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

Optimal. Leaf size=785 \[ \frac{b \sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2} \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2} \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{g^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{d-c^2 d x^2} \left (1-\frac{c^2 f^2}{g^2}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{c x-1} \sqrt{c x+1} (f+g x)}-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{c x-1} \sqrt{c x+1} (f+g x)}-\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g \sqrt{c x-1} \sqrt{c x+1}}-\frac{a \sqrt{c^2 x^2-1} \sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2} \tanh ^{-1}\left (\frac{c^2 f x+g}{\sqrt{c^2 x^2-1} \sqrt{c^2 f^2-g^2}}\right )}{g^2 (1-c x) (c x+1)}+\frac{a \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g (1-c x) (c x+1)}+\frac{b \sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2} \cosh ^{-1}(c x) \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}+1\right )}{g^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2} \cosh ^{-1}(c x) \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}+1\right )}{g^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c x \sqrt{d-c^2 d x^2}}{g \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g} \]

[Out]

-((b*c*x*Sqrt[d - c^2*d*x^2])/(g*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (a*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(g*(1
- c*x)*(1 + c*x)) + (b*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/g - (c*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/
(2*b*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((1 - (c^2*f^2)/g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*c
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) - ((1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*c*S
qrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) - (a*Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d - c^2*d*x^2]*ArcTanh
[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c^2*x^2])])/(g^2*(1 - c*x)*(1 + c*x)) + (b*Sqrt[c^2*f^2 - g^2]*S
qrt[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[-1 + c*x]*S
qrt[1 + c*x]) - (b*Sqrt[c^2*f^2 - g^2]*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[-1 + c*x]*Sqrt[1 + c*x]) + (b*Sqrt[c^2*f^2 - g^2]*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[-1 + c*x]*Sqrt[1 + c*x]) - (b*Sqrt[c^2*f^2 - g^2
]*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[-1 + c*x]*Sqrt[
1 + c*x])

________________________________________________________________________________________

Rubi [A]  time = 3.37884, antiderivative size = 785, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 22, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.71, Rules used = {5836, 5824, 683, 5816, 6742, 93, 208, 1610, 1654, 12, 725, 206, 5860, 5858, 5718, 8, 5832, 3320, 2264, 2190, 2279, 2391} \[ \frac{b \sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2} \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2} \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{g^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{d-c^2 d x^2} \left (1-\frac{c^2 f^2}{g^2}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{c x-1} \sqrt{c x+1} (f+g x)}-\frac{\left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c \sqrt{c x-1} \sqrt{c x+1} (f+g x)}-\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g \sqrt{c x-1} \sqrt{c x+1}}-\frac{a \sqrt{c^2 x^2-1} \sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2} \tanh ^{-1}\left (\frac{c^2 f x+g}{\sqrt{c^2 x^2-1} \sqrt{c^2 f^2-g^2}}\right )}{g^2 (1-c x) (c x+1)}+\frac{a \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g (1-c x) (c x+1)}+\frac{b \sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2} \cosh ^{-1}(c x) \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}+1\right )}{g^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2} \cosh ^{-1}(c x) \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}+1\right )}{g^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c x \sqrt{d-c^2 d x^2}}{g \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*c*x*Sqrt[d - c^2*d*x^2])/(g*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (a*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(g*(1
- c*x)*(1 + c*x)) + (b*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/g - (c*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/
(2*b*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((1 - (c^2*f^2)/g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*c
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) - ((1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*c*S
qrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) - (a*Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d - c^2*d*x^2]*ArcTanh
[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c^2*x^2])])/(g^2*(1 - c*x)*(1 + c*x)) + (b*Sqrt[c^2*f^2 - g^2]*S
qrt[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[-1 + c*x]*S
qrt[1 + c*x]) - (b*Sqrt[c^2*f^2 - g^2]*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[-1 + c*x]*Sqrt[1 + c*x]) + (b*Sqrt[c^2*f^2 - g^2]*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[-1 + c*x]*Sqrt[1 + c*x]) - (b*Sqrt[c^2*f^2 - g^2
]*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[-1 + c*x]*Sqrt[
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 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 5816

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.) + (g_.)*(x_) + (h_.)*(x_)^2)^(p_.))/((d_) + (e_.)*(x_))^2
, x_Symbol] :> With[{u = IntHide[(f + g*x + h*x^2)^p/(d + e*x)^2, x]}, Dist[(a + b*ArcCosh[c*x])^n, u, x] - Di
st[b*c*n, Int[SimplifyIntegrand[(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, h}, x] && IGtQ[n, 0] && IGtQ[p, 0] && EqQ[e*g - 2*d*h, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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 1610

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[((
a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1654

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 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 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 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]

