3.74 \(\int \frac{\sinh ^{-1}(a+b x)^2}{x^4} \, dx\)

Optimal. Leaf size=478 \[ \frac{b^3 \text{PolyLog}\left (2,\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right )}{3 \left (a^2+1\right )^{3/2}}-\frac{a^2 b^3 \text{PolyLog}\left (2,\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right )}{\left (a^2+1\right )^{5/2}}-\frac{b^3 \text{PolyLog}\left (2,\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right )}{3 \left (a^2+1\right )^{3/2}}+\frac{a^2 b^3 \text{PolyLog}\left (2,\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right )}{\left (a^2+1\right )^{5/2}}-\frac{b^2}{3 \left (a^2+1\right ) x}-\frac{a b^3 \log (x)}{\left (a^2+1\right )^2}+\frac{a b^2 \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{\left (a^2+1\right )^2 x}+\frac{b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right )}{3 \left (a^2+1\right )^{3/2}}-\frac{a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right )}{\left (a^2+1\right )^{5/2}}-\frac{b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right )}{3 \left (a^2+1\right )^{3/2}}+\frac{a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right )}{\left (a^2+1\right )^{5/2}}-\frac{b \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{3 \left (a^2+1\right ) x^2}-\frac{\sinh ^{-1}(a+b x)^2}{3 x^3} \]

[Out]

-b^2/(3*(1 + a^2)*x) - (b*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(3*(1 + a^2)*x^2) + (a*b^2*Sqrt[1 + (a + b*x
)^2]*ArcSinh[a + b*x])/((1 + a^2)^2*x) - ArcSinh[a + b*x]^2/(3*x^3) - (a^2*b^3*ArcSinh[a + b*x]*Log[1 - E^ArcS
inh[a + b*x]/(a - Sqrt[1 + a^2])])/(1 + a^2)^(5/2) + (b^3*ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a - Sqr
t[1 + a^2])])/(3*(1 + a^2)^(3/2)) + (a^2*b^3*ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])
/(1 + a^2)^(5/2) - (b^3*ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(3*(1 + a^2)^(3/2))
- (a*b^3*Log[x])/(1 + a^2)^2 - (a^2*b^3*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/(1 + a^2)^(5/2) +
(b^3*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/(3*(1 + a^2)^(3/2)) + (a^2*b^3*PolyLog[2, E^ArcSinh[a
+ b*x]/(a + Sqrt[1 + a^2])])/(1 + a^2)^(5/2) - (b^3*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(3*(1
+ a^2)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 1.57321, antiderivative size = 478, normalized size of antiderivative = 1., number of steps used = 40, number of rules used = 16, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.333, Rules used = {5865, 5801, 5831, 3325, 3324, 3322, 2264, 2190, 2279, 2391, 2668, 31, 6741, 12, 6742, 32} \[ \frac{b^3 \text{PolyLog}\left (2,\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right )}{3 \left (a^2+1\right )^{3/2}}-\frac{a^2 b^3 \text{PolyLog}\left (2,\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right )}{\left (a^2+1\right )^{5/2}}-\frac{b^3 \text{PolyLog}\left (2,\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right )}{3 \left (a^2+1\right )^{3/2}}+\frac{a^2 b^3 \text{PolyLog}\left (2,\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right )}{\left (a^2+1\right )^{5/2}}-\frac{b^2}{3 \left (a^2+1\right ) x}-\frac{a b^3 \log (x)}{\left (a^2+1\right )^2}+\frac{a b^2 \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{\left (a^2+1\right )^2 x}+\frac{b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right )}{3 \left (a^2+1\right )^{3/2}}-\frac{a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right )}{\left (a^2+1\right )^{5/2}}-\frac{b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right )}{3 \left (a^2+1\right )^{3/2}}+\frac{a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right )}{\left (a^2+1\right )^{5/2}}-\frac{b \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{3 \left (a^2+1\right ) x^2}-\frac{\sinh ^{-1}(a+b x)^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a + b*x]^2/x^4,x]

[Out]

