3.1.76 \(\int \frac {(f+g x)^2 (a+b \text {ArcSin}(c x))^2}{(d-c^2 d x^2)^{3/2}} \, dx\) [76]

Optimal. Leaf size=513 \[ \frac {2 f g (a+b \text {ArcSin}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 f^2+g^2\right ) x (a+b \text {ArcSin}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {8 i b f g \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 b \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}} \]

[Out]

2*f*g*(a+b*arcsin(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+(c^2*f^2+g^2)*x*(a+b*arcsin(c*x))^2/c^2/d/(-c^2*d*x^2+d)^
(1/2)-I*(c^2*f^2+g^2)*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)-1/3*g^2*(a+b*arcsin(c*
x))^3*(-c^2*x^2+1)^(1/2)/b/c^3/d/(-c^2*d*x^2+d)^(1/2)+8*I*b*f*g*(a+b*arcsin(c*x))*arctan(I*c*x+(-c^2*x^2+1)^(1
/2))*(-c^2*x^2+1)^(1/2)/c^2/d/(-c^2*d*x^2+d)^(1/2)+2*b*(c^2*f^2+g^2)*(a+b*arcsin(c*x))*ln(1+(I*c*x+(-c^2*x^2+1
)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)-4*I*b^2*f*g*polylog(2,-I*(I*c*x+(-c^2*x^2+1)^(1/2)))
*(-c^2*x^2+1)^(1/2)/c^2/d/(-c^2*d*x^2+d)^(1/2)+4*I*b^2*f*g*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2)))*(-c^2*x^2+1
)^(1/2)/c^2/d/(-c^2*d*x^2+d)^(1/2)-I*b^2*(c^2*f^2+g^2)*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*x^2+1)^(
1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.69, antiderivative size = 513, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 13, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {4861, 4859, 4847, 4745, 4765, 3800, 2221, 2317, 2438, 4767, 4749, 4266, 4737} \begin {gather*} \frac {8 i b f g \sqrt {1-c^2 x^2} \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {x \left (c^2 f^2+g^2\right ) (a+b \text {ArcSin}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 f g (a+b \text {ArcSin}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i \sqrt {1-c^2 x^2} \left (c^2 f^2+g^2\right ) (a+b \text {ArcSin}(c x))^2}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {1-c^2 x^2} \left (c^2 f^2+g^2\right ) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \text {ArcSin}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 \sqrt {1-c^2 x^2} \left (c^2 f^2+g^2\right ) \text {Li}_2\left (-e^{2 i \text {ArcSin}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((f + g*x)^2*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]

[Out]

(2*f*g*(a + b*ArcSin[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) + ((c^2*f^2 + g^2)*x*(a + b*ArcSin[c*x])^2)/(c^2*d*S
qrt[d - c^2*d*x^2]) - (I*(c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^3*d*Sqrt[d - c^2*d*x^2])
- (g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(3*b*c^3*d*Sqrt[d - c^2*d*x^2]) + ((8*I)*b*f*g*Sqrt[1 - c^2*x^
2]*(a + b*ArcSin[c*x])*ArcTan[E^(I*ArcSin[c*x])])/(c^2*d*Sqrt[d - c^2*d*x^2]) + (2*b*(c^2*f^2 + g^2)*Sqrt[1 -
c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])])/(c^3*d*Sqrt[d - c^2*d*x^2]) - ((4*I)*b^2*f*g*Sqrt
[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^2*d*Sqrt[d - c^2*d*x^2]) + ((4*I)*b^2*f*g*Sqrt[1 - c^2*x^
2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^2*d*Sqrt[d - c^2*d*x^2]) - (I*b^2*(c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*Pol
yLog[2, -E^((2*I)*ArcSin[c*x])])/(c^3*d*Sqrt[d - c^2*d*x^2])

Rule 2221

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]
- Dist[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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x
] - Dist[2*I, Int[(c + d*x)^m*(E^(2*I*(e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 4266

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] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[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 4737

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

Rule 4745

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcSin[c
*x])^n/(d*Sqrt[d + e*x^2])), x] - Dist[b*c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Int[x*((a + b*ArcSin
[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]

Rule 4749

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[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 4765

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[-e^(-1), Subst[In
t[(a + b*x)^n*Tan[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 4767

