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

Optimal. Leaf size=508 \[ \frac{2 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{9 a^3}-\frac{2 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{9 a^3}+\frac{4 e x \log \left (c x^n\right )}{9 a^2}-\frac{4 e \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{4}{9} n x \left (\frac{2 e}{a^2}+9 d\right )+\frac{2 n \sqrt{a x-1} \sqrt{a x+1} \left (9 a^2 d+2 e\right ) \cosh ^{-1}(a x)}{9 a^3}-\frac{4 n \left (9 a^2 d+2 e\right ) \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{9 a^3}-\frac{2 e n x}{27 a^2}+\frac{2 e n (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{27 a^3}+\frac{4 e n \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{27 a^3}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )-\frac{2 d \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac{1}{3} e x^3 \cosh ^{-1}(a x)^2 \log \left (c x^n\right )-\frac{2 e x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}-d n x \cosh ^{-1}(a x)^2+\frac{2 d n \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{a}-\frac{1}{9} e n x^3 \cosh ^{-1}(a x)^2+\frac{2 e n x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{27 a}+2 d x \log \left (c x^n\right )+\frac{2}{27} e x^3 \log \left (c x^n\right )-2 d n x-\frac{2}{27} e n x^3 \]

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

________________________________________________________________________________________

Rubi [A]  time = 1.54606, antiderivative size = 508, 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 = {5707, 5654, 5718, 8, 5662, 5759, 30, 2387, 6, 5743, 5761, 4180, 2279, 2391} \[ \frac{2 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{9 a^3}-\frac{2 i n \left (9 a^2 d+2 e\right ) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{9 a^3}+\frac{4 e x \log \left (c x^n\right )}{9 a^2}-\frac{4 e \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{4}{9} n x \left (\frac{2 e}{a^2}+9 d\right )+\frac{2 n \sqrt{a x-1} \sqrt{a x+1} \left (9 a^2 d+2 e\right ) \cosh ^{-1}(a x)}{9 a^3}-\frac{4 n \left (9 a^2 d+2 e\right ) \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{9 a^3}-\frac{2 e n x}{27 a^2}+\frac{2 e n (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{27 a^3}+\frac{4 e n \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{27 a^3}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )-\frac{2 d \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac{1}{3} e x^3 \cosh ^{-1}(a x)^2 \log \left (c x^n\right )-\frac{2 e x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}-d n x \cosh ^{-1}(a x)^2+\frac{2 d n \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{a}-\frac{1}{9} e n x^3 \cosh ^{-1}(a x)^2+\frac{2 e n x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{27 a}+2 d x \log \left (c x^n\right )+\frac{2}{27} e x^3 \log \left (c x^n\right )-2 d n x-\frac{2}{27} e n x^3 \]

Antiderivative was successfully verified.

[In]

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

Rule 5707

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcCosh[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
] && (p > 0 || IGtQ[n, 0])

Rule 5654

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

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 8

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

Rule 5662

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

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 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 5743

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*
(x_)], x_Symbol] :> Simp[((f*x)^(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(f*(m + 2)), x
] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/((m + 2)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[((f*x)^m*(a + b*ArcCo
sh[c*x])^n)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 2)*S
qrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] |
| EqQ[n, 1])

Rule 5761

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]
), x_Symbol] :> Dist[1/(c^(m + 1)*Sqrt[-(d1*d2)]), Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /
; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[n, 0] && GtQ[d1, 0] &&
LtQ[d2, 0] && IntegerQ[m]

