3.328 \(\int (c+a^2 c x^2)^3 \sinh ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=359 \[ -\frac{6 c^3 \left (a^2 x^2+1\right )^{7/2}}{2401 a}-\frac{2664 c^3 \left (a^2 x^2+1\right )^{5/2}}{214375 a}-\frac{30256 c^3 \left (a^2 x^2+1\right )^{3/2}}{385875 a}-\frac{413312 c^3 \sqrt{a^2 x^2+1}}{128625 a}+\frac{6}{343} a^6 c^3 x^7 \sinh ^{-1}(a x)+\frac{702 a^4 c^3 x^5 \sinh ^{-1}(a x)}{6125}+\frac{1514 a^2 c^3 x^3 \sinh ^{-1}(a x)}{3675}+\frac{1}{7} c^3 x \left (a^2 x^2+1\right )^3 \sinh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (a^2 x^2+1\right )^2 \sinh ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (a^2 x^2+1\right ) \sinh ^{-1}(a x)^3-\frac{3 c^3 \left (a^2 x^2+1\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}-\frac{18 c^3 \left (a^2 x^2+1\right )^{5/2} \sinh ^{-1}(a x)^2}{175 a}-\frac{8 c^3 \left (a^2 x^2+1\right )^{3/2} \sinh ^{-1}(a x)^2}{35 a}-\frac{48 c^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{35 a}+\frac{16}{35} c^3 x \sinh ^{-1}(a x)^3+\frac{4322 c^3 x \sinh ^{-1}(a x)}{1225} \]

[Out]

(-413312*c^3*Sqrt[1 + a^2*x^2])/(128625*a) - (30256*c^3*(1 + a^2*x^2)^(3/2))/(385875*a) - (2664*c^3*(1 + a^2*x
^2)^(5/2))/(214375*a) - (6*c^3*(1 + a^2*x^2)^(7/2))/(2401*a) + (4322*c^3*x*ArcSinh[a*x])/1225 + (1514*a^2*c^3*
x^3*ArcSinh[a*x])/3675 + (702*a^4*c^3*x^5*ArcSinh[a*x])/6125 + (6*a^6*c^3*x^7*ArcSinh[a*x])/343 - (48*c^3*Sqrt
[1 + a^2*x^2]*ArcSinh[a*x]^2)/(35*a) - (8*c^3*(1 + a^2*x^2)^(3/2)*ArcSinh[a*x]^2)/(35*a) - (18*c^3*(1 + a^2*x^
2)^(5/2)*ArcSinh[a*x]^2)/(175*a) - (3*c^3*(1 + a^2*x^2)^(7/2)*ArcSinh[a*x]^2)/(49*a) + (16*c^3*x*ArcSinh[a*x]^
3)/35 + (8*c^3*x*(1 + a^2*x^2)*ArcSinh[a*x]^3)/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcSinh[a*x]^3)/35 + (c^3*x*(1 +
a^2*x^2)^3*ArcSinh[a*x]^3)/7

