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]

(-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))

________________________________________________________________________________________

Rubi [A]  time = 1.41225, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, 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 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 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 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 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 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 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 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]

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 = 3.72672, size = 558, normalized size = 1.49 \[ e^3 \left (\frac{a e^{-\frac{4 a}{b}} \sqrt{a+b \cosh ^{-1}(c+d x)} \left (\sqrt{\frac{a}{b}+\cosh ^{-1}(c+d x)} \text{Gamma}\left (\frac{3}{2},-\frac{4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+4 \sqrt{2} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c+d x)} \text{Gamma}\left (\frac{3}{2},-\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c+d x)}{b}} \left (4 \sqrt{2} \text{Gamma}\left (\frac{3}{2},\frac{2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+e^{\frac{2 a}{b}} \text{Gamma}\left (\frac{3}{2},\frac{4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )\right )\right )}{128 d \sqrt{-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2}{b^2}}}+\frac{\sqrt{b} \left (8 \left (\sqrt{2 \pi } (4 a-3 b) \left (\sinh \left (\frac{2 a}{b}\right )+\cosh \left (\frac{2 a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )+\sqrt{2 \pi } (4 a+3 b) \left (\cosh \left (\frac{2 a}{b}\right )-\sinh \left (\frac{2 a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )+8 \sqrt{b} \left (4 \cosh ^{-1}(c+d x) \cosh \left (2 \cosh ^{-1}(c+d x)\right )-3 \sinh \left (2 \cosh ^{-1}(c+d x)\right )\right ) \sqrt{a+b \cosh ^{-1}(c+d x)}\right )+\sqrt{\pi } (8 a-3 b) \left (\sinh \left (\frac{4 a}{b}\right )+\cosh \left (\frac{4 a}{b}\right )\right ) \text{Erf}\left (\frac{2 \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )+\sqrt{\pi } (8 a+3 b) \left (\cosh \left (\frac{4 a}{b}\right )-\sinh \left (\frac{4 a}{b}\right )\right ) \text{Erfi}\left (\frac{2 \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )+8 \sqrt{b} \left (8 \cosh ^{-1}(c+d x) \cosh \left (4 \cosh ^{-1}(c+d x)\right )-3 \sinh \left (4 \cosh ^{-1}(c+d x)\right )\right ) \sqrt{a+b \cosh ^{-1}(c+d x)}\right )}{2048 d}\right ) \]

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))

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Maple [F]  time = 0.249, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{3} \left ( a+b{\rm arccosh} \left (dx+c\right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)

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

[Out]

sage0*x