3.340 \(\int \frac{\sinh ^{-1}(a x)^3}{(c+a^2 c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=515 \[ -\frac{8 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{a^2 c x^2+c}}+\frac{4 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{a^2 c x^2+c}}-\frac{1}{20 a c^3 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \log \left (a^2 x^2+1\right )}{2 a c^3 \sqrt{a^2 c x^2+c}}+\frac{8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt{a^2 c x^2+c}}+\frac{8 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{15 a c^3 \sqrt{a^2 c x^2+c}}+\frac{4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (a^2 c x^2+c\right )^{3/2}}+\frac{2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}+\frac{3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (a^2 x^2+1\right )^{3/2} \sqrt{a^2 c x^2+c}}-\frac{x \sinh ^{-1}(a x)}{c^3 \sqrt{a^2 c x^2+c}}-\frac{x \sinh ^{-1}(a x)}{10 c^3 \left (a^2 x^2+1\right ) \sqrt{a^2 c x^2+c}}-\frac{8 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2 \log \left (e^{2 \sinh ^{-1}(a x)}+1\right )}{5 a c^3 \sqrt{a^2 c x^2+c}}+\frac{x \sinh ^{-1}(a x)^3}{5 c \left (a^2 c x^2+c\right )^{5/2}} \]

[Out]

-1/(20*a*c^3*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2]) - (x*ArcSinh[a*x])/(c^3*Sqrt[c + a^2*c*x^2]) - (x*ArcSinh[
a*x])/(10*c^3*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]) + (3*ArcSinh[a*x]^2)/(20*a*c^3*(1 + a^2*x^2)^(3/2)*Sqrt[c + a
^2*c*x^2]) + (2*ArcSinh[a*x]^2)/(5*a*c^3*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2]) + (x*ArcSinh[a*x]^3)/(5*c*(c +
a^2*c*x^2)^(5/2)) + (4*x*ArcSinh[a*x]^3)/(15*c^2*(c + a^2*c*x^2)^(3/2)) + (8*x*ArcSinh[a*x]^3)/(15*c^3*Sqrt[c
+ a^2*c*x^2]) + (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(15*a*c^3*Sqrt[c + a^2*c*x^2]) - (8*Sqrt[1 + a^2*x^2]*Ar
cSinh[a*x]^2*Log[1 + E^(2*ArcSinh[a*x])])/(5*a*c^3*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*Log[1 + a^2*x^2])
/(2*a*c^3*Sqrt[c + a^2*c*x^2]) - (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]*PolyLog[2, -E^(2*ArcSinh[a*x])])/(5*a*c^3*S
qrt[c + a^2*c*x^2]) + (4*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(2*ArcSinh[a*x])])/(5*a*c^3*Sqrt[c + a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.525902, antiderivative size = 515, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.524, Rules used = {5690, 5687, 5714, 3718, 2190, 2531, 2282, 6589, 5717, 260, 261} \[ -\frac{8 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{a^2 c x^2+c}}+\frac{4 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{a^2 c x^2+c}}-\frac{1}{20 a c^3 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \log \left (a^2 x^2+1\right )}{2 a c^3 \sqrt{a^2 c x^2+c}}+\frac{8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt{a^2 c x^2+c}}+\frac{8 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{15 a c^3 \sqrt{a^2 c x^2+c}}+\frac{4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (a^2 c x^2+c\right )^{3/2}}+\frac{2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}+\frac{3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (a^2 x^2+1\right )^{3/2} \sqrt{a^2 c x^2+c}}-\frac{x \sinh ^{-1}(a x)}{c^3 \sqrt{a^2 c x^2+c}}-\frac{x \sinh ^{-1}(a x)}{10 c^3 \left (a^2 x^2+1\right ) \sqrt{a^2 c x^2+c}}-\frac{8 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2 \log \left (e^{2 \sinh ^{-1}(a x)}+1\right )}{5 a c^3 \sqrt{a^2 c x^2+c}}+\frac{x \sinh ^{-1}(a x)^3}{5 c \left (a^2 c x^2+c\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]^3/(c + a^2*c*x^2)^(7/2),x]

[Out]

