3.1.83 \(\int \frac {(a+b \arcsin (c x))^3 \log (h (f+g x)^m)}{\sqrt {1-c^2 x^2}} \, dx\) [83]

3.1.83.1 Optimal result

Integrand size = 35, antiderivative size = 634 \[ \int \frac {(a+b \arcsin (c x))^3 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\frac {i m (a+b \arcsin (c x))^5}{20 b^2 c}-\frac {m (a+b \arcsin (c x))^4 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{4 b c}-\frac {m (a+b \arcsin (c x))^4 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{4 b c}+\frac {(a+b \arcsin (c x))^4 \log \left (h (f+g x)^m\right )}{4 b c}+\frac {i m (a+b \arcsin (c x))^3 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {i m (a+b \arcsin (c x))^3 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {3 b m (a+b \arcsin (c x))^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {3 b m (a+b \arcsin (c x))^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {6 i b^2 m (a+b \arcsin (c x)) \operatorname {PolyLog}\left (4,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {6 i b^2 m (a+b \arcsin (c x)) \operatorname {PolyLog}\left (4,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {6 b^3 m \operatorname {PolyLog}\left (5,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {6 b^3 m \operatorname {PolyLog}\left (5,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c} \]

output

1/20*I*m*(a+b*arcsin(c*x))^5/b^2/c+1/4*(a+b*arcsin(c*x))^4*ln(h*(g*x+f)^m) 
/b/c-1/4*m*(a+b*arcsin(c*x))^4*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c 
^2*f^2-g^2)^(1/2)))/b/c-1/4*m*(a+b*arcsin(c*x))^4*ln(1-I*(I*c*x+(-c^2*x^2+ 
1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/b/c+I*m*(a+b*arcsin(c*x))^3*polylog 
(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/c+I*m*(a+b*ar 
csin(c*x))^3*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^( 
1/2)))/c-3*b*m*(a+b*arcsin(c*x))^2*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2))* 
g/(c*f-(c^2*f^2-g^2)^(1/2)))/c-3*b*m*(a+b*arcsin(c*x))^2*polylog(3,I*(I*c* 
x+(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/c-6*I*b^2*m*(a+b*arcsin 
(c*x))*polylog(4,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2))) 
/c-6*I*b^2*m*(a+b*arcsin(c*x))*polylog(4,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c 
*f+(c^2*f^2-g^2)^(1/2)))/c+6*b^3*m*polylog(5,I*(I*c*x+(-c^2*x^2+1)^(1/2))* 
g/(c*f-(c^2*f^2-g^2)^(1/2)))/c+6*b^3*m*polylog(5,I*(I*c*x+(-c^2*x^2+1)^(1/ 
2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/c
 

3.1.83.2 Mathematica [F]

\[ \int \frac {(a+b \arcsin (c x))^3 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {(a+b \arcsin (c x))^3 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx \]

input

Integrate[((a + b*ArcSin[c*x])^3*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2],x]
 

output

Integrate[((a + b*ArcSin[c*x])^3*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2], x]
 

3.1.83.3 Rubi [A] (verified)

Time = 2.08 (sec) , antiderivative size = 628, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {5278, 5240, 5030, 2620, 3011, 7163, 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.

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

\(\Big \downarrow \) 5278

\(\displaystyle \frac {(a+b \arcsin (c x))^4 \log \left (h (f+g x)^m\right )}{4 b c}-\frac {g m \int \frac {(a+b \arcsin (c x))^4}{f+g x}dx}{4 b c}\)