-b^2/(3*(1 + a^2)*x) - (b*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(3*(1 + a^2)*x^2) + (a*b^2*Sqrt[1 + (a + b*x
)^2]*ArcSinh[a + b*x])/((1 + a^2)^2*x) - ArcSinh[a + b*x]^2/(3*x^3) - (a^2*b^3*ArcSinh[a + b*x]*Log[1 - E^ArcS
inh[a + b*x]/(a - Sqrt[1 + a^2])])/(1 + a^2)^(5/2) + (b^3*ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a - Sqr
t[1 + a^2])])/(3*(1 + a^2)^(3/2)) + (a^2*b^3*ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])
/(1 + a^2)^(5/2) - (b^3*ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(3*(1 + a^2)^(3/2))
- (a*b^3*Log[x])/(1 + a^2)^2 - (a^2*b^3*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/(1 + a^2)^(5/2) +
(b^3*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])])/(3*(1 + a^2)^(3/2)) + (a^2*b^3*PolyLog[2, E^ArcSinh[a
+ b*x]/(a + Sqrt[1 + a^2])])/(1 + a^2)^(5/2) - (b^3*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(3*(1
+ a^2)^(3/2))

Rule 5865

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 5801

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

Rule 5831

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

Rule 3325

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

Rule 3324

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

Rule 3322

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

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 31

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

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 12

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

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 32

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

Rubi steps

\begin{align*} \int \frac{\sinh ^{-1}(a+b x)^2}{x^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sinh ^{-1}(x)^2}{\left (-\frac{a}{b}+\frac{x}{b}\right )^4} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\sinh ^{-1}(a+b x)^2}{3 x^3}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{\sinh ^{-1}(x)}{\left (-\frac{a}{b}+\frac{x}{b}\right )^3 \sqrt{1+x^2}} \, dx,x,a+b x\right )\\ &=-\frac{\sinh ^{-1}(a+b x)^2}{3 x^3}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{x}{\left (-\frac{a}{b}+\frac{\sinh (x)}{b}\right )^3} \, dx,x,\sinh ^{-1}(a+b x)\right )\\ &=-\frac{b \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}-\frac{\sinh ^{-1}(a+b x)^2}{3 x^3}+\frac{b \operatorname{Subst}\left (\int \frac{\cosh (x)}{\left (-\frac{a}{b}+\frac{\sinh (x)}{b}\right )^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}-\frac{b \operatorname{Subst}\left (\int \frac{x \sinh (x)}{\left (-\frac{a}{b}+\frac{\sinh (x)}{b}\right )^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}-\frac{(2 a b) \operatorname{Subst}\left (\int \frac{x}{\left (-\frac{a}{b}+\frac{\sinh (x)}{b}\right )^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}\\ &=-\frac{b \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac{2 a b^2 \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right )^2 x}-\frac{\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac{b \operatorname{Subst}\left (\int \frac{b^2 x \sinh (x)}{(a-\sinh (x))^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}-\frac{\left (2 a b^2\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{-\frac{a}{b}+\frac{\sinh (x)}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^2}+\frac{\left (2 a^2 b^2\right ) \operatorname{Subst}\left (\int \frac{x}{-\frac{a}{b}+\frac{\sinh (x)}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{\left (-\frac{a}{b}+x\right )^2} \, dx,x,\frac{a}{b}+x\right )}{3 \left (1+a^2\right )}\\ &=-\frac{b^2}{3 \left (1+a^2\right ) x}-\frac{b \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac{2 a b^2 \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right )^2 x}-\frac{\sinh ^{-1}(a+b x)^2}{3 x^3}+\frac{\left (4 a^2 b^2\right ) \operatorname{Subst}\left (\int \frac{e^x x}{-\frac{1}{b}-\frac{2 a e^x}{b}+\frac{e^{2 x}}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^2}-\frac{\left (2 a b^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+x} \, dx,x,\frac{a}{b}+x\right )}{3 \left (1+a^2\right )^2}-\frac{b^3 \operatorname{Subst}\left (\int \frac{x \sinh (x)}{(a-\sinh (x))^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}\\ &=-\frac{b^2}{3 \left (1+a^2\right ) x}-\frac{b \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac{2 a b^2 \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right )^2 x}-\frac{\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac{2 a b^3 \log (x)}{3 \left (1+a^2\right )^2}+\frac{\left (4 a^2 b^2\right ) \operatorname{Subst}\left (\int \frac{e^x x}{-\frac{2 a}{b}-\frac{2 \sqrt{1+a^2}}{b}+\frac{2 e^x}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{5/2}}-\frac{\left (4 a^2 b^2\right ) \operatorname{Subst}\left (\int \frac{e^x x}{-\frac{2 a}{b}+\frac{2 \sqrt{1+a^2}}{b}+\frac{2 e^x}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{5/2}}-\frac{b^3 \operatorname{Subst}\left (\int \left (\frac{a x}{(a-\sinh (x))^2}-\frac{x}{a-\sinh (x)}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}\\ &=-\frac{b^2}{3 \left (1+a^2\right ) x}-\frac{b \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac{2 a b^2 \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right )^2 x}-\frac{\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac{2 a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac{2 a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}-\frac{2 a b^3 \log (x)}{3 \left (1+a^2\right )^2}-\frac{\left (2 a^2 b^3\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^x}{\left (-\frac{2 a}{b}-\frac{2 \sqrt{1+a^2}}{b}\right ) b}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{5/2}}+\frac{\left (2 a^2 b^3\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^x}{\left (-\frac{2 a}{b}+\frac{2 \sqrt{1+a^2}}{b}\right ) b}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{5/2}}+\frac{b^3 \operatorname{Subst}\left (\int \frac{x}{a-\sinh (x)} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}-\frac{\left (a b^3\right ) \operatorname{Subst}\left (\int \frac{x}{(a-\sinh (x))^2} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}\\ &=-\frac{b^2}{3 \left (1+a^2\right ) x}-\frac{b \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac{a b^2 \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right )^2 x}-\frac{\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac{2 a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac{2 a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}-\frac{2 a b^3 \log (x)}{3 \left (1+a^2\right )^2}-\frac{\left (2 a^2 