________________________________________________________________________________________

Rubi [A]  time = 0.727717, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 13, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.684, Rules used = {5684, 5653, 5717, 261, 5679, 444, 43, 194, 12, 1247, 698, 1799, 1850} \[ -\frac{6 c^3 \left (a^2 x^2+1\right )^{7/2}}{2401 a}-\frac{2664 c^3 \left (a^2 x^2+1\right )^{5/2}}{214375 a}-\frac{30256 c^3 \left (a^2 x^2+1\right )^{3/2}}{385875 a}-\frac{413312 c^3 \sqrt{a^2 x^2+1}}{128625 a}+\frac{6}{343} a^6 c^3 x^7 \sinh ^{-1}(a x)+\frac{702 a^4 c^3 x^5 \sinh ^{-1}(a x)}{6125}+\frac{1514 a^2 c^3 x^3 \sinh ^{-1}(a x)}{3675}+\frac{1}{7} c^3 x \left (a^2 x^2+1\right )^3 \sinh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (a^2 x^2+1\right )^2 \sinh ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (a^2 x^2+1\right ) \sinh ^{-1}(a x)^3-\frac{3 c^3 \left (a^2 x^2+1\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}-\frac{18 c^3 \left (a^2 x^2+1\right )^{5/2} \sinh ^{-1}(a x)^2}{175 a}-\frac{8 c^3 \left (a^2 x^2+1\right )^{3/2} \sinh ^{-1}(a x)^2}{35 a}-\frac{48 c^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{35 a}+\frac{16}{35} c^3 x \sinh ^{-1}(a x)^3+\frac{4322 c^3 x \sinh ^{-1}(a x)}{1225} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-413312*c^3*Sqrt[1 + a^2*x^2])/(128625*a) - (30256*c^3*(1 + a^2*x^2)^(3/2))/(385875*a) - (2664*c^3*(1 + a^2*x
^2)^(5/2))/(214375*a) - (6*c^3*(1 + a^2*x^2)^(7/2))/(2401*a) + (4322*c^3*x*ArcSinh[a*x])/1225 + (1514*a^2*c^3*
x^3*ArcSinh[a*x])/3675 + (702*a^4*c^3*x^5*ArcSinh[a*x])/6125 + (6*a^6*c^3*x^7*ArcSinh[a*x])/343 - (48*c^3*Sqrt
[1 + a^2*x^2]*ArcSinh[a*x]^2)/(35*a) - (8*c^3*(1 + a^2*x^2)^(3/2)*ArcSinh[a*x]^2)/(35*a) - (18*c^3*(1 + a^2*x^
2)^(5/2)*ArcSinh[a*x]^2)/(175*a) - (3*c^3*(1 + a^2*x^2)^(7/2)*ArcSinh[a*x]^2)/(49*a) + (16*c^3*x*ArcSinh[a*x]^
3)/35 + (8*c^3*x*(1 + a^2*x^2)*ArcSinh[a*x]^3)/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcSinh[a*x]^3)/35 + (c^3*x*(1 +
a^2*x^2)^3*ArcSinh[a*x]^3)/7

Rule 5684

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(x*(d + e*x^2)^p*
(a + b*ArcSinh[c*x])^n)/(2*p + 1), x] + (Dist[(2*d*p)/(2*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)^FracPart[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] && Gt
Q[n, 0] && GtQ[p, 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 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]

Rule 5679

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /;
FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 194

