\(\int \frac {\text {arcsinh}(a+b x)^2}{x^4} \, dx\) [74]

Optimal result

Integrand size = 12, antiderivative size = 478 \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^4} \, dx=-\frac {b^2}{3 \left (1+a^2\right ) x}-\frac {b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{3 \left (1+a^2\right ) x^2}+\frac {a b^2 \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{\left (1+a^2\right )^2 x}-\frac {\text {arcsinh}(a+b x)^2}{3 x^3}-\frac {a^2 b^3 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}+\frac {b^3 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac {a^2 b^3 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}-\frac {b^3 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}-\frac {a b^3 \log (x)}{\left (1+a^2\right )^2}-\frac {a^2 b^3 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}+\frac {b^3 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}}+\frac {a^2 b^3 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}-\frac {b^3 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )}{3 \left (1+a^2\right )^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 1.16 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.00, number of steps used = 40, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {5859, 5828, 5843, 3406, 3405, 3403, 2296, 2221, 2317, 2438, 2747, 31, 6873, 12, 6874, 32} \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^4} \, dx=\frac {b^3 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{3 \left (a^2+1\right )^{3/2}}-\frac {a^2 b^3 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{5/2}}-\frac {b^3 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )}{3 \left (a^2+1\right )^{3/2}}+\frac {a^2 b^3 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{5/2}}+\frac {b^3 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{3 \left (a^2+1\right )^{3/2}}-\frac {a^2 b^3 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )}{\left (a^2+1\right )^{5/2}}-\frac {b^3 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{3 \left (a^2+1\right )^{3/2}}+\frac {a^2 b^3 \text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )}{\left (a^2+1\right )^{5/2}}+\frac {a b^2 \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{\left (a^2+1\right )^2 x}-\frac {b \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{3 \left (a^2+1\right ) x^2}-\frac {a b^3 \log (x)}{\left (a^2+1\right )^2}-\frac {b^2}{3 \left (a^2+1\right ) x}-\frac {\text {arcsinh}(a+b x)^2}{3 x^3} \]

[In]

Int[ArcSinh[a + b*x]^2/x^4,x]

[Out]

-1/3*b^2/((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^Ar
cSinh[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 - S
qrt[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 12

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

Rule 31

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

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]

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2296

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 2317

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 2438

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 2747

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 3403

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 3405

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 3406

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] &&
IGtQ[m, 0]