\(\Big \downarrow \) 5240

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

\(\Big \downarrow \) 5030

\(\displaystyle \frac {(a+b \arcsin (c x))^4 \log \left (h (f+g x)^m\right )}{4 b c}-\frac {g m \left (\int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))^4}{c f-i e^{i \arcsin (c x)} g-\sqrt {c^2 f^2-g^2}}d\arcsin (c x)+\int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))^4}{c f-i e^{i \arcsin (c x)} g+\sqrt {c^2 f^2-g^2}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^5}{5 b g}\right )}{4 b c}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {(a+b \arcsin (c x))^4 \log \left (h (f+g x)^m\right )}{4 b c}-\frac {g m \left (-\frac {4 b \int (a+b \arcsin (c x))^3 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)}{g}-\frac {4 b \int (a+b \arcsin (c x))^3 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)}{g}+\frac {(a+b \arcsin (c x))^4 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}+\frac {(a+b \arcsin (c x))^4 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {i (a+b \arcsin (c x))^5}{5 b g}\right )}{4 b c}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {(a+b \arcsin (c x))^4 \log \left (h (f+g x)^m\right )}{4 b c}-\frac {g m \left (-\frac {4 b \left (i (a+b \arcsin (c x))^3 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-3 i b \int (a+b \arcsin (c x))^2 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)\right )}{g}-\frac {4 b \left (i (a+b \arcsin (c x))^3 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-3 i b \int (a+b \arcsin (c x))^2 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)\right )}{g}+\frac {(a+b \arcsin (c x))^4 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}+\frac {(a+b \arcsin (c x))^4 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {i (a+b \arcsin (c x))^5}{5 b g}\right )}{4 b c}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {(a+b \arcsin (c x))^4 \log \left (h (f+g x)^m\right )}{4 b c}-\frac {g m \left (-\frac {4 b \left (i (a+b \arcsin (c x))^3 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-3 i b \left (2 i b \int (a+b \arcsin (c x)) \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)-i (a+b \arcsin (c x))^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )\right )}{g}-\frac {4 b \left (i (a+b \arcsin (c x))^3 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-3 i b \left (2 i b \int (a+b \arcsin (c x)) \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)-i (a+b \arcsin (c x))^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )\right )}{g}+\frac {(a+b \arcsin (c x))^4 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}+\frac {(a+b \arcsin (c x))^4 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {i (a+b \arcsin (c x))^5}{5 b g}\right )}{4 b c}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {(a+b \arcsin (c x))^4 \log \left (h (f+g x)^m\right )}{4 b c}-\frac {g m \left (-\frac {4 b \left (i (a+b \arcsin (c x))^3 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-3 i b \left (2 i b \left (i b \int \operatorname {PolyLog}\left (4,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)-i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (4,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )-i (a+b \arcsin (c x))^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )\right )}{g}-\frac {4 b \left (i (a+b \arcsin (c x))^3 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-3 i b \left (2 i b \left (i b \int \operatorname {PolyLog}\left (4,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)-i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (4,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )-i (a+b \arcsin (c x))^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )\right )}{g}+\frac {(a+b \arcsin (c x))^4 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}+\frac {(a+b \arcsin (c x))^4 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {i (a+b \arcsin (c x))^5}{5 b g}\right )}{4 b c}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {(a+b \arcsin (c x))^4 \log \left (h (f+g x)^m\right )}{4 b c}-\frac {g m \left (-\frac {4 b \left (i (a+b \arcsin (c x))^3 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-3 i b \left (2 i b \left (b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (4,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )de^{i \arcsin (c x)}-i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (4,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )-i (a+b \arcsin (c x))^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )\right )}{g}-\frac {4 b \left (i (a+b \arcsin (c x))^3 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-3 i b \left (2 i b \left (b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (4,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )de^{i \arcsin (c x)}-i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (4,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )-i (a+b \arcsin (c x))^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )\right )}{g}+\frac {(a+b \arcsin (c x))^4 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}+\frac {(a+b \arcsin (c x))^4 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {i (a+b \arcsin (c x))^5}{5 b g}\right )}{4 b c}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {(a+b \arcsin (c x))^4 \log \left (h (f+g x)^m\right )}{4 b c}-\frac {g m \left (-\frac {4 b \left (i (a+b \arcsin (c x))^3 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-3 i b \left (2 i b \left (b \operatorname {PolyLog}\left (5,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (4,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )-i (a+b \arcsin (c x))^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )\right )}{g}-\frac {4 b \left (i (a+b \arcsin (c x))^3 \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-3 i b \left (2 i b \left (b \operatorname {PolyLog}\left (5,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (4,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )-i (a+b \arcsin (c x))^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )\right )}{g}+\frac {(a+b \arcsin (c x))^4 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}+\frac {(a+b \arcsin (c x))^4 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {i (a+b \arcsin (c x))^5}{5 b g}\right )}{4 b c}\)

input

Int[((a + b*ArcSin[c*x])^3*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2],x]
 

output

((a + b*ArcSin[c*x])^4*Log[h*(f + g*x)^m])/(4*b*c) - (g*m*(((-1/5*I)*(a + 
b*ArcSin[c*x])^5)/(b*g) + ((a + b*ArcSin[c*x])^4*Log[1 - (I*E^(I*ArcSin[c* 
x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g + ((a + b*ArcSin[c*x])^4*Log[1 - (I 
*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g - (4*b*(I*(a + b*Arc 
Sin[c*x])^3*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]) 
] - (3*I)*b*((-I)*(a + b*ArcSin[c*x])^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g) 
/(c*f - Sqrt[c^2*f^2 - g^2])] + (2*I)*b*((-I)*(a + b*ArcSin[c*x])*PolyLog[ 
4, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + b*PolyLog[5, (I* 
E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])]))))/g - (4*b*(I*(a + b*A 
rcSin[c*x])^3*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2 
])] - (3*I)*b*((-I)*(a + b*ArcSin[c*x])^2*PolyLog[3, (I*E^(I*ArcSin[c*x])* 
g)/(c*f + Sqrt[c^2*f^2 - g^2])] + (2*I)*b*((-I)*(a + b*ArcSin[c*x])*PolyLo 
g[4, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] + b*PolyLog[5, ( 
I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))))/g))/(4*b*c)
 

