3.2.83 \(\int \frac {x^4 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx\) [183]

3.2.83.1 Optimal result

Integrand size = 27, antiderivative size = 297 \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\frac {22 b^2 x}{9 c^4 d}+\frac {2 b^2 x^3}{27 c^2 d}-\frac {22 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c^5 d}-\frac {2 b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 c^3 d}-\frac {x (a+b \arcsin (c x))^2}{c^4 d}-\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d}-\frac {2 i (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{c^5 d}+\frac {2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^5 d}-\frac {2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^5 d}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{c^5 d}+\frac {2 b^2 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{c^5 d} \]

output

22/9*b^2*x/c^4/d+2/27*b^2*x^3/c^2/d-x*(a+b*arcsin(c*x))^2/c^4/d-1/3*x^3*(a 
+b*arcsin(c*x))^2/c^2/d-2*I*(a+b*arcsin(c*x))^2*arctan(I*c*x+(-c^2*x^2+1)^ 
(1/2))/c^5/d+2*I*b*(a+b*arcsin(c*x))*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2 
)))/c^5/d-2*I*b*(a+b*arcsin(c*x))*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/ 
c^5/d-2*b^2*polylog(3,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^5/d+2*b^2*polylog(3 
,I*(I*c*x+(-c^2*x^2+1)^(1/2)))/c^5/d-22/9*b*(a+b*arcsin(c*x))*(-c^2*x^2+1) 
^(1/2)/c^5/d-2/9*b*x^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3/d
 

3.2.83.2 Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.71 \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=-\frac {108 a^2 c x-270 b^2 c x+36 a^2 c^3 x^3+264 a b \sqrt {1-c^2 x^2}+24 a b c^2 x^2 \sqrt {1-c^2 x^2}+108 i a b \pi \arcsin (c x)+216 a b c x \arcsin (c x)+72 a b c^3 x^3 \arcsin (c x)+270 b^2 \sqrt {1-c^2 x^2} \arcsin (c x)+135 b^2 c x \arcsin (c x)^2-6 b^2 \arcsin (c x) \cos (3 \arcsin (c x))-108 a b \pi \log \left (1-i e^{i \arcsin (c x)}\right )-216 a b \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-108 b^2 \arcsin (c x)^2 \log \left (1-i e^{i \arcsin (c x)}\right )-108 a b \pi \log \left (1+i e^{i \arcsin (c x)}\right )+216 a b \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+108 b^2 \arcsin (c x)^2 \log \left (1+i e^{i \arcsin (c x)}\right )+54 a^2 \log (1-c x)-54 a^2 \log (1+c x)+108 a b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+108 a b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-216 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+216 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )+216 b^2 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )-216 b^2 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )+2 b^2 \sin (3 \arcsin (c x))-9 b^2 \arcsin (c x)^2 \sin (3 \arcsin (c x))}{108 c^5 d} \]

input

Integrate[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2),x]
 

output

-1/108*(108*a^2*c*x - 270*b^2*c*x + 36*a^2*c^3*x^3 + 264*a*b*Sqrt[1 - c^2* 
x^2] + 24*a*b*c^2*x^2*Sqrt[1 - c^2*x^2] + (108*I)*a*b*Pi*ArcSin[c*x] + 216 
*a*b*c*x*ArcSin[c*x] + 72*a*b*c^3*x^3*ArcSin[c*x] + 270*b^2*Sqrt[1 - c^2*x 
^2]*ArcSin[c*x] + 135*b^2*c*x*ArcSin[c*x]^2 - 6*b^2*ArcSin[c*x]*Cos[3*ArcS 
in[c*x]] - 108*a*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 216*a*b*ArcSin[c*x]*L 
og[1 - I*E^(I*ArcSin[c*x])] - 108*b^2*ArcSin[c*x]^2*Log[1 - I*E^(I*ArcSin[ 
c*x])] - 108*a*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 216*a*b*ArcSin[c*x]*Log 
[1 + I*E^(I*ArcSin[c*x])] + 108*b^2*ArcSin[c*x]^2*Log[1 + I*E^(I*ArcSin[c* 
x])] + 54*a^2*Log[1 - c*x] - 54*a^2*Log[1 + c*x] + 108*a*b*Pi*Log[-Cos[(Pi 
 + 2*ArcSin[c*x])/4]] + 108*a*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (216 
*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (216*I)*b*( 
a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])] + 216*b^2*PolyLog[3, (- 
I)*E^(I*ArcSin[c*x])] - 216*b^2*PolyLog[3, I*E^(I*ArcSin[c*x])] + 2*b^2*Si 
n[3*ArcSin[c*x]] - 9*b^2*ArcSin[c*x]^2*Sin[3*ArcSin[c*x]])/(c^5*d)
 