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1850

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

Rubi steps

\begin{align*} \int \left (c+a^2 c x^2\right )^3 \sinh ^{-1}(a x)^3 \, dx &=\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^3+\frac{1}{7} (6 c) \int \left (c+a^2 c x^2\right )^2 \sinh ^{-1}(a x)^3 \, dx-\frac{1}{7} \left (3 a c^3\right ) \int x \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2 \, dx\\ &=-\frac{3 c^3 \left (1+a^2 x^2\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^3+\frac{1}{35} \left (24 c^2\right ) \int \left (c+a^2 c x^2\right ) \sinh ^{-1}(a x)^3 \, dx+\frac{1}{49} \left (6 c^3\right ) \int \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x) \, dx-\frac{1}{35} \left (18 a c^3\right ) \int x \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2 \, dx\\ &=\frac{6}{49} c^3 x \sinh ^{-1}(a x)+\frac{6}{49} a^2 c^3 x^3 \sinh ^{-1}(a x)+\frac{18}{245} a^4 c^3 x^5 \sinh ^{-1}(a x)+\frac{6}{343} a^6 c^3 x^7 \sinh ^{-1}(a x)-\frac{18 c^3 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{175 a}-\frac{3 c^3 \left (1+a^2 x^2\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^3+\frac{1}{175} \left (36 c^3\right ) \int \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x) \, dx+\frac{1}{35} \left (16 c^3\right ) \int \sinh ^{-1}(a x)^3 \, dx-\frac{1}{49} \left (6 a c^3\right ) \int \frac{x \left (35+35 a^2 x^2+21 a^4 x^4+5 a^6 x^6\right )}{35 \sqrt{1+a^2 x^2}} \, dx-\frac{1}{35} \left (24 a c^3\right ) \int x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2 \, dx\\ &=\frac{402 c^3 x \sinh ^{-1}(a x)}{1225}+\frac{318 a^2 c^3 x^3 \sinh ^{-1}(a x)}{1225}+\frac{702 a^4 c^3 x^5 \sinh ^{-1}(a x)}{6125}+\frac{6}{343} a^6 c^3 x^7 \sinh ^{-1}(a x)-\frac{8 c^3 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{35 a}-\frac{18 c^3 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{175 a}-\frac{3 c^3 \left (1+a^2 x^2\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}+\frac{16}{35} c^3 x \sinh ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^3+\frac{1}{35} \left (16 c^3\right ) \int \left (1+a^2 x^2\right ) \sinh ^{-1}(a x) \, dx-\frac{\left (6 a c^3\right ) \int \frac{x \left (35+35 a^2 x^2+21 a^4 x^4+5 a^6 x^6\right )}{\sqrt{1+a^2 x^2}} \, dx}{1715}-\frac{1}{175} \left (36 a c^3\right ) \int \frac{x \left (15+10 a^2 x^2+3 a^4 x^4\right )}{15 \sqrt{1+a^2 x^2}} \, dx-\frac{1}{35} \left (48 a c^3\right ) \int \frac{x \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx\\ &=\frac{962 c^3 x \sinh ^{-1}(a x)}{1225}+\frac{1514 a^2 c^3 x^3 \sinh ^{-1}(a x)}{3675}+\frac{702 a^4 c^3 x^5 \sinh ^{-1}(a x)}{6125}+\frac{6}{343} a^6 c^3 x^7 \sinh ^{-1}(a x)-\frac{48 c^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{35 a}-\frac{8 c^3 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{35 a}-\frac{18 c^3 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{175 a}-\frac{3 c^3 \left (1+a^2 x^2\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}+\frac{16}{35} c^3 x \sinh ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^3+\frac{1}{35} \left (96 c^3\right ) \int \sinh ^{-1}(a x) \, dx-\frac{\left (3 a c^3\right ) \operatorname{Subst}\left (\int \frac{35+35 a^2 x+21 a^4 x^2+5 a^6 x^3}{\sqrt{1+a^2 x}} \, dx,x,x^2\right )}{1715}-\frac{1}{875} \left (12 a c^3\right ) \int \frac{x \left (15+10 a^2 x^2+3 a^4 x^4\right )}{\sqrt{1+a^2 x^2}} \, dx-\frac{1}{35} \left (16 a c^3\right ) \int \frac{x \left (1+\frac{a^2 x^2}{3}\right )}{\sqrt{1+a^2 x^2}} \, dx\\ &=\frac{4322 c^3 x \sinh ^{-1}(a x)}{1225}+\frac{1514 a^2 c^3 x^3 \sinh ^{-1}(a x)}{3675}+\frac{702 a^4 c^3 x^5 \sinh ^{-1}(a x)}{6125}+\frac{6}{343} a^6 c^3 x^7 \sinh ^{-1}(a x)-\frac{48 c^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{35 