-1/(20*a*c^3*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2]) - (x*ArcSinh[a*x])/(c^3*Sqrt[c + a^2*c*x^2]) - (x*ArcSinh[
a*x])/(10*c^3*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]) + (3*ArcSinh[a*x]^2)/(20*a*c^3*(1 + a^2*x^2)^(3/2)*Sqrt[c + a
^2*c*x^2]) + (2*ArcSinh[a*x]^2)/(5*a*c^3*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2]) + (x*ArcSinh[a*x]^3)/(5*c*(c +
a^2*c*x^2)^(5/2)) + (4*x*ArcSinh[a*x]^3)/(15*c^2*(c + a^2*c*x^2)^(3/2)) + (8*x*ArcSinh[a*x]^3)/(15*c^3*Sqrt[c
+ a^2*c*x^2]) + (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(15*a*c^3*Sqrt[c + a^2*c*x^2]) - (8*Sqrt[1 + a^2*x^2]*Ar
cSinh[a*x]^2*Log[1 + E^(2*ArcSinh[a*x])])/(5*a*c^3*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*Log[1 + a^2*x^2])
/(2*a*c^3*Sqrt[c + a^2*c*x^2]) - (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]*PolyLog[2, -E^(2*ArcSinh[a*x])])/(5*a*c^3*S
qrt[c + a^2*c*x^2]) + (4*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(2*ArcSinh[a*x])])/(5*a*c^3*Sqrt[c + a^2*c*x^2])

Rule 5690

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(x*(d + e*x^2)^(p
+ 1)*(a + b*ArcSinh[c*x])^n)/(2*d*(p + 1)), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a +
b*ArcSinh[c*x])^n, x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*(p + 1)*(1 + c^2*x^2)^FracPar
t[p]), Int[x*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ
[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 5687

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(x*(a + b*ArcSinh
[c*x])^n)/(d*Sqrt[d + e*x^2]), x] - Dist[(b*c*n*Sqrt[1 + c^2*x^2])/(d*Sqrt[d + e*x^2]), Int[(x*(a + b*ArcSinh[
c*x])^(n - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]

