3.2.96 \(\int (d+e x^2) \sinh ^{-1}(a x)^2 \log (c x^n) \, dx\) [196]

Optimal. Leaf size=458 \[ -2 d n x+\frac {2 e n x}{27 a^2}-\frac {4}{9} \left (9 d-\frac {2 e}{a^2}\right ) n x-\frac {2}{27} e n x^3+\frac {2 d n \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{a}+\frac {2 \left (9 a^2 d-2 e\right ) n \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{9 a^3}-\frac {4 e n \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{27 a^3}+\frac {2 e n x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)}{27 a}+\frac {2 e n \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)}{27 a^3}-d n x \sinh ^{-1}(a x)^2-\frac {1}{9} e n x^3 \sinh ^{-1}(a x)^2-\frac {4 \left (9 a^2 d-2 e\right ) n \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )}{9 a^3}+2 d x \log \left (c x^n\right )-\frac {4 e x \log \left (c x^n\right )}{9 a^2}+\frac {2}{27} e x^3 \log \left (c x^n\right )-\frac {2 d \sqrt {1+a^2 x^2} \sinh ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac {4 e \sqrt {1+a^2 x^2} \sinh ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac {2 e x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sinh ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \sinh ^{-1}(a x)^2 \log \left (c x^n\right )-\frac {2 \left (9 a^2 d-2 e\right ) n \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )}{9 a^3}+\frac {2 \left (9 a^2 d-2 e\right ) n \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )}{9 a^3} \]

[Out]

-2*d*n*x+2/27*e*n*x/a^2-4/9*(9*d-2*e/a^2)*n*x-2/27*e*n*x^3+2/27*e*n*(a^2*x^2+1)^(3/2)*arcsinh(a*x)/a^3-d*n*x*a
rcsinh(a*x)^2-1/9*e*n*x^3*arcsinh(a*x)^2-4/9*(9*a^2*d-2*e)*n*arcsinh(a*x)*arctanh(a*x+(a^2*x^2+1)^(1/2))/a^3+2
*d*x*ln(c*x^n)-4/9*e*x*ln(c*x^n)/a^2+2/27*e*x^3*ln(c*x^n)+d*x*arcsinh(a*x)^2*ln(c*x^n)+1/3*e*x^3*arcsinh(a*x)^
2*ln(c*x^n)-2/9*(9*a^2*d-2*e)*n*polylog(2,-a*x-(a^2*x^2+1)^(1/2))/a^3+2/9*(9*a^2*d-2*e)*n*polylog(2,a*x+(a^2*x
^2+1)^(1/2))/a^3+2*d*n*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a+2/9*(9*a^2*d-2*e)*n*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a^3
-4/27*e*n*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a^3+2/27*e*n*x^2*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a-2*d*arcsinh(a*x)*ln
(c*x^n)*(a^2*x^2+1)^(1/2)/a+4/9*e*arcsinh(a*x)*ln(c*x^n)*(a^2*x^2+1)^(1/2)/a^3-2/9*e*x^2*arcsinh(a*x)*ln(c*x^n
)*(a^2*x^2+1)^(1/2)/a

________________________________________________________________________________________

Rubi [A]
time = 0.45, antiderivative size = 458, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5793, 5772, 5798, 8, 5776, 5812, 30, 2434, 6, 5806, 5816, 4267, 2317, 2438} \begin {gather*} -\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}{9} n x \left (9 d-\frac {2 e}{a^2}\right )+\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 x}{27 a^2}+\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 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 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}+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 \end {gather*}

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 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 8

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

Rule 30

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

Rule 2317

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

Rule 2434

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(Px_.)*(F_)[(d_.)*((e_.) + (f_.)*(x_))]^(m_.), x_Symbol] :> With[{u
= IntHide[Px*F[d*(e + f*x)]^m, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist[1/x, u, x], x], x]] /; F
reeQ[{a, b, c, d, e, f, n}, x] && PolynomialQ[Px, x] && IGtQ[m, 0] && MemberQ[{ArcSin, ArcCos, ArcSinh, ArcCos
h}, F]