3.1.83.3.1 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 5030

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

rule 5240

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

rule 5278

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

3.1.83.4 Maple [F]

\[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{3} \ln \left (h \left (g x +f \right )^{m}\right )}{\sqrt {-c^{2} x^{2}+1}}d x\]

input

int((a+b*arcsin(c*x))^3*ln(h*(g*x+f)^m)/(-c^2*x^2+1)^(1/2),x)
 

output

int((a+b*arcsin(c*x))^3*ln(h*(g*x+f)^m)/(-c^2*x^2+1)^(1/2),x)
 

3.1.83.5 Fricas [F]

\[ \int \frac {(a+b \arcsin (c x))^3 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{3} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]

input

integrate((a+b*arcsin(c*x))^3*log(h*(g*x+f)^m)/(-c^2*x^2+1)^(1/2),x, algor 
ithm="fricas")
 

output

integral(-(b^3*arcsin(c*x)^3 + 3*a*b^2*arcsin(c*x)^2 + 3*a^2*b*arcsin(c*x) 
 + a^3)*sqrt(-c^2*x^2 + 1)*log((g*x + f)^m*h)/(c^2*x^2 - 1), x)
 

3.1.83.6 Sympy [F]

\[ \int \frac {(a+b \arcsin (c x))^3 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{3} \log {\left (h \left (f + g x\right )^{m} \right )}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]

input

integrate((a+b*asin(c*x))**3*ln(h*(g*x+f)**m)/(-c**2*x**2+1)**(1/2),x)
 

output

Integral((a + b*asin(c*x))**3*log(h*(f + g*x)**m)/sqrt(-(c*x - 1)*(c*x + 1 
)), x)
 

3.1.83.7 Maxima [F]

\[ \int \frac {(a+b \arcsin (c x))^3 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{3} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]

input

integrate((a+b*arcsin(c*x))^3*log(h*(g*x+f)^m)/(-c^2*x^2+1)^(1/2),x, algor 
ithm="maxima")
 

output

(b^3*c*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^3/(sqrt(c*x + 
1)*sqrt(-c*x + 1)), x)*log(h) + 3*a*b^2*c*integrate(arctan2(c*x, sqrt(c*x 
+ 1)*sqrt(-c*x + 1))^2/(sqrt(c*x + 1)*sqrt(-c*x + 1)), x)*log(h) + 3*a^2*b 
*c*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(sqrt(c*x + 1)*sqr 
t(-c*x + 1)), x)*log(h) + b^3*c*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt( 
-c*x + 1))^3*log((g*x + f)^m)/(sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + 3*a*b^2 
*c*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log((g*x + f)^m) 
/(sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + 3*a^2*b*c*integrate(arctan2(c*x, sqr 
t(c*x + 1)*sqrt(-c*x + 1))*log((g*x + f)^m)/(sqrt(c*x + 1)*sqrt(-c*x + 1)) 
, x) + a^3*c*integrate(log((g*x + f)^m)/(sqrt(c*x + 1)*sqrt(-c*x + 1)), x) 
 + a^3*arctan2(c*x, sqrt(-c^2*x^2 + 1))*log(h))/c
 

3.1.83.8 Giac [F]

\[ \int \frac {(a+b \arcsin (c x))^3 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{3} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]

input

integrate((a+b*arcsin(c*x))^3*log(h*(g*x+f)^m)/(-c^2*x^2+1)^(1/2),x, algor 
ithm="giac")
 

output

integrate((b*arcsin(c*x) + a)^3*log((g*x + f)^m*h)/sqrt(-c^2*x^2 + 1), x)
 

3.1.83.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arcsin (c x))^3 \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^3}{\sqrt {1-c^2\,x^2}} \,d x \]

input

int((log(h*(f + g*x)^m)*(a + b*asin(c*x))^3)/(1 - c^2*x^2)^(1/2),x)
 

output

int((log(h*(f + g*x)^m)*(a + b*asin(c*x))^3)/(1 - c^2*x^2)^(1/2), x)