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] + Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(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 4847

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

Rule 4859

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n/Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; Free
Q[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]

Rule 4861

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

Rubi steps

\begin {align*} \int \frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \left (\frac {\left (c^2 f^2+g^2+2 c^2 f g x\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \left (1-c^2 x^2\right )^{3/2}}-\frac {g^2 \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {1-c^2 x^2}}\right ) \, dx}{d \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \frac {\left (c^2 f^2+g^2+2 c^2 f g x\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (g^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \int \left (\frac {c^2 f^2 \left (1+\frac {g^2}{c^2 f^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}+\frac {2 c^2 f g x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}}\right ) \, dx}{c^2 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 f g \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d \sqrt {d-c^2 d x^2}}+\frac {\left (\left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}}\\ &=\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 f^2+g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (4 b f g \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}}\\ &=\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 f^2+g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (4 b f g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 f^2+g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {8 i b f g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 f g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 f g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 i b \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 f^2+g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {8 i b f g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 b \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (4 i b^2 f g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 i b^2 f g \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 f^2+g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {8 i b f g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 b \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (i b^2 \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ &=\frac {2 f g \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\left (c^2 f^2+g^2\right ) x \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {8 i b f g \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 b \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {4 i b^2 f g \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 \left (c^2 f^2+g^2\right ) \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{c^3 d \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]
time = 1.34, size = 259, normalized size = 0.50 \begin {gather*} \frac {\sqrt {1-c^2 x^2} \left (-\frac {2 g^2 (a+b \text {ArcSin}(c x))^3}{b}+3 (-c f+g)^2 \left (-(a+b \text {ArcSin}(c x))^2 \cot \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )+i \left ((a+b \text {ArcSin}(c x)) \left (a+b \text {ArcSin}(c x)-4 i b \log \left (1+i e^{-i \text {ArcSin}(c x)}\right )\right )+4 b^2 \text {PolyLog}\left (2,-i e^{-i \text {ArcSin}(c x)}\right )\right )\right )-3 (c f+g)^2 \left (i \left ((a+b \text {ArcSin}(c x)) \left (a+b \text {ArcSin}(c x)+4 i b \log \left (1+i e^{i \text {ArcSin}(c x)}\right )\right )+4 b^2 \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )\right )-(a+b \text {ArcSin}(c x))^2 \tan \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )\right )\right )}{6 c^3 d \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((f + g*x)^2*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]

[Out]

(Sqrt[1 - c^2*x^2]*((-2*g^2*(a + b*ArcSin[c*x])^3)/b + 3*(-(c*f) + g)^2*(-((a + b*ArcSin[c*x])^2*Cot[(Pi + 2*A
rcSin[c*x])/4]) + I*((a + b*ArcSin[c*x])*(a + b*ArcSin[c*x] - (4*I)*b*Log[1 + I/E^(I*ArcSin[c*x])]) + 4*b^2*Po
lyLog[2, (-I)/E^(I*ArcSin[c*x])])) - 3*(c*f + g)^2*(I*((a + b*ArcSin[c*x])*(a + b*ArcSin[c*x] + (4*I)*b*Log[1
+ I*E^(I*ArcSin[c*x])]) + 4*b^2*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]) - (a + b*ArcSin[c*x])^2*Tan[(Pi + 2*ArcSin
[c*x])/4])))/(6*c^3*d*Sqrt[d - c^2*d*x^2])

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1860 vs. \(2 (516 ) = 1032\).
time = 0.70, size = 1861, normalized size = 3.63

