3.2.94 \(\int (d+e x^2) \sin ^{-1}(a x)^2 \log (c x^n) \, dx\) [194]

Optimal. Leaf size=482 \[ 2 d n x+\frac {2 e n x}{27 a^2}+\frac {4}{9} \left (9 d+\frac {2 e}{a^2}\right ) n x+\frac {2}{27} e n x^3-\frac {2 d n \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}-\frac {4 e n \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{27 a^3}-\frac {2 \left (9 a^2 d+2 e\right ) n \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a^3}-\frac {2 e n x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{27 a}+\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)}{27 a^3}-d n x \sin ^{-1}(a x)^2-\frac {1}{9} e n x^3 \sin ^{-1}(a x)^2+\frac {4 \left (9 a^2 d+2 e\right ) n \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{9 a^3}-2 d x \log \left (c x^n\right )-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-\frac {2}{27} e x^3 \log \left (c x^n\right )+\frac {2 d \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac {4 e \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac {2 e x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )-\frac {2 i \left (9 a^2 d+2 e\right ) n \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )}{9 a^3}+\frac {2 i \left (9 a^2 d+2 e\right ) n \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )}{9 a^3} \]

[Out]

2*d*n*x+2/27*e*n*x/a^2+4/9*(9*d+2*e/a^2)*n*x+2/27*e*n*x^3+2/27*e*n*(-a^2*x^2+1)^(3/2)*arcsin(a*x)/a^3-d*n*x*ar
csin(a*x)^2-1/9*e*n*x^3*arcsin(a*x)^2+4/9*(9*a^2*d+2*e)*n*arcsin(a*x)*arctanh(I*a*x+(-a^2*x^2+1)^(1/2))/a^3-2*
d*x*ln(c*x^n)-4/9*e*x*ln(c*x^n)/a^2-2/27*e*x^3*ln(c*x^n)+d*x*arcsin(a*x)^2*ln(c*x^n)+1/3*e*x^3*arcsin(a*x)^2*l
n(c*x^n)-2/9*I*(9*a^2*d+2*e)*n*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))/a^3+2/9*I*(9*a^2*d+2*e)*n*polylog(2,I*a*x+
(-a^2*x^2+1)^(1/2))/a^3-2*d*n*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a-4/27*e*n*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a^3-2/9
*(9*a^2*d+2*e)*n*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a^3-2/27*e*n*x^2*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a+2*d*arcsin(a
*x)*ln(c*x^n)*(-a^2*x^2+1)^(1/2)/a+4/9*e*arcsin(a*x)*ln(c*x^n)*(-a^2*x^2+1)^(1/2)/a^3+2/9*e*x^2*arcsin(a*x)*ln
(c*x^n)*(-a^2*x^2+1)^(1/2)/a

________________________________________________________________________________________

Rubi [A]
time = 0.45, antiderivative size = 482, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4757, 4715, 4767, 8, 4723, 4795, 30, 2434, 6, 4783, 4803, 4268, 2317, 2438} \begin {gather*} -\frac {2 i n \left (9 a^2 d+2 e\right ) \text {PolyLog}\left (2,-e^{i \text {ArcSin}(a x)}\right )}{9 a^3}+\frac {2 i n \left (9 a^2 d+2 e\right ) \text {PolyLog}\left (2,e^{i \text {ArcSin}(a x)}\right )}{9 a^3}+\frac {2 d \sqrt {1-a^2 x^2} \text {ArcSin}(a x) \log \left (c x^n\right )}{a}+\frac {2 e x^2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x) \log \left (c x^n\right )}{9 a}-\frac {2 d n \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{a}-\frac {2 e n x^2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{27 a}-\frac {4 e x \log \left (c x^n\right )}{9 a^2}+\frac {4}{9} n x \left (\frac {2 e}{a^2}+9 d\right )+\frac {2 e n x}{27 a^2}+\frac {4 e \sqrt {1-a^2 x^2} \text {ArcSin}(a x) \log \left (c x^n\right )}{9 a^3}-\frac {2 n \sqrt {1-a^2 x^2} \text {ArcSin}(a x) \left (9 a^2 d+2 e\right )}{9 a^3}+\frac {4 n \text {ArcSin}(a x) \left (9 a^2 d+2 e\right ) \tanh ^{-1}\left (e^{i \text {ArcSin}(a x)}\right )}{9 a^3}+\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \text {ArcSin}(a x)}{27 a^3}-\frac {4 e n \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{27 a^3}+d x \text {ArcSin}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {ArcSin}(a x)^2 \log \left (c x^n\right )-d n x \text {ArcSin}(a x)^2-\frac {1}{9} e n x^3 \text {ArcSin}(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 \end {gather*}

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 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 8

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

Rule 30

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

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 2434

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 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 4268

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 4715

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 4723

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 4757

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 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 4783

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[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/S
qrt[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/(f*(m + 2)))*Si
mp[Sqrt[d + e*x^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] && (IGtQ[m, -2] || EqQ[n, 1])

Rule 4795

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] + (Dist[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] + Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(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 4803

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

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.60, size = 456, normalized size = 0.95 \begin {gather*} \frac {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} \sin ^{-1}(a x)-14 e n \sqrt {1-a^2 x^2} \sin ^{-1}(a x)-4 a^2 e n x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)-27 a^3 d n x \sin ^{-1}(a x)^2-3 a^3 e n x^3 \sin ^{-1}(a x)^2-54 a^2 d 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 )+54 a^2 d 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 )-54 a^3 d x \log \left (c x^n\right )-12 a e x \log \left (c x^n\right )-2 a^3 e x^3 \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 )+12 e \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \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 )+27 a^3 d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+9 a^3 e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )-6 i \left (9 a^2 d+2 e\right ) n \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )+6 i \left (9 a^2 d+2 e\right ) n \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )}{27 a^3} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [F]
time = 3.98, size = 0, normalized size = 0.00 \[\int \left (e \,x^{2}+d \right ) \arcsin \left (a x \right )^{2} \ln \left (c \,x^{n}\right )\, dx\]

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)

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

[Out]

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

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d + e x^{2}\right ) \log {\left (c x^{n} \right )} \operatorname {asin}^{2}{\left (a x \right )}\, dx \end {gather*}

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]

Integral((d + e*x**2)*log(c*x**n)*asin(a*x)**2, x)

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \ln \left (c\,x^n\right )\,{\mathrm {asin}\left (a\,x\right )}^2\,\left (e\,x^2+d\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*x^n)*asin(a*x)^2*(d + e*x^2),x)

[Out]

int(log(c*x^n)*asin(a*x)^2*(d + e*x^2), x)

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