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

Optimal. Leaf size=773 \[ -\frac{b g^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{d \sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac{b g^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{d \sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac{g^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}+1\right )}{d \sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac{g^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}+1\right )}{d \sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac{(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2} (c f-g)}+\frac{(c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2} (c f+g)}+\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} \log \left (\sqrt{-\frac{1-c x}{c x+1}}\right )}{d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2} (c f+g)}-\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} \log \left (\frac{2}{c x+1}\right )}{2 d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2} (c f-g)}-\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} \log \left (\frac{2}{c x+1}\right )}{2 d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2} (c f+g)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.85733, antiderivative size = 773, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 17, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.548, Rules used = {5836, 5834, 37, 5848, 12, 6719, 260, 266, 36, 31, 29, 5832, 3320, 2264, 2190, 2279, 2391} \[ -\frac{b g^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{d \sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac{b g^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{d \sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac{g^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}+1\right )}{d \sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac{g^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}+1\right )}{d \sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac{(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2} (c f-g)}+\frac{(c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{2 d \sqrt{d-c^2 d x^2} (c f+g)}+\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} \log \left (\sqrt{-\frac{1-c x}{c x+1}}\right )}{d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2} (c f+g)}-\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} \log \left (\frac{2}{c x+1}\right )}{2 d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2} (c f-g)}-\frac{b \sqrt{(1-c x) (c x+1)} \sqrt{1-c^2 x^2} \log \left (\frac{2}{c x+1}\right )}{2 d \sqrt{-\frac{1-c x}{c x+1}} (c x+1) \sqrt{d-c^2 d x^2} (c f+g)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - c*x)*(a + b*ArcCosh[c*x]))/(2*d*(c*f - g)*Sqrt[d - c^2*d*x^2]) + ((1 + c*x)*(a + b*ArcCosh[c*x]))/(2*d*
(c*f + g)*Sqrt[d - c^2*d*x^2]) - (g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 + (E^ArcCosh[c*x
]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (g^2*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]*(a + b*ArcCosh[c*x])*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d*(c^2*f^2 - g^2)^(3/2)*
Sqrt[d - c^2*d*x^2]) + (b*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[Sqrt[-((1 - c*x)/(1 + c*x))]])/(d*(c
*f + g)*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^
2*x^2]*Log[2/(1 + c*x)])/(2*d*(c*f - g)*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[
(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[2/(1 + c*x)])/(2*d*(c*f + g)*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)
*Sqrt[d - c^2*d*x^2]) - (b*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^
2 - g^2]))])/(d*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]) + (b*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -(
(E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(d*(c^2*f^2 - g^2)^(3/2)*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 5832

