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

Optimal. Leaf size=523 \[ \frac{b c^2 f \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac{b c^2 f \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac{g \sqrt{c x-1} \sqrt{-\frac{1-c x}{c x+1}} (c x+1)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right ) (f+g x)}+\frac{c^2 f \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}+1\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac{c^2 f \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}+1\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} \log (f+g x)}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.836335, antiderivative size = 523, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.323, Rules used = {5836, 5832, 3324, 3320, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac{b c^2 f \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac{b c^2 f \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac{g \sqrt{c x-1} \sqrt{-\frac{1-c x}{c x+1}} (c x+1)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right ) (f+g x)}+\frac{c^2 f \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}+1\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}-\frac{c^2 f \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}+1\right )}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )^{3/2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} \log (f+g x)}{\sqrt{d-c^2 d x^2} \left (c^2 f^2-g^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5836

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol]
:> Dist[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f + g*x
)^m*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d
+ e, 0] && IntegerQ[m] && IntegerQ[p - 1/2]

Rule 5832

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

Rule 3324

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

Rule 3320

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

Rule 2264

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

Rule 2190

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

Rule 2279

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

Rule 2391

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

Rule 2668

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

Rule 31

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

Rubi steps

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

Mathematica [C]  time = 5.80011, size = 1115, normalized size = 2.13 \[ -\frac{a f \log (f+g x) c^2}{\sqrt{d} \left (g^2-c^2 f^2\right )^{3/2}}-\frac{a f \log \left (d \left (f x c^2+g\right )+\sqrt{d} \sqrt{g^2-c^2 f^2} \sqrt{d-c^2 d x^2}\right ) c^2}{\sqrt{d} (c f-g) (c f+g) \sqrt{g^2-c^2 f^2}}+\frac{b \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (-\frac{g \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)}{(c f-g) (c f+g) (c f+c g x)}+\frac{\log \left (\frac{g x}{f}+1\right )}{c^2 f^2-g^2}+\frac{c f \left (2 \cosh ^{-1}(c x) \tan ^{-1}\left (\frac{(c f+g) \coth \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )-2 i \cos ^{-1}\left (-\frac{c f}{g}\right ) \tan ^{-1}\left (\frac{(g-c f) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )+\left (\cos ^{-1}\left (-\frac{c f}{g}\right )+2 \left (\tan ^{-1}\left (\frac{(c f+g) \coth \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )+\tan ^{-1}\left (\frac{(g-c f) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )\right )\right ) \log \left (\frac{e^{-\frac{1}{2} \cosh ^{-1}(c x)} \sqrt{g^2-c^2 f^2}}{\sqrt{2} \sqrt{g} \sqrt{c (f+g x)}}\right )+\left (\cos ^{-1}\left (-\frac{c f}{g}\right )-2 \left (\tan ^{-1}\left (\frac{(c f+g) \coth \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )+\tan ^{-1}\left (\frac{(g-c f) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )\right )\right ) \log \left (\frac{e^{\frac{1}{2} \cosh ^{-1}(c x)} \sqrt{g^2-c^2 f^2}}{\sqrt{2} \sqrt{g} \sqrt{c (f+g x)}}\right )-\left (\cos ^{-1}\left (-\frac{c f}{g}\right )+2 \tan ^{-1}\left (\frac{(g-c f) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )\right ) \log \left (\frac{(c f+g) \left (c f-g+i \sqrt{g^2-c^2 f^2}\right ) \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )-1\right )}{g \left (c f+g+i \sqrt{g^2-c^2 f^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\left (\cos ^{-1}\left (-\frac{c f}{g}\right )-2 \tan ^{-1}\left (\frac{(g-c f) \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )\right ) \log \left (\frac{(c f+g) \left (-c f+g+i \sqrt{g^2-c^2 f^2}\right ) \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )+1\right )}{g \left (c f+g+i \sqrt{g^2-c^2 f^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}\right )+i \left (\text{PolyLog}\left (2,\frac{\left (c f-i \sqrt{g^2-c^2 f^2}\right ) \left (c f+g-i \sqrt{g^2-c^2 f^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}{g \left (c f+g+i \sqrt{g^2-c^2 f^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\text{PolyLog}\left (2,\frac{\left (c f+i \sqrt{g^2-c^2 f^2}\right ) \left (c f+g-i \sqrt{g^2-c^2 f^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}{g \left (c f+g+i \sqrt{g^2-c^2 f^2} \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )}\right )\right )\right )}{\left (g^2-c^2 f^2\right )^{3/2}}\right ) c}{\sqrt{d-c^2 d x^2}}-\frac{a g \sqrt{d-c^2 d x^2}}{d \left (g^2-c^2 f^2\right ) (f+g x)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((a*g*Sqrt[d - c^2*d*x^2])/(d*(-(c^2*f^2) + g^2)*(f + g*x))) - (a*c^2*f*Log[f + g*x])/(Sqrt[d]*(-(c^2*f^2) +
g^2)^(3/2)) - (a*c^2*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[d]*(c*
f - g)*(c*f + g)*Sqrt[-(c^2*f^2) + g^2]) + (b*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-((g*Sqrt[(-1 + c*x)/(1
+ c*x)]*(1 + c*x)*ArcCosh[c*x])/((c*f - g)*(c*f + g)*(c*f + c*g*x))) + Log[1 + (g*x)/f]/(c^2*f^2 - g^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)]*Ar
cTan[((-(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*Sq
rt[-(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*Sq
rt[-(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]))] - P
olyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))/(g*(c*f
+ g + I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))])))/(-(c^2*f^2) + g^2)^(3/2)))/Sqrt[d - c^2*d*x^2]

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Maple [B]  time = 0.223, size = 1978, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{c^{2} d g^{2} x^{4} + 2 \, c^{2} d f g x^{3} - 2 \, d f g x - d f^{2} +{\left (c^{2} d f^{2} - d g^{2}\right )} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError