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

Optimal. Leaf size=490 \[ -\frac{2 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )}{9 a^3}+\frac{2 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )}{9 a^3}-\frac{2 d \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e x \log \left (c x^n\right )}{9 a^2}-\frac{2 e x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}-\frac{4 e \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac{2 n \sqrt{1-a^2 x^2} \left (9 a^2 d+2 e\right ) \cos ^{-1}(a x)}{9 a^3}+\frac{4}{9} n x \left (\frac{2 e}{a^2}+9 d\right )+\frac{4 i n \left (9 a^2 d+2 e\right ) \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )}{9 a^3}+\frac{2 d n \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+\frac{2 e n x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a}-\frac{2 e n \left (1-a^2 x^2\right )^{3/2} \cos ^{-1}(a x)}{27 a^3}+\frac{4 e n \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac{2 e n x}{27 a^2}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-1}(a x)^2 \log \left (c x^n\right )-d n x \cos ^{-1}(a x)^2-\frac{1}{9} e n x^3 \cos ^{-1}(a x)^2-2 d x \log \left (c x^n\right )-\frac{2}{27} e x^3 \log \left (c x^n\right )+2 d n x+\frac{2}{27} e n x^3 \]

[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]*ArcCos[
a*x])/a + (4*e*n*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(27*a^3) + (2*(9*a^2*d + 2*e)*n*Sqrt[1 - a^2*x^2]*ArcCos[a*x])
/(9*a^3) + (2*e*n*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(27*a) - (2*e*n*(1 - a^2*x^2)^(3/2)*ArcCos[a*x])/(27*a^3)
- d*n*x*ArcCos[a*x]^2 - (e*n*x^3*ArcCos[a*x]^2)/9 + (((4*I)/9)*(9*a^2*d + 2*e)*n*ArcCos[a*x]*ArcTan[E^(I*ArcC
os[a*x])])/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]*ArcCos[a*x]*Log[c*x^n])/a - (4*e*Sqrt[1 - a^2*x^2]*ArcCos[a*x]*Log[c*x^n])/(9*a^3) - (2*e*x^2*Sqrt[1 - a^2
*x^2]*ArcCos[a*x]*Log[c*x^n])/(9*a) + d*x*ArcCos[a*x]^2*Log[c*x^n] + (e*x^3*ArcCos[a*x]^2*Log[c*x^n])/3 - (((2
*I)/9)*(9*a^2*d + 2*e)*n*PolyLog[2, (-I)*E^(I*ArcCos[a*x])])/a^3 + (((2*I)/9)*(9*a^2*d + 2*e)*n*PolyLog[2, I*E
^(I*ArcCos[a*x])])/a^3