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

Rule 3320

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol]
:> Dist[2, Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(E^(I*Pi*(k - 1/2))*(b + (2*a*E^(-(I*e) + f*fz*x))/E^(I*Pi*(k
- 1/2)) - (b*E^(2*(-(I*e) + f*fz*x)))/E^(2*I*k*Pi))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{(-1+c x)^{3/2} (1+c x)^{3/2} (f+g x)} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \left (-\frac{c \left (a+b \cosh ^{-1}(c x)\right )}{2 (c f-g) \sqrt{-1+c x} (1+c x)^{3/2}}+\frac{c \left (a+b \cosh ^{-1}(c x)\right )}{2 (c f+g) (-1+c x)^{3/2} \sqrt{1+c x}}+\frac{g^2 \left (a+b \cosh ^{-1}(c x)\right )}{(c f-g) (c f+g) \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}\right ) \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=\frac{\left (c \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 d (c f-g) \sqrt{d-c^2 d x^2}}-\frac{\left (c \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 d (c f+g) \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} (f+g x)} \, dx}{d \left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}\\ &=-\frac{(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{(1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f+g) \sqrt{d-c^2 d x^2}}-\frac{\left (g^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{a+b x}{c f+g \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{d \left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}-\frac{\left (b c^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 d (c f-g) \sqrt{d-c^2 d x^2}}-\frac{\left (b c^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 d (c f+g) \sqrt{d-c^2 d x^2}}\\ &=-\frac{(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{(1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f+g) \sqrt{d-c^2 d x^2}}-\frac{\left (2 g^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{2 c e^x f+g+e^{2 x} g} \, dx,x,\cosh ^{-1}(c x)\right )}{d \left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}}-\frac{\left (b c \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 d (c f-g) \sqrt{d-c^2 d x^2}}-\frac{\left (b c \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 d (c f+g) \sqrt{d-c^2 d x^2}}\\ &=-\frac{(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{(1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f+g) \sqrt{d-c^2 d x^2}}-\frac{\left (2 g^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{2 c f+2 e^x g-2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{\left (2 g^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{2 c f+2 e^x g+2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{\left (b \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 )}{d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{\left (b \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 )}{d (c f+g) \sqrt{d-c^2 d x^2}}\\ &=-\frac{(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{(1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f+g) \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 ) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \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 ) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{\left (b g^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^x g}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{\left (b g^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^x g}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{\left (b \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 )}{d (c f-g) \sqrt{\frac{-1+c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (b \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 )}{d (c f+g) \sqrt{\frac{-1+c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}\\ &=-\frac{(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{(1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f+g) \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 ) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \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 ) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{(1-c x) (1+c x)} \sqrt{1-c^2 x^2} \log (1+c x)}{2 d (c f-g) \sqrt{-\frac{1-c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}+\frac{\left (b g^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 g x}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{\left (b g^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 g x}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{\left (b \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 d (c f+g) \sqrt{\frac{-1+c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}\\ &=-\frac{(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{(1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f+g) \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 ) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \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 ) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{(1-c x) (1+c x)} \sqrt{1-c^2 x^2} \log (1+c x)}{2 d (c f-g) \sqrt{-\frac{1-c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}-\frac{b g^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{b g^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}-\frac{\left (b \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 d (c f+g) \sqrt{\frac{-1+c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}+\frac{\left (b \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 d (c f+g) \sqrt{\frac{-1+c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}\\ &=-\frac{(1-c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{(1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{2 d (c f+g) \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 ) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \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 ) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{(1-c x) (1+c x)} \sqrt{1-c^2 x^2} \log \left (-\frac{1-c x}{1+c x}\right )}{2 d (c f+g) \sqrt{-\frac{1-c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{(1-c x) (1+c x)} \sqrt{1-c^2 x^2} \log (1+c x)}{2 d (c f-g) \sqrt{-\frac{1-c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{(1-c x) (1+c x)} \sqrt{1-c^2 x^2} \log (1+c x)}{2 d (c f+g) \sqrt{-\frac{1-c x}{1+c x}} (1+c x) \sqrt{d-c^2 d x^2}}-\frac{b g^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}+\frac{b g^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [C]  time = 9.71484, size = 1203, normalized size = 1.56 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-(a*g) + a*c^2*f*x)*Sqrt[-(d*(-1 + c^2*x^2))])/(d^2*(-(c^2*f^2) + g^2)*(-1 + c^2*x^2)) + (a*g^2*Log[f + g*x]
)/(d^(3/2)*(-(c*f) + g)*(c*f + g)*Sqrt[-(c^2*f^2) + g^2]) - (a*g^2*Log[d*g + c^2*d*f*x + Sqrt[d]*Sqrt[-(c^2*f^
2) + g^2]*Sqrt[-(d*(-1 + c^2*x^2))]])/(d^(3/2)*(-(c*f) + g)*(c*f + g)*Sqrt[-(c^2*f^2) + g^2]) - (b*Sqrt[(-1 +
c*x)/(1 + c*x)]*(1 + c*x)*(-((ArcCosh[c*x]*Coth[ArcCosh[c*x]/2])/(c*f + g)) + (2*c*f*Log[Sqrt[(-1 + c*x)/(1 +
c*x)]*(1 + c*x)])/(c^2*f^2 - g^2) + (2*g*Log[Tanh[ArcCosh[c*x]/2]])/(-(c^2*f^2) + g^2) + (2*g^2*(2*ArcCosh[c*x
]*ArcTan[((c*f + g)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - (2*I)*ArcCos[-((c*f)/g)]*ArcTan[((-(c*f) +
g)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)] + 2*(ArcTan[((c*f + g)*Coth[ArcCosh[c*
x]/2])/Sqrt[-(c^2*f^2) + g^2]] + ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[Sqrt
[-(c^2*f^2) + g^2]/(Sqrt[2]*E^(ArcCosh[c*x]/2)*Sqrt[g]*Sqrt[c*f + c*g*x])] + (ArcCos[-((c*f)/g)] - 2*(ArcTan[(
(c*f + g)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^
2*f^2) + g^2]]))*Log[(E^(ArcCosh[c*x]/2)*Sqrt[-(c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*f + c*g*x])] - (ArcCo
s[-((c*f)/g)] + 2*ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*(c*f - g
+ I*Sqrt[-(c^2*f^2) + g^2])*(-1 + Tanh[ArcCosh[c*x]/2]))/(g*(c*f + g + I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c
*x]/2]))] - (ArcCos[-((c*f)/g)] - 2*ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((
c*f + g)*(-(c*f) + g + I*Sqrt[-(c^2*f^2) + g^2])*(1 + Tanh[ArcCosh[c*x]/2]))/(g*(c*f + g + I*Sqrt[-(c^2*f^2) +
g^2]*Tanh[ArcCosh[c*x]/2]))] + I*(PolyLog[2, ((c*f - I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - I*Sqrt[-(c^2*f^2) +
g^2]*Tanh[ArcCosh[c*x]/2]))/(g*(c*f + g + I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))] - PolyLog[2, ((c*f
+ I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))/(g*(c*f + g + I*Sqrt[-
(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))])))/((-(c*f) + g)*(c*f + g)*Sqrt[-(c^2*f^2) + g^2]) - (ArcCosh[c*x]*Ta
nh[ArcCosh[c*x]/2])/(c*f - g)))/(2*d*Sqrt[-(d*(-1 + c*x)*(1 + c*x))])

