3.224 \(\int \frac{(d+c^2 d x^2)^3 (a+b \sinh ^{-1}(c x))^2}{x^4} \, dx\)

Optimal. Leaf size=326 \[ -\frac{17}{3} b^2 c^3 d^3 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+\frac{17}{3} b^2 c^3 d^3 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )+\frac{8}{3} c^4 d^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} b c^3 d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-5 b c^3 d^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 c^2 d^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{b c d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{34}{3} b c^3 d^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac{2}{27} b^2 c^6 d^3 x^3+\frac{50}{9} b^2 c^4 d^3 x-\frac{b^2 c^2 d^3}{3 x} \]

[Out]

-(b^2*c^2*d^3)/(3*x) + (50*b^2*c^4*d^3*x)/9 + (2*b^2*c^6*d^3*x^3)/27 - 5*b*c^3*d^3*Sqrt[1 + c^2*x^2]*(a + b*Ar
cSinh[c*x]) + (b*c^3*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/9 - (b*c*d^3*(1 + c^2*x^2)^(5/2)*(a + b*Arc
Sinh[c*x]))/(3*x^2) + (16*c^4*d^3*x*(a + b*ArcSinh[c*x])^2)/3 + (8*c^4*d^3*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]
)^2)/3 - (2*c^2*d^3*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/x - (d^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/(
3*x^3) - (34*b*c^3*d^3*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/3 - (17*b^2*c^3*d^3*PolyLog[2, -E^ArcSinh
[c*x]])/3 + (17*b^2*c^3*d^3*PolyLog[2, E^ArcSinh[c*x]])/3

________________________________________________________________________________________

Rubi [A]  time = 1.02137, antiderivative size = 326, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 12, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {5739, 5684, 5653, 5717, 8, 5744, 5742, 5760, 4182, 2279, 2391, 270} \[ -\frac{17}{3} b^2 c^3 d^3 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+\frac{17}{3} b^2 c^3 d^3 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )+\frac{8}{3} c^4 d^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{9} b c^3 d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-5 b c^3 d^3 \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac{2 c^2 d^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{b c d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{34}{3} b c^3 d^3 \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+\frac{2}{27} b^2 c^6 d^3 x^3+\frac{50}{9} b^2 c^4 d^3 x-\frac{b^2 c^2 d^3}{3 x} \]

Antiderivative was successfully verified.

[In]

Int[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2)/x^4,x]

[Out]

-(b^2*c^2*d^3)/(3*x) + (50*b^2*c^4*d^3*x)/9 + (2*b^2*c^6*d^3*x^3)/27 - 5*b*c^3*d^3*Sqrt[1 + c^2*x^2]*(a + b*Ar
cSinh[c*x]) + (b*c^3*d^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/9 - (b*c*d^3*(1 + c^2*x^2)^(5/2)*(a + b*Arc
Sinh[c*x]))/(3*x^2) + (16*c^4*d^3*x*(a + b*ArcSinh[c*x])^2)/3 + (8*c^4*d^3*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]
)^2)/3 - (2*c^2*d^3*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/x - (d^3*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/(
3*x^3) - (34*b*c^3*d^3*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/3 - (17*b^2*c^3*d^3*PolyLog[2, -E^ArcSinh
[c*x]])/3 + (17*b^2*c^3*d^3*PolyLog[2, E^ArcSinh[c*x]])/3

Rule 5739

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

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 8

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

Rule 5744

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

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]

