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

Optimal. Leaf size=366 \[ -\frac{8 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(a x)}\right )}{15 a c^3 \sqrt{a^2 c x^2+c}}-\frac{x}{3 c^3 \sqrt{a^2 c x^2+c}}-\frac{x}{30 c^3 \left (a^2 x^2+1\right ) \sqrt{a^2 c x^2+c}}+\frac{8 x \sinh ^{-1}(a x)^2}{15 c^3 \sqrt{a^2 c x^2+c}}+\frac{8 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{15 a c^3 \sqrt{a^2 c x^2+c}}+\frac{4 x \sinh ^{-1}(a x)^2}{15 c^2 \left (a^2 c x^2+c\right )^{3/2}}+\frac{4 \sinh ^{-1}(a x)}{15 a c^3 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}+\frac{\sinh ^{-1}(a x)}{10 a c^3 \left (a^2 x^2+1\right )^{3/2} \sqrt{a^2 c x^2+c}}-\frac{16 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \log \left (e^{2 \sinh ^{-1}(a x)}+1\right )}{15 a c^3 \sqrt{a^2 c x^2+c}}+\frac{x \sinh ^{-1}(a x)^2}{5 c \left (a^2 c x^2+c\right )^{5/2}} \]

[Out]

-x/(3*c^3*Sqrt[c + a^2*c*x^2]) - x/(30*c^3*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]) + ArcSinh[a*x]/(10*a*c^3*(1 + a^
2*x^2)^(3/2)*Sqrt[c + a^2*c*x^2]) + (4*ArcSinh[a*x])/(15*a*c^3*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2]) + (x*Arc
Sinh[a*x]^2)/(5*c*(c + a^2*c*x^2)^(5/2)) + (4*x*ArcSinh[a*x]^2)/(15*c^2*(c + a^2*c*x^2)^(3/2)) + (8*x*ArcSinh[
a*x]^2)/(15*c^3*Sqrt[c + a^2*c*x^2]) + (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(15*a*c^3*Sqrt[c + a^2*c*x^2]) - (
16*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]*Log[1 + E^(2*ArcSinh[a*x])])/(15*a*c^3*Sqrt[c + a^2*c*x^2]) - (8*Sqrt[1 + a^
2*x^2]*PolyLog[2, -E^(2*ArcSinh[a*x])])/(15*a*c^3*Sqrt[c + a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.352804, antiderivative size = 366, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {5690, 5687, 5714, 3718, 2190, 2279, 2391, 5717, 191, 192} \[ -\frac{8 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(a x)}\right )}{15 a c^3 \sqrt{a^2 c x^2+c}}-\frac{x}{3 c^3 \sqrt{a^2 c x^2+c}}-\frac{x}{30 c^3 \left (a^2 x^2+1\right ) \sqrt{a^2 c x^2+c}}+\frac{8 x \sinh ^{-1}(a x)^2}{15 c^3 \sqrt{a^2 c x^2+c}}+\frac{8 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{15 a c^3 \sqrt{a^2 c x^2+c}}+\frac{4 x \sinh ^{-1}(a x)^2}{15 c^2 \left (a^2 c x^2+c\right )^{3/2}}+\frac{4 \sinh ^{-1}(a x)}{15 a c^3 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}+\frac{\sinh ^{-1}(a x)}{10 a c^3 \left (a^2 x^2+1\right )^{3/2} \sqrt{a^2 c x^2+c}}-\frac{16 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \log \left (e^{2 \sinh ^{-1}(a x)}+1\right )}{15 a c^3 \sqrt{a^2 c x^2+c}}+\frac{x \sinh ^{-1}(a x)^2}{5 c \left (a^2 c x^2+c\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x/(3*c^3*Sqrt[c + a^2*c*x^2]) - x/(30*c^3*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]) + ArcSinh[a*x]/(10*a*c^3*(1 + a^
2*x^2)^(3/2)*Sqrt[c + a^2*c*x^2]) + (4*ArcSinh[a*x])/(15*a*c^3*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2]) + (x*Arc
Sinh[a*x]^2)/(5*c*(c + a^2*c*x^2)^(5/2)) + (4*x*ArcSinh[a*x]^2)/(15*c^2*(c + a^2*c*x^2)^(3/2)) + (8*x*ArcSinh[
a*x]^2)/(15*c^3*Sqrt[c + a^2*c*x^2]) + (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(15*a*c^3*Sqrt[c + a^2*c*x^2]) - (
16*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]*Log[1 + E^(2*ArcSinh[a*x])])/(15*a*c^3*Sqrt[c + a^2*c*x^2]) - (8*Sqrt[1 + a^
2*x^2]*PolyLog[2, -E^(2*ArcSinh[a*x])])/(15*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 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]

