\(\int \frac {(a+b \arccos (c x))^2 \log (h (f+g x)^m)}{\sqrt {1-c^2 x^2}} \, dx\) [20]

Optimal result

Integrand size = 35, antiderivative size = 496 \[ \int \frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=-\frac {i m (a+b \arccos (c x))^4}{12 b^2 c}+\frac {m (a+b \arccos (c x))^3 \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{3 b c}+\frac {m (a+b \arccos (c x))^3 \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{3 b c}-\frac {(a+b \arccos (c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {i m (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {i m (a+b \arccos (c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {2 b m (a+b \arccos (c x)) \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {2 b m (a+b \arccos (c x)) \operatorname {PolyLog}\left (3,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {2 i b^2 m \operatorname {PolyLog}\left (4,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {2 i b^2 m \operatorname {PolyLog}\left (4,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c} \] Output:

-1/12*I*m*(a+b*arccos(c*x))^4/b^2/c+1/3*m*(a+b*arccos(c*x))^3*ln(1+(c*x+I* 
(-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*arccos(c*x 
))^3*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/b/c-1/3* 
(a+b*arccos(c*x))^3*ln(h*(g*x+f)^m)/b/c-I*m*(a+b*arccos(c*x))^2*polylog(2, 
-(c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/c-I*m*(a+b*arccos 
(c*x))^2*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)) 
)/c+2*b*m*(a+b*arccos(c*x))*polylog(3,-(c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f-( 
c^2*f^2-g^2)^(1/2)))/c+2*b*m*(a+b*arccos(c*x))*polylog(3,-(c*x+I*(-c^2*x^2 
+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/c+2*I*b^2*m*polylog(4,-(c*x+I*(-c^ 
2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/c+2*I*b^2*m*polylog(4,-(c*x+I 
*(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/c
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2330\) vs. \(2(496)=992\).

Time = 24.43 (sec) , antiderivative size = 2330, normalized size of antiderivative = 4.70 \[ \int \frac {(a+b \arccos (c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

((-6*I)*a^2*m*ArcCos[c*x]^2 - (4*I)*a*b*m*ArcCos[c*x]^3 - I*b^2*m*ArcCos[c 
*x]^4 + (48*I)*a^2*m*ArcSin[Sqrt[1 + (c*f)/g]/Sqrt[2]]*ArcTan[((c*f - g)*T 
an[ArcCos[c*x]/2])/Sqrt[c^2*f^2 - g^2]] + 12*a*b*m*ArcCos[c*x]^2*Log[1 + ( 
E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + 8*b^2*m*ArcCos[c*x]^3* 
Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + 12*a^2*m*ArcC 
os[c*x]*Log[1 + (E^(I*ArcCos[c*x])*(c*f - Sqrt[c^2*f^2 - g^2]))/g] + 12*a* 
b*m*ArcCos[c*x]^2*Log[1 + (E^(I*ArcCos[c*x])*(c*f - Sqrt[c^2*f^2 - g^2]))/ 
g] + 4*b^2*m*ArcCos[c*x]^3*Log[1 + (E^(I*ArcCos[c*x])*(c*f - Sqrt[c^2*f^2 
- g^2]))/g] + 24*a^2*m*ArcSin[Sqrt[1 + (c*f)/g]/Sqrt[2]]*Log[1 + (E^(I*Arc 
Cos[c*x])*(c*f - Sqrt[c^2*f^2 - g^2]))/g] + 48*a*b*m*ArcCos[c*x]*ArcSin[Sq 
rt[1 + (c*f)/g]/Sqrt[2]]*Log[1 + (E^(I*ArcCos[c*x])*(c*f - Sqrt[c^2*f^2 - 
g^2]))/g] + 24*b^2*m*ArcCos[c*x]^2*ArcSin[Sqrt[1 + (c*f)/g]/Sqrt[2]]*Log[1 
 + (E^(I*ArcCos[c*x])*(c*f - Sqrt[c^2*f^2 - g^2]))/g] + 12*a*b*m*ArcCos[c* 
x]^2*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] + 8*b^2*m* 
ArcCos[c*x]^3*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] + 
 12*a^2*m*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*(c*f + Sqrt[c^2*f^2 - g^2 
]))/g] + 12*a*b*m*ArcCos[c*x]^2*Log[1 + (E^(I*ArcCos[c*x])*(c*f + Sqrt[c^2 
*f^2 - g^2]))/g] + 4*b^2*m*ArcCos[c*x]^3*Log[1 + (E^(I*ArcCos[c*x])*(c*f + 
 Sqrt[c^2*f^2 - g^2]))/g] - 24*a^2*m*ArcSin[Sqrt[1 + (c*f)/g]/Sqrt[2]]*Log 
[1 + (E^(I*ArcCos[c*x])*(c*f + Sqrt[c^2*f^2 - g^2]))/g] - 48*a*b*m*ArcC...
 