Rule 270

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

Rubi steps

\begin{align*} \int \frac{\left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\left (2 c^2 d\right ) \int \frac{\left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x^2} \, dx+\frac{1}{3} \left (2 b c d^3\right ) \int \frac{\left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{x^3} \, dx\\ &=-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac{2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\left (8 c^4 d^2\right ) \int \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac{1}{3} \left (b^2 c^2 d^3\right ) \int \frac{\left (1+c^2 x^2\right )^2}{x^2} \, dx+\frac{1}{3} \left (5 b c^3 d^3\right ) \int \frac{\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\left (4 b c^3 d^3\right ) \int \frac{\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx\\ &=\frac{17}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (b^2 c^2 d^3\right ) \int \left (2 c^2+\frac{1}{x^2}+c^4 x^2\right ) \, dx+\frac{1}{3} \left (5 b c^3 d^3\right ) \int \frac{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\left (4 b c^3 d^3\right ) \int \frac{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\frac{1}{3} \left (16 c^4 d^3\right ) \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac{1}{9} \left (5 b^2 c^4 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac{1}{3} \left (4 b^2 c^4 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac{1}{3} \left (16 b c^5 d^3\right ) \int x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac{b^2 c^2 d^3}{3 x}-\frac{11}{9} b^2 c^4 d^3 x-\frac{14}{27} b^2 c^6 d^3 x^3+\frac{17}{3} b c^3 d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (5 b c^3 d^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx+\left (4 b c^3 d^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx-\frac{1}{3} \left (5 b^2 c^4 d^3\right ) \int 1 \, dx+\frac{1}{9} \left (16 b^2 c^4 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\left (4 b^2 c^4 d^3\right ) \int 1 \, dx-\frac{1}{3} \left (32 b c^5 d^3\right ) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx\\ &=-\frac{b^2 c^2 d^3}{3 x}-\frac{46}{9} b^2 c^4 d^3 x+\frac{2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (5 b c^3 d^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )+\left (4 b c^3 d^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )+\frac{1}{3} \left (32 b^2 c^4 d^3\right ) \int 1 \, dx\\ &=-\frac{b^2 c^2 d^3}{3 x}+\frac{50}{9} b^2 c^4 d^3 x+\frac{2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac{34}{3} b c^3 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\frac{1}{3} \left (5 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\frac{1}{3} \left (5 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )-\left (4 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )+\left (4 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=-\frac{b^2 c^2 d^3}{3 x}+\frac{50}{9} b^2 c^4 d^3 x+\frac{2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac{34}{3} b c^3 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\frac{1}{3} \left (5 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )+\frac{1}{3} \left (5 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )-\left (4 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )+\left (4 b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )\\ &=-\frac{b^2 c^2 d^3}{3 x}+\frac{50}{9} b^2 c^4 d^3 x+\frac{2}{27} b^2 c^6 d^3 x^3-5 b c^3 d^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{9} b c^3 d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{b c d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac{16}{3} c^4 d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{8}{3} c^4 d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{2 c^2 d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-\frac{d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac{34}{3} b c^3 d^3 \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-\frac{17}{3} b^2 c^3 d^3 \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+\frac{17}{3} b^2 c^3 d^3 \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 1.11144, size = 461, normalized size = 1.41 \[ \frac{d^3 \left (153 b^2 c^3 x^3 \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-153 b^2 c^3 x^3 \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+9 a^2 c^6 x^6+81 a^2 c^4 x^4-81 a^2 c^2 x^2-9 a^2-6 a b c^5 x^5 \sqrt{c^2 x^2+1}-150 a b c^3 x^3 \sqrt{c^2 x^2+1}-9 a b c x \sqrt{c^2 x^2+1}+18 a b c^6 x^6 \sinh ^{-1}(c x)+162 a b c^4 x^4 \sinh ^{-1}(c x)-162 a b c^2 x^2 \sinh ^{-1}(c x)-153 a b c^3 x^3 \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )-18 a b \sinh ^{-1}(c x)+2 b^2 c^6 x^6+150 b^2 c^4 x^4-9 b^2 c^2 x^2+9 b^2 c^6 x^6 \sinh ^{-1}(c x)^2-6 b^2 c^5 x^5 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+81 b^2 c^4 x^4 \sinh ^{-1}(c x)^2-150 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-81 b^2 c^2 x^2 \sinh ^{-1}(c x)^2-9 b^2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+153 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-153 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )-9 b^2 \sinh ^{-1}(c x)^2\right )}{27 x^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2)/x^4,x]

[Out]