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 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 192

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

Rubi steps

\begin{align*} \int \frac{\sinh ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{7/2}} \, dx &=\frac{x \sinh ^{-1}(a x)^2}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac{4 \int \frac{\sinh ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{5 c}-\frac{\left (2 a \sqrt{1+a^2 x^2}\right ) \int \frac{x \sinh ^{-1}(a x)}{\left (1+a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt{c+a^2 c x^2}}\\ &=\frac{\sinh ^{-1}(a x)}{10 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)^2}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac{4 x \sinh ^{-1}(a x)^2}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac{8 \int \frac{\sinh ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{15 c^2}-\frac{\sqrt{1+a^2 x^2} \int \frac{1}{\left (1+a^2 x^2\right )^{5/2}} \, 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)}{\left (1+a^2 x^2\right )^2} \, dx}{15 c^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x}{30 c^3 \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2}}+\frac{\sinh ^{-1}(a x)}{10 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2}}+\frac{4 \sinh ^{-1}(a x)}{15 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^2}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac{4 x \sinh ^{-1}(a x)^2}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac{8 x \sinh ^{-1}(a x)^2}{15 c^3 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \int \frac{1}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{15 c^3 \sqrt{c+a^2 c x^2}}-\frac{\left (4 \sqrt{1+a^2 x^2}\right ) \int \frac{1}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{15 c^3 \sqrt{c+a^2 c x^2}}-\frac{\left (16 a \sqrt{1+a^2 x^2}\right ) \int \frac{x \sinh ^{-1}(a x)}{1+a^2 x^2} \, dx}{15 c^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x}{3 c^3 \sqrt{c+a^2 c x^2}}-\frac{x}{30 c^3 \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2}}+\frac{\sinh ^{-1}(a x)}{10 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2}}+\frac{4 \sinh ^{-1}(a x)}{15 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^2}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac{4 x \sinh ^{-1}(a x)^2}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac{8 x \sinh ^{-1}(a x)^2}{15 c^3 \sqrt{c+a^2 c x^2}}-\frac{\left (16 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \tanh (x) \, dx,x,\sinh ^{-1}(a x)\right )}{15 a c^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x}{3 c^3 \sqrt{c+a^2 c x^2}}-\frac{x}{30 c^3 \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2}}+\frac{\sinh ^{-1}(a x)}{10 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2}}+\frac{4 \sinh ^{-1}(a x)}{15 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^2}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac{4 x \sinh ^{-1}(a x)^2}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac{8 x \sinh ^{-1}(a x)^2}{15 c^3 \sqrt{c+a^2 c x^2}}+\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{15 a c^3 \sqrt{c+a^2 c x^2}}-\frac{\left (32 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(a x)\right )}{15 a c^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x}{3 c^3 \sqrt{c+a^2 c x^2}}-\frac{x}{30 c^3 \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2}}+\frac{\sinh ^{-1}(a x)}{10 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2}}+\frac{4 \sinh ^{-1}(a x)}{15 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^2}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac{4 x \sinh ^{-1}(a x)^2}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac{8 x \sinh ^{-1}(a x)^2}{15 c^3 \sqrt{c+a^2 c x^2}}+\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{15 a c^3 \sqrt{c+a^2 c x^2}}-\frac{16 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x) \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c+a^2 c x^2}}+\frac{\left (16 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{15 a c^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x}{3 c^3 \sqrt{c+a^2 c x^2}}-\frac{x}{30 c^3 \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2}}+\frac{\sinh ^{-1}(a x)}{10 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2}}+\frac{4 \sinh ^{-1}(a x)}{15 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^2}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac{4 x \sinh ^{-1}(a x)^2}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac{8 x \sinh ^{-1}(a x)^2}{15 c^3 \sqrt{c+a^2 c x^2}}+\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{15 a c^3 \sqrt{c+a^2 c x^2}}-\frac{16 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x) \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c+a^2 c x^2}}+\frac{\left (8 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x}{3 c^3 \sqrt{c+a^2 c x^2}}-\frac{x}{30 c^3 \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2}}+\frac{\sinh ^{-1}(a x)}{10 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2}}+\frac{4 \sinh ^{-1}(a x)}{15 a c^3 \sqrt{1+a^2 x^2} \sqrt{c+a^2 c x^2}}+\frac{x \sinh ^{-1}(a x)^2}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac{4 x \sinh ^{-1}(a x)^2}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac{8 x \sinh ^{-1}(a x)^2}{15 c^3 \sqrt{c+a^2 c x^2}}+\frac{8 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{15 a c^3 \sqrt{c+a^2 c x^2}}-\frac{16 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x) \log \left (1+e^{2 \sinh ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c+a^2 c x^2}}-\frac{8 \sqrt{1+a^2 x^2} \text{Li}_2\left (-e^{2 \sinh ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.761946, size = 178, normalized size = 0.49 \[ \frac{16 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-e^{-2 \sinh ^{-1}(a x)}\right )+a x \left (-\frac{1}{a^2 x^2+1}-10\right )+\left (\frac{2 a x \left (8 a^4 x^4+20 a^2 x^2+15\right )}{\left (a^2 x^2+1\right )^2}-16 \sqrt{a^2 x^2+1}\right ) \sinh ^{-1}(a x)^2+\frac{\sinh ^{-1}(a x) \left (8 a^2 x^2-32 \left (a^2 x^2+1\right )^2 \log \left (e^{-2 \sinh ^{-1}(a x)}+1\right )+11\right )}{\left (a^2 x^2+1\right )^{3/2}}}{30 a c^3 \sqrt{a^2 c x^2+c}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*x*(-10 - (1 + a^2*x^2)^(-1)) + (-16*Sqrt[1 + a^2*x^2] + (2*a*x*(15 + 20*a^2*x^2 + 8*a^4*x^4))/(1 + a^2*x^2)
^2)*ArcSinh[a*x]^2 + (ArcSinh[a*x]*(11 + 8*a^2*x^2 - 32*(1 + a^2*x^2)^2*Log[1 + E^(-2*ArcSinh[a*x])]))/(1 + a^