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, 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 ) \cosh ^{-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 x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-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 x} \sqrt{1+a x} \cosh ^{-1}(a x)}{a x}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a^3 x}-\frac{2 e x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a}+d \cosh ^{-1}(a x)^2+\frac{1}{3} e x^2 \cosh ^{-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 x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-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 x} \sqrt{1+a x} \cosh ^{-1}(a x)}{x}-\frac{2 e x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a}+d \cosh ^{-1}(a x)^2+\frac{1}{3} e x^2 \cosh ^{-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 x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-1}(a x)^2 \log \left (c x^n\right )-(d n) \int \cosh ^{-1}(a x)^2 \, dx-\frac{1}{3} (e n) \int x^2 \cosh ^{-1}(a x)^2 \, dx+\frac{(2 e n) \int x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \, dx}{9 a}+\frac{\left (2 \left (9 a^2 d+2 e\right ) n\right ) \int \frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-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 x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a^3}+\frac{2 e n (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{27 a^3}-d n x \cosh ^{-1}(a x)^2-\frac{1}{9} e n x^3 \cosh ^{-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 x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+(2 a d n) \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, 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 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx-\frac{\left (2 \left (9 a^2 d+2 e\right ) n\right ) \int \frac{\cosh ^{-1}(a x)}{x \sqrt{-1+a x} \sqrt{1+a x}} \, 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 x} \sqrt{1+a x} \cosh ^{-1}(a x)}{a}+\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a^3}+\frac{2 e n x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a}+\frac{2 e n (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{27 a^3}-d n x \cosh ^{-1}(a x)^2-\frac{1}{9} e n x^3 \cosh ^{-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 x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-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 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{27 a}-\frac{\left (2 \left (9 a^2 d+2 e\right ) n\right ) \operatorname{Subst}\left (\int x \text{sech}(x) \, dx,x,\cosh ^{-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 x} \sqrt{1+a x} \cosh ^{-1}(a x)}{a}+\frac{4 e n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a^3}+\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a^3}+\frac{2 e n x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a}+\frac{2 e n (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{27 a^3}-d n x \cosh ^{-1}(a x)^2-\frac{1}{9} e n x^3 \cosh ^{-1}(a x)^2-\frac{4 \left (9 a^2 d+2 e\right ) n \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-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 x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-1}(a x)^2 \log \left (c x^n\right )-\frac{(4 e n) \int 1 \, dx}{27 a^2}+\frac{\left (2 i \left (9 a^2 d+2 e\right ) n\right ) \operatorname{Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-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 \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-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 x} \sqrt{1+a x} \cosh ^{-1}(a x)}{a}+\frac{4 e n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a^3}+\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a^3}+\frac{2 e n x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a}+\frac{2 e n (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{27 a^3}-d n x \cosh ^{-1}(a x)^2-\frac{1}{9} e n x^3 \cosh ^{-1}(a x)^2-\frac{4 \left (9 a^2 d+2 e\right ) n \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-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 x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-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^{\cosh ^{-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^{\cosh ^{-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 x} \sqrt{1+a x} \cosh ^{-1}(a x)}{a}+\frac{4 e n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a^3}+\frac{2 \left (9 a^2 d+2 e\right ) n \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{9 a^3}+\frac{2 e n x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{27 a}+\frac{2 e n (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{27 a^3}-d n x \cosh ^{-1}(a x)^2-\frac{1}{9} e n x^3 \cosh ^{-1}(a x)^2-\frac{4 \left (9 a^2 d+2 e\right ) n \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-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 x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac{2 e x^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \cosh ^{-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^{\cosh ^{-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^{\cosh ^{-1}(a x)}\right )}{9 a^3}\\ \end{align*}

Mathematica [A]  time = 3.5644, size = 770, normalized size = 1.52 \[ \frac{4 e n \sqrt{a x-1} \sqrt{a x+1} \left (\frac{i \left (\text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )-\text{PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )\right )}{\sqrt{\frac{a x-1}{a x+1}} (a x+1)}-\frac{a x}{\sqrt{\frac{a x-1}{a x+1}} (a x+1)}+\cosh ^{-1}(a x)+\frac{i \cosh ^{-1}(a x) \left (\log \left (1-i e^{-\cosh ^{-1}(a x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(a x)}\right )\right )}{\sqrt{\frac{a x-1}{a x+1}} (a x+1)}\right )}{9 a^3}+\frac{2 d n \sqrt{a x-1} \sqrt{a x+1} \left (\frac{i \left (\text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )-\text{PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )\right )}{\sqrt{\frac{a x-1}{a x+1}} (a x+1)}-\frac{a x}{\sqrt{\frac{a x-1}{a x+1}} (a x+1)}+\cosh ^{-1}(a x)+\frac{i \cosh ^{-1}(a x) \left (\log \left (1-i e^{-\cosh ^{-1}(a x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(a x)}\right )\right )}{\sqrt{\frac{a x-1}{a x+1}} (a x+1)}\right )}{a}+\frac{e \left (27 a x \left (\cosh ^{-1}(a x)^2+2\right )+\left (9 \cosh ^{-1}(a x)^2+2\right ) \cosh \left (3 \cosh ^{-1}(a x)\right )-6 \cosh ^{-1}(a x) \left (9 \sqrt{\frac{a x-1}{a x+1}} (a x+1)+\sinh \left (3 \cosh ^{-1}(a x)\right )\right )\right ) \left (3 \left (\log \left (c x^n\right )-n \log (x)\right )-n\right )}{324 a^3}+\frac{e n \log (x) \left (2 a^3 x^3+9 a^3 x^3 \cosh ^{-1}(a x)^2-6 a^2 x^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)+12 a x-12 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)\right )}{27 a^3}-\frac{4 e n x}{9 a^2}-\frac{e n \sqrt{a x-1} \left (-9 a x-12 \left (\frac{a x-1}{a x+1}\right )^{3/2} (a x+1)^3 \cosh ^{-1}(a x)+\cosh \left (3 \cosh ^{-1}(a x)\right )\right )}{162 a^3 \sqrt{\frac{a x-1}{a x+1}} \sqrt{a x+1}}+\frac{d \left (a x \left (\cosh ^{-1}(a x)^2+2\right )-2 \sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)\right ) \left (\log \left (c x^n\right )+n (-\log (x))-n\right )}{a}+\frac{d n \log (x) \left (2 a x+a x \cosh ^{-1}(a x)^2-2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)\right )}{a}-2 d n x-\frac{2}{81} e n x^3 \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(d + e*x^2)*ArcCosh[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*Sqrt[-1 + a*x]*(-9*a*x - 12*((-1 + a*x)/(1 + a*x))^(3/2)*
(1 + a*x)^3*ArcCosh[a*x] + Cosh[3*ArcCosh[a*x]]))/(162*a^3*Sqrt[(-1 + a*x)/(1 + a*x)]*Sqrt[1 + a*x]) + (d*n*(2
*a*x - 2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x] + a*x*ArcCosh[a*x]^2)*Log[x])/a + (e*n*(12*a*x + 2*a^3*x^3
- 12*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x] - 6*a^2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x] + 9*a^3*x
^3*ArcCosh[a*x]^2)*Log[x])/(27*a^3) + (d*(-2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x] + a*x*(2 + ArcC
osh[a*x]^2))*(-n - n*Log[x] + Log[c*x^n]))/a + (2*d*n*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-((a*x)/(Sqrt[(-1 + a*x)/(
1 + a*x)]*(1 + a*x))) + ArcCosh[a*x] + (I*ArcCosh[a*x]*(Log[1 - I/E^ArcCosh[a*x]] - Log[1 + I/E^ArcCosh[a*x]])
)/(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)) + (I*(PolyLog[2, (-I)/E^ArcCosh[a*x]] - PolyLog[2, I/E^ArcCosh[a*x]])
)/(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))))/a + (4*e*n*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-((a*x)/(Sqrt[(-1 + a*x)/(
1 + a*x)]*(1 + a*x))) + ArcCosh[a*x] + (I*ArcCosh[a*x]*(Log[1 - I/E^ArcCosh[a*x]] - Log[1 + I/E^ArcCosh[a*x]])
)/(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)) + (I*(PolyLog[2, (-I)/E^ArcCosh[a*x]] - PolyLog[2, I/E^ArcCosh[a*x]])
)/(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))))/(9*a^3) + (e*(-n + 3*(-(n*Log[x]) + Log[c*x^n]))*(27*a*x*(2 + ArcCo
sh[a*x]^2) + (2 + 9*ArcCosh[a*x]^2)*Cosh[3*ArcCosh[a*x]] - 6*ArcCosh[a*x]*(9*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a
*x) + Sinh[3*ArcCosh[a*x]])))/(324*a^3)

