\(\int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{(f+g x)^2} \, dx\) [38]

Optimal result

Integrand size = 30, antiderivative size = 781 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{(f+g x)^2} \, dx=-\frac {a \sqrt {d+c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}+\frac {b c^3 f^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^2}{2 g^2 \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2}}-\frac {\left (g-c^2 f x\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c \left (c^2 f^2+g^2\right ) (f+g x)^2 \sqrt {1+c^2 x^2}}+\frac {\sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)^2}+\frac {a c^2 f \sqrt {d+c^2 d x^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}-\frac {b c^2 f \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {b c^2 f \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {b c \sqrt {d+c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1+c^2 x^2}}-\frac {b c^2 f \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {b c^2 f \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}} \]

[Out]

-a*(c^2*d*x^2+d)^(1/2)/g/(g*x+f)-b*arcsinh(c*x)*(c^2*d*x^2+d)^(1/2)/g/(g*x+f)+a*c^3*f^2*arcsinh(c*x)*(c^2*d*x^
2+d)^(1/2)/g^2/(c^2*f^2+g^2)/(c^2*x^2+1)^(1/2)+1/2*b*c^3*f^2*arcsinh(c*x)^2*(c^2*d*x^2+d)^(1/2)/g^2/(c^2*f^2+g
^2)/(c^2*x^2+1)^(1/2)-1/2*(-c^2*f*x+g)^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/c/(c^2*f^2+g^2)/(g*x+f)^2/
(c^2*x^2+1)^(1/2)+b*c*ln(g*x+f)*(c^2*d*x^2+d)^(1/2)/g^2/(c^2*x^2+1)^(1/2)+a*c^2*f*arctanh((-c^2*f*x+g)/(c^2*f^
2+g^2)^(1/2)/(c^2*x^2+1)^(1/2))*(c^2*d*x^2+d)^(1/2)/g^2/(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2)-b*c^2*f*arcsinh(
c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2+g^2)^(1/2)))*(c^2*d*x^2+d)^(1/2)/g^2/(c^2*f^2+g^2)^(1/2)/(c^
2*x^2+1)^(1/2)+b*c^2*f*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2+g^2)^(1/2)))*(c^2*d*x^2+d)^(1
/2)/g^2/(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2)-b*c^2*f*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2+g^2)^
(1/2)))*(c^2*d*x^2+d)^(1/2)/g^2/(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2)+b*c^2*f*polylog(2,-(c*x+(c^2*x^2+1)^(1/2
))*g/(c*f+(c^2*f^2+g^2)^(1/2)))*(c^2*d*x^2+d)^(1/2)/g^2/(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2)+1/2*(a+b*arcsinh
(c*x))^2*(c^2*x^2+1)^(1/2)*(c^2*d*x^2+d)^(1/2)/b/c/(g*x+f)^2

Rubi [A] (verified)

Time = 1.84 (sec) , antiderivative size = 781, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {5845, 5839, 37, 5834, 12, 1665, 858, 221, 739, 212, 5856, 5855, 5783, 5843, 3405, 3403, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{(f+g x)^2} \, dx=-\frac {\sqrt {c^2 d x^2+d} \left (g-c^2 f x\right )^2 (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {c^2 x^2+1} \left (c^2 f^2+g^2\right ) (f+g x)^2}+\frac {\sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)^2}+\frac {a c^3 f^2 \text {arcsinh}(c x) \sqrt {c^2 d x^2+d}}{g^2 \sqrt {c^2 x^2+1} \left (c^2 f^2+g^2\right )}+\frac {a c^2 f \sqrt {c^2 d x^2+d} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}-\frac {a \sqrt {c^2 d x^2+d}}{g (f+g x)}-\frac {b c^2 f \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}+\frac {b c^2 f \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^2 \sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}-\frac {b c^2 f \text {arcsinh}(c x) \sqrt {c^2 d x^2+d} \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g^2 \sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}+\frac {b c^2 f \text {arcsinh}(c x) \sqrt {c^2 d x^2+d} \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g^2 \sqrt {c^2 x^2+1} \sqrt {c^2 f^2+g^2}}-\frac {b \text {arcsinh}(c x) \sqrt {c^2 d x^2+d}}{g (f+g x)}+\frac {b c^3 f^2 \text {arcsinh}(c x)^2 \sqrt {c^2 d x^2+d}}{2 g^2 \sqrt {c^2 x^2+1} \left (c^2 f^2+g^2\right )}+\frac {b c \sqrt {c^2 d x^2+d} \log (f+g x)}{g^2 \sqrt {c^2 x^2+1}} \]