a}-\frac{8 c^3 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{35 a}-\frac{18 c^3 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{175 a}-\frac{3 c^3 \left (1+a^2 x^2\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}+\frac{16}{35} c^3 x \sinh ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^3-\frac{\left (3 a c^3\right ) \operatorname{Subst}\left (\int \left (\frac{16}{\sqrt{1+a^2 x}}+8 \sqrt{1+a^2 x}+6 \left (1+a^2 x\right )^{3/2}+5 \left (1+a^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )}{1715}-\frac{1}{875} \left (6 a c^3\right ) \operatorname{Subst}\left (\int \frac{15+10 a^2 x+3 a^4 x^2}{\sqrt{1+a^2 x}} \, dx,x,x^2\right )-\frac{1}{35} \left (8 a c^3\right ) \operatorname{Subst}\left (\int \frac{1+\frac{a^2 x}{3}}{\sqrt{1+a^2 x}} \, dx,x,x^2\right )-\frac{1}{35} \left (96 a c^3\right ) \int \frac{x}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{960 c^3 \sqrt{1+a^2 x^2}}{343 a}-\frac{16 c^3 \left (1+a^2 x^2\right )^{3/2}}{1715 a}-\frac{36 c^3 \left (1+a^2 x^2\right )^{5/2}}{8575 a}-\frac{6 c^3 \left (1+a^2 x^2\right )^{7/2}}{2401 a}+\frac{4322 c^3 x \sinh ^{-1}(a x)}{1225}+\frac{1514 a^2 c^3 x^3 \sinh ^{-1}(a x)}{3675}+\frac{702 a^4 c^3 x^5 \sinh ^{-1}(a x)}{6125}+\frac{6}{343} a^6 c^3 x^7 \sinh ^{-1}(a x)-\frac{48 c^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{35 a}-\frac{8 c^3 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{35 a}-\frac{18 c^3 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{175 a}-\frac{3 c^3 \left (1+a^2 x^2\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}+\frac{16}{35} c^3 x \sinh ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^3-\frac{1}{875} \left (6 a c^3\right ) \operatorname{Subst}\left (\int \left (\frac{8}{\sqrt{1+a^2 x}}+4 \sqrt{1+a^2 x}+3 \left (1+a^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )-\frac{1}{35} \left (8 a c^3\right ) \operatorname{Subst}\left (\int \left (\frac{2}{3 \sqrt{1+a^2 x}}+\frac{1}{3} \sqrt{1+a^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac{413312 c^3 \sqrt{1+a^2 x^2}}{128625 a}-\frac{30256 c^3 \left (1+a^2 x^2\right )^{3/2}}{385875 a}-\frac{2664 c^3 \left (1+a^2 x^2\right )^{5/2}}{214375 a}-\frac{6 c^3 \left (1+a^2 x^2\right )^{7/2}}{2401 a}+\frac{4322 c^3 x \sinh ^{-1}(a x)}{1225}+\frac{1514 a^2 c^3 x^3 \sinh ^{-1}(a x)}{3675}+\frac{702 a^4 c^3 x^5 \sinh ^{-1}(a x)}{6125}+\frac{6}{343} a^6 c^3 x^7 \sinh ^{-1}(a x)-\frac{48 c^3 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^2}{35 a}-\frac{8 c^3 \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x)^2}{35 a}-\frac{18 c^3 \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x)^2}{175 a}-\frac{3 c^3 \left (1+a^2 x^2\right )^{7/2} \sinh ^{-1}(a x)^2}{49 a}+\frac{16}{35} c^3 x \sinh ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1+a^2 x^2\right )^3 \sinh ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.258246, size = 169, normalized size = 0.47 \[ \frac{c^3 \left (-2 \sqrt{a^2 x^2+1} \left (16875 a^6 x^6+134541 a^4 x^4+747937 a^2 x^2+22329151\right )+385875 a x \left (5 a^6 x^6+21 a^4 x^4+35 a^2 x^2+35\right ) \sinh ^{-1}(a x)^3-11025 \sqrt{a^2 x^2+1} \left (75 a^6 x^6+351 a^4 x^4+757 a^2 x^2+2161\right ) \sinh ^{-1}(a x)^2+210 a x \left (1125 a^6 x^6+7371 a^4 x^4+26495 a^2 x^2+226905\right ) \sinh ^{-1}(a x)\right )}{13505625 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*(-2*Sqrt[1 + a^2*x^2]*(22329151 + 747937*a^2*x^2 + 134541*a^4*x^4 + 16875*a^6*x^6) + 210*a*x*(226905 + 26
495*a^2*x^2 + 7371*a^4*x^4 + 1125*a^6*x^6)*ArcSinh[a*x] - 11025*Sqrt[1 + a^2*x^2]*(2161 + 757*a^2*x^2 + 351*a^
4*x^4 + 75*a^6*x^6)*ArcSinh[a*x]^2 + 385875*a*x*(35 + 35*a^2*x^2 + 21*a^4*x^4 + 5*a^6*x^6)*ArcSinh[a*x]^3))/(1
3505625*a)