(d^3*(-9*a^2 - 81*a^2*c^2*x^2 - 9*b^2*c^2*x^2 + 81*a^2*c^4*x^4 + 150*b^2*c^4*x^4 + 9*a^2*c^6*x^6 + 2*b^2*c^6*x
^6 - 9*a*b*c*x*Sqrt[1 + c^2*x^2] - 150*a*b*c^3*x^3*Sqrt[1 + c^2*x^2] - 6*a*b*c^5*x^5*Sqrt[1 + c^2*x^2] - 18*a*
b*ArcSinh[c*x] - 162*a*b*c^2*x^2*ArcSinh[c*x] + 162*a*b*c^4*x^4*ArcSinh[c*x] + 18*a*b*c^6*x^6*ArcSinh[c*x] - 9
*b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 150*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 6*b^2*c^5*x^5*Sqrt[
1 + c^2*x^2]*ArcSinh[c*x] - 9*b^2*ArcSinh[c*x]^2 - 81*b^2*c^2*x^2*ArcSinh[c*x]^2 + 81*b^2*c^4*x^4*ArcSinh[c*x]
^2 + 9*b^2*c^6*x^6*ArcSinh[c*x]^2 - 153*a*b*c^3*x^3*ArcTanh[Sqrt[1 + c^2*x^2]] + 153*b^2*c^3*x^3*ArcSinh[c*x]*
Log[1 - E^(-ArcSinh[c*x])] - 153*b^2*c^3*x^3*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 153*b^2*c^3*x^3*PolyLog
[2, -E^(-ArcSinh[c*x])] - 153*b^2*c^3*x^3*PolyLog[2, E^(-ArcSinh[c*x])]))/(27*x^3)

________________________________________________________________________________________

Maple [A]  time = 0.331, size = 528, normalized size = 1.6 \begin{align*} -{\frac{17\,{c}^{3}{d}^{3}ab}{3}{\it Artanh} \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}} \right ) }-{\frac{2\,{d}^{3}ab{\it Arcsinh} \left ( cx \right ) }{3\,{x}^{3}}}+{\frac{{c}^{6}{d}^{3}{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}{x}^{3}}{3}}+3\,{c}^{4}{d}^{3}{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}x-3\,{\frac{{d}^{3}{b}^{2}{c}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{x}}-{\frac{50\,{d}^{3}{b}^{2}{c}^{3}{\it Arcsinh} \left ( cx \right ) }{9}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{17\,{d}^{3}{b}^{2}{c}^{3}{\it Arcsinh} \left ( cx \right ) }{3}\ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }+{\frac{17\,{d}^{3}{b}^{2}{c}^{3}{\it Arcsinh} \left ( cx \right ) }{3}\ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }-{\frac{{d}^{3}{b}^{2}{c}^{2}}{3\,x}}+{\frac{50\,{b}^{2}{c}^{4}{d}^{3}x}{9}}-{\frac{2\,{c}^{5}{d}^{3}ab{x}^{2}}{9}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{c{d}^{3}ab}{3\,{x}^{2}}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{{d}^{3}{b}^{2}c{\it Arcsinh} \left ( cx \right ) }{3\,{x}^{2}}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{2\,{c}^{5}{d}^{3}{b}^{2}{\it Arcsinh} \left ( cx \right ){x}^{2}}{9}\sqrt{{c}^{2}{x}^{2}+1}}+{\frac{2\,{c}^{6}{d}^{3}ab{\it Arcsinh} \left ( cx \right ){x}^{3}}{3}}+6\,{c}^{4}{d}^{3}ab{\it Arcsinh} \left ( cx \right ) x-6\,{\frac{{d}^{3}ab{c}^{2}{\it Arcsinh} \left ( cx \right ) }{x}}+{\frac{2\,{b}^{2}{c}^{6}{d}^{3}{x}^{3}}{27}}-{\frac{17\,{d}^{3}{b}^{2}{c}^{3}}{3}{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) }+{\frac{17\,{d}^{3}{b}^{2}{c}^{3}}{3}{\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ) }-{\frac{50\,{c}^{3}{d}^{3}ab}{9}\sqrt{{c}^{2}{x}^{2}+1}}-{\frac{{d}^{3}{a}^{2}}{3\,{x}^{3}}}+{\frac{{c}^{6}{d}^{3}{a}^{2}{x}^{3}}{3}}+3\,{c}^{4}{d}^{3}{a}^{2}x-3\,{\frac{{c}^{2}{d}^{3}{a}^{2}}{x}}-{\frac{{d}^{3}{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2/x^4,x)