3.2.83.3 Rubi [A] (verified)

Time = 1.79 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.06, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5210, 27, 5210, 15, 5164, 3042, 4669, 3011, 2720, 5182, 24, 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.

\(\displaystyle \int \frac {x^4 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {\int \frac {x^2 (a+b \arcsin (c x))^2}{d \left (1-c^2 x^2\right )}dx}{c^2}+\frac {2 b \int \frac {x^3 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c d}-\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {x^2 (a+b \arcsin (c x))^2}{1-c^2 x^2}dx}{c^2 d}+\frac {2 b \int \frac {x^3 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c d}-\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {2 b \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {b \int x^2dx}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}\right )}{3 c d}+\frac {\frac {2 b \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c}+\frac {\int \frac {(a+b \arcsin (c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \arcsin (c x))^2}{c^2}}{c^2 d}-\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {\frac {2 b \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c}+\frac {\int \frac {(a+b \arcsin (c x))^2}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \arcsin (c x))^2}{c^2}}{c^2 d}+\frac {2 b \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {b x^3}{9 c}\right )}{3 c d}-\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 5164

\(\displaystyle \frac {2 b \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {b x^3}{9 c}\right )}{3 c d}+\frac {\frac {2 b \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c}+\frac {\int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{c^3}-\frac {x (a+b \arcsin (c x))^2}{c^2}}{c^2 d}-\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 b \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {b x^3}{9 c}\right )}{3 c d}+\frac {\frac {\int (a+b \arcsin (c x))^2 \csc \left (\arcsin (c x)+\frac {\pi }{2}\right )d\arcsin (c x)}{c^3}+\frac {2 b \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {x (a+b \arcsin (c x))^2}{c^2}}{c^2 d}-\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 4669

\(\displaystyle \frac {\frac {-2 b \int (a+b \arcsin (c x)) \log \left (1-i e^{i \arcsin (c x)}\right )d\arcsin (c x)+2 b \int (a+b \arcsin (c x)) \log \left (1+i e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{c^3}+\frac {2 b \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {x (a+b \arcsin (c x))^2}{c^2}}{c^2 d}+\frac {2 b \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {b x^3}{9 c}\right )}{3 c d}-\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-i b \int \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )d\arcsin (c x)\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{c^3}+\frac {2 b \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {x (a+b \arcsin (c x))^2}{c^2}}{c^2 d}+\frac {2 b \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {b x^3}{9 c}\right )}{3 c d}-\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{c^3}+\frac {2 b \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{c}-\frac {x (a+b \arcsin (c x))^2}{c^2}}{c^2 d}+\frac {2 b \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {b x^3}{9 c}\right )}{3 c d}-\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 5182

\(\displaystyle \frac {\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{c^3}+\frac {2 b \left (\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{c}-\frac {x (a+b \arcsin (c x))^2}{c^2}}{c^2 d}+\frac {2 b \left (\frac {2 \left (\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {b x^3}{9 c}\right )}{3 c d}-\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {\frac {2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}\right )-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{c^3}+\frac {2 b \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{c}-\frac {x (a+b \arcsin (c x))^2}{c^2}}{c^2 d}-\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 b \left (-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {2 \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{3 c^2}+\frac {b x^3}{9 c}\right )}{3 c d}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\frac {-2 i \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))-b \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )\right )}{c^3}+\frac {2 b \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{c}-\frac {x (a+b \arcsin (c x))^2}{c^2}}{c^2 d}-\frac {x^3 (a+b \arcsin (c x))^2}{3 c^2 d}+\frac {2 b \left (-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {2 \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{3 c^2}+\frac {b x^3}{9 c}\right )}{3 c d}\)

input

Int[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2),x]
 

