3.7.63 \(\int \frac {(a+b \arcsin (c x))^2}{d+e x^2} \, dx\) [663]

3.7.63.1 Optimal result

Integrand size = 20, antiderivative size = 821 \[ \int \frac {(a+b \arcsin (c x))^2}{d+e x^2} \, dx=\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \arcsin (c x))^2 \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(a+b \arcsin (c x))^2 \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {b^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{\sqrt {-d} \sqrt {e}} \]

output

1/2*(a+b*arcsin(c*x))^2*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^ 
(1/2)-(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)-1/2*(a+b*arcsin(c*x))^2*ln(1+(I 
*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(1 
/2)/e^(1/2)+1/2*(a+b*arcsin(c*x))^2*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2 
)/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)-1/2*(a+b*arcsin(c*x 
))^2*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/ 
2)))/(-d)^(1/2)/e^(1/2)+I*b*(a+b*arcsin(c*x))*polylog(2,-(I*c*x+(-c^2*x^2+ 
1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)-I*b 
*(a+b*arcsin(c*x))*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^ 
(1/2)-(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)+I*b*(a+b*arcsin(c*x))*polylog(2 
,-(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d 
)^(1/2)/e^(1/2)-I*b*(a+b*arcsin(c*x))*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2)) 
*e^(1/2)/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)-b^2*polylog( 
3,-(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/(- 
d)^(1/2)/e^(1/2)+b^2*polylog(3,(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d 
)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)-b^2*polylog(3,-(I*c*x+(-c^2*x 
^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(1/2)+ 
b^2*polylog(3,(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)+(c^2*d+e) 
^(1/2)))/(-d)^(1/2)/e^(1/2)
 

3.7.63.2 Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 1101, normalized size of antiderivative = 1.34 \[ \int \frac {(a+b \arcsin (c x))^2}{d+e x^2} \, dx=\frac {2 a^2 \sqrt {-d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-2 a b \sqrt {d} \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )-b^2 \sqrt {d} \arcsin (c x)^2 \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )+2 a b \sqrt {d} \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )+b^2 \sqrt {d} \arcsin (c x)^2 \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )+2 a b \sqrt {d} \arcsin (c x) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )+b^2 \sqrt {d} \arcsin (c x)^2 \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )-2 a b \sqrt {d} \arcsin (c x) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )-b^2 \sqrt {d} \arcsin (c x)^2 \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )-2 i b \sqrt {d} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )+2 i b \sqrt {d} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )+2 i a b \sqrt {d} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )+2 i b^2 \sqrt {d} \arcsin (c x) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )-2 i a b \sqrt {d} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )-2 i b^2 \sqrt {d} \arcsin (c x) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )+2 b^2 \sqrt {d} \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )-2 b^2 \sqrt {d} \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )-2 b^2 \sqrt {d} \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )+2 b^2 \sqrt {d} \operatorname {PolyLog}\left (3,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 \sqrt {-d^2} \sqrt {e}} \]

input

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

output

(2*a^2*Sqrt[-d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - 2*a*b*Sqrt[d]*ArcSin[c*x]*Lo 
g[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])] - b^2* 
Sqrt[d]*ArcSin[c*x]^2*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - 
Sqrt[c^2*d + e])] + 2*a*b*Sqrt[d]*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin 
[c*x]))/((-I)*c*Sqrt[-d] + Sqrt[c^2*d + e])] + b^2*Sqrt[d]*ArcSin[c*x]^2*L 
og[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/((-I)*c*Sqrt[-d] + Sqrt[c^2*d + e])] + 
2*a*b*Sqrt[d]*ArcSin[c*x]*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d 
] + Sqrt[c^2*d + e])] + b^2*Sqrt[d]*ArcSin[c*x]^2*Log[1 - (Sqrt[e]*E^(I*Ar 
cSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])] - 2*a*b*Sqrt[d]*ArcSin[c*x]* 
Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])] - b^ 
2*Sqrt[d]*ArcSin[c*x]^2*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] 
+ Sqrt[c^2*d + e])] - (2*I)*b*Sqrt[d]*(a + b*ArcSin[c*x])*PolyLog[2, (Sqrt 
[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])] + (2*I)*b*Sqrt[d] 
*(a + b*ArcSin[c*x])*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/((-I)*c*Sqrt[- 
d] + Sqrt[c^2*d + e])] + (2*I)*a*b*Sqrt[d]*PolyLog[2, -((Sqrt[e]*E^(I*ArcS 
in[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))] + (2*I)*b^2*Sqrt[d]*ArcSin[c* 
x]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e 
]))] - (2*I)*a*b*Sqrt[d]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[ 
-d] + Sqrt[c^2*d + e])] - (2*I)*b^2*Sqrt[d]*ArcSin[c*x]*PolyLog[2, (Sqrt[e 
]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])] + 2*b^2*Sqrt[d]*...
 

3.7.63.3 Rubi [A] (verified)

Time = 1.78 (sec) , antiderivative size = 821, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5172, 2009}

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.

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

\(\Big \downarrow \) 5172

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

\(\Big \downarrow \) 2009

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

input

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

output

((a + b*ArcSin[c*x])^2*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - 
 Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcSin[c*x])^2*Log[1 + 
(Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*Sqrt[-d] 
*Sqrt[e]) + ((a + b*ArcSin[c*x])^2*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I* 
c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcSin[c*x] 
)^2*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])]) 
/(2*Sqrt[-d]*Sqrt[e]) + (I*b*(a + b*ArcSin[c*x])*PolyLog[2, -((Sqrt[e]*E^( 
I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(Sqrt[-d]*Sqrt[e]) - ( 
I*b*(a + b*ArcSin[c*x])*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[- 
d] - Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e]) + (I*b*(a + b*ArcSin[c*x])*Poly 
Log[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/( 
Sqrt[-d]*Sqrt[e]) - (I*b*(a + b*ArcSin[c*x])*PolyLog[2, (Sqrt[e]*E^(I*ArcS 
in[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e]) - (b^2*Pol 
yLog[3, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/ 
(Sqrt[-d]*Sqrt[e]) + (b^2*PolyLog[3, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt 
[-d] - Sqrt[c^2*d + e])])/(Sqrt[-d]*Sqrt[e]) - (b^2*PolyLog[3, -((Sqrt[e]* 
E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(Sqrt[-d]*Sqrt[e]) 
+ (b^2*PolyLog[3, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + 
 e])])/(Sqrt[-d]*Sqrt[e])
 

3.7.63.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5172

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x 
] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (G 
tQ[p, 0] || IGtQ[n, 0])
 

3.7.63.4 Maple [F]

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

input

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

output

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

3.7.63.5 Fricas [F]

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

input

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

output

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

3.7.63.6 Sympy [F]

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

input

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

output

Integral((a + b*asin(c*x))**2/(d + e*x**2), x)
 

3.7.63.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]

input

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

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

3.7.63.8 Giac [F]

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

input

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

output

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

3.7.63.9 Mupad [F(-1)]

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

input

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

output

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