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{\left (-\frac{2 a}{b}-\frac{2 \sqrt{1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac{\left (2 a^2 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{\left (-\frac{2 a}{b}+\frac{2 \sqrt{1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac{\left (a b^3\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{a-\sinh (x)} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^2}-\frac{\left (a^2 b^3\right ) \operatorname{Subst}\left (\int \frac{x}{a-\sinh (x)} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^2}+\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{e^x x}{1+2 a e^x-e^{2 x}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )}\\ &=-\frac{b^2}{3 \left (1+a^2\right ) x}-\frac{b \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac{a b^2 \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right )^2 x}-\frac{\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac{2 a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac{2 a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}-\frac{2 a b^3 \log (x)}{3 \left (1+a^2\right )^2}-\frac{2 a^2 b^3 \text{Li}_2\left (\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac{2 a^2 b^3 \text{Li}_2\left (\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}-\frac{\left (a b^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,-a-b x\right )}{3 \left (1+a^2\right )^2}-\frac{\left (2 a^2 b^3\right ) \operatorname{Subst}\left (\int \frac{e^x x}{1+2 a e^x-e^{2 x}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^2}-\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 a-2 \sqrt{1+a^2}-2 e^x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{3/2}}+\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 a+2 \sqrt{1+a^2}-2 e^x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{3/2}}\\ &=-\frac{b^2}{3 \left (1+a^2\right ) x}-\frac{b \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac{a b^2 \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right )^2 x}-\frac{\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac{2 a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac{b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac{2 a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}-\frac{b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}-\frac{a b^3 \log (x)}{\left (1+a^2\right )^2}-\frac{2 a^2 b^3 \text{Li}_2\left (\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac{2 a^2 b^3 \text{Li}_2\left (\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac{\left (2 a^2 b^3\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 a-2 \sqrt{1+a^2}-2 e^x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{5/2}}-\frac{\left (2 a^2 b^3\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 a+2 \sqrt{1+a^2}-2 e^x} \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{5/2}}-\frac{b^3 \operatorname{Subst}\left (\int \log \left (1-\frac{2 e^x}{2 a-2 \sqrt{1+a^2}}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{3/2}}+\frac{b^3 \operatorname{Subst}\left (\int \log \left (1-\frac{2 e^x}{2 a+2 \sqrt{1+a^2}}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{3/2}}\\ &=-\frac{b^2}{3 \left (1+a^2\right ) x}-\frac{b \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac{a b^2 \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right )^2 x}-\frac{\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac{a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}+\frac{b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac{a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}-\frac{b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}-\frac{a b^3 \log (x)}{\left (1+a^2\right )^2}-\frac{2 a^2 b^3 \text{Li}_2\left (\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac{2 a^2 b^3 \text{Li}_2\left (\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac{\left (a^2 b^3\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{2 e^x}{2 a-2 \sqrt{1+a^2}}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{5/2}}-\frac{\left (a^2 b^3\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{2 e^x}{2 a+2 \sqrt{1+a^2}}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 \left (1+a^2\right )^{5/2}}-\frac{b^3 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 x}{2 a-2 \sqrt{1+a^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 x}{2 a+2 \sqrt{1+a^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{3 \left (1+a^2\right )^{3/2}}\\ &=-\frac{b^2}{3 \left (1+a^2\right ) x}-\frac{b \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac{a b^2 \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right )^2 x}-\frac{\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac{a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}+\frac{b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac{a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}-\frac{b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}-\frac{a b^3 \log (x)}{\left (1+a^2\right )^2}-\frac{2 a^2 b^3 \text{Li}_2\left (\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}+\frac{b^3 \text{Li}_2\left (\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac{2 a^2 b^3 \text{Li}_2\left (\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{5/2}}-\frac{b^3 \text{Li}_2\left (\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac{\left (a^2 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 x}{2 a-2 \sqrt{1+a^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{3 \left (1+a^2\right )^{5/2}}-\frac{\left (a^2 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 x}{2 a+2 \sqrt{1+a^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )}{3 \left (1+a^2\right )^{5/2}}\\ &=-\frac{b^2}{3 \left (1+a^2\right ) x}-\frac{b \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac{a b^2 \sqrt{1+(a+b x)^2} \sinh ^{-1}(a+b x)}{\left (1+a^2\right )^2 x}-\frac{\sinh ^{-1}(a+b x)^2}{3 x^3}-\frac{a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}+\frac{b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac{a^2 b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}-\frac{b^3 \sinh ^{-1}(a+b x) \log \left (1-\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}-\frac{a b^3 \log (x)}{\left (1+a^2\right )^2}-\frac{a^2 b^3 \text{Li}_2\left (\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}+\frac{b^3 \text{Li}_2\left (\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac{a^2 b^3 \text{Li}_2\left (\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}-\frac{b^3 \text{Li}_2\left (\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}\\ \end{align*}

