3.2.97 \(\int (d+e x^2) \cosh ^{-1}(a x)^2 \log (c x^n) \, dx\) [197]

Optimal. Leaf size=508 \[ -2 d n x-\frac {2 e n x}{27 a^2}-\frac {4}{9} \left (9 d+\frac {2 e}{a^2}\right ) n x-\frac {2}{27} e n x^3+\frac {2 d n \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{a}+\frac {4 e n \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{27 a^3}+\frac {2 \left (9 a^2 d+2 e\right ) n \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{9 a^3}+\frac {2 e n x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{27 a}+\frac {2 e n (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{27 a^3}-d n x \cosh ^{-1}(a x)^2-\frac {1}{9} e n x^3 \cosh ^{-1}(a x)^2-\frac {4 \left (9 a^2 d+2 e\right ) n \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{9 a^3}+2 d x \log \left (c x^n\right )+\frac {4 e x \log \left (c x^n\right )}{9 a^2}+\frac {2}{27} e x^3 \log \left (c x^n\right )-\frac {2 d \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac {4 e \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac {2 e x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \cosh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {2 i \left (9 a^2 d+2 e\right ) n \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{9 a^3}-\frac {2 i \left (9 a^2 d+2 e\right ) n \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{9 a^3} \]

[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*x-1)^(3/2)*(a*x+1)^(3/2)*arccosh(a*x)/a
^3-d*n*x*arccosh(a*x)^2-1/9*e*n*x^3*arccosh(a*x)^2-4/9*(9*a^2*d+2*e)*n*arccosh(a*x)*arctan(a*x+(a*x-1)^(1/2)*(
a*x+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*arccosh(a*x)^2*ln(c*x^n)+1/3*
e*x^3*arccosh(a*x)^2*ln(c*x^n)-2/9*I*(9*a^2*d+2*e)*n*polylog(2,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))/a^3+2/9*I*
(9*a^2*d+2*e)*n*polylog(2,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))/a^3+2*d*n*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(
1/2)/a+4/27*e*n*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3+2/9*(9*a^2*d+2*e)*n*arccosh(a*x)*(a*x-1)^(1/2)*(a
*x+1)^(1/2)/a^3+2/27*e*n*x^2*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a-2*d*arccosh(a*x)*ln(c*x^n)*(a*x-1)^(1/
2)*(a*x+1)^(1/2)/a-4/9*e*arccosh(a*x)*ln(c*x^n)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3-2/9*e*x^2*arccosh(a*x)*ln(c*x^
n)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a

________________________________________________________________________________________

Rubi [A]
time = 0.98, antiderivative size = 508, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 15, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5909, 5879, 5915, 8, 5883, 5939, 30, 2434, 6, 5927, 5947, 4265, 2317, 2438, 41} \begin {gather*} \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 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\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}+\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 n \left (9 a^2 d+2 e\right ) \cosh ^{-1}(a x) \text {ArcTan}\left (e^{\cosh ^{-1}(a x)}\right )}{9 a^3}+\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}+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 \end {gather*}

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 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 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 2317

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

Rule 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 4265

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 5879

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 5883

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)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5909

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 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> 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/(2*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[(1 + c*x)^(p + 1/2)*(-
1 + c*x)^(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] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]

Rule 5927

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[(1/(m + 2))*Simp[Sqrt[d1 + e1*x]/Sqrt[1 + c*x]]*Simp[Sqrt[d2 + e2*x]/Sqrt[-1 + c*x]], Int[(f*x)^m*(
(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] - Dist[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d1 + e1*x]
/Sqrt[1 + c*x]]*Simp[Sqrt[d2 + e2*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, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && (IGtQ[m, -2
] || EqQ[n, 1])

Rule 5939

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

Rule 5947

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]
), x_Symbol] :> Dist[(1/c^(m + 1))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], S
ubst[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*d
1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && IntegerQ[m]

