3.71 \(\int \frac{(f+g x)^3 (a+b \cosh ^{-1}(c x))}{(d-c^2 d x^2)^{3/2}} \, dx\)

Optimal. Leaf size=549 \[ -\frac{3 f g^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}-\frac{(1-c x) (c f-g)^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{(c x+1) (c f+g)^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{g^3 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} (c f-g)^3 \log \left (\frac{2}{c x+1}\right )}{2 c^4 d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} (c f+g)^3 \log \left (\sqrt{-\frac{1-c x}{c x+1}}\right )}{c^4 d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} (c f+g)^3 \log \left (\frac{2}{c x+1}\right )}{2 c^4 d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2}}+\frac{b g^3 x \sqrt{c x-1} \sqrt{c x+1}}{c^3 d \sqrt{d-c^2 d x^2}} \]

[Out]

(b*g^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c^3*d*Sqrt[d - c^2*d*x^2]) - ((c*f - g)^3*(1 - c*x)*(a + b*ArcCosh[c*x
]))/(2*c^4*d*Sqrt[d - c^2*d*x^2]) + ((c*f + g)^3*(1 + c*x)*(a + b*ArcCosh[c*x]))/(2*c^4*d*Sqrt[d - c^2*d*x^2])
+ (g^3*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(c^4*d*Sqrt[d - c^2*d*x^2]) - (3*f*g^2*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]*(a + b*ArcCosh[c*x])^2)/(2*b*c^3*d*Sqrt[d - c^2*d*x^2]) + (b*(c*f + g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqr
t[1 - c^2*x^2]*Log[Sqrt[-((1 - c*x)/(1 + c*x))]])/(c^4*d*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d
*x^2]) - (b*(c*f - g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[2/(1 + c*x)])/(2*c^4*d*Sqrt[-((1 - c*x
)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (b*(c*f + g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[
2/(1 + c*x)])/(2*c^4*d*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.59949, antiderivative size = 549, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 14, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.452, Rules used = {5836, 5834, 37, 5848, 12, 6719, 260, 266, 36, 31, 29, 5676, 5718, 8} \[ -\frac{3 f g^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}-\frac{(1-c x) (c f-g)^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{(c x+1) (c f+g)^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{g^3 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} (c f-g)^3 \log \left (\frac{2}{c x+1}\right )}{2 c^4 d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} (c f+g)^3 \log \left (\sqrt{-\frac{1-c x}{c x+1}}\right )}{c^4 d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} (c f+g)^3 \log \left (\frac{2}{c x+1}\right )}{2 c^4 d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2}}+\frac{b g^3 x \sqrt{c x-1} \sqrt{c x+1}}{c^3 d \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*g^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c^3*d*Sqrt[d - c^2*d*x^2]) - ((c*f - g)^3*(1 - c*x)*(a + b*ArcCosh[c*x
]))/(2*c^4*d*Sqrt[d - c^2*d*x^2]) + ((c*f + g)^3*(1 + c*x)*(a + b*ArcCosh[c*x]))/(2*c^4*d*Sqrt[d - c^2*d*x^2])
+ (g^3*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(c^4*d*Sqrt[d - c^2*d*x^2]) - (3*f*g^2*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]*(a + b*ArcCosh[c*x])^2)/(2*b*c^3*d*Sqrt[d - c^2*d*x^2]) + (b*(c*f + g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqr
t[1 - c^2*x^2]*Log[Sqrt[-((1 - c*x)/(1 + c*x))]])/(c^4*d*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d
*x^2]) - (b*(c*f - g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[2/(1 + c*x)])/(2*c^4*d*Sqrt[-((1 - c*x
)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (b*(c*f + g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[
2/(1 + c*x)])/(2*c^4*d*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d*x^2])

Rule 5836

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol]
:> Dist[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f + g*x
)^m*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d
+ e, 0] && IntegerQ[m] && IntegerQ[p - 1/2]

