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

Optimal. Leaf size=459 \[ -\frac{(1-c x) (c f-g)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt{d-c^2 d x^2}}+\frac{(c x+1) (c f+g)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt{d-c^2 d x^2}}-\frac{g^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} (c f-g)^2 \log \left (\frac{2}{c x+1}\right )}{2 c^3 d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} (c f+g)^2 \log \left (\sqrt{-\frac{1-c x}{c x+1}}\right )}{c^3 d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} (c f+g)^2 \log \left (\frac{2}{c x+1}\right )}{2 c^3 d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.26664, antiderivative size = 459, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.387, Rules used = {5836, 5834, 37, 5848, 12, 6719, 260, 266, 36, 31, 29, 5676} \[ -\frac{(1-c x) (c f-g)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt{d-c^2 d x^2}}+\frac{(c x+1) (c f+g)^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^3 d \sqrt{d-c^2 d x^2}}-\frac{g^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} (c f-g)^2 \log \left (\frac{2}{c x+1}\right )}{2 c^3 d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} (c f+g)^2 \log \left (\sqrt{-\frac{1-c x}{c x+1}}\right )}{c^3 d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} (c f+g)^2 \log \left (\frac{2}{c x+1}\right )}{2 c^3 d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.07785, size = 281, normalized size = 0.61 \[ \frac{2 \sqrt{d} \left (a c \left (c^2 f^2 x+2 f g+g^2 x\right )-b \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (c^2 f^2+g^2\right ) \log \left (\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right )-2 b c f g \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )\right )+2 a g^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )+2 b c \sqrt{d} \cosh ^{-1}(c x) \left (c^2 f^2 x+2 f g+g^2 x\right )-b \sqrt{d} g^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2}{2 c^3 d^{3/2} \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.276, size = 879, normalized size = 1.9 \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)^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x)

[Out]

a*g^2*x/c^2/d/(-c^2*d*x^2+d)^(1/2)-a*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))+2*a*
f*g/c^2/d/(-c^2*d*x^2+d)^(1/2)+a*f^2/d*x/(-c^2*d*x^2+d)^(1/2)+1/2*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)*g^2*arccosh(c*x)^2-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)*arccosh(c*x)*f^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
)*g^2+2*b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/c^2/(c^2*x^2-1)*(c*x-1)*(c*x+1)*f*g-2*b*(-d*(c^2*x^2-1))^(1/
2)*arccosh(c*x)/d^2/(c^2*x^2-1)*x^2*f*g-b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/(c^2*x^2-1)*x*f^2-b*(-d*(c^2
*x^2-1))^(1/2)*arccosh(c*x)/d^2/c^2/(c^2*x^2-1)*x*g^2+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)*ln(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)-1)*f^2+2*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2
)/d^2/c^2/(c^2*x^2-1)*ln(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)-1)*f*g+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)*ln(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)-1)*g^2+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)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*f^2-2*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/
2)*(c*x+1)^(1/2)/d^2/c^2/(c^2*x^2-1)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*f*g+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)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*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)^2*(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{\sqrt{-c^{2} d x^{2} + d}{\left (a g^{2} x^{2} + 2 \, a f g x + a f^{2} +{\left (b g^{2} x^{2} + 2 \, b f g x + b f^{2}\right )} \operatorname{arcosh}\left (c x\right )\right )}}{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)^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

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

[Out]

Integral((a + b*acosh(c*x))*(f + g*x)**2/(-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 )}^{2}{\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)^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="giac")

[Out]

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