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Maple [F]  time = 0.849, size = 0, normalized size = 0. \begin{align*} \int \left ( e{x}^{2}+d \right ) \left ({\rm arccosh} \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)*arccosh(a*x)^2*ln(c*x^n),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\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 - 3 \,{\left (e x^{3} + 3 \, d x\right )} \log \left (x^{n}\right )\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{2} - \int -\frac{2 \,{\left ({\left (e n - 3 \, e \log \left (c\right )\right )} a^{3} x^{5} +{\left (9 \,{\left (d n - d \log \left (c\right )\right )} a^{3} -{\left (e n - 3 \, e \log \left (c\right )\right )} a\right )} x^{3} - 9 \,{\left (d n - d \log \left (c\right )\right )} a x +{\left ({\left (e n - 3 \, e \log \left (c\right )\right )} a^{2} x^{4} + 9 \,{\left (d n - d \log \left (c\right )\right )} a^{2} x^{2} - 3 \,{\left (a^{2} e x^{4} + 3 \, a^{2} d x^{2}\right )} \log \left (x^{n}\right )\right )} \sqrt{a x + 1} \sqrt{a x - 1} - 3 \,{\left (a^{3} e x^{5} +{\left (3 \, a^{3} d - a e\right )} x^{3} - 3 \, a d x\right )} \log \left (x^{n}\right )\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )}{9 \,{\left (a^{3} x^{3} +{\left (a^{2} x^{2} - 1\right )} \sqrt{a x + 1} \sqrt{a x - 1} - a x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*((e*n - 3*e*log(c))*x^3 + 9*(d*n - d*log(c))*x - 3*(e*x^3 + 3*d*x)*log(x^n))*log(a*x + sqrt(a*x + 1)*sqrt
(a*x - 1))^2 - integrate(-2/9*((e*n - 3*e*log(c))*a^3*x^5 + (9*(d*n - d*log(c))*a^3 - (e*n - 3*e*log(c))*a)*x^
3 - 9*(d*n - d*log(c))*a*x + ((e*n - 3*e*log(c))*a^2*x^4 + 9*(d*n - d*log(c))*a^2*x^2 - 3*(a^2*e*x^4 + 3*a^2*d
*x^2)*log(x^n))*sqrt(a*x + 1)*sqrt(a*x - 1) - 3*(a^3*e*x^5 + (3*a^3*d - a*e)*x^3 - 3*a*d*x)*log(x^n))*log(a*x
+ sqrt(a*x + 1)*sqrt(a*x - 1))/(a^3*x^3 + (a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1) - a*x), x)

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

[Out]

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

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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)*acosh(a*x)**2*ln(c*x**n),x)

[Out]

Timed out

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

[Out]

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