Rule 5714

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

Rule 3718

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

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

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

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 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\sinh ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{7/2}} \, dx &=\frac{x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac{4 \int \frac{\sinh ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{5 c}-\frac{\left (3 a \sqrt{1+a^2 x^2}\right ) \int \frac{x \sinh ^{-1}(a x)^2}{\left (1+a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt{c+a^2 c x^2}}\\ &=\frac{3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac{4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac{8 \int \frac{\sinh ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{15 c^2}-\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \int \frac{\sinh ^{-1}(a x)}{\left (1+a^2 x^2\right )^{5/2}} \, dx}{10 c^3 \sqrt{c+a^2 c x^2}}-\frac{\left (4 a \sqrt{1+a^2 x^2}\right ) \int \frac{x \sinh ^{-1}(a x)^2}{\left (1+a^2 x^2\right )^2} \, dx}{5 c^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x \sinh ^{-1}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2}}+\frac{3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2}}+\frac{2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac{4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac{8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \int \frac{\sinh ^{-1}(a x)}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{5 c^3 \sqrt{c+a^2 c x^2}}-\frac{\left (4 \sqrt{1+a^2 x^2}\right ) \int \frac{\sinh ^{-1}(a x)}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{5 c^3 \sqrt{c+a^2 c x^2}}+\frac{\left (a \sqrt{1+a^2 x^2}\right ) \int \frac{x}{\left (1+a^2 x^2\right )^2} \, dx}{10 c^3 \sqrt{c+a^2 c x^2}}-\frac{\left (8 a \sqrt{1+a^2 x^2}\right ) \int \frac{x \sinh ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{5 c^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{20 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}-\frac{x \sinh ^{-1}(a x)}{c^3 \sqrt{c+a^2 c x^2}}-\frac{x \sinh ^{-1}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2}}+\frac{3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2}}+\frac{2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac{4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac{8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt{c+a^2 c x^2}}-\frac{\left (8 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \tanh (x) \, dx,x,\sinh ^{-1}(a x)\right )}{5 a c^3 \sqrt{c+a^2 c x^2}}+\frac{\left (a \sqrt{1+a^2 x^2}\right ) \int \frac{x}{1+a^2 x^2} \, dx}{5 c^3 \sqrt{c+a^2 c x^2}}+\frac{\left (4 a \sqrt{1+a^2 x^2}\right ) \int \frac{x}{1+a^2 x^2} \, dx}{5 c^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{20 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}-\frac{x \sinh ^{-1}(a x)}{c^3 \sqrt{c+a^2 c x^2}}-\frac{x \sinh ^{-1}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2}}+\frac{3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2}}+\frac{2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac{4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac{8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt{c+a^2 c x^2}}+\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{15 a c^3 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^3 \sqrt{c+a^2 c x^2}}-\frac{\left (16 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x^2}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(a x)\right )}{5 a c^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{20 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}-\frac{x \sinh ^{-1}(a x)}{c^3 \sqrt{c+a^2 c x^2}}-\frac{x \sinh ^{-1}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2}}+\frac{3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2}}+\frac{2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac{4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac{8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt{c+a^2 c x^2}}+\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{15 a c^3 \sqrt{c+a^2 c x^2}}-\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^3 \sqrt{c+a^2 c x^2}}+\frac{\left (16 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{5 a c^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{20 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}-\frac{x \sinh ^{-1}(a x)}{c^3 \sqrt{c+a^2 c x^2}}-\frac{x \sinh ^{-1}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2}}+\frac{3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2}}+\frac{2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac{4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac{8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt{c+a^2 c x^2}}+\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{15 a c^3 \sqrt{c+a^2 c x^2}}-\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^3 \sqrt{c+a^2 c x^2}}-\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x) \text{Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c+a^2 c x^2}}+\frac{\left (8 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{5 a c^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{20 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}-\frac{x \sinh ^{-1}(a x)}{c^3 \sqrt{c+a^2 c x^2}}-\frac{x \sinh ^{-1}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2}}+\frac{3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2}}+\frac{2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac{4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac{8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt{c+a^2 c x^2}}+\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{15 a c^3 \sqrt{c+a^2 c x^2}}-\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^3 \sqrt{c+a^2 c x^2}}-\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x) \text{Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c+a^2 c x^2}}+\frac{\left (4 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{20 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}-\frac{x \sinh ^{-1}(a x)}{c^3 \sqrt{c+a^2 c x^2}}-\frac{x \sinh ^{-1}(a x)}{10 c^3 \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2}}+\frac{3 \sinh ^{-1}(a x)^2}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2}}+\frac{2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^3}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac{4 x \sinh ^{-1}(a x)^3}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac{8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt{c+a^2 c x^2}}+\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{15 a c^3 \sqrt{c+a^2 c x^2}}-\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2 \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \log \left (1+a^2 x^2\right )}{2 a c^3 \sqrt{c+a^2 c x^2}}-\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x) \text{Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c+a^2 c x^2}}+\frac{4 \sqrt{1+a^2 x^2} \text{Li}_3\left (-e^{2 \sinh ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.659682, size = 297, normalized size = 0.58 \[ \frac{96 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-2 \sinh ^{-1}(a x)}\right )+48 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{-2 \sinh ^{-1}(a x)}\right )-\frac{3}{\sqrt{a^2 x^2+1}}+30 \sqrt{a^2 x^2+1} \log \left (a^2 x^2+1\right )-32 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3+\frac{16 a x \sinh ^{-1}(a x)^3}{a^2 x^2+1}+\frac{12 a x \sinh ^{-1}(a x)^3}{\left (a^2 x^2+1\right )^2}+\frac{24 \sinh ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{9 \sinh ^{-1}(a x)^2}{\left (a^2 x^2+1\right )^{3/2}}-\frac{6 a x \sinh ^{-1}(a x)}{a^2 x^2+1}-96 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2 \log \left (e^{-2 \sinh ^{-1}(a x)}+1\right )+32 a x \sinh ^{-1}(a x)^3-60 a x \sinh ^{-1}(a x)}{60 a c^3 \sqrt{a^2 c x^2+c}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcSinh[a*x]^3/(c + a^2*c*x^2)^(7/2),x]

[Out]

(-3/Sqrt[1 + a^2*x^2] - 60*a*x*ArcSinh[a*x] - (6*a*x*ArcSinh[a*x])/(1 + a^2*x^2) + (9*ArcSinh[a*x]^2)/(1 + a^2
*x^2)^(3/2) + (24*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2] + 32*a*x*ArcSinh[a*x]^3 + (12*a*x*ArcSinh[a*x]^3)/(1 + a^2
*x^2)^2 + (16*a*x*ArcSinh[a*x]^3)/(1 + a^2*x^2) - 32*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3 - 96*Sqrt[1 + a^2*x^2]*A
rcSinh[a*x]^2*Log[1 + E^(-2*ArcSinh[a*x])] + 30*Sqrt[1 + a^2*x^2]*Log[1 + a^2*x^2] + 96*Sqrt[1 + a^2*x^2]*ArcS
inh[a*x]*PolyLog[2, -E^(-2*ArcSinh[a*x])] + 48*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(-2*ArcSinh[a*x])])/(60*a*c^3*S
qrt[c + a^2*c*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.226, size = 888, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a*x)^3/(a^2*c*x^2+c)^(7/2),x)

