3.194 \(\int (d+e x^2) \sin ^{-1}(a x)^2 \log (c x^n) \, dx\)

Optimal. Leaf size=482 \[ -\frac{2 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )}{9 a^3}+\frac{2 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )}{9 a^3}+\frac{2 d \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e x \log \left (c x^n\right )}{9 a^2}+\frac{2 e x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+\frac{4 e \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 n \sqrt{1-a^2 x^2} \left (9 a^2 d+2 e\right ) \sin ^{-1}(a x)}{9 a^3}+\frac{4}{9} n x \left (\frac{2 e}{a^2}+9 d\right )+\frac{4 n \left (9 a^2 d+2 e\right ) \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{9 a^3}-\frac{2 d n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a}-\frac{2 e n x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{27 a}+\frac{2 e n \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)}{27 a^3}-\frac{4 e n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{27 a^3}+\frac{2 e n x}{27 a^2}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )-d n x \sin ^{-1}(a x)^2-\frac{1}{9} e n x^3 \sin ^{-1}(a x)^2-2 d x \log \left (c x^n\right )-\frac{2}{27} e x^3 \log \left (c x^n\right )+2 d n x+\frac{2}{27} e n x^3 \]

[Out]

2*d*n*x + (2*e*n*x)/(27*a^2) + (4*(9*d + (2*e)/a^2)*n*x)/9 + (2*e*n*x^3)/27 - (2*d*n*Sqrt[1 - a^2*x^2]*ArcSin[
a*x])/a - (4*e*n*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(27*a^3) - (2*(9*a^2*d + 2*e)*n*Sqrt[1 - a^2*x^2]*ArcSin[a*x])
/(9*a^3) - (2*e*n*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(27*a) + (2*e*n*(1 - a^2*x^2)^(3/2)*ArcSin[a*x])/(27*a^3)
- d*n*x*ArcSin[a*x]^2 - (e*n*x^3*ArcSin[a*x]^2)/9 + (4*(9*a^2*d + 2*e)*n*ArcSin[a*x]*ArcTanh[E^(I*ArcSin[a*x]
)])/(9*a^3) - 2*d*x*Log[c*x^n] - (4*e*x*Log[c*x^n])/(9*a^2) - (2*e*x^3*Log[c*x^n])/27 + (2*d*Sqrt[1 - a^2*x^2]
*ArcSin[a*x]*Log[c*x^n])/a + (4*e*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*Log[c*x^n])/(9*a^3) + (2*e*x^2*Sqrt[1 - a^2*x^
2]*ArcSin[a*x]*Log[c*x^n])/(9*a) + d*x*ArcSin[a*x]^2*Log[c*x^n] + (e*x^3*ArcSin[a*x]^2*Log[c*x^n])/3 - (((2*I)
/9)*(9*a^2*d + 2*e)*n*PolyLog[2, -E^(I*ArcSin[a*x])])/a^3 + (((2*I)/9)*(9*a^2*d + 2*e)*n*PolyLog[2, E^(I*ArcSi
n[a*x])])/a^3

