3.1.57 \(\int \frac {\sqrt {d-c^2 d x^2} (a+b \cosh ^{-1}(c x))}{(f+g x)^2} \, dx\) [57]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 2.44, antiderivative size = 918, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 22, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.710, Rules used = {5972, 5976, 37, 5969, 12, 186, 54, 98, 95, 214, 5993, 5992, 5893, 5980, 3405, 3401, 2296, 2221, 2317, 2438, 2747, 31} \begin {gather*} \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 {Li}_2\left (-\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 {Li}_2\left (-\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} \end {gather*}

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 12

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

Rule 31

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

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 54

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 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 98

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 186

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 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 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 2747

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 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 3405

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 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

Rule 5969

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 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]

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \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 (b c \sqrt {d-c^2 d x^2}\right ) \text {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 ) \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 \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 ) \text {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 ) \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 (4 b c^2 f \left (-c^2 f^2+g^2\right ) \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 \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 ) \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 \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 ) \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^2 f \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^2 f \left (-c^2 f^2+g^2\right ) \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 \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 ) \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 \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 ) \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^2 f \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 (2 b c^2 f \left (-c^2 f^2+g^2\right ) \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 \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 ) \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 \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 ) \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^2 f \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 {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 4.68, size = 1139, normalized size = 1.24 \begin {gather*} \frac {-\frac {2 a g \sqrt {d-c^2 d x^2}}{f+g x}+2 a c \sqrt {d} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {2 a c^2 \sqrt {d} f \log (f+g x)}{\sqrt {-c^2 f^2+g^2}}-\frac {2 a c^2 \sqrt {d} f \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 )}{\sqrt {-c^2 f^2+g^2}}+b c \sqrt {d-c^2 d x^2} \left (-\frac {2 g \cosh ^{-1}(c x)}{c f+c g x}+\frac {\cosh ^{-1}(c x)^2}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {2 \log \left (1+\frac {g x}{f}\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {2 c f \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)^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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1955\) vs. \(2(864)=1728\).
time = 10.36, size = 1956, normalized size = 2.13

method	result	size
default	\(\text {Expression too large to display}\)	\(1956\)

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,method=_RETURNVERBOSE)

[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^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2/(c^2*f^2-g^2)^(1/2)*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^2/(c*x-1)^(1/2
)/(c*x+1)^(1/2)/g^2/(c^2*f^2-g^2)^(1/2)*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^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2/(c^2*f^2-g
^2)*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^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2/(c^2*
f^2-g^2)*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^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2/(c^2*f^2-g^2)^(1/2)*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^2/(c*x-1)^(1/2)/(c*x+1)^(1
/2)/g^2/(c^2*f^2-g^2)^(1/2)*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/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(c^2*f^2-g^2)*ln(c*x+(c*x-1)^(1/2)*(c*
x+1)^(1/2))-b*(-d*(c^2*x^2-1))^(1/2)*c/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(c^2*f^2-g^2)*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)

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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)^2,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)^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)

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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 )}{\left (f + g x\right )^{2}}\, 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)**2,x)

[Out]

Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*acosh(c*x))/(f + g*x)**2, 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)^2,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}}{{\left (f+g\,x\right )}^2} \,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)^2,x)

[Out]

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

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