[In]

Int[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(f + g*x)^2,x]

[Out]

-((a*Sqrt[d + c^2*d*x^2])/(g*(f + g*x))) - (b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(g*(f + g*x)) + (a*c^3*f^2*Sqr
t[d + c^2*d*x^2]*ArcSinh[c*x])/(g^2*(c^2*f^2 + g^2)*Sqrt[1 + c^2*x^2]) + (b*c^3*f^2*Sqrt[d + c^2*d*x^2]*ArcSin
h[c*x]^2)/(2*g^2*(c^2*f^2 + g^2)*Sqrt[1 + c^2*x^2]) - ((g - c^2*f*x)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]
)^2)/(2*b*c*(c^2*f^2 + g^2)*(f + g*x)^2*Sqrt[1 + c^2*x^2]) + (Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*Arc
Sinh[c*x])^2)/(2*b*c*(f + g*x)^2) + (a*c^2*f*Sqrt[d + c^2*d*x^2]*ArcTanh[(g - c^2*f*x)/(Sqrt[c^2*f^2 + g^2]*Sq
rt[1 + c^2*x^2])])/(g^2*Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2]) - (b*c^2*f*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log
[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(g^2*Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2]) + (b*c^2*f*S
qrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/(g^2*Sqrt[c^2*f^2 + g
^2]*Sqrt[1 + c^2*x^2]) + (b*c*Sqrt[d + c^2*d*x^2]*Log[f + g*x])/(g^2*Sqrt[1 + c^2*x^2]) - (b*c^2*f*Sqrt[d + c^
2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/(g^2*Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*
x^2]) + (b*c^2*f*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/(g^2*Sqrt[
c^2*f^2 + g^2]*Sqrt[1 + c^2*x^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 37

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1665

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[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 5783

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

Rule 5834

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(p_.), x_Symbol] :>
With[{u = IntHide[(f + g*x)^p*(d + e*x)^m, x]}, Dist[(a + b*ArcSinh[c*x])^n, u, x] - Dist[b*c*n, Int[Simplify
Integrand[u*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]

Rule 5839

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

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 5845

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

Rule 5855

Int[ArcSinh[(c_.)*(x_)]^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = ExpandIntegrand[(d + e
*x^2)^p*ArcSinh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] &&
IGtQ[n, 0] && EqQ[e, c^2*d] && IntegerQ[p - 1/2]

Rule 5856

Int[(ArcSinh[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x^2)^p, RFx*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x]
&& IGtQ[n, 0] && EqQ[e, c^2*d] && IntegerQ[p - 1/2]