Rule 2438

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

Rule 4267

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 5772

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 5776

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 5793

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 5798

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^p], Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*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 5806

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[(1/(m + 2))*Simp[Sqrt[d + e*x^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/(f*(m + 2)))
*Simp[Sqrt[d + e*x^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] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])

Rule 5812

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

Rule 5816

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

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

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Mathematica [A]
time = 0.51, size = 516, normalized size = 1.13 \begin {gather*} -2 d n x+\frac {4 e n x}{9 a^2}-\frac {2}{81} e n x^3+\frac {2 e n \left (-\frac {a x}{3}-\frac {a^3 x^3}{9}+\frac {1}{3} \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)\right )}{9 a^3}+\frac {d n \left (2 a x-2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)+a x \sinh ^{-1}(a x)^2\right ) \log (x)}{a}+\frac {e n \left (-12 a x+2 a^3 x^3+12 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)-6 a^2 x^2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)+9 a^3 x^3 \sinh ^{-1}(a x)^2\right ) \log (x)}{27 a^3}+\frac {d \left (-2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)+a x \left (2+\sinh ^{-1}(a x)^2\right )\right ) \left (-n-n \log (x)+\log \left (c x^n\right )\right )}{a}+\frac {e \left (27 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)+a x \left (-26-9 \sinh ^{-1}(a x)^2+\left (2+9 \sinh ^{-1}(a x)^2\right ) \cosh \left (2 \sinh ^{-1}(a x)\right )\right )-3 \sinh ^{-1}(a x) \cosh \left (3 \sinh ^{-1}(a x)\right )\right ) \left (-n+3 \left (-n \log (x)+\log \left (c x^n\right )\right )\right )}{162 a^3}+\frac {2 d n \left (-a x+\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)+\sinh ^{-1}(a x) \log \left (1-e^{-\sinh ^{-1}(a x)}\right )-\sinh ^{-1}(a x) \log \left (1+e^{-\sinh ^{-1}(a x)}\right )+\text {Li}_2\left (-e^{-\sinh ^{-1}(a x)}\right )-\text {Li}_2\left (e^{-\sinh ^{-1}(a x)}\right )\right )}{a}-\frac {4 e n \left (-a x+\sqrt {1+a^2 x^2} \sinh ^{-1}(a x)+\sinh ^{-1}(a x) \log \left (1-e^{-\sinh ^{-1}(a x)}\right )-\sinh ^{-1}(a x) \log \left (1+e^{-\sinh ^{-1}(a x)}\right )+\text {Li}_2\left (-e^{-\sinh ^{-1}(a x)}\right )-\text {Li}_2\left (e^{-\sinh ^{-1}(a x)}\right )\right )}{9 a^3} \end {gather*}

Antiderivative was successfully verified.

[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*(-1/3*(a*x) - (a^3*x^3)/9 + ((1 + a^2*x^2)^(3/2)*ArcSin
h[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*
ArcSinh[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*L
og[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^(-Ar
cSinh[a*x])] + PolyLog[2, -E^(-ArcSinh[a*x])] - PolyLog[2, E^(-ArcSinh[a*x])]))/(9*a^3)

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Maple [F]
time = 1.64, size = 0, normalized size = 0.00 \[\int \left (e \,x^{2}+d \right ) \arcsinh \left (a x \right )^{2} \ln \left (c \,x^{n}\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*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 = 0.00, 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)*arcsinh(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^2*x^2 + 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*l
og(c))*a*x - 3*(a^3*x^5*e + (3*a^3*d + a*e)*x^3 + 3*a*d*x)*log(x^n) + (a^2*(n - 3*log(c))*x^4*e + 9*(d*n - d*l
og(c))*a^2*x^2 - 3*(a^2*x^4*e + 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 = 0.00, 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)*arcsinh(a*x)^2*log(c*x^n),x, algorithm="fricas")

[Out]

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

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

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]

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

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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)*arcsinh(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

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

[Out]

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

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