[Out]

1/60*(c*(a^2*x^2+1))^(1/2)*(8*x^5*a^5-8*a^4*x^4*(a^2*x^2+1)^(1/2)+20*x^3*a^3-16*a^2*x^2*(a^2*x^2+1)^(1/2)+15*a
*x-8*(a^2*x^2+1)^(1/2))*(24-1590*a^4*x^4*arcsinh(a*x)-1410*a^2*x^2*arcsinh(a*x)+105*a^3*x^3*(a^2*x^2+1)^(1/2)+
45*a*x*(a^2*x^2+1)^(1/2)-495*arcsinh(a*x)^2*a*x*(a^2*x^2+1)^(1/2)-372*arcsinh(a*x)*(a^2*x^2+1)^(1/2)*a*x+96*a^
2*x^2+256*arcsinh(a*x)^3-480*arcsinh(a*x)-264*arcsinh(a*x)^2-1368*arcsinh(a*x)^2*x^4*a^4-984*arcsinh(a*x)^2*a^
2*x^2-936*arcsinh(a*x)*(a^2*x^2+1)^(1/2)*a^3*x^3-192*arcsinh(a*x)*x^8*a^8-852*arcsinh(a*x)*x^6*a^6-1020*a^3*x^
3*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)-192*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)*x^7*a^7-744*arcsinh(a*x)^2*(a^2*x^2+1)
^(1/2)*x^5*a^5+24*x^8*a^8+96*x^6*a^6+144*x^4*a^4+24*(a^2*x^2+1)^(1/2)*x^7*a^7+84*(a^2*x^2+1)^(1/2)*x^5*a^5-840
*arcsinh(a*x)^2*x^6*a^6+160*arcsinh(a*x)^3*x^4*a^4+380*arcsinh(a*x)^3*x^2*a^2-192*arcsinh(a*x)^2*x^8*a^8-192*(
a^2*x^2+1)^(1/2)*arcsinh(a*x)*x^7*a^7-756*(a^2*x^2+1)^(1/2)*arcsinh(a*x)*x^5*a^5)/(40*a^10*x^10+215*a^8*x^8+46
9*a^6*x^6+517*a^4*x^4+287*a^2*x^2+64)/a/c^4-2/(a^2*x^2+1)^(1/2)*(c*(a^2*x^2+1))^(1/2)/a/c^4*ln(a*x+(a^2*x^2+1)
^(1/2))+1/(a^2*x^2+1)^(1/2)*(c*(a^2*x^2+1))^(1/2)/a/c^4*ln(1+(a*x+(a^2*x^2+1)^(1/2))^2)+16/15/(a^2*x^2+1)^(1/2
)*(c*(a^2*x^2+1))^(1/2)/a/c^4*arcsinh(a*x)^3-8/5/(a^2*x^2+1)^(1/2)*(c*(a^2*x^2+1))^(1/2)/a/c^4*arcsinh(a*x)^2*
ln(1+(a*x+(a^2*x^2+1)^(1/2))^2)-8/5/(a^2*x^2+1)^(1/2)*(c*(a^2*x^2+1))^(1/2)/a/c^4*arcsinh(a*x)*polylog(2,-(a*x
+(a^2*x^2+1)^(1/2))^2)+4/5/(a^2*x^2+1)^(1/2)*(c*(a^2*x^2+1))^(1/2)/a/c^4*polylog(3,-(a*x+(a^2*x^2+1)^(1/2))^2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^3/(a^2*c*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

integrate(arcsinh(a*x)^3/(a^2*c*x^2 + c)^(7/2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c} \operatorname{arsinh}\left (a x\right )^{3}}{a^{8} c^{4} x^{8} + 4 \, a^{6} c^{4} x^{6} + 6 \, a^{4} c^{4} x^{4} + 4 \, a^{2} c^{4} x^{2} + c^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^3/(a^2*c*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*arcsinh(a*x)^3/(a^8*c^4*x^8 + 4*a^6*c^4*x^6 + 6*a^4*c^4*x^4 + 4*a^2*c^4*x^2 + c^4
), x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a*x)**3/(a**2*c*x**2+c)**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arsinh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a*x)^3/(a^2*c*x^2+c)^(7/2),x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)^3/(a^2*c*x^2 + c)^(7/2), x)