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 2.48, size = 619, normalized size = 1.22 \begin {gather*} \frac {-648 a^3 d n x-144 a e n x-8 a^3 e n x^3+2 e n \left (9 a x+12 \left (\frac {-1+a x}{1+a x}\right )^{3/2} (1+a x)^3 \cosh ^{-1}(a x)-\cosh \left (3 \cosh ^{-1}(a x)\right )\right )+324 a^2 d n \left (2 a x-2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)+a x \cosh ^{-1}(a x)^2\right ) \log (x)+12 e n \left (2 a x \left (6+a^2 x^2\right )-6 \sqrt {-1+a x} \sqrt {1+a x} \left (2+a^2 x^2\right ) \cosh ^{-1}(a x)+9 a^3 x^3 \cosh ^{-1}(a x)^2\right ) \log (x)+324 a^2 d \left (2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \cosh ^{-1}(a x)-a x \left (2+\cosh ^{-1}(a x)^2\right )\right ) \left (n+n \log (x)-\log \left (c x^n\right )\right )+648 a^2 d n \left (-a x+\sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x)+a x \sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x)+i \cosh ^{-1}(a x) \log \left (1-i e^{-\cosh ^{-1}(a x)}\right )-i \cosh ^{-1}(a x) \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+i \text {Li}_2\left (-i e^{-\cosh ^{-1}(a x)}\right )-i \text {Li}_2\left (i e^{-\cosh ^{-1}(a x)}\right )\right )+144 e n \left (-a x+\sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x)+a x \sqrt {\frac {-1+a x}{1+a x}} \cosh ^{-1}(a x)+i \cosh ^{-1}(a x) \log \left (1-i e^{-\cosh ^{-1}(a x)}\right )-i \cosh ^{-1}(a x) \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+i \text {Li}_2\left (-i e^{-\cosh ^{-1}(a x)}\right )-i \text {Li}_2\left (i e^{-\cosh ^{-1}(a x)}\right )\right )-e \left (n+3 n \log (x)-3 \log \left (c x^n\right )\right ) \left (27 a x \left (2+\cosh ^{-1}(a x)^2\right )+\left (2+9 \cosh ^{-1}(a x)^2\right ) \cosh \left (3 \cosh ^{-1}(a x)\right )-6 \cosh ^{-1}(a x) \left (9 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)+\sinh \left (3 \cosh ^{-1}(a x)\right )\right )\right )}{324 a^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-648*a^3*d*n*x - 144*a*e*n*x - 8*a^3*e*n*x^3 + 2*e*n*(9*a*x + 12*((-1 + a*x)/(1 + a*x))^(3/2)*(1 + a*x)^3*Arc
Cosh[a*x] - Cosh[3*ArcCosh[a*x]]) + 324*a^2*d*n*(2*a*x - 2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x] + a*x*Arc
Cosh[a*x]^2)*Log[x] + 12*e*n*(2*a*x*(6 + a^2*x^2) - 6*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(2 + a^2*x^2)*ArcCosh[a*x]
+ 9*a^3*x^3*ArcCosh[a*x]^2)*Log[x] + 324*a^2*d*(2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x] - a*x*(2 +
ArcCosh[a*x]^2))*(n + n*Log[x] - Log[c*x^n]) + 648*a^2*d*n*(-(a*x) + Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]
+ a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x] + I*ArcCosh[a*x]*Log[1 - I/E^ArcCosh[a*x]] - I*ArcCosh[a*x]*Log[
1 + I/E^ArcCosh[a*x]] + I*PolyLog[2, (-I)/E^ArcCosh[a*x]] - I*PolyLog[2, I/E^ArcCosh[a*x]]) + 144*e*n*(-(a*x)
+ Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x] + a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x] + I*ArcCosh[a*x]*Log[1
- I/E^ArcCosh[a*x]] - I*ArcCosh[a*x]*Log[1 + I/E^ArcCosh[a*x]] + I*PolyLog[2, (-I)/E^ArcCosh[a*x]] - I*PolyLo
g[2, I/E^ArcCosh[a*x]]) - e*(n + 3*n*Log[x] - 3*Log[c*x^n])*(27*a*x*(2 + ArcCosh[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]])))/(3
24*a^3)

________________________________________________________________________________________

Maple [F]
time = 2.07, size = 0, normalized size = 0.00 \[\int \left (e \,x^{2}+d \right ) \mathrm {arccosh}\left (a x \right )^{2} \ln \left (c \,x^{n}\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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)*arccosh(a*x)^2*log(c*x^n),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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 {acosh}^{2}{\left (a x \right )}\, dx \end {gather*}

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]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________