________________________________________________________________________________________

Rubi [A]  time = 0.70234, antiderivative size = 490, 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 = {4668, 4620, 4678, 8, 4628, 4708, 30, 2387, 6, 4698, 4710, 4181, 2279, 2391} \[ -\frac{2 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )}{9 a^3}+\frac{2 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )}{9 a^3}-\frac{2 d \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e x \log \left (c x^n\right )}{9 a^2}-\frac{2 e x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}-\frac{4 e \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac{2 n \sqrt{1-a^2 x^2} \left (9 a^2 d+2 e\right ) \cos ^{-1}(a x)}{9 a^3}+\frac{4}{9} n x \left (\frac{2 e}{a^2}+9 d\right )+\frac{4 i n \left (9 a^2 d+2 e\right ) \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )}{9 a^3}+\frac{2 d n \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+\frac{2 e n x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a}-\frac{2 e n \left (1-a^2 x^2\right )^{3/2} \cos ^{-1}(a x)}{27 a^3}+\frac{4 e n \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac{2 e n x}{27 a^2}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-1}(a x)^2 \log \left (c x^n\right )-d n x \cos ^{-1}(a x)^2-\frac{1}{9} e n x^3 \cos ^{-1}(a x)^2-2 d x \log \left (c x^n\right )-\frac{2}{27} e x^3 \log \left (c x^n\right )+2 d n x+\frac{2}{27} e n x^3 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)*ArcCos[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]*ArcCos[
a*x])/a + (4*e*n*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(27*a^3) + (2*(9*a^2*d + 2*e)*n*Sqrt[1 - a^2*x^2]*ArcCos[a*x])
/(9*a^3) + (2*e*n*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(27*a) - (2*e*n*(1 - a^2*x^2)^(3/2)*ArcCos[a*x])/(27*a^3)
- d*n*x*ArcCos[a*x]^2 - (e*n*x^3*ArcCos[a*x]^2)/9 + (((4*I)/9)*(9*a^2*d + 2*e)*n*ArcCos[a*x]*ArcTan[E^(I*ArcC
os[a*x])])/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]*ArcCos[a*x]*Log[c*x^n])/a - (4*e*Sqrt[1 - a^2*x^2]*ArcCos[a*x]*Log[c*x^n])/(9*a^3) - (2*e*x^2*Sqrt[1 - a^2
*x^2]*ArcCos[a*x]*Log[c*x^n])/(9*a) + d*x*ArcCos[a*x]^2*Log[c*x^n] + (e*x^3*ArcCos[a*x]^2*Log[c*x^n])/3 - (((2
*I)/9)*(9*a^2*d + 2*e)*n*PolyLog[2, (-I)*E^(I*ArcCos[a*x])])/a^3 + (((2*I)/9)*(9*a^2*d + 2*e)*n*PolyLog[2, I*E
^(I*ArcCos[a*x])])/a^3

Rule 4668

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcCos[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 4620

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

Rule 4678

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcCos[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*ArcCos[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 4628

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCo
s[c*x])^n)/(d*(m + 1)), x] + Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCos[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 4708

Int[(((a_.) + ArcCos[(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*ArcCos[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcCos[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*ArcCos[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 4698

Int[((a_.) + ArcCos[(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*ArcCos[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*ArcCos[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*ArcCos[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 4710

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> -Dist[(c^(m +
1)*Sqrt[d])^(-1), Subst[Int[(a + b*x)^n*Cos[x]^m, x], x, ArcCos[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 4181

