\(\int \frac {(a+b \text {arccosh}(c x))^2 \log (h (f+g x)^m)}{\sqrt {1-c^2 x^2}} \, dx\) [67]

Optimal result

Integrand size = 35, antiderivative size = 774 \[ \int \frac {(a+b \text {arccosh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\frac {m \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^4}{12 b^2 c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3 \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{3 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3 \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{3 b c \sqrt {1-c^2 x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}+\frac {2 b m \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}+\frac {2 b m \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}-\frac {2 b^2 m \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (4,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}-\frac {2 b^2 m \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (4,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}} \] Output:

1/12*m*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^4/b^2/c/(-c^2*x^2+1) 
^(1/2)-1/3*m*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^3*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)))/b/c/(-c^2*x^2+1)^( 
1/2)-1/3*m*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^3*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)))/b/c/(-c^2*x^2+1)^(1/ 
2)+1/3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^3*ln(h*(g*x+f)^m)/b/ 
c/(-c^2*x^2+1)^(1/2)-m*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^2*po 
lylog(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/ 
(-c^2*x^2+1)^(1/2)-m*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^2*poly 
log(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/(- 
c^2*x^2+1)^(1/2)+2*b*m*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))*poly 
log(3,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/c/(- 
c^2*x^2+1)^(1/2)+2*b*m*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))*poly 
log(3,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/c/(- 
c^2*x^2+1)^(1/2)-2*b^2*m*(c*x-1)^(1/2)*(c*x+1)^(1/2)*polylog(4,-(c*x+(c*x- 
1)^(1/2)*(c*x+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/c/(-c^2*x^2+1)^(1/2)- 
2*b^2*m*(c*x-1)^(1/2)*(c*x+1)^(1/2)*polylog(4,-(c*x+(c*x-1)^(1/2)*(c*x+1)^ 
(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/c/(-c^2*x^2+1)^(1/2)
 

Mathematica [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {(a+b \text {arccosh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.28 (sec) , antiderivative size = 475, normalized size of antiderivative = 0.61, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {6388, 6398, 6377, 6096, 2620, 3011, 7163, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx\)

\(\Big \downarrow \) 6388

\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int \frac {(a+b \text {arccosh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 6398

\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {(a+b \text {arccosh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \int \frac {(a+b \text {arccosh}(c x))^3}{f+g x}dx}{3 b c}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 6377

\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {(a+b \text {arccosh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \int \frac {\sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))^3}{c f+c g x}d\text {arccosh}(c x)}{3 b c}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 6096

\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {(a+b \text {arccosh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (\int \frac {e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))^3}{c f+e^{\text {arccosh}(c x)} g-\sqrt {c^2 f^2-g^2}}d\text {arccosh}(c x)+\int \frac {e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))^3}{c f+e^{\text {arccosh}(c x)} g+\sqrt {c^2 f^2-g^2}}d\text {arccosh}(c x)-\frac {(a+b \text {arccosh}(c x))^4}{4 b g}\right )}{3 b c}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {(a+b \text {arccosh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (-\frac {3 b \int (a+b \text {arccosh}(c x))^2 \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right )d\text {arccosh}(c x)}{g}-\frac {3 b \int (a+b \text {arccosh}(c x))^2 \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right )d\text {arccosh}(c x)}{g}+\frac {(a+b \text {arccosh}(c x))^3 \log \left (\frac {g e^{\text {arccosh}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g}+\frac {(a+b \text {arccosh}(c x))^3 \log \left (\frac {g e^{\text {arccosh}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{g}-\frac {(a+b \text {arccosh}(c x))^4}{4 b g}\right )}{3 b c}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {(a+b \text {arccosh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (-\frac {3 b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )d\text {arccosh}(c x)-(a+b \text {arccosh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )}{g}-\frac {3 b \left (2 b \int (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )d\text {arccosh}(c x)-(a+b \text {arccosh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{g}+\frac {(a+b \text {arccosh}(c x))^3 \log \left (\frac {g e^{\text {arccosh}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g}+\frac {(a+b \text {arccosh}(c x))^3 \log \left (\frac {g e^{\text {arccosh}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{g}-\frac {(a+b \text {arccosh}(c x))^4}{4 b g}\right )}{3 b c}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {(a+b \text {arccosh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (-\frac {3 b \left (2 b \left ((a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-b \int \operatorname {PolyLog}\left (3,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )d\text {arccosh}(c x)\right )-(a+b \text {arccosh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )}{g}-\frac {3 b \left (2 b \left ((a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-b \int \operatorname {PolyLog}\left (3,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )d\text {arccosh}(c x)\right )-(a+b \text {arccosh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{g}+\frac {(a+b \text {arccosh}(c x))^3 \log \left (\frac {g e^{\text {arccosh}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g}+\frac {(a+b \text {arccosh}(c x))^3 \log \left (\frac {g e^{\text {arccosh}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{g}-\frac {(a+b \text {arccosh}(c x))^4}{4 b g}\right )}{3 b c}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {(a+b \text {arccosh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (-\frac {3 b \left (2 b \left ((a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (3,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )de^{\text {arccosh}(c x)}\right )-(a+b \text {arccosh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )}{g}-\frac {3 b \left (2 b \left ((a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (3,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )de^{\text {arccosh}(c x)}\right )-(a+b \text {arccosh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{g}+\frac {(a+b \text {arccosh}(c x))^3 \log \left (\frac {g e^{\text {arccosh}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g}+\frac {(a+b \text {arccosh}(c x))^3 \log \left (\frac {g e^{\text {arccosh}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{g}-\frac {(a+b \text {arccosh}(c x))^4}{4 b g}\right )}{3 b c}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {(a+b \text {arccosh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (-\frac {3 b \left (2 b \left ((a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-b \operatorname {PolyLog}\left (4,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )-(a+b \text {arccosh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )}{g}-\frac {3 b \left (2 b \left ((a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-b \operatorname {PolyLog}\left (4,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )-(a+b \text {arccosh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{g}+\frac {(a+b \text {arccosh}(c x))^3 \log \left (\frac {g e^{\text {arccosh}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g}+\frac {(a+b \text {arccosh}(c x))^3 \log \left (\frac {g e^{\text {arccosh}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{g}-\frac {(a+b \text {arccosh}(c x))^4}{4 b g}\right )}{3 b c}\right )}{\sqrt {1-c^2 x^2}}\)

