\(\int \frac {(a+b \arcsin (c x))^2}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx\) [146]

Optimal result

Integrand size = 33, antiderivative size = 589 \[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=-\frac {i \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {i \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}} \] Output:

-I*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2) 
)*g/(c*f-(c^2*f^2-g^2)^(1/2)))/(c^2*f^2-g^2)^(1/2)/(-c^2*d*x^2+d)^(1/2)+I* 
(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*g 
/(c*f+(c^2*f^2-g^2)^(1/2)))/(c^2*f^2-g^2)^(1/2)/(-c^2*d*x^2+d)^(1/2)-2*b*( 
-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))*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^2*f^2-g^2)^(1/2)/(-c^2*d*x^2+d)^(1/2)+2*b 
*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))*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^2*f^2-g^2)^(1/2)/(-c^2*d*x^2+d)^(1/2)-2 
*I*b^2*(-c^2*x^2+1)^(1/2)*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^2*f^2-g^2)^(1/2)/(-c^2*d*x^2+d)^(1/2)+2*I*b^2*(-c^2 
*x^2+1)^(1/2)*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^2*f^2-g^2)^(1/2)/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 357, normalized size of antiderivative = 0.61 \[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=-\frac {i \sqrt {1-c^2 x^2} \left ((a+b \arcsin (c x))^2 \log \left (1+\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-(a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )+2 b^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-2 b^2 \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.71 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.68, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {5276, 5272, 3042, 3804, 2694, 27, 2620, 3011, 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.

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

\(\Big \downarrow \) 5276

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

\(\Big \downarrow \) 5272

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3804

\(\displaystyle \frac {2 \sqrt {1-c^2 x^2} \int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))^2}{2 c e^{i \arcsin (c x)} f-i e^{2 i \arcsin (c x)} g+i g}d\arcsin (c x)}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

(2*Sqrt[1 - c^2*x^2]*(((-1/2*I)*g*(((a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I* 
ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g - (2*b*(I*(a + b*ArcSin[c* 
x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] - b*Po 
lyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])]))/g))/Sqrt[c 
^2*f^2 - g^2] + ((I/2)*g*(((a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c* 
x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g - (2*b*(I*(a + b*ArcSin[c*x])*PolyL 
og[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] - b*PolyLog[3, 
(I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))/g))/Sqrt[c^2*f^2 - 
g^2]))/Sqrt[d - c^2*d*x^2]
 

Defintions of rubi rules used

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 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 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && 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 3042

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

rule 3804

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

rule 5272

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sq 
rt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[1/(c^(m + 1)*Sqrt[d])   Subst[In 
t[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c 
, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (G 
tQ[m, 0] || IGtQ[n, 0])
 

rule 5276

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

Maple [F]

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

Input:

int((a+b*arcsin(c*x))^2/(g*x+f)/(-c^2*d*x^2+d)^(1/2),x)
 

Output:

int((a+b*arcsin(c*x))^2/(g*x+f)/(-c^2*d*x^2+d)^(1/2),x)
 

Fricas [F]

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

integrate((a+b*arcsin(c*x))^2/(g*x+f)/(-c^2*d*x^2+d)^(1/2),x, algorithm="f 
ricas")
 

Output:

integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^ 
2)/(c^2*d*g*x^3 + c^2*d*f*x^2 - d*g*x - d*f), x)
 

Sympy [F]

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

integrate((a+b*asin(c*x))**2/(g*x+f)/(-c**2*d*x**2+d)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((a+b*arcsin(c*x))^2/(g*x+f)/(-c^2*d*x^2+d)^(1/2),x, algorithm="m 
axima")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*arcsin(c*x))^2/(g*x+f)/(-c^2*d*x^2+d)^(1/2),x, algorithm="g 
iac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

int((a + b*asin(c*x))^2/((f + g*x)*(d - c^2*d*x^2)^(1/2)),x)
                                                                                    
                                                                                    
 

Output:

int((a + b*asin(c*x))^2/((f + g*x)*(d - c^2*d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=\frac {2 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a^{2}+2 \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, f +\sqrt {-c^{2} x^{2}+1}\, g x}d x \right ) a b \,c^{2} f^{2}-2 \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, f +\sqrt {-c^{2} x^{2}+1}\, g x}d x \right ) a b \,g^{2}+\left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, f +\sqrt {-c^{2} x^{2}+1}\, g x}d x \right ) b^{2} c^{2} f^{2}-\left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, f +\sqrt {-c^{2} x^{2}+1}\, g x}d x \right ) b^{2} g^{2}}{\sqrt {d}\, \left (c^{2} f^{2}-g^{2}\right )} \] Input:

int((a+b*asin(c*x))^2/(g*x+f)/(-c^2*d*x^2+d)^(1/2),x)
 

Output:

(2*sqrt(c**2*f**2 - g**2)*atan((tan(asin(c*x)/2)*c*f + g)/sqrt(c**2*f**2 - 
 g**2))*a**2 + 2*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*f + sqrt( - c**2*x* 
*2 + 1)*g*x),x)*a*b*c**2*f**2 - 2*int(asin(c*x)/(sqrt( - c**2*x**2 + 1)*f 
+ sqrt( - c**2*x**2 + 1)*g*x),x)*a*b*g**2 + int(asin(c*x)**2/(sqrt( - c**2 
*x**2 + 1)*f + sqrt( - c**2*x**2 + 1)*g*x),x)*b**2*c**2*f**2 - int(asin(c* 
x)**2/(sqrt( - c**2*x**2 + 1)*f + sqrt( - c**2*x**2 + 1)*g*x),x)*b**2*g**2 
)/(sqrt(d)*(c**2*f**2 - g**2))