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Maple [A]  time = 0.055, size = 270, normalized size = 0.8 \begin{align*}{\frac{{c}^{3}}{13505625\,a} \left ( 1929375\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}{a}^{7}{x}^{7}-826875\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\sqrt{{a}^{2}{x}^{2}+1}{a}^{6}{x}^{6}+8103375\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}{a}^{5}{x}^{5}+236250\,{\it Arcsinh} \left ( ax \right ){a}^{7}{x}^{7}-3869775\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\sqrt{{a}^{2}{x}^{2}+1}{a}^{4}{x}^{4}-33750\,{a}^{6}{x}^{6}\sqrt{{a}^{2}{x}^{2}+1}[phone]\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}{a}^{3}{x}^{3}+1547910\,{\it Arcsinh} \left ( ax \right ){a}^{5}{x}^{5}-8345925\,{a}^{2}{x}^{2} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\sqrt{{a}^{2}{x}^{2}+1}-269082\,{a}^{4}{x}^{4}\sqrt{{a}^{2}{x}^{2}+1}[phone]\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}ax+5563950\,{\it Arcsinh} \left ( ax \right ){a}^{3}{x}^{3}-23825025\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\sqrt{{a}^{2}{x}^{2}+1}-1495874\,{a}^{2}{x}^{2}\sqrt{{a}^{2}{x}^{2}+1}[phone]\,ax{\it Arcsinh} \left ( ax \right ) -44658302\,\sqrt{{a}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13505625/a*c^3*(1929375*arcsinh(a*x)^3*a^7*x^7-826875*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)*a^6*x^6+8103375*arcsi
nh(a*x)^3*a^5*x^5+236250*arcsinh(a*x)*a^7*x^7-3869775*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)*a^4*x^4-33750*a^6*x^6*(
a^2*x^2+1)^(1/2)[phone]*arcsinh(a*x)^3*a^3*x^3+1547910*arcsinh(a*x)*a^5*x^5-8345925*a^2*x^2*arcsinh(a*x)^2*(
a^2*x^2+1)^(1/2)-269082*a^4*x^4*(a^2*x^2+1)^(1/2)[phone]*arcsinh(a*x)^3*a*x+5563950*arcsinh(a*x)*a^3*x^3-238
25025*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)-1495874*a^2*x^2*(a^2*x^2+1)^(1/2)[phone]*a*x*arcsinh(a*x)-44658302*(a
^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.22839, size = 373, normalized size = 1.04 \begin{align*} -\frac{1}{1225} \,{\left (75 \, \sqrt{a^{2} x^{2} + 1} a^{4} c^{3} x^{6} + 351 \, \sqrt{a^{2} x^{2} + 1} a^{2} c^{3} x^{4} + 757 \, \sqrt{a^{2} x^{2} + 1} c^{3} x^{2} + \frac{2161 \, \sqrt{a^{2} x^{2} + 1} c^{3}}{a^{2}}\right )} a \operatorname{arsinh}\left (a x\right )^{2} + \frac{1}{35} \,{\left (5 \, a^{6} c^{3} x^{7} + 21 \, a^{4} c^{3} x^{5} + 35 \, a^{2} c^{3} x^{3} + 35 \, c^{3} x\right )} \operatorname{arsinh}\left (a x\right )^{3} - \frac{2}{13505625} \,{\left (16875 \, \sqrt{a^{2} x^{2} + 1} a^{4} c^{3} x^{6} + 134541 \, \sqrt{a^{2} x^{2} + 1} a^{2} c^{3} x^{4} + 747937 \, \sqrt{a^{2} x^{2} + 1} c^{3} x^{2} + \frac{22329151 \, \sqrt{a^{2} x^{2} + 1} c^{3}}{a^{2}} - \frac{105 \,{\left (1125 \, a^{6} c^{3} x^{7} + 7371 \, a^{4} c^{3} x^{5} + 26495 \, a^{2} c^{3} x^{3} + 226905 \, c^{3} x\right )} \operatorname{arsinh}\left (a x\right )}{a}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1225*(75*sqrt(a^2*x^2 + 1)*a^4*c^3*x^6 + 351*sqrt(a^2*x^2 + 1)*a^2*c^3*x^4 + 757*sqrt(a^2*x^2 + 1)*c^3*x^2
+ 2161*sqrt(a^2*x^2 + 1)*c^3/a^2)*a*arcsinh(a*x)^2 + 1/35*(5*a^6*c^3*x^7 + 21*a^4*c^3*x^5 + 35*a^2*c^3*x^3 + 3
5*c^3*x)*arcsinh(a*x)^3 - 2/13505625*(16875*sqrt(a^2*x^2 + 1)*a^4*c^3*x^6 + 134541*sqrt(a^2*x^2 + 1)*a^2*c^3*x
^4 + 747937*sqrt(a^2*x^2 + 1)*c^3*x^2 + 22329151*sqrt(a^2*x^2 + 1)*c^3/a^2 - 105*(1125*a^6*c^3*x^7 + 7371*a^4*
c^3*x^5 + 26495*a^2*c^3*x^3 + 226905*c^3*x)*arcsinh(a*x)/a)*a