Input:

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

Output:

(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(((a + b*ArcCosh[c*x])^3*Log[h*(f + g*x)^m]) 
/(3*b*c) - (g*m*(-1/4*(a + b*ArcCosh[c*x])^4/(b*g) + ((a + b*ArcCosh[c*x]) 
^3*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g + ((a + b*Ar 
cCosh[c*x])^3*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g - 
 (3*b*(-((a + b*ArcCosh[c*x])^2*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqr 
t[c^2*f^2 - g^2]))]) + 2*b*((a + b*ArcCosh[c*x])*PolyLog[3, -((E^ArcCosh[c 
*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))] - b*PolyLog[4, -((E^ArcCosh[c*x]*g)/( 
c*f - Sqrt[c^2*f^2 - g^2]))])))/g - (3*b*(-((a + b*ArcCosh[c*x])^2*PolyLog 
[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))]) + 2*b*((a + b*ArcC 
osh[c*x])*PolyLog[3, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))] - 
b*PolyLog[4, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])))/g))/(3* 
b*c)))/Sqrt[1 - c^2*x^2]
 

Defintions of rubi rules used

rule 2620

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] - Si 
mp[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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{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 3011

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] + Simp[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 6096

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 6377

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbo 
l] :> 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 6388

Int[Log[(h_.)*((f_.) + (g_.)*(x_))^(m_.)]*((a_.) + ArcCosh[(c_.)*(x_)]*(b_. 
))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-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] && Integer 
Q[p - 1/2]
 

rule 6398

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)*d 
2]*(n + 1))), x] - Simp[g*(m/(b*c*Sqrt[(-d1)*d2]*(n + 1)))   Int[(a + b*Arc 
Cosh[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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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]
 

rule 7163

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] - Simp[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]
 

Maple [F]

\[\int \frac {\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} \ln \left (h \left (g x +f \right )^{m}\right )}{\sqrt {-c^{2} x^{2}+1}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \] Input:

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

Output:

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)
 

Sympy [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2} \log {\left (h \left (f + g x\right )^{m} \right )}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \text {arccosh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{36,[0,2,1,1,1,3,0,0]%%%}+%%%{-24,[0,2,1,1,1,2,1,0]%%%}+%%% 
{-12,[0,2
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{\sqrt {1-c^2\,x^2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \text {arccosh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\left (\int \frac {\mathrm {log}\left (\left (g x +f \right )^{m} h \right )}{\sqrt {-c^{2} x^{2}+1}}d x \right ) a^{2}+2 \left (\int \frac {\mathit {acosh} \left (c x \right ) \mathrm {log}\left (\left (g x +f \right )^{m} h \right )}{\sqrt {-c^{2} x^{2}+1}}d x \right ) a b +\left (\int \frac {\mathit {acosh} \left (c x \right )^{2} \mathrm {log}\left (\left (g x +f \right )^{m} h \right )}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} \] Input:

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

Output:

int(log((f + g*x)**m*h)/sqrt( - c**2*x**2 + 1),x)*a**2 + 2*int((acosh(c*x) 
*log((f + g*x)**m*h))/sqrt( - c**2*x**2 + 1),x)*a*b + int((acosh(c*x)**2*l 
og((f + g*x)**m*h))/sqrt( - c**2*x**2 + 1),x)*b**2