3.79 \(\int \frac{(a+b \cosh ^{-1}(c x))^2 \log (h (f+g x)^m)}{\sqrt{1-c^2 x^2}} \, dx\)

Optimal. Leaf size=774 \[ -\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{c \sqrt{1-c^2 x^2}}+\frac{2 b m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (3,-\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}+\frac{2 b m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (3,-\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{c \sqrt{1-c^2 x^2}}-\frac{2 b^2 m \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (4,-\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}-\frac{2 b^2 m \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (4,-\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{c \sqrt{1-c^2 x^2}}+\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^4}{12 b^2 c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}+1\right )}{3 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}+1\right )}{3 b c \sqrt{1-c^2 x^2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (h (f+g x)^m\right )}{3 b c \sqrt{1-c^2 x^2}} \]

[Out]

(m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^4)/(12*b^2*c*Sqrt[1 - c^2*x^2]) - (m*Sqrt[-1 + c*x]*Sqrt[
1 + c*x]*(a + b*ArcCosh[c*x])^3*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(3*b*c*Sqrt[1 - c^2*x
^2]) - (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 -
g^2])])/(3*b*c*Sqrt[1 - c^2*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3*Log[h*(f + g*x)^m])/
(3*b*c*Sqrt[1 - c^2*x^2]) - (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*PolyLog[2, -((E^ArcCosh[c*x
]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) - (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x
])^2*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) + (2*b*m*Sqrt[-1 + c
*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[3, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[
1 - c^2*x^2]) + (2*b*m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[3, -((E^ArcCosh[c*x]*g)/(c*f
+ Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) - (2*b^2*m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[4, -((E^ArcCos
h[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) - (2*b^2*m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog
[4, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.59875, antiderivative size = 774, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.314, Rules used = {5713, 5676, 5841, 5839, 5800, 5562, 2190, 2531, 6609, 2282, 6589} \[ -\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \text{PolyLog}\left (2,-\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{c \sqrt{1-c^2 x^2}}+\frac{2 b m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (3,-\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}+\frac{2 b m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (3,-\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{c \sqrt{1-c^2 x^2}}-\frac{2 b^2 m \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (4,-\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}-\frac{2 b^2 m \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (4,-\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )}{c \sqrt{1-c^2 x^2}}+\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^4}{12 b^2 c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}+1\right )}{3 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (\frac{g e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}+1\right )}{3 b c \sqrt{1-c^2 x^2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (h (f+g x)^m\right )}{3 b c \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*ArcCosh[c*x])^2*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2],x]

[Out]

(m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^4)/(12*b^2*c*Sqrt[1 - c^2*x^2]) - (m*Sqrt[-1 + c*x]*Sqrt[
1 + c*x]*(a + b*ArcCosh[c*x])^3*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(3*b*c*Sqrt[1 - c^2*x
^2]) - (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 -
g^2])])/(3*b*c*Sqrt[1 - c^2*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3*Log[h*(f + g*x)^m])/
(3*b*c*Sqrt[1 - c^2*x^2]) - (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*PolyLog[2, -((E^ArcCosh[c*x
]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) - (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x
])^2*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) + (2*b*m*Sqrt[-1 + c
*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[3, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[
1 - c^2*x^2]) + (2*b*m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[3, -((E^ArcCosh[c*x]*g)/(c*f
+ Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) - (2*b^2*m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[4, -((E^ArcCos
h[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) - (2*b^2*m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog
[4, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2])

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((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[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 5841

Int[Log[(h_.)*((f_.) + (g_.)*(x_))^(m_.)]*((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((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[Log[h*(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,
h, m, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]

Rule 5839

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

Rule 5800

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Subst[Int[((a + b*x)^n*Sinh[x
])/(c*d + e*Cosh[x]), x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 5562

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 - b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 6609

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

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{\sqrt{1-c^2 x^2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (h (f+g x)^m\right )}{3 b c \sqrt{1-c^2 x^2}}-\frac{\left (g m \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^3}{f+g x} \, dx}{3 b c \sqrt{1-c^2 x^2}}\\ &=\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (h (f+g x)^m\right )}{3 b c \sqrt{1-c^2 x^2}}-\frac{\left (g m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^3 \sinh (x)}{c f+g \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{3 b c \sqrt{1-c^2 x^2}}\\ &=\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^4}{12 b^2 c \sqrt{1-c^2 x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (h (f+g x)^m\right )}{3 b c \sqrt{1-c^2 x^2}}-\frac{\left (g m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{e^x (a+b x)^3}{c f+e^x g-\sqrt{c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{3 b c \sqrt{1-c^2 x^2}}-\frac{\left (g m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{e^x (a+b x)^3}{c f+e^x g+\sqrt{c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{3 b c \sqrt{1-c^2 x^2}}\\ &=\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^4}{12 b^2 c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{3 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{3 b c \sqrt{1-c^2 x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (h (f+g x)^m\right )}{3 b c \sqrt{1-c^2 x^2}}+\frac{\left (m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \log \left (1+\frac{e^x g}{c f-\sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt{1-c^2 x^2}}+\frac{\left (m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \log \left (1+\frac{e^x g}{c f+\sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt{1-c^2 x^2}}\\ &=\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^4}{12 b^2 c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{3 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{3 b c \sqrt{1-c^2 x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (h (f+g x)^m\right )}{3 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}+\frac{\left (2 b m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{Li}_2\left (-\frac{e^x g}{c f-\sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt{1-c^2 x^2}}+\frac{\left (2 b m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{Li}_2\left (-\frac{e^x g}{c f+\sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt{1-c^2 x^2}}\\ &=\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^4}{12 b^2 c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{3 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{3 b c \sqrt{1-c^2 x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (h (f+g x)^m\right )}{3 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}+\frac{2 b m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_3\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}+\frac{2 b m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_3\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-\frac{e^x g}{c f-\sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-\frac{e^x g}{c f+\sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt{1-c^2 x^2}}\\ &=\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^4}{12 b^2 c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{3 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{3 b c \sqrt{1-c^2 x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (h (f+g x)^m\right )}{3 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}+\frac{2 b m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_3\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}+\frac{2 b m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_3\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{g x}{-c f+\sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 m \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{g x}{c f+\sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c \sqrt{1-c^2 x^2}}\\ &=\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^4}{12 b^2 c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{3 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{3 b c \sqrt{1-c^2 x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3 \log \left (h (f+g x)^m\right )}{3 b c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}-\frac{m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}+\frac{2 b m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_3\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}+\frac{2 b m \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_3\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}-\frac{2 b^2 m \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_4\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}-\frac{2 b^2 m \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_4\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{c \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [F]  time = 4.24758, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{\sqrt{1-c^2 x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((a + b*ArcCosh[c*x])^2*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2],x]

[Out]

Integrate[((a + b*ArcCosh[c*x])^2*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2], x]

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Maple [F]  time = 1.726, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}\ln \left ( h \left ( gx+f \right ) ^{m} \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccosh(c*x) + a)^2*log((g*x + f)^m*h)/sqrt(-c^2*x^2 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} x^{2} + 1}{\left (b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}\right )} \log \left ({\left (g x + f\right )}^{m} h\right )}{c^{2} x^{2} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-c^2*x^2 + 1)*(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)*log((g*x + f)^m*h)/(c^2*x^2 - 1),
x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccosh(c*x) + a)^2*log((g*x + f)^m*h)/sqrt(-c^2*x^2 + 1), x)