Rule 5834

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 5848

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHide[u, x]}, Dist[a + b*ArcCosh[c*x],
v, x] - Dist[(b*c*Sqrt[1 - c^2*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), Int[SimplifyIntegrand[v/Sqrt[1 - c^2*x^2]
, x], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, 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 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{(f+g x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{(f+g x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \left (-\frac{(c f-g)^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 \sqrt{-1+c x} (1+c x)^{3/2}}+\frac{(c f+g)^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 (-1+c x)^{3/2} \sqrt{1+c x}}+\frac{3 f g^2 \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{g^3 x \left (a+b \cosh ^{-1}(c x)\right )}{c^2 \sqrt{-1+c x} \sqrt{1+c x}}\right ) \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{\left ((c f-g)^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} (1+c x)^{3/2}} \, dx}{2 c^3 d \sqrt{d-c^2 d x^2}}-\frac{\left (3 f g^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\left (g^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\left ((c f+g)^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{(-1+c x)^{3/2} \sqrt{1+c x}} \, dx}{2 c^3 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{(c f-g)^3 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}+\frac{\left (b g^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int 1 \, dx}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f-g)^3 \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{\frac{-1+c x}{1+c x}}}{c \sqrt{1-c^2 x^2}} \, dx}{2 c^2 d \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f+g)^3 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{c \sqrt{\frac{-1+c x}{1+c x}} \sqrt{1-c^2 x^2}} \, dx}{2 c^2 d \sqrt{d-c^2 d x^2}}\\ &=\frac{b g^3 x \sqrt{-1+c x} \sqrt{1+c x}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^3 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f-g)^3 \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{\frac{-1+c x}{1+c x}}}{\sqrt{1-c^2 x^2}} \, dx}{2 c^3 d \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f+g)^3 \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\sqrt{\frac{-1+c x}{1+c x}} \sqrt{1-c^2 x^2}} \, dx}{2 c^3 d \sqrt{d-c^2 d x^2}}\\ &=\frac{b g^3 x \sqrt{-1+c x} \sqrt{1+c x}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^3 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}+\frac{\left (b (c f-g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \sqrt{-\frac{x^2}{\left (-1+x^2\right )^2}} \, dx,x,\sqrt{\frac{-1+c x}{1+c x}}\right )}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{\left (b (c f+g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{-\frac{x^2}{\left (-1+x^2\right )^2}}}{x^2} \, dx,x,\sqrt{\frac{-1+c x}{1+c x}}\right )}{c^4 d \sqrt{d-c^2 d x^2}}\\ &=\frac{b g^3 x \sqrt{-1+c x} \sqrt{1+c x}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^3 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f-g)^3 \sqrt{-(-1+c x) (1+c x)} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{-1+x^2} \, dx,x,\sqrt{\frac{-1+c x}{1+c x}}\right )}{c^4 d \sqrt{\frac{-1+c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f+g)^3 \sqrt{-(-1+c x) (1+c x)} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-1+x^2\right )} \, dx,x,\sqrt{\frac{-1+c x}{1+c x}}\right )}{c^4 d \sqrt{\frac{-1+c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}\\ &=\frac{b g^3 x \sqrt{-1+c x} \sqrt{1+c x}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^3 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}+\frac{b (c f-g)^3 \sqrt{(1-c x) (1+c x)} \sqrt{1-c^2 x^2} \log (1+c x)}{2 c^4 d \sqrt{-\frac{1-c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f+g)^3 \sqrt{-(-1+c x) (1+c x)} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{(-1+x) x} \, dx,x,\frac{-1+c x}{1+c x}\right )}{2 c^4 d \sqrt{\frac{-1+c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}\\ &=\frac{b g^3 x \sqrt{-1+c x} \sqrt{1+c x}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^3 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}+\frac{b (c f-g)^3 \sqrt{(1-c x) (1+c x)} \sqrt{1-c^2 x^2} \log (1+c x)}{2 c^4 d \sqrt{-\frac{1-c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f+g)^3 \sqrt{-(-1+c x) (1+c x)} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,\frac{-1+c x}{1+c x}\right )}{2 c^4 d \sqrt{\frac{-1+c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}+\frac{\left (b (c f+g)^3 \sqrt{-(-1+c x) (1+c x)} \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\frac{-1+c x}{1+c x}\right )}{2 c^4 d \sqrt{\frac{-1+c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}\\ &=\frac{b g^3 x \sqrt{-1+c x} \sqrt{1+c x}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^3 (1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{g^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}+\frac{b (c f+g)^3 \sqrt{(1-c x) (1+c x)} \sqrt{1-c^2 x^2} \log \left (-\frac{1-c x}{1+c x}\right )}{2 c^4 d \sqrt{-\frac{1-c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}+\frac{b (c f-g)^3 \sqrt{(1-c x) (1+c x)} \sqrt{1-c^2 x^2} \log (1+c x)}{2 c^4 d \sqrt{-\frac{1-c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}+\frac{b (c f+g)^3 \sqrt{(1-c x) (1+c x)} \sqrt{1-c^2 x^2} \log (1+c x)}{2 c^4 d \sqrt{-\frac{1-c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 1.85331, size = 353, normalized size = 0.64 \[ \frac{-2 \sqrt{d} \left (-a \left (c^2 g \left (3 f^2+3 f g x-g^2 x^2\right )+c^4 f^3 x+2 g^3\right )+b c f \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (c^2 f^2+3 g^2\right ) \log \left (\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right )+b g \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (3 c^2 f^2+g^2\right ) \log \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )\right )+6 a c f g^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )+b \sqrt{d} \cosh ^{-1}(c x) \left (2 c^4 f^3 x+6 c^2 f g (f+g x)-g^3 \cosh \left (2 \cosh ^{-1}(c x)\right )+3 g^3\right )-3 b c \sqrt{d} f g^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2+b \sqrt{d} g^3 \sinh \left (2 \cosh ^{-1}(c x)\right )}{2 c^4 d^{3/2} \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*b*c*Sqrt[d]*f*g^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2 + 6*a*c*f*g^2*Sqrt[d - c^2*d*x^2]*Ar
cTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + b*Sqrt[d]*ArcCosh[c*x]*(3*g^3 + 2*c^4*f^3*x + 6*c^2
*f*g*(f + g*x) - g^3*Cosh[2*ArcCosh[c*x]]) - 2*Sqrt[d]*(-(a*(2*g^3 + c^4*f^3*x + c^2*g*(3*f^2 + 3*f*g*x - g^2*
x^2))) + b*c*f*(c^2*f^2 + 3*g^2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)
] + b*g*(3*c^2*f^2 + g^2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c*x]/2]]) + b*Sqrt[d]*g^3*Sinh
[2*ArcCosh[c*x]])/(2*c^4*d^(3/2)*Sqrt[d - c^2*d*x^2])