[Out]

-2/3*d^3*a*b*arcsinh(c*x)/x^3-1/3*b^2*c^2*d^3/x+50/9*b^2*c^4*d^3*x-2/9*c^5*d^3*a*b*x^2*(c^2*x^2+1)^(1/2)-1/3*c
*d^3*a*b/x^2*(c^2*x^2+1)^(1/2)-1/3*c*d^3*b^2/x^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)-2/9*c^5*d^3*b^2*arcsinh(c*x)*(
c^2*x^2+1)^(1/2)*x^2+2/3*c^6*d^3*a*b*arcsinh(c*x)*x^3+6*c^4*d^3*a*b*arcsinh(c*x)*x-6*c^2*d^3*a*b*arcsinh(c*x)/
x+2/27*b^2*c^6*d^3*x^3-17/3*b^2*c^3*d^3*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+17/3*b^2*c^3*d^3*polylog(2,c*x+(c^2*
x^2+1)^(1/2))+1/3*c^6*d^3*b^2*arcsinh(c*x)^2*x^3+3*c^4*d^3*b^2*arcsinh(c*x)^2*x-3*c^2*d^3*b^2*arcsinh(c*x)^2/x
-50/9*c^3*d^3*a*b*(c^2*x^2+1)^(1/2)-17/3*c^3*d^3*a*b*arctanh(1/(c^2*x^2+1)^(1/2))-50/9*c^3*d^3*b^2*arcsinh(c*x
)*(c^2*x^2+1)^(1/2)-17/3*c^3*d^3*b^2*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))+17/3*c^3*d^3*b^2*arcsinh(c*x)*ln
(1-c*x-(c^2*x^2+1)^(1/2))-1/3*d^3*a^2/x^3+1/3*c^6*d^3*a^2*x^3+3*c^4*d^3*a^2*x-3*c^2*d^3*a^2/x-1/3*d^3*b^2/x^3*
arcsinh(c*x)^2

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a^{2} c^{6} d^{3} x^{3} + \frac{2}{9} \,{\left (3 \, x^{3} \operatorname{arsinh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac{2 \, \sqrt{c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{6} d^{3} + 3 \, b^{2} c^{4} d^{3} x \operatorname{arsinh}\left (c x\right )^{2} + 6 \, b^{2} c^{4} d^{3}{\left (x - \frac{\sqrt{c^{2} x^{2} + 1} \operatorname{arsinh}\left (c x\right )}{c}\right )} + 3 \, a^{2} c^{4} d^{3} x + 6 \,{\left (c x \operatorname{arsinh}\left (c x\right ) - \sqrt{c^{2} x^{2} + 1}\right )} a b c^{3} d^{3} - 6 \,{\left (c \operatorname{arsinh}\left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) + \frac{\operatorname{arsinh}\left (c x\right )}{x}\right )} a b c^{2} d^{3} + \frac{1}{3} \,{\left ({\left (c^{2} \operatorname{arsinh}\left (\frac{1}{\sqrt{c^{2}}{\left | x \right |}}\right ) - \frac{\sqrt{c^{2} x^{2} + 1}}{x^{2}}\right )} c - \frac{2 \, \operatorname{arsinh}\left (c x\right )}{x^{3}}\right )} a b d^{3} - \frac{3 \, a^{2} c^{2} d^{3}}{x} - \frac{a^{2} d^{3}}{3 \, x^{3}} + \frac{{\left (b^{2} c^{6} d^{3} x^{6} - 9 \, b^{2} c^{2} d^{3} x^{2} - b^{2} d^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{3 \, x^{3}} - \int \frac{2 \,{\left (b^{2} c^{9} d^{3} x^{8} + b^{2} c^{7} d^{3} x^{6} - 9 \, b^{2} c^{5} d^{3} x^{4} - 10 \, b^{2} c^{3} d^{3} x^{2} - b^{2} c d^{3} +{\left (b^{2} c^{8} d^{3} x^{7} - 9 \, b^{2} c^{4} d^{3} x^{3} - b^{2} c^{2} d^{3} x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{3 \,{\left (c^{3} x^{6} + c x^{4} +{\left (c^{2} x^{5} + x^{3}\right )} \sqrt{c^{2} x^{2} + 1}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2/x^4,x, algorithm="maxima")