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 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 ) \cos ^{-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} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-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} \cos ^{-1}(a x)}{a x}-\frac{4 e \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a^3 x}-\frac{2 e x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a}+d \cos ^{-1}(a x)^2+\frac{1}{3} e x^2 \cos ^{-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} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-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} \cos ^{-1}(a x)}{x}-\frac{2 e x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a}+d \cos ^{-1}(a x)^2+\frac{1}{3} e x^2 \cos ^{-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} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-1}(a x)^2 \log \left (c x^n\right )-(d n) \int \cos ^{-1}(a x)^2 \, dx-\frac{1}{3} (e n) \int x^2 \cos ^{-1}(a x)^2 \, dx+\frac{(2 e n) \int x \sqrt{1-a^2 x^2} \cos ^{-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} \cos ^{-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} \cos ^{-1}(a x)}{9 a^3}-\frac{2 e n \left (1-a^2 x^2\right )^{3/2} \cos ^{-1}(a x)}{27 a^3}-d n x \cos ^{-1}(a x)^2-\frac{1}{9} e n x^3 \cos ^{-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} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-1}(a x)^2 \log \left (c x^n\right )-(2 a d n) \int \frac{x \cos ^{-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 \cos ^{-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{\cos ^{-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} \cos ^{-1}(a x)}{a}+\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a^3}+\frac{2 e n x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a}-\frac{2 e n \left (1-a^2 x^2\right )^{3/2} \cos ^{-1}(a x)}{27 a^3}-d n x \cos ^{-1}(a x)^2-\frac{1}{9} e n x^3 \cos ^{-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} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-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 \cos ^{-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 \sec (x) \, dx,x,\cos ^{-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} \cos ^{-1}(a x)}{a}+\frac{4 e n \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a^3}+\frac{2 e n x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a}-\frac{2 e n \left (1-a^2 x^2\right )^{3/2} \cos ^{-1}(a x)}{27 a^3}-d n x \cos ^{-1}(a x)^2-\frac{1}{9} e n x^3 \cos ^{-1}(a x)^2+\frac{4 i \left (9 a^2 d+2 e\right ) n \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-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} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-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-i e^{i x}\right ) \, dx,x,\cos ^{-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+i e^{i x}\right ) \, dx,x,\cos ^{-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} \cos ^{-1}(a x)}{a}+\frac{4 e n \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a^3}+\frac{2 e n x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a}-\frac{2 e n \left (1-a^2 x^2\right )^{3/2} \cos ^{-1}(a x)}{27 a^3}-d n x \cos ^{-1}(a x)^2-\frac{1}{9} e n x^3 \cos ^{-1}(a x)^2+\frac{4 i \left (9 a^2 d+2 e\right ) n \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-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} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-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-i x)}{x} \, dx,x,e^{i \cos ^{-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+i x)}{x} \, dx,x,e^{i \cos ^{-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} \cos ^{-1}(a x)}{a}+\frac{4 e n \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a^3}+\frac{2 e n x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a}-\frac{2 e n \left (1-a^2 x^2\right )^{3/2} \cos ^{-1}(a x)}{27 a^3}-d n x \cos ^{-1}(a x)^2-\frac{1}{9} e n x^3 \cos ^{-1}(a x)^2+\frac{4 i \left (9 a^2 d+2 e\right ) n \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-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} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cos ^{-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 (-i e^{i \cos ^{-1}(a x)}\right )}{9 a^3}+\frac{2 i \left (9 a^2 d+2 e\right ) n \text{Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )}{9 a^3}\\ \end{align*}

Mathematica [A]  time = 0.800832, size = 564, normalized size = 1.15 \[ \frac{2 d n \left (-i \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )+i \text{PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )+\sqrt{1-a^2 x^2} \cos ^{-1}(a x)+a x-\cos ^{-1}(a x) \log \left (1-i e^{i \cos ^{-1}(a x)}\right )+\cos ^{-1}(a x) \log \left (1+i e^{i \cos ^{-1}(a x)}\right )\right )}{a}+\frac{4 e n \left (-i \text{PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )+i \text{PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )+\sqrt{1-a^2 x^2} \cos ^{-1}(a x)+a x-\cos ^{-1}(a x) \log \left (1-i e^{i \cos ^{-1}(a x)}\right )+\cos ^{-1}(a x) \log \left (1+i e^{i \cos ^{-1}(a x)}\right )\right )}{9 a^3}+\frac{d \left (a x \left (\cos ^{-1}(a x)^2-2\right )-2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)\right ) \left (\log \left (c x^n\right )+n (-\log (x))-n\right )}{a}+\frac{e \left (-6 \cos ^{-1}(a x) \left (9 \sqrt{1-a^2 x^2}+\sin \left (3 \cos ^{-1}(a x)\right )\right )+27 a x \left (\cos ^{-1}(a x)^2-2\right )-\left (2-9 \cos ^{-1}(a x)^2\right ) \cos \left (3 \cos ^{-1}(a x)\right )\right ) \left (3 \left (\log \left (c x^n\right )-n \log (x)\right )-n\right )}{324 a^3}+\frac{d n \log (x) \left (-2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)-2 a x+a x \cos ^{-1}(a x)^2\right )}{a}+\frac{e n \left (-12 \left (1-a^2 x^2\right )^{3/2} \cos ^{-1}(a x)-9 a x+\cos \left (3 \cos ^{-1}(a x)\right )\right )}{162 a^3}+\frac{e n \log (x) \left (-2 a^3 x^3+9 a^3 x^3 \cos ^{-1}(a x)^2-6 a^2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)-12 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)-12 a x\right )}{27 a^3}+\frac{4 e n x}{9 a^2}+2 d n x+\frac{2}{81} e n x^3 \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