________________________________________________________________________________________

Rubi [A]  time = 0.734442, antiderivative size = 482, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 14, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {4667, 4619, 4677, 8, 4627, 4707, 30, 2387, 6, 4697, 4709, 4183, 2279, 2391} \[ -\frac{2 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )}{9 a^3}+\frac{2 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )}{9 a^3}+\frac{2 d \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e x \log \left (c x^n\right )}{9 a^2}+\frac{2 e x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+\frac{4 e \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 n \sqrt{1-a^2 x^2} \left (9 a^2 d+2 e\right ) \sin ^{-1}(a x)}{9 a^3}+\frac{4}{9} n x \left (\frac{2 e}{a^2}+9 d\right )+\frac{4 n \left (9 a^2 d+2 e\right ) \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{9 a^3}-\frac{2 d n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a}-\frac{2 e n x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{27 a}+\frac{2 e n \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)}{27 a^3}-\frac{4 e n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{27 a^3}+\frac{2 e n x}{27 a^2}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )-d n x \sin ^{-1}(a x)^2-\frac{1}{9} e n x^3 \sin ^{-1}(a x)^2-2 d x \log \left (c x^n\right )-\frac{2}{27} e x^3 \log \left (c x^n\right )+2 d n x+\frac{2}{27} e n x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*d*n*x + (2*e*n*x)/(27*a^2) + (4*(9*d + (2*e)/a^2)*n*x)/9 + (2*e*n*x^3)/27 - (2*d*n*Sqrt[1 - a^2*x^2]*ArcSin[
a*x])/a - (4*e*n*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(27*a^3) - (2*(9*a^2*d + 2*e)*n*Sqrt[1 - a^2*x^2]*ArcSin[a*x])
/(9*a^3) - (2*e*n*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(27*a) + (2*e*n*(1 - a^2*x^2)^(3/2)*ArcSin[a*x])/(27*a^3)
- d*n*x*ArcSin[a*x]^2 - (e*n*x^3*ArcSin[a*x]^2)/9 + (4*(9*a^2*d + 2*e)*n*ArcSin[a*x]*ArcTanh[E^(I*ArcSin[a*x]
)])/(9*a^3) - 2*d*x*Log[c*x^n] - (4*e*x*Log[c*x^n])/(9*a^2) - (2*e*x^3*Log[c*x^n])/27 + (2*d*Sqrt[1 - a^2*x^2]
*ArcSin[a*x]*Log[c*x^n])/a + (4*e*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*Log[c*x^n])/(9*a^3) + (2*e*x^2*Sqrt[1 - a^2*x^
2]*ArcSin[a*x]*Log[c*x^n])/(9*a) + d*x*ArcSin[a*x]^2*Log[c*x^n] + (e*x^3*ArcSin[a*x]^2*Log[c*x^n])/3 - (((2*I)
/9)*(9*a^2*d + 2*e)*n*PolyLog[2, -E^(I*ArcSin[a*x])])/a^3 + (((2*I)/9)*(9*a^2*d + 2*e)*n*PolyLog[2, E^(I*ArcSi
n[a*x])])/a^3

Rule 4667

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]
&& (GtQ[p, 0] || IGtQ[n, 0])

Rule 4619

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

Rule 4677

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[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 8

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

Rule 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4707

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

Rule 30

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

Rule 2387

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(Px_.)*(F_)[(d_.)*((e_.) + (f_.)*(x_))]^(m_.), x_Symbol] :> With[{u
= IntHide[Px*F[d*(e + f*x)]^m, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist[1/x, u, x], x], x]] /; F
reeQ[{a, b, c, d, e, f, n}, x] && PolynomialQ[Px, x] && IGtQ[m, 0] && MemberQ[{ArcSin, ArcCos, ArcSinh, ArcCos
h}, F]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 4697

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

Rule 4709

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

Rule 4183

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

Rule 2279

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 2391

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]