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{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{f+g x} \, 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} \, 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)}-\frac{\sqrt{d-c^2 d x^2} \int \frac{\left (g+2 c^2 f x+c^2 g x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{(f+g x)^2} \, dx}{2 b c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (1-\frac{c^2 f^2}{g^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)}-\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)}+\frac{\sqrt{d-c^2 d x^2} \int \frac{\left (\frac{1}{f+g x}-\frac{c^2 \left (g x+\frac{f^2}{f+g x}\right )}{g^2}\right ) \left (-a-b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (1-\frac{c^2 f^2}{g^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)}-\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)}+\frac{\sqrt{d-c^2 d x^2} \int \left (\frac{a \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{b \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \cosh ^{-1}(c x)}{g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (1-\frac{c^2 f^2}{g^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)}-\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)}+\frac{\left (a \sqrt{d-c^2 d x^2}\right ) \int \frac{c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{\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 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)} \, dx}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (1-\frac{c^2 f^2}{g^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)}-\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)}+\frac{\left (b \sqrt{d-c^2 d x^2}\right ) \int \left (\frac{c^2 g x \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^2 f^2-g^2\right ) \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)}\right ) \, dx}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (a \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \int \frac{c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt{-1+c^2 x^2}} \, dx}{g^2 (-1+c x) (1+c x)}\\ &=\frac{a \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g (1-c x) (1+c x)}-\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (1-\frac{c^2 f^2}{g^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)}-\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)}+\frac{\left (b c^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b (c f-g) (c f+g) \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 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (a \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \int \frac{c^4 f^2 g^2-c^2 g^4}{(f+g x) \sqrt{-1+c^2 x^2}} \, dx}{c^2 g^4 (-1+c x) (1+c x)}\\ &=\frac{a \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g (1-c x) (1+c x)}+\frac{b \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g}-\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (1-\frac{c^2 f^2}{g^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)}-\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)}-\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int 1 \, dx}{g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b (c f-g) (c f+g) \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 (a (c f-g) (c f+g) \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{(f+g x) \sqrt{-1+c^2 x^2}} \, dx}{g^2 (-1+c x) (1+c x)}\\ &=-\frac{b c x \sqrt{d-c^2 d x^2}}{g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g (1-c x) (1+c x)}+\frac{b \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g}-\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (1-\frac{c^2 f^2}{g^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)}-\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)}+\frac{\left (2 b (c f-g) (c f+g) \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 (a (c f-g) (c f+g) \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{c^2 f^2-g^2-x^2} \, dx,x,\frac{-g-c^2 f x}{\sqrt{-1+c^2 x^2}}\right )}{g^2 (-1+c x) (1+c x)}\\ &=-\frac{b c x \sqrt{d-c^2 d x^2}}{g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g (1-c x) (1+c x)}+\frac{b \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g}-\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (1-\frac{c^2 f^2}{g^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)}-\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)}-\frac{a (c f-g) (c f+g) \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{-1+c^2 x^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} (1-c x) (1+c x)}+\frac{\left (2 b (c f-g) (c f+g) \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 f-g) (c f+g) \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{b c x \sqrt{d-c^2 d x^2}}{g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g (1-c x) (1+c x)}+\frac{b \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g}-\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (1-\frac{c^2 f^2}{g^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)}-\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)}-\frac{a (c f-g) (c f+g) \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{-1+c^2 x^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} (1-c x) (1+c x)}+\frac{b (c f-g) (c f+g) \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 f-g) (c f+g) \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{\left (b (c f-g) (c f+g) \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 f-g) (c f+g) \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{b c x \sqrt{d-c^2 d x^2}}{g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g (1-c x) (1+c x)}+\frac{b \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g}-\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (1-\frac{c^2 f^2}{g^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)}-\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)}-\frac{a (c f-g) (c f+g) \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{-1+c^2 x^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} (1-c x) (1+c x)}+\frac{b (c f-g) (c f+g) \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 f-g) (c f+g) \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{\left (b (c f-g) (c f+g) \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 f-g) (c f+g) \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{b c x \sqrt{d-c^2 d x^2}}{g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g (1-c x) (1+c x)}+\frac{b \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g}-\frac{c x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (1-\frac{c^2 f^2}{g^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)}-\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)}-\frac{a (c f-g) (c f+g) \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{-1+c^2 x^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} (1-c x) (1+c x)}+\frac{b (c f-g) (c f+g) \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 f-g) (c f+g) \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 f-g) (c f+g) \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 f-g) (c f+g) \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 = 4.05017, size = 1121, normalized size = 1.43 \[ \frac{2 a \sqrt{d-c^2 d x^2} g-2 a c \sqrt{d} f \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )+2 a \sqrt{d} \sqrt{g^2-c^2 f^2} \log (f+g x)-2 a \sqrt{d} \sqrt{g^2-c^2 f^2} \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 )+b \sqrt{d-c^2 d x^2} \left (\frac{c f \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)^2}{1-c x}+2 g \cosh ^{-1}(c x)+\frac{2 (g-c f) (c f+g) \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)}+\frac{2 c g x \sqrt{\frac{c x-1}{c x+1}}}{1-c x}\right )}{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),x]

[Out]

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

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Maple [A]  time = 0.347, size = 1072, normalized size = 1.4 \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),x)

[Out]

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

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

[Out]

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

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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 )}{f + g x}\, 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),x)

[Out]

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

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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 )}}{g x + f}\,{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),x, algorithm="giac")

[Out]

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