Rule 5828

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 5843

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 5859

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 6873

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

Rule 6874

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.00 (sec) , antiderivative size = 1830, normalized size of antiderivative = 3.83 \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^4} \, dx=\frac {1}{3} \left (-\frac {b \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{\left (1+a^2\right ) x^2}-\frac {\text {arcsinh}(a+b x)^2}{x^3}-\frac {b^2 \left (1+a^2-3 a \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)\right )}{\left (1+a^2\right )^2 x}+\frac {i b^3 \pi \text {arctanh}\left (\frac {-1-a \tanh \left (\frac {1}{2} \text {arcsinh}(a+b x)\right )}{\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}-\frac {2 i a^2 b^3 \pi \text {arctanh}\left (\frac {-1-a \tanh \left (\frac {1}{2} \text {arcsinh}(a+b x)\right )}{\sqrt {1+a^2}}\right )}{\left (1+a^2\right )^{5/2}}-\frac {3 a b^3 \log \left (-\frac {b x}{a}\right )}{\left (1+a^2\right )^2}+\frac {b^3 \left (-2 \arccos (i a) \text {arctanh}\left (\frac {(-i+a) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}{\sqrt {-1-a^2}}\right )-(\pi -2 i \text {arcsinh}(a+b x)) \text {arctanh}\left (\frac {(i+a) \tan \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}{\sqrt {-1-a^2}}\right )+\left (\arccos (i a)+2 i \text {arctanh}\left (\frac {(-i+a) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}{\sqrt {-1-a^2}}\right )+2 i \text {arctanh}\left (\frac {(i+a) \tan \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}{\sqrt {-1-a^2}}\right )\right ) \log \left (\frac {\sqrt {-1-a^2} e^{-\frac {1}{2} \text {arcsinh}(a+b x)}}{\sqrt {2} \sqrt {b x}}\right )+\left (\arccos (i a)-2 i \left (\text {arctanh}\left (\frac {(-i+a) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}{\sqrt {-1-a^2}}\right )+\text {arctanh}\left (\frac {(i+a) \tan \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}{\sqrt {-1-a^2}}\right )\right )\right ) \log \left (\frac {i \sqrt {-1-a^2} e^{\frac {1}{2} \text {arcsinh}(a+b x)}}{\sqrt {2} \sqrt {b x}}\right )-\left (\arccos (i a)+2 i \text {arctanh}\left (\frac {(-i+a) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}{\sqrt {-1-a^2}}\right )\right ) \log \left (\frac {(i+a) \left (a+i \left (-1+\sqrt {-1-a^2}\right )\right ) \left (i+\cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )\right )}{i+a-\sqrt {-1-a^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}\right )-\left (\arccos (i a)-2 i \text {arctanh}\left (\frac {(-i+a) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}{\sqrt {-1-a^2}}\right )\right ) \log \left (\frac {(i+a) \left (a-i \left (1+\sqrt {-1-a^2}\right )\right ) \left (-i+\cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )\right )}{-i-a+\sqrt {-1-a^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}\right )+i \left (\operatorname {PolyLog}\left (2,-\frac {\left (-i a+\sqrt {-1-a^2}\right ) \left (i+a+\sqrt {-1-a^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )\right )}{-i-a+\sqrt {-1-a^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (i a+\sqrt {-1-a^2}\right ) \left (i+a+\sqrt {-1-a^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )\right )}{-i-a+\sqrt {-1-a^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}\right )\right )\right )}{\left (-1-a^2\right )^{5/2}}-\frac {2 a^2 b^3 \left (-2 \arccos (i a) \text {arctanh}\left (\frac {(-i+a) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}{\sqrt {-1-a^2}}\right )-(\pi -2 i \text {arcsinh}(a+b x)) \text {arctanh}\left (\frac {(i+a) \tan \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}{\sqrt {-1-a^2}}\right )+\left (\arccos (i a)+2 i \text {arctanh}\left (\frac {(-i+a) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}{\sqrt {-1-a^2}}\right )+2 i \text {arctanh}\left (\frac {(i+a) \tan \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}{\sqrt {-1-a^2}}\right )\right ) \log \left (\frac {\sqrt {-1-a^2} e^{-\frac {1}{2} \text {arcsinh}(a+b x)}}{\sqrt {2} \sqrt {b x}}\right )+\left (\arccos (i a)-2 i \left (\text {arctanh}\left (\frac {(-i+a) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}{\sqrt {-1-a^2}}\right )+\text {arctanh}\left (\frac {(i+a) \tan \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}{\sqrt {-1-a^2}}\right )\right )\right ) \log \left (\frac {i \sqrt {-1-a^2} e^{\frac {1}{2} \text {arcsinh}(a+b x)}}{\sqrt {2} \sqrt {b x}}\right )-\left (\arccos (i a)+2 i \text {arctanh}\left (\frac {(-i+a) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}{\sqrt {-1-a^2}}\right )\right ) \log \left (\frac {(i+a) \left (a+i \left (-1+\sqrt {-1-a^2}\right )\right ) \left (i+\cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )\right )}{i+a-\sqrt {-1-a^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}\right )-\left (\arccos (i a)-2 i \text {arctanh}\left (\frac {(-i+a) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}{\sqrt {-1-a^2}}\right )\right ) \log \left (\frac {(i+a) \left (a-i \left (1+\sqrt {-1-a^2}\right )\right ) \left (-i+\cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )\right )}{-i-a+\sqrt {-1-a^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}\right )+i \left (\operatorname {PolyLog}\left (2,-\frac {\left (-i a+\sqrt {-1-a^2}\right ) \left (i+a+\sqrt {-1-a^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )\right )}{-i-a+\sqrt {-1-a^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (i a+\sqrt {-1-a^2}\right ) \left (i+a+\sqrt {-1-a^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )\right )}{-i-a+\sqrt {-1-a^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(a+b x))\right )}\right )\right )\right )}{\left (-1-a^2\right )^{5/2}}\right ) \]