output

-1/3*(x^3*(a + b*ArcSin[c*x])^2)/(c^2*d) + (2*b*((b*x^3)/(9*c) - (x^2*Sqrt 
[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*c^2) + (2*((b*x)/c - (Sqrt[1 - c^2*x 
^2]*(a + b*ArcSin[c*x]))/c^2))/(3*c^2)))/(3*c*d) + (-((x*(a + b*ArcSin[c*x 
])^2)/c^2) + (2*b*((b*x)/c - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^2)) 
/c + ((-2*I)*(a + b*ArcSin[c*x])^2*ArcTan[E^(I*ArcSin[c*x])] + 2*b*(I*(a + 
 b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - b*PolyLog[3, (-I)*E^( 
I*ArcSin[c*x])]) - 2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x 
])] - b*PolyLog[3, I*E^(I*ArcSin[c*x])]))/c^3)/(c^2*d)
 

3.2.83.3.1 Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4669

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si 
mp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], 
 x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x 
))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 5164

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> Simp[1/(c*d)   Subst[Int[(a + b*x)^n*Sec[x], x], x, ArcSin[c*x]], x] 
/; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
 

rule 5182

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ 
.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 
1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   I 
nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, 
 b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
 

rule 5210

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + 
 b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p 
 + 1)))   Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S 
imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f* 
x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; 
FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m 
, 1] && NeQ[m + 2*p + 1, 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]
 

3.2.83.4 Maple [F]

\[\int \frac {x^{4} \left (a +b \arcsin \left (c x \right )\right )^{2}}{-c^{2} d \,x^{2}+d}d x\]

input

int(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d),x)
 

output

int(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d),x)
 

3.2.83.5 Fricas [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{4}}{c^{2} d x^{2} - d} \,d x } \]

input

integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d),x, algorithm="fricas")
 

output

integral(-(b^2*x^4*arcsin(c*x)^2 + 2*a*b*x^4*arcsin(c*x) + a^2*x^4)/(c^2*d 
*x^2 - d), x)
 

3.2.83.6 Sympy [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=- \frac {\int \frac {a^{2} x^{4}}{c^{2} x^{2} - 1}\, dx + \int \frac {b^{2} x^{4} \operatorname {asin}^{2}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx + \int \frac {2 a b x^{4} \operatorname {asin}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \]

input

integrate(x**4*(a+b*asin(c*x))**2/(-c**2*d*x**2+d),x)
 

output

-(Integral(a**2*x**4/(c**2*x**2 - 1), x) + Integral(b**2*x**4*asin(c*x)**2 
/(c**2*x**2 - 1), x) + Integral(2*a*b*x**4*asin(c*x)/(c**2*x**2 - 1), x))/ 
d
 

3.2.83.7 Maxima [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{4}}{c^{2} d x^{2} - d} \,d x } \]

input

integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d),x, algorithm="maxima")
 

output

-1/6*a^2*(2*(c^2*x^3 + 3*x)/(c^4*d) - 3*log(c*x + 1)/(c^5*d) + 3*log(c*x - 
 1)/(c^5*d)) + 1/6*(6*c^5*d*integrate(-1/3*(6*a*b*c^4*x^4*arctan2(c*x, sqr 
t(c*x + 1)*sqrt(-c*x + 1)) - (3*b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 
 1))*log(c*x + 1) - 3*b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))*log(- 
c*x + 1) - 2*(b^2*c^3*x^3 + 3*b^2*c*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c* 
x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1))/(c^6*d*x^2 - c^4*d), x) + 3*b^2*arc 
tan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(c*x + 1) - 3*b^2*arctan2(c*x 
, sqrt(c*x + 1)*sqrt(-c*x + 1))^2*log(-c*x + 1) - 2*(b^2*c^3*x^3 + 3*b^2*c 
*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2)/(c^5*d)
 

3.2.83.8 Giac [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\int { -\frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{4}}{c^{2} d x^{2} - d} \,d x } \]

input

integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d),x, algorithm="giac")
 

output

integrate(-(b*arcsin(c*x) + a)^2*x^4/(c^2*d*x^2 - d), x)
 

3.2.83.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{d-c^2\,d\,x^2} \,d x \]

input

int((x^4*(a + b*asin(c*x))^2)/(d - c^2*d*x^2),x)
 

output

int((x^4*(a + b*asin(c*x))^2)/(d - c^2*d*x^2), x)