________________________________________________________________________________________

Maple [B]  time = 0.255, size = 2484, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*g/d/(c^2*f^2-g^2)/(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)+a*f/(c^2*f^2-g^2)/d/(-(x
+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)*c^2*x+a*g/d/(c^2*f^2-g^2)/(-d*(c^2*f^2-g^2)/g^2)^
(1/2)*ln((-2*d*(c^2*f^2-g^2)/g^2+2*c^2*d*f/g*(x+f/g)+2*(-d*(c^2*f^2-g^2)/g^2)^(1/2)*(-(x+f/g)^2*c^2*d+2*c^2*d*
f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))/(x+f/g))-b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/(c^2*x^2-1)/(c^2*f^
2-g^2)*(c*x-1)*(c*x+1)*g+b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/(c^2*x^2-1)/(c^2*f^2-g^2)*x^2*c^2*g+b*(-d*(
c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/(c^2*x^2-1)/(c^2*f^2-g^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*f-b*(-d*(c^2*x^2-1)
)^(1/2)*arccosh(c*x)/d^2/(c^2*x^2-1)/(c^2*f^2-g^2)*x*c^2*f-2*b*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1
/2)/d^2/(c^6*f^4*x^2-2*c^4*f^2*g^2*x^2-c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2-g^4)*ln(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/
2))*c^3*f^3+b*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^6*f^4*x^2-2*c^4*f^2*g^2*x^2-c^4*f^4+c^
2*g^4*x^2+2*c^2*f^2*g^2-g^4)*ln(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)-1)*c^3*f^3+b*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(
c^2*x^2-1))^(1/2)/d^2/(c^6*f^4*x^2-2*c^4*f^2*g^2*x^2-c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2-g^4)*ln(1+c*x+(c*x-1)^(
1/2)*(c*x+1)^(1/2))*c^3*f^3-b*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^6*f^4*x^2-2*c^4*f^2*g^
2*x^2-c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2-g^4)*ln(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)-1)*c^2*f^2*g+b*(c*x+1)^(1/2)*(
c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^6*f^4*x^2-2*c^4*f^2*g^2*x^2-c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2-g^4)*
ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*c^2*f^2*g+b*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^6*
f^4*x^2-2*c^4*f^2*g^2*x^2-c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2-g^4)*arccosh(c*x)*ln((-(c*x+(c*x-1)^(1/2)*(c*x+1)^
(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))*(c^2*f^2-g^2)^(1/2)*g^2-b*(c*x+1)^(1/2)*(c*x-1)^
(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^6*f^4*x^2-2*c^4*f^2*g^2*x^2-c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2-g^4)*arccosh
(c*x)*ln(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+c*f+(c^2*f^2-g^2)^(1/2))/(c*f+(c^2*f^2-g^2)^(1/2)))*(c^2*f^2-g^2
)^(1/2)*g^2+2*b*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^6*f^4*x^2-2*c^4*f^2*g^2*x^2-c^4*f^4+
c^2*g^4*x^2+2*c^2*f^2*g^2-g^4)*ln(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*c*f*g^2-b*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(
c^2*x^2-1))^(1/2)/d^2/(c^6*f^4*x^2-2*c^4*f^2*g^2*x^2-c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2-g^4)*ln(c*x+(c*x-1)^(1/
2)*(c*x+1)^(1/2)-1)*c*f*g^2-b*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^6*f^4*x^2-2*c^4*f^2*g^
2*x^2-c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2-g^4)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*c*f*g^2+b*(c*x+1)^(1/2)*(c*
x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^6*f^4*x^2-2*c^4*f^2*g^2*x^2-c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2-g^4)*ln
(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)-1)*g^3-b*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^6*f^4*x^2-
2*c^4*f^2*g^2*x^2-c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2-g^4)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g^3+b*(c*x+1)^(
1/2)*(c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^6*f^4*x^2-2*c^4*f^2*g^2*x^2-c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2
-g^4)*dilog((-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))*(c^2*f^
2-g^2)^(1/2)*g^2-b*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/(c^6*f^4*x^2-2*c^4*f^2*g^2*x^2-c^4*f
^4+c^2*g^4*x^2+2*c^2*f^2*g^2-g^4)*dilog(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+c*f+(c^2*f^2-g^2)^(1/2))/(c*f+(c^
2*f^2-g^2)^(1/2)))*(c^2*f^2-g^2)^(1/2)*g^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{c^{4} d^{2} g x^{5} + c^{4} d^{2} f x^{4} - 2 \, c^{2} d^{2} g x^{3} - 2 \, c^{2} d^{2} f x^{2} + d^{2} g x + d^{2} f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^4*d^2*g*x^5 + c^4*d^2*f*x^4 - 2*c^2*d^2*g*x^3 - 2*c^2*d^
2*f*x^2 + d^2*g*x + d^2*f), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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