Mathematica [C]  time = 10.3128, size = 1830, normalized size = 3.83 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcSinh[a + b*x]^2/x^4,x]

[Out]

(-((b*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/((1 + a^2)*x^2)) - ArcSinh[a + b*x]^2/x^3 - (b^2*(1 + a^2 - 3*a*
Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]))/((1 + a^2)^2*x) + (I*b^3*Pi*ArcTanh[(-1 - a*Tanh[ArcSinh[a + b*x]/2])
/Sqrt[1 + a^2]])/(1 + a^2)^(5/2) - ((2*I)*a^2*b^3*Pi*ArcTanh[(-1 - a*Tanh[ArcSinh[a + b*x]/2])/Sqrt[1 + a^2]])
/(1 + a^2)^(5/2) - (3*a*b^3*Log[-((b*x)/a)])/(1 + a^2)^2 + (b^3*(-2*ArcCos[I*a]*ArcTanh[((-I + a)*Cot[(Pi + (2
*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]] - (Pi - (2*I)*ArcSinh[a + b*x])*ArcTanh[((I + a)*Tan[(Pi + (2*I)*Arc
Sinh[a + b*x])/4])/Sqrt[-1 - a^2]] + (ArcCos[I*a] + (2*I)*ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/
4])/Sqrt[-1 - a^2]] + (2*I)*ArcTanh[((I + a)*Tan[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]])*Log[Sqrt[-
1 - a^2]/(Sqrt[2]*E^(ArcSinh[a + b*x]/2)*Sqrt[b*x])] + (ArcCos[I*a] - (2*I)*(ArcTanh[((-I + a)*Cot[(Pi + (2*I)
*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]] + ArcTanh[((I + a)*Tan[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]
]))*Log[(I*Sqrt[-1 - a^2]*E^(ArcSinh[a + b*x]/2))/(Sqrt[2]*Sqrt[b*x])] - (ArcCos[I*a] + (2*I)*ArcTanh[((-I + a
)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]])*Log[((I + a)*(a + I*(-1 + Sqrt[-1 - a^2]))*(I + Cot[(
Pi + (2*I)*ArcSinh[a + b*x])/4]))/(I + a - Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])] - (ArcCos[I*a
] - (2*I)*ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]])*Log[((I + a)*(a - I*(1 + Sq
rt[-1 - a^2]))*(-I + Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4]))/(-I - a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a
+ b*x])/4])] + I*(PolyLog[2, -((((-I)*a + Sqrt[-1 - a^2])*(I + a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a +
b*x])/4]))/(-I - a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4]))] - PolyLog[2, ((I*a + Sqrt[-1 - a^
2])*(I + a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4]))/(-I - a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*Ar
cSinh[a + b*x])/4])])))/(-1 - a^2)^(5/2) - (2*a^2*b^3*(-2*ArcCos[I*a]*ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSin
h[a + b*x])/4])/Sqrt[-1 - a^2]] - (Pi - (2*I)*ArcSinh[a + b*x])*ArcTanh[((I + a)*Tan[(Pi + (2*I)*ArcSinh[a + b
*x])/4])/Sqrt[-1 - a^2]] + (ArcCos[I*a] + (2*I)*ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-
1 - a^2]] + (2*I)*ArcTanh[((I + a)*Tan[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]])*Log[Sqrt[-1 - a^2]/(
Sqrt[2]*E^(ArcSinh[a + b*x]/2)*Sqrt[b*x])] + (ArcCos[I*a] - (2*I)*(ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a
+ b*x])/4])/Sqrt[-1 - a^2]] + ArcTanh[((I + a)*Tan[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]]))*Log[(I
*Sqrt[-1 - a^2]*E^(ArcSinh[a + b*x]/2))/(Sqrt[2]*Sqrt[b*x])] - (ArcCos[I*a] + (2*I)*ArcTanh[((-I + a)*Cot[(Pi
+ (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]])*Log[((I + a)*(a + I*(-1 + Sqrt[-1 - a^2]))*(I + Cot[(Pi + (2*I)
*ArcSinh[a + b*x])/4]))/(I + a - Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])] - (ArcCos[I*a] - (2*I)*
ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]])*Log[((I + a)*(a - I*(1 + Sqrt[-1 - a^
2]))*(-I + Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4]))/(-I - a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4
])] + I*(PolyLog[2, -((((-I)*a + Sqrt[-1 - a^2])*(I + a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])
)/(-I - a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4]))] - PolyLog[2, ((I*a + Sqrt[-1 - a^2])*(I + a
+ Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4]))/(-I - a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a +
b*x])/4])])))/(-1 - a^2)^(5/2))/3