Rubi [A] (verified)

Time = 1.86 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {5279, 5241, 5031, 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.

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

\(\Big \downarrow \) 5279

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

\(\Big \downarrow \) 5241

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

\(\Big \downarrow \) 5031

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-1/3*((a + b*ArcCos[c*x])^3*Log[h*(f + g*x)^m])/(b*c) - (g*m*(((I/4)*(a + 
b*ArcCos[c*x])^4)/(b*g) - I*(((-I)*(a + b*ArcCos[c*x])^3*Log[1 + (E^(I*Arc 
Cos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g + ((3*I)*b*(I*(a + b*ArcCos[c 
*x])^2*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))] - 
(2*I)*b*((-I)*(a + b*ArcCos[c*x])*PolyLog[3, -((E^(I*ArcCos[c*x])*g)/(c*f 
- Sqrt[c^2*f^2 - g^2]))] + b*PolyLog[4, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqr 
t[c^2*f^2 - g^2]))])))/g) - I*(((-I)*(a + b*ArcCos[c*x])^3*Log[1 + (E^(I*A 
rcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g + ((3*I)*b*(I*(a + b*ArcCos 
[c*x])^2*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))] 
- (2*I)*b*((-I)*(a + b*ArcCos[c*x])*PolyLog[3, -((E^(I*ArcCos[c*x])*g)/(c* 
f + Sqrt[c^2*f^2 - g^2]))] + b*PolyLog[4, -((E^(I*ArcCos[c*x])*g)/(c*f + S 
qrt[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 5031

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

rule 5241

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

rule 5279

Int[(Log[(h_.)*((f_.) + (g_.)*(x_))^(m_.)]*((a_.) + ArcCos[(c_.)*(x_)]*(b_. 
))^(n_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(-Log[h*(f + g*x)^m] 
)*((a + b*ArcCos[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1))), x] + Simp[g*(m/(b*c* 
Sqrt[d]*(n + 1)))   Int[(a + b*ArcCos[c*x])^(n + 1)/(f + g*x), x], x] /; Fr 
eeQ[{a, b, c, d, e, f, g, h, m}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && IG 
tQ[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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

(b^2*c*integrate(arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2/(sqrt(c*x + 
1)*sqrt(-c*x + 1)), x)*log(h) + 2*a*b*c*integrate(arctan2(sqrt(c*x + 1)*sq 
rt(-c*x + 1), c*x)/(sqrt(c*x + 1)*sqrt(-c*x + 1)), x)*log(h) + b^2*c*integ 
rate(arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2*log((g*x + f)^m)/(sqrt(c 
*x + 1)*sqrt(-c*x + 1)), x) + 2*a*b*c*integrate(arctan2(sqrt(c*x + 1)*sqrt 
(-c*x + 1), c*x)*log((g*x + f)^m)/(sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + a^2 
*c*integrate(log((g*x + f)^m)/(sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + a^2*arc 
tan2(c*x, sqrt(-c^2*x^2 + 1))*log(h))/c
                                                                                    
                                                                                    
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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