2*d*n*x + (4*e*n*x)/(9*a^2) + (2*e*n*x^3)/81 + (e*n*(-9*a*x - 12*(1 - a^2*x^2)^(3/2)*ArcCos[a*x] + Cos[3*ArcCo
s[a*x]]))/(162*a^3) + (d*n*(-2*a*x - 2*Sqrt[1 - a^2*x^2]*ArcCos[a*x] + a*x*ArcCos[a*x]^2)*Log[x])/a + (e*n*(-1
2*a*x - 2*a^3*x^3 - 12*Sqrt[1 - a^2*x^2]*ArcCos[a*x] - 6*a^2*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x] + 9*a^3*x^3*Arc
Cos[a*x]^2)*Log[x])/(27*a^3) + (d*(-2*Sqrt[1 - a^2*x^2]*ArcCos[a*x] + a*x*(-2 + ArcCos[a*x]^2))*(-n - n*Log[x]
+ Log[c*x^n]))/a + (2*d*n*(a*x + Sqrt[1 - a^2*x^2]*ArcCos[a*x] - ArcCos[a*x]*Log[1 - I*E^(I*ArcCos[a*x])] + A
rcCos[a*x]*Log[1 + I*E^(I*ArcCos[a*x])] - I*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] + I*PolyLog[2, I*E^(I*ArcCos[a*
x])]))/a + (4*e*n*(a*x + Sqrt[1 - a^2*x^2]*ArcCos[a*x] - ArcCos[a*x]*Log[1 - I*E^(I*ArcCos[a*x])] + ArcCos[a*x
]*Log[1 + I*E^(I*ArcCos[a*x])] - I*PolyLog[2, (-I)*E^(I*ArcCos[a*x])] + I*PolyLog[2, I*E^(I*ArcCos[a*x])]))/(9
*a^3) + (e*(-n + 3*(-(n*Log[x]) + Log[c*x^n]))*(27*a*x*(-2 + ArcCos[a*x]^2) - (2 - 9*ArcCos[a*x]^2)*Cos[3*ArcC
os[a*x]] - 6*ArcCos[a*x]*(9*Sqrt[1 - a^2*x^2] + Sin[3*ArcCos[a*x]])))/(324*a^3)

________________________________________________________________________________________

Maple [F]  time = 2.036, size = 0, normalized size = 0. \begin{align*} \int \left ( e{x}^{2}+d \right ) \left ( \arccos \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)*arccos(a*x)^2*ln(c*x^n),x)

[Out]

int((e*x^2+d)*arccos(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 (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\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 (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{2} - \int \frac{2 \,{\left (3 \,{\left (a e x^{3} + 3 \, a d x\right )} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\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 (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\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)*arccos(a*x)^2*log(c*x^n),x, algorithm="maxima")

[Out]

1/3*(e*x^3 + 3*d*x)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2*log(x^n) - 1/9*((e*n - 3*e*log(c))*x^3 + 9*(d
*n - d*log(c))*x)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2 - integrate(2/9*(3*(a*e*x^3 + 3*a*d*x)*arctan2(
sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)*log(x^n) - ((a*e*n - 3*a*e*log(c))*x^3 + 9*(a*d*n - a*d*log(c))*x)*arctan2(
sqrt(a*x + 1)*sqrt(-a*x + 1), a*x))*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 )} \arccos \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)*arccos(a*x)^2*log(c*x^n),x, algorithm="fricas")

[Out]

integral((e*x^2 + d)*arccos(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)*acos(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 )} \arccos \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)*arccos(a*x)^2*log(c*x^n),x, algorithm="giac")

[Out]

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