2*x^2)^(3/2) + 16*Sqrt[1 + a^2*x^2]*PolyLog[2, -E^(-2*ArcSinh[a*x])])/(30*a*c^3*Sqrt[c + a^2*c*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.178, size = 570, normalized size = 1.6 \begin{align*}{\frac{1}{ \left ( 1200\,{a}^{10}{x}^{10}+6450\,{x}^{8}{a}^{8}+14070\,{x}^{6}{a}^{6}+15510\,{x}^{4}{a}^{4}+8610\,{a}^{2}{x}^{2}+1920 \right ) a{c}^{4}}\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) } \left ( 8\,{x}^{5}{a}^{5}-8\,{a}^{4}{x}^{4}\sqrt{{a}^{2}{x}^{2}+1}+20\,{x}^{3}{a}^{3}-16\,{a}^{2}{x}^{2}\sqrt{{a}^{2}{x}^{2}+1}+15\,ax-8\,\sqrt{{a}^{2}{x}^{2}+1} \right ) \left ( -64\,{\it Arcsinh} \left ( ax \right ){x}^{8}{a}^{8}-64\,\sqrt{{a}^{2}{x}^{2}+1}{\it Arcsinh} \left ( ax \right ){x}^{7}{a}^{7}-32\,{x}^{8}{a}^{8}-32\,\sqrt{{a}^{2}{x}^{2}+1}{x}^{7}{a}^{7}-280\,{\it Arcsinh} \left ( ax \right ){x}^{6}{a}^{6}-248\,\sqrt{{a}^{2}{x}^{2}+1}{\it Arcsinh} \left ( ax \right ){x}^{5}{a}^{5}-142\,{x}^{6}{a}^{6}-126\,\sqrt{{a}^{2}{x}^{2}+1}{x}^{5}{a}^{5}+80\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}{x}^{4}{a}^{4}-456\,{a}^{4}{x}^{4}{\it Arcsinh} \left ( ax \right ) -340\,{\it Arcsinh} \left ( ax \right ) \sqrt{{a}^{2}{x}^{2}+1}{a}^{3}{x}^{3}-265\,{x}^{4}{a}^{4}-156\,{a}^{3}{x}^{3}\sqrt{{a}^{2}{x}^{2}+1}+190\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}{a}^{2}{x}^{2}-328\,{a}^{2}{x}^{2}{\it Arcsinh} \left ( ax \right ) -165\,{\it Arcsinh} \left ( ax \right ) \sqrt{{a}^{2}{x}^{2}+1}ax-235\,{a}^{2}{x}^{2}-62\,ax\sqrt{{a}^{2}{x}^{2}+1}+128\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}-88\,{\it Arcsinh} \left ( ax \right ) -80 \right ) }+{\frac{16\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{15\,{c}^{4}a}\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-{\frac{16\,{\it Arcsinh} \left ( ax \right ) }{15\,{c}^{4}a}\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) }\ln \left ( 1+ \left ( ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) ^{2} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-{\frac{8}{15\,{c}^{4}a}\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) }{\it polylog} \left ( 2,- \left ( ax+\sqrt{{a}^{2}{x}^{2}+1} \right ) ^{2} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(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))*(-64*arcsinh(a*x)*x^8*a^8-64*(a^2*x^2+1)^(1/2)*arcsinh(a*x)*x^7*a^7-32*x^8*a^8-32*(a^2
*x^2+1)^(1/2)*x^7*a^7-280*arcsinh(a*x)*x^6*a^6-248*(a^2*x^2+1)^(1/2)*arcsinh(a*x)*x^5*a^5-142*x^6*a^6-126*(a^2
*x^2+1)^(1/2)*x^5*a^5+80*arcsinh(a*x)^2*x^4*a^4-456*a^4*x^4*arcsinh(a*x)-340*arcsinh(a*x)*(a^2*x^2+1)^(1/2)*a^
3*x^3-265*x^4*a^4-156*a^3*x^3*(a^2*x^2+1)^(1/2)+190*arcsinh(a*x)^2*a^2*x^2-328*a^2*x^2*arcsinh(a*x)-165*arcsin
h(a*x)*(a^2*x^2+1)^(1/2)*a*x-235*a^2*x^2-62*a*x*(a^2*x^2+1)^(1/2)+128*arcsinh(a*x)^2-88*arcsinh(a*x)-80)/(40*a
^10*x^10+215*a^8*x^8+469*a^6*x^6+517*a^4*x^4+287*a^2*x^2+64)/a/c^4+16/15/(a^2*x^2+1)^(1/2)*(c*(a^2*x^2+1))^(1/
2)/a/c^4*arcsinh(a*x)^2-16/15/(a^2*x^2+1)^(1/2)*(c*(a^2*x^2+1))^(1/2)/a/c^4*arcsinh(a*x)*ln(1+(a*x+(a^2*x^2+1)
^(1/2))^2)-8/15/(a^2*x^2+1)^(1/2)*(c*(a^2*x^2+1))^(1/2)/a/c^4*polylog(2,-(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 )^{2}}{{\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)^2/(a^2*c*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

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

[Out]

integral(sqrt(a^2*c*x^2 + c)*arcsinh(a*x)^2/(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)**2/(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 )^{2}}{{\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)^2/(a^2*c*x^2+c)^(7/2),x, algorithm="giac")

[Out]

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