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

Optimal. Leaf size=458 \[ -\frac{2 n \left (9 a^2 d-2 e\right ) \text{PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )}{9 a^3}+\frac{2 n \left (9 a^2 d-2 e\right ) \text{PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )}{9 a^3}-\frac{2 d \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e x \log \left (c x^n\right )}{9 a^2}-\frac{2 e x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+\frac{4 e \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac{2 n \sqrt{a^2 x^2+1} \left (9 a^2 d-2 e\right ) \sinh ^{-1}(a x)}{9 a^3}-\frac{4}{9} n x \left (9 d-\frac{2 e}{a^2}\right )-\frac{4 n \left (9 a^2 d-2 e\right ) \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{9 a^3}+\frac{2 d n \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a}+\frac{2 e n x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{27 a}+\frac{2 e n \left (a^2 x^2+1\right )^{3/2} \sinh ^{-1}(a x)}{27 a^3}-\frac{4 e n \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{27 a^3}+\frac{2 e n x}{27 a^2}+d x \sinh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \sinh ^{-1}(a x)^2 \log \left (c x^n\right )-d n x \sinh ^{-1}(a x)^2-\frac{1}{9} e n x^3 \sinh ^{-1}(a x)^2+2 d x \log \left (c x^n\right )+\frac{2}{27} e x^3 \log \left (c x^n\right )-2 d n x-\frac{2}{27} e n x^3 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.702017, antiderivative size = 458, 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 = {5706, 5653, 5717, 8, 5661, 5758, 30, 2387, 6, 5742, 5760, 4182, 2279, 2391} \[ -\frac{2 n \left (9 a^2 d-2 e\right ) \text{PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )}{9 a^3}+\frac{2 n \left (9 a^2 d-2 e\right ) \text{PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )}{9 a^3}-\frac{2 d \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac{4 e x \log \left (c x^n\right )}{9 a^2}-\frac{2 e x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+\frac{4 e \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac{2 n \sqrt{a^2 x^2+1} \left (9 a^2 d-2 e\right ) \sinh ^{-1}(a x)}{9 a^3}-\frac{4}{9} n x \left (9 d-\frac{2 e}{a^2}\right )-\frac{4 n \left (9 a^2 d-2 e\right ) \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{9 a^3}+\frac{2 d n \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a}+\frac{2 e n x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{27 a}+\frac{2 e n \left (a^2 x^2+1\right )^{3/2} \sinh ^{-1}(a x)}{27 a^3}-\frac{4 e n \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{27 a^3}+\frac{2 e n x}{27 a^2}+d x \sinh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac{1}{3} e x^3 \sinh ^{-1}(a x)^2 \log \left (c x^n\right )-d n x \sinh ^{-1}(a x)^2-\frac{1}{9} e n x^3 \sinh ^{-1}(a x)^2+2 d x \log \left (c x^n\right )+\frac{2}{27} e x^3 \log \left (c x^n\right )-2 d n x-\frac{2}{27} e n x^3 \]

Antiderivative was successfully verified.

[In]

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

Rule 5706

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] &&
(p > 0 || IGtQ[n, 0])

Rule 5653

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

Rule 5717

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)
^(p + 1)*(a + b*ArcSinh[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*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a,
b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]

Rule 8

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

Rule 5661

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcS
inh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSinh[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 5758

Int[(((a_.) + ArcSinh[(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*ArcSinh[c*x])^n)/(e*m), x] + (-Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)
^(m - 2)*(a + b*ArcSinh[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]
), Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] &&
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 5742

Int[((a_.) + ArcSinh[(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*ArcSinh[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*ArcSinh[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*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f
, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])