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Fricas [A]  time = 2.24818, size = 603, normalized size = 1.68 \begin{align*} \frac{385875 \,{\left (5 \, a^{7} c^{3} x^{7} + 21 \, a^{5} c^{3} x^{5} + 35 \, a^{3} c^{3} x^{3} + 35 \, a c^{3} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} - 11025 \,{\left (75 \, a^{6} c^{3} x^{6} + 351 \, a^{4} c^{3} x^{4} + 757 \, a^{2} c^{3} x^{2} + 2161 \, c^{3}\right )} \sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} + 210 \,{\left (1125 \, a^{7} c^{3} x^{7} + 7371 \, a^{5} c^{3} x^{5} + 26495 \, a^{3} c^{3} x^{3} + 226905 \, a c^{3} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - 2 \,{\left (16875 \, a^{6} c^{3} x^{6} + 134541 \, a^{4} c^{3} x^{4} + 747937 \, a^{2} c^{3} x^{2} + 22329151 \, c^{3}\right )} \sqrt{a^{2} x^{2} + 1}}{13505625 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13505625*(385875*(5*a^7*c^3*x^7 + 21*a^5*c^3*x^5 + 35*a^3*c^3*x^3 + 35*a*c^3*x)*log(a*x + sqrt(a^2*x^2 + 1))
^3 - 11025*(75*a^6*c^3*x^6 + 351*a^4*c^3*x^4 + 757*a^2*c^3*x^2 + 2161*c^3)*sqrt(a^2*x^2 + 1)*log(a*x + sqrt(a^
2*x^2 + 1))^2 + 210*(1125*a^7*c^3*x^7 + 7371*a^5*c^3*x^5 + 26495*a^3*c^3*x^3 + 226905*a*c^3*x)*log(a*x + sqrt(
a^2*x^2 + 1)) - 2*(16875*a^6*c^3*x^6 + 134541*a^4*c^3*x^4 + 747937*a^2*c^3*x^2 + 22329151*c^3)*sqrt(a^2*x^2 +
1))/a