Rubi steps

\begin{align*} \int \left (d+e x^2\right ) \sin ^{-1}(a x)^2 \log \left (c x^n\right ) \, dx &=-2 d x \log \left (c x^n\right )-\frac{4 e x \log \left (c x^n\right )}{9 a^2}-\frac{2}{27} e x^3 \log \left (c x^n\right )+\frac{2 d \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac{4 e \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac{2 e x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )-n \int \left (-2 d-\frac{4 e}{9 a^2}-\frac{2 e x^2}{27}+\frac{2 d \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a x}+\frac{4 e \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^3 x}+\frac{2 e x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a}+d \sin ^{-1}(a x)^2+\frac{1}{3} e x^2 \sin ^{-1}(a x)^2\right ) \, dx\\ &=-2 d x \log \left (c x^n\right )-\frac{4 e x \log \left (c x^n\right )}{9 a^2}-\frac{2}{27} e x^3 \log \left (c x^n\right )+\frac{2 d \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac{4 e \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac{2 e x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )-n \int \left (-2 d-\frac{4 e}{9 a^2}-\frac{2 e x^2}{27}+\frac{\left (\frac{2 d}{a}+\frac{4 e}{9 a^3}\right ) \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{x}+\frac{2 e x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a}+d \sin ^{-1}(a x)^2+\frac{1}{3} e x^2 \sin ^{-1}(a x)^2\right ) \, dx\\ &=\frac{2}{9} \left (9 d+\frac{2 e}{a^2}\right ) n x+\frac{2}{81} e n x^3-2 d x \log \left (c x^n\right )-\frac{4 e x \log \left (c x^n\right )}{9 a^2}-\frac{2}{27} e x^3 \log \left (c x^n\right )+\frac{2 d \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac{4 e \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac{2 e x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )-(d n) \int \sin ^{-1}(a x)^2 \, dx-\frac{1}{3} (e n) \int x^2 \sin ^{-1}(a x)^2 \, dx-\frac{(2 e n) \int x \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \, dx}{9 a}-\frac{\left (2 \left (9 a^2 d+2 e\right ) n\right ) \int \frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{x} \, dx}{9 a^3}\\ &=\frac{2}{9} \left (9 d+\frac{2 e}{a^2}\right ) n x+\frac{2}{81} e n x^3-\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^3}+\frac{2 e n \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)}{27 a^3}-d n x \sin ^{-1}(a x)^2-\frac{1}{9} e n x^3 \sin ^{-1}(a x)^2-2 d x \log \left (c x^n\right )-\frac{4 e x \log \left (c x^n\right )}{9 a^2}-\frac{2}{27} e x^3 \log \left (c x^n\right )+\frac{2 d \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac{4 e \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac{2 e x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )+(2 a d n) \int \frac{x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx-\frac{(2 e n) \int \left (1-a^2 x^2\right ) \, dx}{27 a^2}+\frac{1}{9} (2 a e n) \int \frac{x^3 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx-\frac{\left (2 \left (9 a^2 d+2 e\right ) n\right ) \int \frac{\sin ^{-1}(a x)}{x \sqrt{1-a^2 x^2}} \, dx}{9 a^3}+\frac{\left (2 \left (9 a^2 d+2 e\right ) n\right ) \int 1 \, dx}{9 a^2}\\ &=-\frac{2 e n x}{27 a^2}+\frac{2 \left (9 a^2 d+2 e\right ) n x}{9 a^2}+\frac{2}{9} \left (9 d+\frac{2 e}{a^2}\right ) n x+\frac{4}{81} e n x^3-\frac{2 d n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a}-\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^3}-\frac{2 e n x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{27 a}+\frac{2 e n \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)}{27 a^3}-d n x \sin ^{-1}(a x)^2-\frac{1}{9} e n x^3 \sin ^{-1}(a x)^2-2 d x \log \left (c x^n\right )-\frac{4 e x \log \left (c x^n\right )}{9 a^2}-\frac{2}{27} e x^3 \log \left (c x^n\right )+\frac{2 d \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac{4 e \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac{2 e x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )+(2 d n) \int 1 \, dx+\frac{1}{27} (2 e n) \int x^2 \, dx+\frac{(4 e n) \int \frac{x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{27 a}-\frac{\left (2 \left (9 a^2 d+2 e\right ) n\right ) \operatorname{Subst}\left (\int x \csc (x) \, dx,x,\sin ^{-1}(a x)\right )}{9 a^3}\\ &=2 d n x-\frac{2 e n x}{27 a^2}+\frac{2 \left (9 a^2 d+2 e\right ) n x}{9 a^2}+\frac{2}{9} \left (9 d+\frac{2 e}{a^2}\right ) n x+\frac{2}{27} e n x^3-\frac{2 d n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a}-\frac{4 e n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{27 a^3}-\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^3}-\frac{2 e n x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{27 a}+\frac{2 e n \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)}{27 a^3}-d n x \sin ^{-1}(a x)^2-\frac{1}{9} e n x^3 \sin ^{-1}(a x)^2+\frac{4 \left (9 a^2 d+2 e\right ) n \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{9 a^3}-2 d x \log \left (c x^n\right )-\frac{4 e x \log \left (c x^n\right )}{9 a^2}-\frac{2}{27} e x^3 \log \left (c x^n\right )+\frac{2 d \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac{4 e \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac{2 e x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{(4 e n) \int 1 \, dx}{27 a^2}+\frac{\left (2 \left (9 a^2 d+2 e\right ) n\right ) \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{9 a^3}-\frac{\left (2 \left (9 a^2 d+2 e\right ) n\right ) \operatorname{Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{9 a^3}\\ &=2 d n x+\frac{2 e n x}{27 a^2}+\frac{2 \left (9 a^2 d+2 e\right ) n x}{9 a^2}+\frac{2}{9} \left (9 d+\frac{2 e}{a^2}\right ) n x+\frac{2}{27} e n x^3-\frac{2 d n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a}-\frac{4 e n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{27 a^3}-\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^3}-\frac{2 e n x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{27 a}+\frac{2 e n \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)}{27 a^3}-d n x \sin ^{-1}(a x)^2-\frac{1}{9} e n x^3 \sin ^{-1}(a x)^2+\frac{4 \left (9 a^2 d+2 e\right ) n \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{9 a^3}-2 d x \log \left (c x^n\right )-\frac{4 e x \log \left (c x^n\right )}{9 a^2}-\frac{2}{27} e x^3 \log \left (c x^n\right )+\frac{2 d \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac{4 e \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac{2 e x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )-\frac{\left (2 i \left (9 a^2 d+2 e\right ) n\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{9 a^3}+\frac{\left (2 i \left (9 a^2 d+2 e\right ) n\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{9 a^3}\\ &=2 d n x+\frac{2 e n x}{27 a^2}+\frac{2 \left (9 a^2 d+2 e\right ) n x}{9 a^2}+\frac{2}{9} \left (9 d+\frac{2 e}{a^2}\right ) n x+\frac{2}{27} e n x^3-\frac{2 d n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a}-\frac{4 e n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{27 a^3}-\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^3}-\frac{2 e n x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{27 a}+\frac{2 e n \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)}{27 a^3}-d n x \sin ^{-1}(a x)^2-\frac{1}{9} e n x^3 \sin ^{-1}(a x)^2+\frac{4 \left (9 a^2 d+2 e\right ) n \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{9 a^3}-2 d x \log \left (c x^n\right )-\frac{4 e x \log \left (c x^n\right )}{9 a^2}-\frac{2}{27} e x^3 \log \left (c x^n\right )+\frac{2 d \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac{4 e \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac{2 e x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )-\frac{2 i \left (9 a^2 d+2 e\right ) n \text{Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )}{9 a^3}+\frac{2 i \left (9 a^2 d+2 e\right ) n \text{Li}_2\left (e^{i \sin ^{-1}(a x)}\right )}{9 a^3}\\ \end{align*}