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Maple [A]  time = 0.39, size = 730, normalized size = 1.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^3/(a^2+1)^2*arcsinh(b*x+a)*a+a*b^2*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)/(a^2+1)^2/x-1/3/(a^2+1)^2/x^3*arcsinh
(b*x+a)^2*a^4-1/3*b/(a^2+1)^2/x^2*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)*a^2-1/3*b^2/(a^2+1)^2/x*a^2-2/3/(a^2+1)^2
/x^3*arcsinh(b*x+a)^2*a^2-1/3*b/(a^2+1)^2/x^2*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)-1/3*b^2/(a^2+1)^2/x-1/3/(a^2+
1)^2/x^3*arcsinh(b*x+a)^2+2*b^3/(a^2+1)^2*a*ln(b*x+a+(1+(b*x+a)^2)^(1/2))-b^3/(a^2+1)^2*a*ln((b*x+a+(1+(b*x+a)
^2)^(1/2))^2-2*a*(b*x+a+(1+(b*x+a)^2)^(1/2))-1)-1/3*b^3/(a^2+1)^(5/2)*arcsinh(b*x+a)*ln(((a^2+1)^(1/2)-b*x-(1+
(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))+1/3*b^3/(a^2+1)^(5/2)*arcsinh(b*x+a)*ln(((a^2+1)^(1/2)+b*x+(1+(b*x+a)^2)^
(1/2))/(-a+(a^2+1)^(1/2)))-1/3*b^3/(a^2+1)^(5/2)*dilog(((a^2+1)^(1/2)-b*x-(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2
)))+1/3*b^3/(a^2+1)^(5/2)*dilog(((a^2+1)^(1/2)+b*x+(1+(b*x+a)^2)^(1/2))/(-a+(a^2+1)^(1/2)))+2/3*b^3/(a^2+1)^(5
/2)*a^2*arcsinh(b*x+a)*ln(((a^2+1)^(1/2)-b*x-(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))-2/3*b^3/(a^2+1)^(5/2)*a^2
*arcsinh(b*x+a)*ln(((a^2+1)^(1/2)+b*x+(1+(b*x+a)^2)^(1/2))/(-a+(a^2+1)^(1/2)))+2/3*b^3/(a^2+1)^(5/2)*a^2*dilog
(((a^2+1)^(1/2)-b*x-(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))-2/3*b^3/(a^2+1)^(5/2)*a^2*dilog(((a^2+1)^(1/2)+b*x
+(1+(b*x+a)^2)^(1/2))/(-a+(a^2+1)^(1/2)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arcsinh(b*x + a)^2/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asinh(a + b*x)**2/x**4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsinh(b*x + a)^2/x^4, x)