method	result	size
default	\(\text {Expression too large to display}\)	\(1861\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^2*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d^2/(c^2*x^2-1)*g^2*arcsin(c*x)^2-4*a*b*(-d*(c^2*x^2-1))^(1/
2)*arcsin(c*x)/d^2/(c^2*x^2-1)*x^2*f*g-2*a*b*(-d*(c^2*x^2-1))^(1/2)*arcsin(c*x)/c^2/d^2/(c^2*x^2-1)*x*g^2-2*a*
b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)*f^2-2*a*b*(-d*(c^
2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)*g^2-2*a*b*(-d*(c^2*x^2-1
))^(1/2)*(-c^2*x^2+1)^(1/2)/c/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)*f^2-2*b^2*(-d*(c^2*x^2-1))^(1/2)*
arcsin(c*x)^2/c^2/d^2/(c^2*x^2-1)*(-c^2*x^2+1)*f*g+a^2*f^2*x/d/(-c^2*d*x^2+d)^(1/2)+I*b^2*(-d*(c^2*x^2-1))^(1/
2)*arcsin(c*x)^2/c/d^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*f^2+I*b^2*(-d*(c^2*x^2-1))^(1/2)*arcsin(c*x)^2/c^3/d^2/(
c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*g^2-2*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/d^2/(c^2*x^2-1)*f^2*arcsin
(c*x)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)+I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/d^2/(c^2*x^2-1)*f^2
*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-2*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d^2/(c^2*x^2-1)*
g^2*arcsin(c*x)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)-4*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/d^2/(c^
2*x^2-1)*f*g*arcsin(c*x)*ln(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))+4*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^
2/d^2/(c^2*x^2-1)*f*g*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+4*I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+
1)^(1/2)/c^2/d^2/(c^2*x^2-1)*f*g*dilog(1+I*(I*c*x+(-c^2*x^2+1)^(1/2)))-4*I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^
2+1)^(1/2)/c^2/d^2/(c^2*x^2-1)*f*g*dilog(1-I*(I*c*x+(-c^2*x^2+1)^(1/2)))+2*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x
^2-1))^(1/2)/c/d^2/(c^2*x^2-1)*arcsin(c*x)*f^2+2*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/c^3/d^2/(c^2*
x^2-1)*arcsin(c*x)*g^2-4*a*b*(-d*(c^2*x^2-1))^(1/2)*arcsin(c*x)/c^2/d^2/(c^2*x^2-1)*(-c^2*x^2+1)*f*g-4*a*b*(-d
*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)-I)*f*g+4*a*b*(-d*(c^2*x
^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/d^2/(c^2*x^2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)*f*g-a^2*g^2/c^2/d/(c^2*d)^(
1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+2*a^2*f*g/c^2/d/(-c^2*d*x^2+d)^(1/2)+a^2*g^2*x/c^2/d/(-c^2*d
*x^2+d)^(1/2)-b^2*(-d*(c^2*x^2-1))^(1/2)*arcsin(c*x)^2/d^2/(c^2*x^2-1)*x*f^2+I*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^
2*x^2+1)^(1/2)/c^3/d^2/(c^2*x^2-1)*g^2*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)
*(-c^2*x^2+1)^(1/2)/c^3/d^2/(c^2*x^2-1)*g^2*arcsin(c*x)^3-2*b^2*(-d*(c^2*x^2-1))^(1/2)*arcsin(c*x)^2/d^2/(c^2*
x^2-1)*x^2*f*g-b^2*(-d*(c^2*x^2-1))^(1/2)*arcsin(c*x)^2/c^2/d^2/(c^2*x^2-1)*x*g^2-2*a*b*(-d*(c^2*x^2-1))^(1/2)
*arcsin(c*x)/d^2/(c^2*x^2-1)*x*f^2-2*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d^2/(c^2*x^2-1)*ln(I*c*
x+(-c^2*x^2+1)^(1/2)+I)*g^2

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxima")

[Out]

a^2*g^2*(x/(sqrt(-c^2*d*x^2 + d)*c^2*d) - arcsin(c*x)/(c^3*d^(3/2))) + 2*a*b*f^2*x*arcsin(c*x)/(sqrt(-c^2*d*x^
2 + d)*d) + a^2*f^2*x/(sqrt(-c^2*d*x^2 + d)*d) - a*b*f^2*log(x^2 - 1/c^2)/(c*d^(3/2)) - sqrt(d)*integrate(((b^
2*g^2*x^2 + 2*b^2*f*g*x + b^2*f^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*g^2*x^2 + 2*a*b*f*g*x
)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/((c^2*d^2*x^2 - d^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + 2*a^2*f
*g/(sqrt(-c^2*d*x^2 + d)*c^2*d)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**2*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**(3/2),x)

[Out]

Integral((a + b*asin(c*x))**2*(f + g*x)**2/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^2*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(t_

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (f+g\,x\right )}^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)^2*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(3/2),x)

[Out]

int(((f + g*x)^2*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(3/2), x)

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