Rule 5760

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

Rule 4182

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

Rule 2279

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

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

Mathematica [A]  time = 0.705591, size = 516, normalized size = 1.13 \[ \frac{2 d n \left (\text{PolyLog}\left (2,-e^{-\sinh ^{-1}(a x)}\right )-\text{PolyLog}\left (2,e^{-\sinh ^{-1}(a x)}\right )+\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)-a x+\sinh ^{-1}(a x) \log \left (1-e^{-\sinh ^{-1}(a x)}\right )-\sinh ^{-1}(a x) \log \left (e^{-\sinh ^{-1}(a x)}+1\right )\right )}{a}-\frac{4 e n \left (\text{PolyLog}\left (2,-e^{-\sinh ^{-1}(a x)}\right )-\text{PolyLog}\left (2,e^{-\sinh ^{-1}(a x)}\right )+\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)-a x+\sinh ^{-1}(a x) \log \left (1-e^{-\sinh ^{-1}(a x)}\right )-\sinh ^{-1}(a x) \log \left (e^{-\sinh ^{-1}(a x)}+1\right )\right )}{9 a^3}+\frac{d \left (a x \left (\sinh ^{-1}(a x)^2+2\right )-2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)\right ) \left (\log \left (c x^n\right )+n (-\log (x))-n\right )}{a}+\frac{e \left (27 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)-3 \sinh ^{-1}(a x) \cosh \left (3 \sinh ^{-1}(a x)\right )+a x \left (-9 \sinh ^{-1}(a x)^2+\left (9 \sinh ^{-1}(a x)^2+2\right ) \cosh \left (2 \sinh ^{-1}(a x)\right )-26\right )\right ) \left (3 \left (\log \left (c x^n\right )-n \log (x)\right )-n\right )}{162 a^3}+\frac{d n \log (x) \left (-2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)+2 a x+a x \sinh ^{-1}(a x)^2\right )}{a}+\frac{2 e n \left (-\frac{1}{9} a^3 x^3+\frac{1}{3} \left (a^2 x^2+1\right )^{3/2} \sinh ^{-1}(a x)-\frac{a x}{3}\right )}{9 a^3}+\frac{e n \log (x) \left (2 a^3 x^3+9 a^3 x^3 \sinh ^{-1}(a x)^2-6 a^2 x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)+12 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)-12 a x\right )}{27 a^3}+\frac{4 e n x}{9 a^2}-2 d n x-\frac{2}{81} e n x^3 \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(d + e*x^2)*ArcSinh[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 + (2*e*n*(-(a*x)/3 - (a^3*x^3)/9 + ((1 + a^2*x^2)^(3/2)*ArcSinh[
a*x])/3))/(9*a^3) + (d*n*(2*a*x - 2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + a*x*ArcSinh[a*x]^2)*Log[x])/a + (e*n*(-12
*a*x + 2*a^3*x^3 + 12*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] - 6*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + 9*a^3*x^3*Ar
cSinh[a*x]^2)*Log[x])/(27*a^3) + (d*(-2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + a*x*(2 + ArcSinh[a*x]^2))*(-n - n*Log
[x] + Log[c*x^n]))/a + (e*(27*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + a*x*(-26 - 9*ArcSinh[a*x]^2 + (2 + 9*ArcSinh[a*
x]^2)*Cosh[2*ArcSinh[a*x]]) - 3*ArcSinh[a*x]*Cosh[3*ArcSinh[a*x]])*(-n + 3*(-(n*Log[x]) + Log[c*x^n])))/(162*a
^3) + (2*d*n*(-(a*x) + Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + ArcSinh[a*x]*Log[1 - E^(-ArcSinh[a*x])] - ArcSinh[a*x]
*Log[1 + E^(-ArcSinh[a*x])] + PolyLog[2, -E^(-ArcSinh[a*x])] - PolyLog[2, E^(-ArcSinh[a*x])]))/a - (4*e*n*(-(a
*x) + Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + ArcSinh[a*x]*Log[1 - E^(-ArcSinh[a*x])] - ArcSinh[a*x]*Log[1 + E^(-ArcS
inh[a*x])] + PolyLog[2, -E^(-ArcSinh[a*x])] - PolyLog[2, E^(-ArcSinh[a*x])]))/(9*a^3)

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Maple [F]  time = 0.892, size = 0, normalized size = 0. \begin{align*} \int \left ( e{x}^{2}+d \right ) \left ({\it Arcsinh} \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)*arcsinh(a*x)^2*ln(c*x^n),x)

[Out]

int((e*x^2+d)*arcsinh(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^{2} x^{2} + 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 - 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 ) +{\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^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{9 \,{\left (a^{3} x^{3} + a x +{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*arcsinh(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^2*x^2 + 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 - 3*(a^3*e*x^5 + (3*a^3*d + a*e)*x^3 + 3*a*d*x)*log(x^n) + ((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^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))
/(a^3*x^3 + a*x + (a^2*x^2 + 1)^(3/2)), 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{arsinh}\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)*arcsinh(a*x)^2*log(c*x^n),x, algorithm="fricas")

[Out]

integral((e*x^2 + d)*arcsinh(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)*asinh(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{arsinh}\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)*arcsinh(a*x)^2*log(c*x^n),x, algorithm="giac")

[Out]

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