Mathematica [A]  time = 0.828152, size = 456, normalized size = 0.95 \[ \frac{-6 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )+6 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-54 a^3 d x \log \left (c x^n\right )+27 a^3 d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+54 a^2 d \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )-2 a^3 e x^3 \log \left (c x^n\right )+9 a^3 e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )+6 a^2 e x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )+12 e \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )-108 a^2 d n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+162 a^3 d n x-27 a^3 d n x \sin ^{-1}(a x)^2-54 a^2 d n \sin ^{-1}(a x) \log \left (1-e^{i \sin ^{-1}(a x)}\right )+54 a^2 d n \sin ^{-1}(a x) \log \left (1+e^{i \sin ^{-1}(a x)}\right )+2 a^3 e n x^3-3 a^3 e n x^3 \sin ^{-1}(a x)^2-4 a^2 e n x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-14 e n \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-12 a e x \log \left (c x^n\right )+26 a e n x-12 e n \sin ^{-1}(a x) \log \left (1-e^{i \sin ^{-1}(a x)}\right )+12 e n \sin ^{-1}(a x) \log \left (1+e^{i \sin ^{-1}(a x)}\right )}{27 a^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(162*a^3*d*n*x + 26*a*e*n*x + 2*a^3*e*n*x^3 - 108*a^2*d*n*Sqrt[1 - a^2*x^2]*ArcSin[a*x] - 14*e*n*Sqrt[1 - a^2*
x^2]*ArcSin[a*x] - 4*a^2*e*n*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x] - 27*a^3*d*n*x*ArcSin[a*x]^2 - 3*a^3*e*n*x^3*Ar
cSin[a*x]^2 - 54*a^2*d*n*ArcSin[a*x]*Log[1 - E^(I*ArcSin[a*x])] - 12*e*n*ArcSin[a*x]*Log[1 - E^(I*ArcSin[a*x])
] + 54*a^2*d*n*ArcSin[a*x]*Log[1 + E^(I*ArcSin[a*x])] + 12*e*n*ArcSin[a*x]*Log[1 + E^(I*ArcSin[a*x])] - 54*a^3
*d*x*Log[c*x^n] - 12*a*e*x*Log[c*x^n] - 2*a^3*e*x^3*Log[c*x^n] + 54*a^2*d*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*Log[c*
x^n] + 12*e*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*Log[c*x^n] + 6*a^2*e*x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*Log[c*x^n] +
27*a^3*d*x*ArcSin[a*x]^2*Log[c*x^n] + 9*a^3*e*x^3*ArcSin[a*x]^2*Log[c*x^n] - (6*I)*(9*a^2*d + 2*e)*n*PolyLog[2
, -E^(I*ArcSin[a*x])] + (6*I)*(9*a^2*d + 2*e)*n*PolyLog[2, E^(I*ArcSin[a*x])])/(27*a^3)