[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

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.60

method	result	size
derivativedivides	\(b^{3} \left (-\frac {a^{4} \operatorname {arcsinh}\left (b x +a \right )^{2}-4 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, a^{3}+7 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, a^{2} \left (b x +a \right )-3 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, a \left (b x +a \right )^{2}-3 \,\operatorname {arcsinh}\left (b x +a \right ) a^{4}+9 \,\operatorname {arcsinh}\left (b x +a \right ) a^{3} \left (b x +a \right )-9 \,\operatorname {arcsinh}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}+3 \,\operatorname {arcsinh}\left (b x +a \right ) a \left (b x +a \right )^{3}+2 a^{2} \operatorname {arcsinh}\left (b x +a \right )^{2}+a^{4}-2 a^{3} \left (b x +a \right )+a^{2} \left (b x +a \right )^{2}-\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, a +\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )+\operatorname {arcsinh}\left (b x +a \right )^{2}+a^{2}-2 a \left (b x +a \right )+\left (b x +a \right )^{2}}{3 \left (a^{2}+1\right )^{2} b^{3} x^{3}}+\frac {2 a \ln \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )}{\left (a^{2}+1\right )^{2}}-\frac {a \ln \left (2 a \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )-\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}+1\right )}{\left (a^{2}+1\right )^{2}}-\frac {\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}-\frac {\operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {\operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {2 a^{2} \operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}-\frac {2 a^{2} \operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {2 a^{2} \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}-\frac {2 a^{2} \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}\right )\)	\(764\)
default	\(b^{3} \left (-\frac {a^{4} \operatorname {arcsinh}\left (b x +a \right )^{2}-4 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, a^{3}+7 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, a^{2} \left (b x +a \right )-3 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, a \left (b x +a \right )^{2}-3 \,\operatorname {arcsinh}\left (b x +a \right ) a^{4}+9 \,\operatorname {arcsinh}\left (b x +a \right ) a^{3} \left (b x +a \right )-9 \,\operatorname {arcsinh}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}+3 \,\operatorname {arcsinh}\left (b x +a \right ) a \left (b x +a \right )^{3}+2 a^{2} \operatorname {arcsinh}\left (b x +a \right )^{2}+a^{4}-2 a^{3} \left (b x +a \right )+a^{2} \left (b x +a \right )^{2}-\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, a +\operatorname {arcsinh}\left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )+\operatorname {arcsinh}\left (b x +a \right )^{2}+a^{2}-2 a \left (b x +a \right )+\left (b x +a \right )^{2}}{3 \left (a^{2}+1\right )^{2} b^{3} x^{3}}+\frac {2 a \ln \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )}{\left (a^{2}+1\right )^{2}}-\frac {a \ln \left (2 a \left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )-\left (b x +a +\sqrt {1+\left (b x +a \right )^{2}}\right )^{2}+1\right )}{\left (a^{2}+1\right )^{2}}-\frac {\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}-\frac {\operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {\operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {2 a^{2} \operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}-\frac {2 a^{2} \operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}+\frac {2 a^{2} \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}-\frac {2 a^{2} \operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{3 \left (a^{2}+1\right )^{\frac {5}{2}}}\right )\)	\(764\)

[In]

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

[Out]

b^3*(-1/3*(a^4*arcsinh(b*x+a)^2-4*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)*a^3+7*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)*
a^2*(b*x+a)-3*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)*a*(b*x+a)^2-3*arcsinh(b*x+a)*a^4+9*arcsinh(b*x+a)*a^3*(b*x+a)
-9*arcsinh(b*x+a)*a^2*(b*x+a)^2+3*arcsinh(b*x+a)*a*(b*x+a)^3+2*a^2*arcsinh(b*x+a)^2+a^4-2*a^3*(b*x+a)+a^2*(b*x
+a)^2-arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)*a+arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)*(b*x+a)+arcsinh(b*x+a)^2+a^2-2*a
*(b*x+a)+(b*x+a)^2)/(a^2+1)^2/b^3/x^3+2/(a^2+1)^2*a*ln(b*x+a+(1+(b*x+a)^2)^(1/2))-1/(a^2+1)^2*a*ln(2*a*(b*x+a+
(1+(b*x+a)^2)^(1/2))-(b*x+a+(1+(b*x+a)^2)^(1/2))^2+1)-1/3/(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/(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/(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/(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/(a^2+1)^(5/2)*a^2*arcsin
h(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/(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/(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/(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))))

Fricas [F]

\[ \int \frac {\text {arcsinh}(a+b x)^2}{x^4} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x^{4}} \,d x } \]

[In]

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

[Out]

integral(arcsinh(b*x + a)^2/x^4, x)

Sympy [F]

\[ \int \frac {\text {arcsinh}(a+b x)^2}{x^4} \, dx=\int \frac {\operatorname {asinh}^{2}{\left (a + b x \right )}}{x^{4}}\, dx \]

[In]

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

[Out]

Integral(asinh(a + b*x)**2/x**4, x)

Maxima [F]

\[ \int \frac {\text {arcsinh}(a+b x)^2}{x^4} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x^{4}} \,d x } \]

[In]

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

[Out]

-1/3*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2/x^3 + integrate(2/3*(b^3*x^2 + 2*a*b^2*x + a^2*b + sqr
t(b^2*x^2 + 2*a*b*x + a^2 + 1)*(b^2*x + a*b) + b)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/(b^3*x^6 +
3*a*b^2*x^5 + (3*a^2*b + b)*x^4 + (a^3 + a)*x^3 + (b^2*x^5 + 2*a*b*x^4 + (a^2 + 1)*x^3)*sqrt(b^2*x^2 + 2*a*b*x
+ a^2 + 1)), x)

Giac [F]

\[ \int \frac {\text {arcsinh}(a+b x)^2}{x^4} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )^{2}}{x^{4}} \,d x } \]

[In]

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

[Out]

integrate(arcsinh(b*x + a)^2/x^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a+b x)^2}{x^4} \, dx=\int \frac {{\mathrm {asinh}\left (a+b\,x\right )}^2}{x^4} \,d x \]

[In]

int(asinh(a + b*x)^2/x^4,x)

[Out]

int(asinh(a + b*x)^2/x^4, x)