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

Optimal. Leaf size=918 \[ \frac{b f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)^2 c^3}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{c x-1} \sqrt{c x+1}}+\frac{a f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) c^3}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 a f \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{\sqrt{c f+g} \sqrt{c x+1}}{\sqrt{c f-g} \sqrt{c x-1}}\right ) c^2}{\sqrt{c f-g} g^2 \sqrt{c f+g} \sqrt{c x-1} \sqrt{c x+1}}-\frac{b f \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}+1\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b f \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}+1\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{c x-1} \sqrt{c x+1}}-\frac{b f \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b f \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \sqrt{d-c^2 d x^2} \log (f+g x) c}{g^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{-\frac{1-c x}{c x+1}} \sqrt{c x+1} \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g \sqrt{c x-1} (f+g x)}-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b \sqrt{c x-1} \sqrt{c x+1} (f+g x)^2 c}-\frac{\left (f x c^2+g\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b \left (c^2 f^2-g^2\right ) \sqrt{c x-1} \sqrt{c x+1} (f+g x)^2 c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 3.56108, antiderivative size = 918, normalized size of antiderivative = 1., number of steps used = 38, number of rules used = 22, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.71, Rules used = {5836, 5824, 37, 5814, 12, 180, 52, 96, 93, 208, 5860, 5858, 5676, 5832, 3324, 3320, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac{b f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)^2 c^3}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{c x-1} \sqrt{c x+1}}+\frac{a f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) c^3}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 a f \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{\sqrt{c f+g} \sqrt{c x+1}}{\sqrt{c f-g} \sqrt{c x-1}}\right ) c^2}{\sqrt{c f-g} g^2 \sqrt{c f+g} \sqrt{c x-1} \sqrt{c x+1}}-\frac{b f \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}+1\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b f \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}+1\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{c x-1} \sqrt{c x+1}}-\frac{b f \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b f \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \sqrt{d-c^2 d x^2} \log (f+g x) c}{g^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{-\frac{1-c x}{c x+1}} \sqrt{c x+1} \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g \sqrt{c x-1} (f+g x)}-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b \sqrt{c x-1} \sqrt{c x+1} (f+g x)^2 c}-\frac{\left (f x c^2+g\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b \left (c^2 f^2-g^2\right ) \sqrt{c x-1} \sqrt{c x+1} (f+g x)^2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 5824

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]*((f_) + (g_.
)*(x_))^(m_), x_Symbol] :> Simp[((f + g*x)^m*(d1*d2 + e1*e2*x^2)*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-(d1*
d2)]*(n + 1)), x] - Dist[1/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[(d1*d2*g*m + 2*e1*e2*f*x + e1*e2*g*(m + 2)*x^2)*(
f + g*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 -
c*d1, 0] && EqQ[e2 + c*d2, 0] && ILtQ[m, 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 5814

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(p_.), x_Symbol] :>
With[{u = IntHide[(f + g*x)^p*(d + e*x)^m, x]}, Dist[(a + b*ArcCosh[c*x])^n, u, x] - Dist[b*c*n, Int[Simplify
Integrand[(u*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e
, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 180

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_))^(q_), x
_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] /; FreeQ[{a, b, c, d,
e, f, g, h, m, n}, x] && IntegersQ[p, q]

Rule 52

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

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 5860

Int[(ArcCosh[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_S
ymbol] :> Int[ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p, RFx*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b,
c, d1, e1, d2, e2}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] &&
IntegerQ[p - 1/2]

Rule 5858

Int[ArcCosh[(c_.)*(x_)]^(n_.)*(RFx_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> With[
{u = ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p*ArcCosh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c,
d1, e1, d2, e2}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && In
tegerQ[p - 1/2]

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 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 3324

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*(c + d*x)^m*Cos[
e + f*x])/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[(b*d*m)/(f*(a^2 - b^2)), Int[((c + d*x)^(m - 1)*Cos[e + f*x])/(a + b*Sin[e + f*x]), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 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]