[Out]

1/3*a^2*c^6*d^3*x^3 + 2/9*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*a*b*c
^6*d^3 + 3*b^2*c^4*d^3*x*arcsinh(c*x)^2 + 6*b^2*c^4*d^3*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + 3*a^2*c^4*d^3
*x + 6*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*a*b*c^3*d^3 - 6*(c*arcsinh(1/(sqrt(c^2)*abs(x))) + arcsinh(c*x)/
x)*a*b*c^2*d^3 + 1/3*((c^2*arcsinh(1/(sqrt(c^2)*abs(x))) - sqrt(c^2*x^2 + 1)/x^2)*c - 2*arcsinh(c*x)/x^3)*a*b*
d^3 - 3*a^2*c^2*d^3/x - 1/3*a^2*d^3/x^3 + 1/3*(b^2*c^6*d^3*x^6 - 9*b^2*c^2*d^3*x^2 - b^2*d^3)*log(c*x + sqrt(c
^2*x^2 + 1))^2/x^3 - integrate(2/3*(b^2*c^9*d^3*x^8 + b^2*c^7*d^3*x^6 - 9*b^2*c^5*d^3*x^4 - 10*b^2*c^3*d^3*x^2
- b^2*c*d^3 + (b^2*c^8*d^3*x^7 - 9*b^2*c^4*d^3*x^3 - b^2*c^2*d^3*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2
+ 1))/(c^3*x^6 + c*x^4 + (c^2*x^5 + x^3)*sqrt(c^2*x^2 + 1)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2/x^4,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} d^{3} \left (\int 3 a^{2} c^{4}\, dx + \int \frac{a^{2}}{x^{4}}\, dx + \int \frac{3 a^{2} c^{2}}{x^{2}}\, dx + \int a^{2} c^{6} x^{2}\, dx + \int 3 b^{2} c^{4} \operatorname{asinh}^{2}{\left (c x \right )}\, dx + \int \frac{b^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int 6 a b c^{4} \operatorname{asinh}{\left (c x \right )}\, dx + \int \frac{2 a b \operatorname{asinh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{3 b^{2} c^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int b^{2} c^{6} x^{2} \operatorname{asinh}^{2}{\left (c x \right )}\, dx + \int \frac{6 a b c^{2} \operatorname{asinh}{\left (c x \right )}}{x^{2}}\, dx + \int 2 a b c^{6} x^{2} \operatorname{asinh}{\left (c x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c**2*d*x**2+d)**3*(a+b*asinh(c*x))**2/x**4,x)

[Out]

d**3*(Integral(3*a**2*c**4, x) + Integral(a**2/x**4, x) + Integral(3*a**2*c**2/x**2, x) + Integral(a**2*c**6*x
**2, x) + Integral(3*b**2*c**4*asinh(c*x)**2, x) + Integral(b**2*asinh(c*x)**2/x**4, x) + Integral(6*a*b*c**4*
asinh(c*x), x) + Integral(2*a*b*asinh(c*x)/x**4, x) + Integral(3*b**2*c**2*asinh(c*x)**2/x**2, x) + Integral(b
**2*c**6*x**2*asinh(c*x)**2, x) + Integral(6*a*b*c**2*asinh(c*x)/x**2, x) + Integral(2*a*b*c**6*x**2*asinh(c*x
), x))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c^{2} d x^{2} + d\right )}^{3}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2/x^4,x, algorithm="giac")

[Out]

integrate((c^2*d*x^2 + d)^3*(b*arcsinh(c*x) + a)^2/x^4, x)