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Sympy [A]  time = 25.4759, size = 355, normalized size = 0.99 \begin{align*} \begin{cases} \frac{a^{6} c^{3} x^{7} \operatorname{asinh}^{3}{\left (a x \right )}}{7} + \frac{6 a^{6} c^{3} x^{7} \operatorname{asinh}{\left (a x \right )}}{343} - \frac{3 a^{5} c^{3} x^{6} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{49} - \frac{6 a^{5} c^{3} x^{6} \sqrt{a^{2} x^{2} + 1}}{2401} + \frac{3 a^{4} c^{3} x^{5} \operatorname{asinh}^{3}{\left (a x \right )}}{5} + \frac{702 a^{4} c^{3} x^{5} \operatorname{asinh}{\left (a x \right )}}{6125} - \frac{351 a^{3} c^{3} x^{4} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{1225} - \frac{29898 a^{3} c^{3} x^{4} \sqrt{a^{2} x^{2} + 1}}{1500625} + a^{2} c^{3} x^{3} \operatorname{asinh}^{3}{\left (a x \right )} + \frac{1514 a^{2} c^{3} x^{3} \operatorname{asinh}{\left (a x \right )}}{3675} - \frac{757 a c^{3} x^{2} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{1225} - \frac{1495874 a c^{3} x^{2} \sqrt{a^{2} x^{2} + 1}}{13505625} + c^{3} x \operatorname{asinh}^{3}{\left (a x \right )} + \frac{4322 c^{3} x \operatorname{asinh}{\left (a x \right )}}{1225} - \frac{2161 c^{3} \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{2}{\left (a x \right )}}{1225 a} - \frac{44658302 c^{3} \sqrt{a^{2} x^{2} + 1}}{13505625 a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**6*c**3*x**7*asinh(a*x)**3/7 + 6*a**6*c**3*x**7*asinh(a*x)/343 - 3*a**5*c**3*x**6*sqrt(a**2*x**2
+ 1)*asinh(a*x)**2/49 - 6*a**5*c**3*x**6*sqrt(a**2*x**2 + 1)/2401 + 3*a**4*c**3*x**5*asinh(a*x)**3/5 + 702*a**
4*c**3*x**5*asinh(a*x)/6125 - 351*a**3*c**3*x**4*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/1225 - 29898*a**3*c**3*x**4
*sqrt(a**2*x**2 + 1)/1500625 + a**2*c**3*x**3*asinh(a*x)**3 + 1514*a**2*c**3*x**3*asinh(a*x)/3675 - 757*a*c**3
*x**2*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/1225 - 1495874*a*c**3*x**2*sqrt(a**2*x**2 + 1)/13505625 + c**3*x*asinh
(a*x)**3 + 4322*c**3*x*asinh(a*x)/1225 - 2161*c**3*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/(1225*a) - 44658302*c**3*
sqrt(a**2*x**2 + 1)/(13505625*a), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.67381, size = 333, normalized size = 0.93 \begin{align*} \frac{1}{13505625} \,{\left (210 \,{\left (1125 \, a^{6} x^{7} + 7371 \, a^{4} x^{5} + 26495 \, a^{2} x^{3} + 226905 \, x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \frac{11025 \,{\left (75 \,{\left (a^{2} x^{2} + 1\right )}^{\frac{7}{2}} + 126 \,{\left (a^{2} x^{2} + 1\right )}^{\frac{5}{2}} + 280 \,{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 1680 \, \sqrt{a^{2} x^{2} + 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{a} - \frac{2 \,{\left (16875 \,{\left (a^{2} x^{2} + 1\right )}^{\frac{7}{2}} + 83916 \,{\left (a^{2} x^{2} + 1\right )}^{\frac{5}{2}} + 529480 \,{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 21698880 \, \sqrt{a^{2} x^{2} + 1}\right )}}{a}\right )} c^{3} + \frac{1}{35} \,{\left (5 \, a^{6} c^{3} x^{7} + 21 \, a^{4} c^{3} x^{5} + 35 \, a^{2} c^{3} x^{3} + 35 \, c^{3} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13505625*(210*(1125*a^6*x^7 + 7371*a^4*x^5 + 26495*a^2*x^3 + 226905*x)*log(a*x + sqrt(a^2*x^2 + 1)) - 11025*
(75*(a^2*x^2 + 1)^(7/2) + 126*(a^2*x^2 + 1)^(5/2) + 280*(a^2*x^2 + 1)^(3/2) + 1680*sqrt(a^2*x^2 + 1))*log(a*x
+ sqrt(a^2*x^2 + 1))^2/a - 2*(16875*(a^2*x^2 + 1)^(7/2) + 83916*(a^2*x^2 + 1)^(5/2) + 529480*(a^2*x^2 + 1)^(3/
2) + 21698880*sqrt(a^2*x^2 + 1))/a)*c^3 + 1/35*(5*a^6*c^3*x^7 + 21*a^4*c^3*x^5 + 35*a^2*c^3*x^3 + 35*c^3*x)*lo
g(a*x + sqrt(a^2*x^2 + 1))^3