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 31

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

Rubi steps

\begin{align*} \int \frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{(f+g x)^2} \, dx &=\frac{\sqrt{d-c^2 d x^2} \int \frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{(f+g x)^2} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{\sqrt{d-c^2 d x^2} \int \frac{\left (2 g+2 c^2 f x\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{(f+g x)^3} \, dx}{2 b c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}+\frac{\sqrt{d-c^2 d x^2} \int \frac{\left (g+c^2 f x\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}+\frac{\sqrt{d-c^2 d x^2} \int \frac{\left (g+c^2 f x\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2} \, dx}{\left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}+\frac{\sqrt{d-c^2 d x^2} \int \left (\frac{a \left (g+c^2 f x\right )^2}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}+\frac{b \left (g+c^2 f x\right )^2 \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}\right ) \, dx}{\left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}+\frac{\left (a \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (g+c^2 f x\right )^2}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2} \, dx}{\left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (g+c^2 f x\right )^2 \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2} \, dx}{\left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}+\frac{\left (a \sqrt{d-c^2 d x^2}\right ) \int \left (\frac{c^4 f^2}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (-c^2 f^2+g^2\right )^2}{g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}+\frac{2 c^2 f \left (-c^2 f^2+g^2\right )}{g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}\right ) \, dx}{\left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b \sqrt{d-c^2 d x^2}\right ) \int \left (\frac{c^4 f^2 \cosh ^{-1}(c x)}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (-c^2 f^2+g^2\right )^2 \cosh ^{-1}(c x)}{g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}+\frac{2 c^2 f \left (-c^2 f^2+g^2\right ) \cosh ^{-1}(c x)}{g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}\right ) \, dx}{\left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}+\frac{\left (a c^4 f^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c^4 f^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (a \left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2} \, dx}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b \left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int \frac{\cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2} \, dx}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 a c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)} \, dx}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int \frac{\cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)} \, dx}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}+\frac{a c^3 f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}+\frac{\left (a c^2 f \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)} \, dx}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c \left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{(c f+g \cosh (x))^2} \, dx,x,\cosh ^{-1}(c x)\right )}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (4 a c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{c f-g-(c f+g) x^2} \, dx,x,\frac{\sqrt{1+c x}}{\sqrt{-1+c x}}\right )}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{c f+g \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}+\frac{a c^3 f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \sqrt{-\frac{1-c x}{1+c x}} \sqrt{1+c x} \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g \sqrt{-1+c x} (f+g x)}+\frac{b c^3 f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{4 a c^2 f \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{\sqrt{c f+g} \sqrt{1+c x}}{\sqrt{c f-g} \sqrt{-1+c x}}\right )}{\sqrt{c f-g} g^2 \sqrt{c f+g} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 a c^2 f \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{c f-g-(c f+g) x^2} \, dx,x,\frac{\sqrt{1+c x}}{\sqrt{-1+c x}}\right )}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c^2 f \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{c f+g \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{c f+g \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (4 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c e^x f+g+e^{2 x} g} \, dx,x,\cosh ^{-1}(c x)\right )}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}+\frac{a c^3 f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \sqrt{-\frac{1-c x}{1+c x}} \sqrt{1+c x} \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g \sqrt{-1+c x} (f+g x)}+\frac{b c^3 f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{2 a c^2 f \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{\sqrt{c f+g} \sqrt{1+c x}}{\sqrt{c f-g} \sqrt{-1+c x}}\right )}{\sqrt{c f-g} g^2 \sqrt{c f+g} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{c