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Maple [B]  time = 0.319, size = 1238, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*g^3*x^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+2*a*g^3/d/c^4/(-c^2*d*x^2+d)^(1/2)+3*a*f*g^2*x/c^2/d/(-c^2*d*x^2+d)^(1/2
)-3*a*f*g^2/c^2/d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+3*a*f^2*g/c^2/d/(-c^2*d*x^2+d)^(1
/2)+a*f^3/d*x/(-c^2*d*x^2+d)^(1/2)-b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/(c^2*x^2-1)*x*f^3-b*(-d*(c^2*x^2-
1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c/(c^2*x^2-1)*f^3*arccosh(c*x)-3*b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*
x)/d^2/(c^2*x^2-1)*x^2*f^2*g+b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/c^4/d^2/(c^2*x^2-1)*(c*x+1)*(c*x-1)*g^3-3*b
*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/c^2/d^2/(c^2*x^2-1)*x*f*g^2+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1
)^(1/2)/c/d^2/(c^2*x^2-1)*ln(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)-1)*f^3+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*
x+1)^(1/2)/c^4/d^2/(c^2*x^2-1)*ln(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)-1)*g^3+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2
)*(c*x+1)^(1/2)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d^2/(c^2*x^2-1)*f^3-b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(
1/2)*(c*x+1)^(1/2)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^4/d^2/(c^2*x^2-1)*g^3-b*(-d*(c^2*x^2-1))^(1/2)*g^3/
c^3/d^2/(c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x+3/2*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2
/c^3/(c^2*x^2-1)*arccosh(c*x)^2*f*g^2-b*(-d*(c^2*x^2-1))^(1/2)*g^3/c^4/d^2/(c^2*x^2-1)*arccosh(c*x)-3*b*(-d*(c
^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c^3/(c^2*x^2-1)*f*arccosh(c*x)*g^2+3*b*(-d*(c^2*x^2-1))^(1/2)
*arccosh(c*x)/c^2/d^2/(c^2*x^2-1)*(c*x+1)*(c*x-1)*f^2*g+3*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)
/c^2/d^2/(c^2*x^2-1)*ln(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)-1)*f^2*g+3*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x
+1)^(1/2)/c^3/d^2/(c^2*x^2-1)*ln(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)-1)*f*g^2-3*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(
1/2)*(c*x+1)^(1/2)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^2/d^2/(c^2*x^2-1)*f^2*g+3*b*(-d*(c^2*x^2-1))^(1/2)*
(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c^3/d^2/(c^2*x^2-1)*f*g^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a g^{3} x^{3} + 3 \, a f g^{2} x^{2} + 3 \, a f^{2} g x + a f^{3} +{\left (b g^{3} x^{3} + 3 \, b f g^{2} x^{2} + 3 \, b f^{2} g x + b f^{3}\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a*g^3*x^3 + 3*a*f*g^2*x^2 + 3*a*f^2*g*x + a*f^3 + (b*g^3*x^3 + 3*b*f*g^2*x^2 + 3*b*f^2*g*x + b*f^3)*
arccosh(c*x))*sqrt(-c^2*d*x^2 + d)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acosh}{\left (c x \right )}\right ) \left (f + g x\right )^{3}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**3*(a+b*acosh(c*x))/(-c**2*d*x**2+d)**(3/2),x)

[Out]

Integral((a + b*acosh(c*x))*(f + g*x)**3/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x + f\right )}^{3}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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