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

Optimal. Leaf size=212 \[ -\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{3 b e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x) \sqrt{a+b \cosh ^{-1}(c+d x)}}{8 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d} \]

[Out]

(-3*b*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(8*d) - (e*(a + b*ArcCosh
[c + d*x])^(3/2))/(4*d) + (e*(c + d*x)^2*(a + b*ArcCosh[c + d*x])^(3/2))/(2*d) - (3*b^(3/2)*e*E^((2*a)/b)*Sqrt
[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(64*d) + (3*b^(3/2)*e*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqr
t[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(64*d*E^((2*a)/b))

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Rubi [A]  time = 0.687308, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {5866, 12, 5664, 5759, 5676, 5670, 5448, 3308, 2180, 2204, 2205} \[ -\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{3 b e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x) \sqrt{a+b \cosh ^{-1}(c+d x)}}{8 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]])/(8*d) - (e*(a + b*ArcCosh
[c + d*x])^(3/2))/(4*d) + (e*(c + d*x)^2*(a + b*ArcCosh[c + d*x])^(3/2))/(2*d) - (3*b^(3/2)*e*E^((2*a)/b)*Sqrt
[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(64*d) + (3*b^(3/2)*e*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqr
t[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(64*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) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int e x \left (a+b \cosh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e \operatorname{Subst}\left (\int x \left (a+b \cosh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{(3 b e) \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 )}{4 d}\\ &=-\frac{3 b e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{8 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{(3 b e) \operatorname{Subst}\left (\int \frac{\sqrt{a+b \cosh ^{-1}(x)}}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,c+d x\right )}{8 d}+\frac{\left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{16 d}\\ &=-\frac{3 b e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{8 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{\left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{16 d}\\ &=-\frac{3 b e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{8 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{\left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{16 d}\\ &=-\frac{3 b e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{8 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{\left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{32 d}\\ &=-\frac{3 b e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{8 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{\left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{64 d}+\frac{\left (3 b^2 e\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{64 d}\\ &=-\frac{3 b e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{8 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{(3 b e) \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 )}{32 d}+\frac{(3 b e) \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 )}{32 d}\\ &=-\frac{3 b e \sqrt{-1+c+d x} (c+d x) \sqrt{1+c+d x} \sqrt{a+b \cosh ^{-1}(c+d x)}}{8 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac{3 b^{3/2} e 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 )}{64 d}+\frac{3 b^{3/2} e 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 )}{64 d}\\ \end{align*}

Mathematica [B]  time = 7.85377, size = 1144, normalized size = 5.4 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

e*((a*c*Sqrt[-1 + c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]]*((E^((2*a)/b)*Gamma[3/2, a/b + ArcCosh[c + d*x]])/Sqrt
[a/b + ArcCosh[c + d*x]] + Gamma[3/2, -((a + b*ArcCosh[c + d*x])/b)]/Sqrt[-((a + b*ArcCosh[c + d*x])/b)]))/(2*
d*E^(a/b)*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*Sqrt[1 + c + d*x]) + (b*c*Sqrt[-1 + c + d*x]*(-12*Sqrt[(-1 + c +
d*x)/(1 + c + d*x)]*(1 + c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]] + 8*(c + d*x)*ArcCosh[c + d*x]*Sqrt[a + b*ArcCo
sh[c + d*x]] + ((2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]))/Sqrt[
b] + ((2*a - 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]))/Sqrt[b]))/(8*d*S
qrt[(-1 + c + d*x)/(1 + c + d*x)]*Sqrt[1 + c + d*x]) + (a*Sqrt[-1 + c + d*x]*(-32*c*(c + d*x)*Sqrt[a + b*ArcCo
sh[c + d*x]] + 8*Sqrt[a + b*ArcCosh[c + d*x]]*Cosh[2*ArcCosh[c + d*x]] + 8*Sqrt[b]*c*Sqrt[Pi]*Cosh[a/b]*Erfi[S
qrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]] - Sqrt[b]*Sqrt[2*Pi]*Cosh[(2*a)/b]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d
*x]])/Sqrt[b]] - 8*Sqrt[b]*c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*Sinh[a/b] + 8*Sqrt[b]*c*Sqrt[
Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]) + Sqrt[b]*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[
a + b*ArcCosh[c + d*x]])/Sqrt[b]]*Sinh[(2*a)/b] - Sqrt[b]*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]]
)/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b])))/(32*d*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*Sqrt[1 + c + d*x]) + (Sq
rt[-1 + c + d*x]*(-16*c*(-12*b*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]] +
8*b*(c + d*x)*ArcCosh[c + d*x]*Sqrt[a + b*ArcCosh[c + d*x]] + Sqrt[b]*(2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*Ar
cCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (2*a - 3*b)*Sqrt[b]*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x
]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + Sqrt[b]*(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[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*b*Sqrt[a + b*ArcCosh[c + d*x]]*(4*ArcCosh[c + d*x]*Cos
h[2*ArcCosh[c + d*x]] - 3*Sinh[2*ArcCosh[c + d*x]])))/(128*d*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*Sqrt[1 + c + d
*x]))

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Maple [F]  time = 0.127, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) \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)*(a+b*arccosh(d*x+c))^(3/2),x)

[Out]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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

[Out]

e*(Integral(a*c*sqrt(a + b*acosh(c + d*x)), x) + Integral(a*d*x*sqrt(a + b*acosh(c + d*x)), x) + Integral(b*c*
sqrt(a + b*acosh(c + d*x))*acosh(c + d*x), x) + Integral(b*d*x*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x), x))

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

[Out]

sage0*x