\(\int \frac {(a+b \text {arcsinh}(c x))^2 \log (h (f+g x)^m)}{\sqrt {1+c^2 x^2}} \, dx\) [54]

Optimal result

Integrand size = 34, antiderivative size = 438 \[ \int \frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\frac {m (a+b \text {arcsinh}(c x))^4}{12 b^2 c}-\frac {m (a+b \text {arcsinh}(c x))^3 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{3 b c}-\frac {m (a+b \text {arcsinh}(c x))^3 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{3 b c}+\frac {(a+b \text {arcsinh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {m (a+b \text {arcsinh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m (a+b \text {arcsinh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {2 b m (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {2 b m (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {2 b^2 m \operatorname {PolyLog}\left (4,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {2 b^2 m \operatorname {PolyLog}\left (4,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c} \] Output:

1/12*m*(a+b*arcsinh(c*x))^4/b^2/c-1/3*m*(a+b*arcsinh(c*x))^3*ln(1+(c*x+(c^ 
2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2+g^2)^(1/2)))/b/c-1/3*m*(a+b*arcsinh(c*x))^ 
3*ln(1+(c*x+(c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2+g^2)^(1/2)))/b/c+1/3*(a+b*a 
rcsinh(c*x))^3*ln(h*(g*x+f)^m)/b/c-m*(a+b*arcsinh(c*x))^2*polylog(2,-(c*x+ 
(c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2+g^2)^(1/2)))/c-m*(a+b*arcsinh(c*x))^2*p 
olylog(2,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2+g^2)^(1/2)))/c+2*b*m*(a+ 
b*arcsinh(c*x))*polylog(3,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2+g^2)^(1 
/2)))/c+2*b*m*(a+b*arcsinh(c*x))*polylog(3,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f 
+(c^2*f^2+g^2)^(1/2)))/c-2*b^2*m*polylog(4,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f 
-(c^2*f^2+g^2)^(1/2)))/c-2*b^2*m*polylog(4,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f 
+(c^2*f^2+g^2)^(1/2)))/c
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 397, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=-\frac {-\frac {m (a+b \text {arcsinh}(c x))^4}{4 b}+m (a+b \text {arcsinh}(c x))^3 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )+m (a+b \text {arcsinh}(c x))^3 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )-(a+b \text {arcsinh}(c x))^3 \log \left (h (f+g x)^m\right )+3 b m \left ((a+b \text {arcsinh}(c x))^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(c x)} g}{-c f+\sqrt {c^2 f^2+g^2}}\right )-2 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(c x)} g}{-c f+\sqrt {c^2 f^2+g^2}}\right )+2 b^2 \operatorname {PolyLog}\left (4,\frac {e^{\text {arcsinh}(c x)} g}{-c f+\sqrt {c^2 f^2+g^2}}\right )\right )+3 b m \left ((a+b \text {arcsinh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )-2 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )+2 b^2 \operatorname {PolyLog}\left (4,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )}{3 b c} \] Input:

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

Output:

-1/3*(-1/4*(m*(a + b*ArcSinh[c*x])^4)/b + m*(a + b*ArcSinh[c*x])^3*Log[1 + 
 (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])] + m*(a + b*ArcSinh[c*x])^ 
3*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])] - (a + b*ArcSinh 
[c*x])^3*Log[h*(f + g*x)^m] + 3*b*m*((a + b*ArcSinh[c*x])^2*PolyLog[2, (E^ 
ArcSinh[c*x]*g)/(-(c*f) + Sqrt[c^2*f^2 + g^2])] - 2*b*(a + b*ArcSinh[c*x]) 
*PolyLog[3, (E^ArcSinh[c*x]*g)/(-(c*f) + Sqrt[c^2*f^2 + g^2])] + 2*b^2*Pol 
yLog[4, (E^ArcSinh[c*x]*g)/(-(c*f) + Sqrt[c^2*f^2 + g^2])]) + 3*b*m*((a + 
b*ArcSinh[c*x])^2*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^ 
2]))] - 2*b*(a + b*ArcSinh[c*x])*PolyLog[3, -((E^ArcSinh[c*x]*g)/(c*f + Sq 
rt[c^2*f^2 + g^2]))] + 2*b^2*PolyLog[4, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c 
^2*f^2 + g^2]))]))/(b*c)
 

Rubi [A] (verified)

Time = 1.76 (sec) , antiderivative size = 426, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6261, 6242, 6095, 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.

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

\(\Big \downarrow \) 6261

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

\(\Big \downarrow \) 6242

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

((a + b*ArcSinh[c*x])^3*Log[h*(f + g*x)^m])/(3*b*c) - (g*m*(-1/4*(a + b*Ar 
cSinh[c*x])^4/(b*g) + ((a + b*ArcSinh[c*x])^3*Log[1 + (E^ArcSinh[c*x]*g)/( 
c*f - Sqrt[c^2*f^2 + g^2])])/g + ((a + b*ArcSinh[c*x])^3*Log[1 + (E^ArcSin 
h[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/g - (3*b*(-((a + b*ArcSinh[c*x])^2 
*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))]) + 2*b*((a 
+ b*ArcSinh[c*x])*PolyLog[3, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^ 
2]))] - b*PolyLog[4, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))]))) 
/g - (3*b*(-((a + b*ArcSinh[c*x])^2*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + 
 Sqrt[c^2*f^2 + g^2]))]) + 2*b*((a + b*ArcSinh[c*x])*PolyLog[3, -((E^ArcSi 
nh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))] - b*PolyLog[4, -((E^ArcSinh[c*x]* 
g)/(c*f + Sqrt[c^2*f^2 + g^2]))])))/g))/(3*b*c)
 

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 6095

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin 
h[(c_.) + (d_.)*(x_)]), 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 6242

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

rule 6261

Int[(Log[(h_.)*((f_.) + (g_.)*(x_))^(m_.)]*((a_.) + ArcSinh[(c_.)*(x_)]*(b_ 
.))^(n_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[Log[h*(f + g*x)^m]* 
((a + b*ArcSinh[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1))), x] - Simp[g*(m/(b*c*S 
qrt[d]*(n + 1)))   Int[(a + b*ArcSinh[c*x])^(n + 1)/(f + g*x), x], x] /; Fr 
eeQ[{a, b, c, d, e, f, g, h, m}, x] && EqQ[e, c^2*d] && GtQ[d, 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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((a+b*arcsinh(c*x))^2*log(h*(g*x+f)^m)/(c^2*x^2+1)^(1/2),x, algor 
ithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((a+b*arcsinh(c*x))^2*log(h*(g*x+f)^m)/(c^2*x^2+1)^(1/2),x, algor 
ithm="maxima")
 

Output:

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

Giac [F(-2)]

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

integrate((a+b*arcsinh(c*x))^2*log(h*(g*x+f)^m)/(c^2*x^2+1)^(1/2),x, algor 
ithm="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%%%{1,[0,1,1,1,0,0]%%%}+%%%{-1,[0,0,1,1,0,0]%%%} / %%%{1,[0,0, 
0,0,1,1]%
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((a+b*asinh(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((asinh(c*x)*lo 
g((f + g*x)**m*h))/sqrt(c**2*x**2 + 1),x)*a*b + int((asinh(c*x)**2*log((f 
+ g*x)**m*h))/sqrt(c**2*x**2 + 1),x)*b**2