frac {\left (9 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{128 d}\\ &=-\frac {9 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{64 d}-\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}-\frac {3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {\left (3 b^2 e^3\right ) \operatorname {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 \sqrt {a+b x}}+\frac {\sinh (4 x)}{8 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{64 d}+\frac {\left (9 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{128 d}\\ &=-\frac {9 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{64 d}-\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}-\frac {3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {\left (3 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{512 d}+\frac {\left (3 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{256 d}+\frac {\left (9 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{128 d}\\ &=-\frac {9 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{64 d}-\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}-\frac {3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {\left (3 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{1024 d}+\frac {\left (3 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {e^{4 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{1024 d}-\frac {\left (3 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{512 d}+\frac {\left (3 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{512 d}+\frac {\left (9 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{256 d}\\ &=-\frac {9 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{64 d}-\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}-\frac {3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {\left (3 b e^3\right ) \operatorname {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{512 d}+\frac {\left (3 b e^3\right ) \operatorname {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{512 d}-\frac {\left (3 b e^3\right ) \operatorname {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{256 d}+\frac {\left (3 b e^3\right ) \operatorname {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{256 d}-\frac {\left (9 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{512 d}+\frac {\left (9 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{512 d}\\ &=-\frac {9 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{64 d}-\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}-\frac {3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {3 b^{3/2} e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2048 d}-\frac {3 b^{3/2} e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{512 d}+\frac {3 b^{3/2} e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2048 d}+\frac {3 b^{3/2} e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{512 d}-\frac {\left (9 b e^3\right ) \operatorname {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{256 d}+\frac {\left (9 b e^3\right ) \operatorname {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{256 d}\\ &=-\frac {9 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{64 d}-\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{32 d}-\frac {3 e^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {3 b^{3/2} e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2048 d}-\frac {3 b^{3/2} e^3 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}+\frac {3 b^{3/2} e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2048 d}+\frac {3 b^{3/2} e^3 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 3.95, size = 558, normalized size = 1.49 \[ e^3 \left (\frac {a e^{-\frac {4 a}{b}} \sqrt {a+b \cosh ^{-1}(c+d x)} \left (\sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+4 \sqrt {2} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c+d x)}{b}} \left (4 \sqrt {2} \Gamma \left (\frac {3}{2},\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )\right )\right )}{128 d \sqrt {-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{b^2}}}+\frac {\sqrt {b} \left (8 \left (\sqrt {2 \pi } (4 a-3 b) \left (\sinh \left (\frac {2 a}{b}\right )+\cosh \left (\frac {2 a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } (4 a+3 b) \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )+8 \sqrt {b} \left (4 \cosh ^{-1}(c+d x) \cosh \left (2 \cosh ^{-1}(c+d x)\right )-3 \sinh \left (2 \cosh ^{-1}(c+d x)\right )\right ) \sqrt {a+b \cosh ^{-1}(c+d x)}\right )+\sqrt {\pi } (8 a-3 b) \left (\sinh \left (\frac {4 a}{b}\right )+\cosh \left (\frac {4 a}{b}\right )\right ) \text {erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )+\sqrt {\pi } (8 a+3 b) \left (\cosh \left (\frac {4 a}{b}\right )-\sinh \left (\frac {4 a}{b}\right )\right ) \text {erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )+8 \sqrt {b} \left (8 \cosh ^{-1}(c+d x) \cosh \left (4 \cosh ^{-1}(c+d x)\right )-3 \sinh \left (4 \cosh ^{-1}(c+d x)\right )\right ) \sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{2048 d}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

e^3*((a*Sqrt[a + b*ArcCosh[c + d*x]]*(Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[3/2, (-4*(a + b*ArcCosh[c + d*x]))/b]
+ 4*Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c + d*x]]*Gamma[3/2, (-2*(a + b*ArcCosh[c + d*x]))/b] + E^((6*a)/b
)*Sqrt[-((a + b*ArcCosh[c + d*x])/b)]*(4*Sqrt[2]*Gamma[3/2, (2*(a + b*ArcCosh[c + d*x]))/b] + E^((2*a)/b)*Gamm
a[3/2, (4*(a + b*ArcCosh[c + d*x]))/b])))/(128*d*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c + d*x])^2/b^2)]) + (Sqrt[
b]*((8*a + 3*b)*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(4*a)/b] - Sinh[(4*a)/b]) + (8*a
- 3*b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(4*a)/b] + Sinh[(4*a)/b]) + 8*((4*a + 3*b
)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] - Sinh[(2*a)/b]) + (4*a - 3*b
)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + 8*Sqrt[b]*S
qrt[a + b*ArcCosh[c + d*x]]*(4*ArcCosh[c + d*x]*Cosh[2*ArcCosh[c + d*x]] - 3*Sinh[2*ArcCosh[c + d*x]])) + 8*Sq
rt[b]*Sqrt[a + b*ArcCosh[c + d*x]]*(8*ArcCosh[c + d*x]*Cosh[4*ArcCosh[c + d*x]] - 3*Sinh[4*ArcCosh[c + d*x]]))
)/(2048*d))

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right )^{3} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e**3*(Integral(a*c**3*sqrt(a + b*acosh(c + d*x)), x) + Integral(a*d**3*x**3*sqrt(a + b*acosh(c + d*x)), x) + I
ntegral(b*c**3*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x), x) + Integral(3*a*c*d**2*x**2*sqrt(a + b*acosh(c + d
*x)), x) + Integral(3*a*c**2*d*x*sqrt(a + b*acosh(c + d*x)), x) + Integral(b*d**3*x**3*sqrt(a + b*acosh(c + d*
x))*acosh(c + d*x), x) + Integral(3*b*c*d**2*x**2*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x), x) + Integral(3*b
*c**2*d*x*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x), x))

________________________________________________________________________________________