________________________________________________________________________________________

Maple [F]  time = 2.362, size = 0, normalized size = 0. \begin{align*} \int \left ( e{x}^{2}+d \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{2}\ln \left ( c{x}^{n} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*arcsin(a*x)^2*ln(c*x^n),x)

[Out]

int((e*x^2+d)*arcsin(a*x)^2*ln(c*x^n),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \,{\left (e x^{3} + 3 \, d x\right )} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{2} \log \left (x^{n}\right ) - \frac{1}{9} \,{\left ({\left (e n - 3 \, e \log \left (c\right )\right )} x^{3} + 9 \,{\left (d n - d \log \left (c\right )\right )} x\right )} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{2} + \int \frac{2 \,{\left (3 \,{\left (a e x^{3} + 3 \, a d x\right )} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right ) \log \left (x^{n}\right ) -{\left ({\left (a e n - 3 \, a e \log \left (c\right )\right )} x^{3} + 9 \,{\left (a d n - a d \log \left (c\right )\right )} x\right )} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )\right )} \sqrt{a x + 1} \sqrt{-a x + 1}}{9 \,{\left (a^{2} x^{2} - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(e*x^3 + 3*d*x)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2*log(x^n) - 1/9*((e*n - 3*e*log(c))*x^3 + 9*(d
*n - d*log(c))*x)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2 + integrate(2/9*(3*(a*e*x^3 + 3*a*d*x)*arctan2(
a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))*log(x^n) - ((a*e*n - 3*a*e*log(c))*x^3 + 9*(a*d*n - a*d*log(c))*x)*arctan2(
a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)))*sqrt(a*x + 1)*sqrt(-a*x + 1)/(a^2*x^2 - 1), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (e x^{2} + d\right )} \arcsin \left (a x\right )^{2} \log \left (c x^{n}\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((e*x^2 + d)*arcsin(a*x)^2*log(c*x^n), x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)*asin(a*x)**2*ln(c*x**n),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )} \arcsin \left (a x\right )^{2} \log \left (c x^{n}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x^2 + d)*arcsin(a*x)^2*log(c*x^n), x)