f+x} \, dx,x,c g x\right )}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b c^2 f \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c e^x f+g+e^{2 x} g} \, dx,x,\cosh ^{-1}(c x)\right )}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (4 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c f+2 e^x g-2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{g \left (c^2 f^2-g^2\right )^{3/2} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (4 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c f+2 e^x g+2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{g \left (c^2 f^2-g^2\right )^{3/2} \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}+\frac{a c^3 f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \sqrt{-\frac{1-c x}{1+c x}} \sqrt{1+c x} \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g \sqrt{-1+c x} (f+g x)}+\frac{b c^3 f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{2 a c^2 f \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{\sqrt{c f+g} \sqrt{1+c x}}{\sqrt{c f-g} \sqrt{-1+c x}}\right )}{\sqrt{c f-g} g^2 \sqrt{c f+g} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^2 f \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^2 f \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c \sqrt{d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b c^2 f \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c f+2 e^x g-2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{g \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b c^2 f \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c f+2 e^x g+2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{g \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\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 )}{g^2 \left (c^2 f^2-g^2\right )^{3/2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\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 )}{g^2 \left (c^2 f^2-g^2\right )^{3/2} \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}+\frac{a c^3 f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \sqrt{-\frac{1-c x}{1+c x}} \sqrt{1+c x} \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g \sqrt{-1+c x} (f+g x)}+\frac{b c^3 f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{2 a c^2 f \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{\sqrt{c f+g} \sqrt{1+c x}}{\sqrt{c f-g} \sqrt{-1+c x}}\right )}{\sqrt{c f-g} g^2 \sqrt{c f+g} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^2 f \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^2 f \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c \sqrt{d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c^2 f \sqrt{d-c^2 d x^2}\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 )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c^2 f \sqrt{d-c^2 d x^2}\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 )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\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 )}{g^2 \left (c^2 f^2-g^2\right )^{3/2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\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 )}{g^2 \left (c^2 f^2-g^2\right )^{3/2} \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}+\frac{a c^3 f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \sqrt{-\frac{1-c x}{1+c x}} \sqrt{1+c x} \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g \sqrt{-1+c x} (f+g x)}+\frac{b c^3 f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{2 a c^2 f \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{\sqrt{c f+g} \sqrt{1+c x}}{\sqrt{c f-g} \sqrt{-1+c x}}\right )}{\sqrt{c f-g} g^2 \sqrt{c f+g} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^2 f \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^2 f \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c \sqrt{d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{2 b c^2 f \sqrt{d-c^2 d x^2} \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^2 f \sqrt{d-c^2 d x^2} \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c^2 f \sqrt{d-c^2 d x^2}\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 )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c^2 f \sqrt{d-c^2 d x^2}\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 )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}+\frac{a c^3 f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \sqrt{-\frac{1-c x}{1+c x}} \sqrt{1+c x} \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g \sqrt{-1+c x} (f+g x)}+\frac{b c^3 f^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{-1+c x} \sqrt{1+c x} (f+g x)^2}-\frac{2 a c^2 f \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{\sqrt{c f+g} \sqrt{1+c x}}{\sqrt{c f-g} \sqrt{-1+c x}}\right )}{\sqrt{c f-g} g^2 \sqrt{c f+g} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^2 f \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^2 f \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c \sqrt{d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^2 f \sqrt{d-c^2 d x^2} \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^2 