3.1.56 \(\int \frac {\sqrt {d-c^2 d x^2} (a+b \cosh ^{-1}(c x))}{f+g x} \, dx\) [56]

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

[Out]

a*(-c^2*x^2+1)*(-c^2*d*x^2+d)^(1/2)/g/(-c*x+1)/(c*x+1)+b*arccosh(c*x)*(-c^2*d*x^2+d)^(1/2)/g-b*c*x*(-c^2*d*x^2
+d)^(1/2)/g/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*c*x*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/g/(c*x-1)^(1/2)/(c
*x+1)^(1/2)+1/2*(1-c^2*f^2/g^2)*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/(g*x+f)/(c*x-1)^(1/2)/(c*x+1)^(1
/2)-1/2*(-c^2*x^2+1)*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/(g*x+f)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+b*arcco
sh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)
^(1/2)/g^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-b*arccosh(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f+(c^2*f^2-g
^2)^(1/2)))*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)/g^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+b*polylog(2,-(c*x+(c*x-1)
^(1/2)*(c*x+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))*(c^2*f^2-g^2)^(1/2)*(-c^2*d*x^2+d)^(1/2)/g^2/(c*x-1)^(1/2)/
(c*x+1)^(1/2)-b*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))*(c^2*f^2-g^2)^(1/2)*
(-c^2*d*x^2+d)^(1/2)/g^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)-a*arctanh((c^2*f*x+g)/(c^2*f^2-g^2)^(1/2)/(c^2*x^2-1)^(1/
2))*(c^2*f^2-g^2)^(1/2)*(c^2*x^2-1)^(1/2)*(-c^2*d*x^2+d)^(1/2)/g^2/(-c*x+1)/(c*x+1)

________________________________________________________________________________________

Rubi [A]
time = 2.30, antiderivative size = 785, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 22, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.710, Rules used = {5972, 5976, 697, 5970, 6874, 95, 214, 1624, 1668, 12, 739, 212, 5993, 5992, 5915, 8, 5980, 3401, 2296, 2221, 2317, 2438} \begin {gather*} \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} \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 x-1} \sqrt {c x+1}}-\frac {b \sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^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 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)}}{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} \end {gather*}

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 95

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 212

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

Rule 214

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

Rule 697

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 739

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 1624

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 1668

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2296

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 2317

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 2438

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 3401

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)/(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))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> 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/(2*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[(1 + c*x)^(p + 1/2)*(-
1 + c*x)^(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] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]

Rule 5970

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 5972

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[p - 1/2] && IntegerQ[m]

Rule 5976

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 5980

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 5992

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 5993

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 6874

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

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.63, size = 1121, normalized size = 1.43 \begin {gather*} \frac {2 a g \sqrt {d-c^2 d x^2}-2 a c \sqrt {d} f \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+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 \left (d \left (g+c^2 f x\right )+\sqrt {d} \sqrt {-c^2 f^2+g^2} \sqrt {d-c^2 d x^2}\right )+b \sqrt {d-c^2 d x^2} \left (\frac {2 c g x \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}+2 g \cosh ^{-1}(c x)+\frac {c f \sqrt {\frac {-1+c x}{1+c x}} \cosh ^{-1}(c x)^2}{1-c x}+\frac {2 (-c f+g) (c f+g) \left (2 \cosh ^{-1}(c x) \text {ArcTan}\left (\frac {(c f+g) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )-2 i \text {ArcCos}\left (-\frac {c f}{g}\right ) \text {ArcTan}\left (\frac {(-c f+g) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )+\left (\text {ArcCos}\left (-\frac {c f}{g}\right )+2 \left (\text {ArcTan}\left (\frac {(c f+g) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )+\text {ArcTan}\left (\frac {(-c f+g) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right )\right ) \log \left (\frac {e^{-\frac {1}{2} \cosh ^{-1}(c x)} \sqrt {-c^2 f^2+g^2}}{\sqrt {2} \sqrt {g} \sqrt {c (f+g x)}}\right )+\left (\text {ArcCos}\left (-\frac {c f}{g}\right )-2 \left (\text {ArcTan}\left (\frac {(c f+g) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )+\text {ArcTan}\left (\frac {(-c f+g) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right )\right ) \log \left (\frac {e^{\frac {1}{2} \cosh ^{-1}(c x)} \sqrt {-c^2 f^2+g^2}}{\sqrt {2} \sqrt {g} \sqrt {c (f+g x)}}\right )-\left (\text {ArcCos}\left (-\frac {c f}{g}\right )+2 \text {ArcTan}\left (\frac {(-c f+g) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right ) \log \left (\frac {(c f+g) \left (c f-g+i \sqrt {-c^2 f^2+g^2}\right ) \left (-1+\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{g \left (c f+g+i \sqrt {-c^2 f^2+g^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\left (\text {ArcCos}\left (-\frac {c f}{g}\right )-2 \text {ArcTan}\left (\frac {(-c f+g) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right ) \log \left (\frac {(c f+g) \left (-c f+g+i \sqrt {-c^2 f^2+g^2}\right ) \left (1+\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{g \left (c f+g+i \sqrt {-c^2 f^2+g^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 {-c^2 f^2+g^2}\right ) \left (c f+g-i \sqrt {-c^2 f^2+g^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{g \left (c f+g+i \sqrt {-c^2 f^2+g^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\text {PolyLog}\left (2,\frac {\left (c f+i \sqrt {-c^2 f^2+g^2}\right ) \left (c f+g-i \sqrt {-c^2 f^2+g^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{g \left (c f+g+i \sqrt {-c^2 f^2+g^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )\right )\right )}{\sqrt {-c^2 f^2+g^2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right )}{2 g^2} \end {gather*}

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 = 6.77, size = 1072, normalized size = 1.37

method	result	size
default	\(\frac {a \sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}{g}+\frac {a \,c^{2} d f \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}\right )}{g^{2} \sqrt {c^{2} d}}+\frac {a d \ln \left (\frac {-\frac {2 d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}+\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}\, \sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right ) c^{2} f^{2}}{g^{3} \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}-\frac {a d \ln \left (\frac {-\frac {2 d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}+\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}\, \sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \mathrm {arccosh}\left (c x \right )^{2} c}{2 \sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x^{2} c^{2}}{\left (c x +1\right ) \left (c x -1\right ) g}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x c}{\sqrt {c x +1}\, \sqrt {c x -1}\, g}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{\left (c x +1\right ) \left (c x -1\right ) g}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \dilog \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \dilog \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{\sqrt {c x -1}\, \sqrt {c x +1}\, g^{2}}\)	\(1072\)

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,method=_RETURNVERBOSE)

[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
*x*c-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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(g-c*f>0)', see `assume?` for m
ore details)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{f+g\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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