f \sqrt{d-c^2 d x^2} \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [C]  time = 7.0517, size = 1139, normalized size = 1.24 \[ \frac{\frac{2 a \sqrt{d} f \log (f+g x) c^2}{\sqrt{g^2-c^2 f^2}}-\frac{2 a \sqrt{d} f \log \left (d \left (f x c^2+g\right )+\sqrt{d} \sqrt{g^2-c^2 f^2} \sqrt{d-c^2 d x^2}\right ) c^2}{\sqrt{g^2-c^2 f^2}}+2 a \sqrt{d} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right ) c+b \sqrt{d-c^2 d x^2} \left (\frac{\cosh ^{-1}(c x)^2}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{2 g \cosh ^{-1}(c x)}{c f+c g x}+\frac{2 \log \left (\frac{g x}{f}+1\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{2 c f \left (2 \cosh ^{-1}(c x) \tan ^{-1}\left (\frac{(c f+g) \coth \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )-2 i \cos ^{-1}\left (-\frac{c f}{g}\right ) \tan ^{-1}\left (\frac{(g-c f) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )+\left (\cos ^{-1}\left (-\frac{c f}{g}\right )+2 \left (\tan ^{-1}\left (\frac{(c f+g) \coth \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )+\tan ^{-1}\left (\frac{(g-c f) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )\right )\right ) \log \left (\frac{e^{-\frac{1}{2} \cosh ^{-1}(c x)} \sqrt{g^2-c^2 f^2}}{\sqrt{2} \sqrt{g} \sqrt{c (f+g x)}}\right )+\left (\cos ^{-1}\left (-\frac{c f}{g}\right )-2 \left (\tan ^{-1}\left (\frac{(c f+g) \coth \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )+\tan ^{-1}\left (\frac{(g-c f) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )\right )\right ) \log \left (\frac{e^{\frac{1}{2} \cosh ^{-1}(c x)} \sqrt{g^2-c^2 f^2}}{\sqrt{2} \sqrt{g} \sqrt{c (f+g x)}}\right )-\left (\cos ^{-1}\left (-\frac{c f}{g}\right )+2 \tan ^{-1}\left (\frac{(g-c f) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )\right ) \log \left (\frac{(c f+g) \left (c f-g+i \sqrt{g^2-c^2 f^2}\right ) \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )-1\right )}{g \left (c f+g+i \sqrt{g^2-c^2 f^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\left (\cos ^{-1}\left (-\frac{c f}{g}\right )-2 \tan ^{-1}\left (\frac{(g-c f) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )\right ) \log \left (\frac{(c f+g) \left (-c f+g+i \sqrt{g^2-c^2 f^2}\right ) \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )+1\right )}{g \left (c f+g+i \sqrt{g^2-c^2 f^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}\right )+i \left (\text{PolyLog}\left (2,\frac{\left (c f-i \sqrt{g^2-c^2 f^2}\right ) \left (c f+g-i \sqrt{g^2-c^2 f^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}{g \left (c f+g+i \sqrt{g^2-c^2 f^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\text{PolyLog}\left (2,\frac{\left (c f+i \sqrt{g^2-c^2 f^2}\right ) \left (c f+g-i \sqrt{g^2-c^2 f^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}{g \left (c f+g+i \sqrt{g^2-c^2 f^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}\right )\right )\right )}{\sqrt{g^2-c^2 f^2} \sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right ) c-\frac{2 a g \sqrt{d-c^2 d x^2}}{f+g x}}{2 g^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*a*g*Sqrt[d - c^2*d*x^2])/(f + g*x) + 2*a*c*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^
2))] + (2*a*c^2*Sqrt[d]*f*Log[f + g*x])/Sqrt[-(c^2*f^2) + g^2] - (2*a*c^2*Sqrt[d]*f*Log[d*(g + c^2*f*x) + Sqrt
[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt[d - c^2*d*x^2]])/Sqrt[-(c^2*f^2) + g^2] + b*c*Sqrt[d - c^2*d*x^2]*((-2*g*ArcCo
sh[c*x])/(c*f + c*g*x) + ArcCosh[c*x]^2/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (2*Log[1 + (g*x)/f])/(Sqrt[(-
1 + c*x)/(1 + c*x)]*(1 + c*x)) + (2*c*f*(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]] + (Arc
Cos[-((c*f)/g)] + 2*(ArcTan[((c*f + g)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + ArcTan[((-(c*f) + g)*Ta
nh[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]*S
qrt[c*(f + 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 + g*x)])] - (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]))] - (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 + Ta
nh[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]))])))/(Sqrt[-(c^2*
f^2) + g^2]*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))))/(2*g^2)

________________________________________________________________________________________

Maple [B]  time = 0.309, size = 1956, normalized size = 2.1 \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))*(-c^2*d*x^2+d)^(1/2)/(g*x+f)^2,x)

[Out]

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

________________________________________________________________________________________

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((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(g*x+f)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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 )}}{g^{2} x^{2} + 2 \, f g x + f^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(g*x+f)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(g*x+f)^2,x, algorithm="giac")

[Out]

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