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.67 (sec) , antiderivative size = 1384, normalized size of antiderivative = 1.77 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{(f+g x)^2} \, dx=\frac {-\frac {2 a g \sqrt {d+c^2 d x^2}}{f+g x}-\frac {2 a c^2 \sqrt {d} f \log (f+g x)}{\sqrt {c^2 f^2+g^2}}+2 a c \sqrt {d} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {2 a c^2 \sqrt {d} f \log \left (d \left (g-c^2 f x\right )+\sqrt {d} \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}\right )}{\sqrt {c^2 f^2+g^2}}+b c \sqrt {d+c^2 d x^2} \left (-\frac {2 g \text {arcsinh}(c x)}{c f+c g x}+\frac {\text {arcsinh}(c x)^2}{\sqrt {1+c^2 x^2}}+\frac {2 i c f \pi \text {arctanh}\left (\frac {-g+c f \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}{\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}+\frac {2 \log \left (1+\frac {g x}{f}\right )}{\sqrt {1+c^2 x^2}}+\frac {2 c f \left (2 \arccos \left (-\frac {i c f}{g}\right ) \text {arctanh}\left (\frac {(c f+i g) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )}{\sqrt {-c^2 f^2-g^2}}\right )+(\pi -2 i \text {arcsinh}(c x)) \text {arctanh}\left (\frac {(c f-i g) \tan \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )}{\sqrt {-c^2 f^2-g^2}}\right )+\left (\arccos \left (-\frac {i c f}{g}\right )-2 i \text {arctanh}\left (\frac {(c f+i g) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )}{\sqrt {-c^2 f^2-g^2}}\right )-2 i \text {arctanh}\left (\frac {(c f-i g) \tan \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )}{\sqrt {-c^2 f^2-g^2}}\right )\right ) \log \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) e^{-\frac {1}{2} \text {arcsinh}(c x)} \sqrt {-c^2 f^2-g^2}}{\sqrt {-i g} \sqrt {c (f+g x)}}\right )+\left (\arccos \left (-\frac {i c f}{g}\right )+2 i \left (\text {arctanh}\left (\frac {(c f+i g) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )}{\sqrt {-c^2 f^2-g^2}}\right )+\text {arctanh}\left (\frac {(c f-i g) \tan \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )}{\sqrt {-c^2 f^2-g^2}}\right )\right )\right ) \log \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) e^{\frac {1}{2} \text {arcsinh}(c x)} \sqrt {-c^2 f^2-g^2}}{\sqrt {-i g} \sqrt {c (f+g x)}}\right )-\left (\arccos \left (-\frac {i c f}{g}\right )+2 i \text {arctanh}\left (\frac {(c f+i g) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )}{\sqrt {-c^2 f^2-g^2}}\right )\right ) \log \left (\frac {(i c f+g) \left (-i c f+g+\sqrt {-c^2 f^2-g^2}\right ) \left (1+i \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )}{g \left (i c f+g+i \sqrt {-c^2 f^2-g^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )}\right )-\left (\arccos \left (-\frac {i c f}{g}\right )-2 i \text {arctanh}\left (\frac {(c f+i g) \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )}{\sqrt {-c^2 f^2-g^2}}\right )\right ) \log \left (\frac {(i c f+g) \left (i c f-g+\sqrt {-c^2 f^2-g^2}\right ) \left (i+\cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )}{g \left (c f-i g+\sqrt {-c^2 f^2-g^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (i c f+\sqrt {-c^2 f^2-g^2}\right ) \left (i c f+g-i \sqrt {-c^2 f^2-g^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )}{g \left (i c f+g+i \sqrt {-c^2 f^2-g^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (c f+i \sqrt {-c^2 f^2-g^2}\right ) \left (-c f+i g+\sqrt {-c^2 f^2-g^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )}{g \left (i c f+g+i \sqrt {-c^2 f^2-g^2} \cot \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )}\right )\right )\right )}{\sqrt {-c^2 f^2-g^2} \sqrt {1+c^2 x^2}}\right )}{2 g^2} \]

[In]

Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(f + g*x)^2,x]

[Out]

((-2*a*g*Sqrt[d + c^2*d*x^2])/(f + g*x) - (2*a*c^2*Sqrt[d]*f*Log[f + g*x])/Sqrt[c^2*f^2 + g^2] + 2*a*c*Sqrt[d]
*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (2*a*c^2*Sqrt[d]*f*Log[d*(g - c^2*f*x) + Sqrt[d]*Sqrt[c^2*f^2 + g^
2]*Sqrt[d + c^2*d*x^2]])/Sqrt[c^2*f^2 + g^2] + b*c*Sqrt[d + c^2*d*x^2]*((-2*g*ArcSinh[c*x])/(c*f + c*g*x) + Ar
cSinh[c*x]^2/Sqrt[1 + c^2*x^2] + ((2*I)*c*f*Pi*ArcTanh[(-g + c*f*Tanh[ArcSinh[c*x]/2])/Sqrt[c^2*f^2 + g^2]])/(
Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2]) + (2*Log[1 + (g*x)/f])/Sqrt[1 + c^2*x^2] + (2*c*f*(2*ArcCos[((-I)*c*f)/
g]*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (Pi - (2*I)*ArcSinh[c*x])*
ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (ArcCos[((-I)*c*f)/g] - (2*I)
*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] - (2*I)*ArcTanh[((c*f - I*g)*T
an[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((1/2 - I/2)*Sqrt[-(c^2*f^2) - g^2])/(E^(ArcSinh
[c*x]/2)*Sqrt[(-I)*g]*Sqrt[c*(f + g*x)])] + (ArcCos[((-I)*c*f)/g] + (2*I)*(ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I
)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^
2*f^2) - g^2]]))*Log[((1/2 + I/2)*E^(ArcSinh[c*x]/2)*Sqrt[-(c^2*f^2) - g^2])/(Sqrt[(-I)*g]*Sqrt[c*(f + g*x)])]
- (ArcCos[((-I)*c*f)/g] + (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]
])*Log[((I*c*f + g)*((-I)*c*f + g + Sqrt[-(c^2*f^2) - g^2])*(1 + I*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*
f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))] - (ArcCos[((-I)*c*f)/g] - (2*I)*ArcTanh[(
(c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((I*c*f + g)*(I*c*f - g + Sqrt[-(c^
2*f^2) - g^2])*(I + Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(c*f - I*g + Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*
ArcSinh[c*x])/4]))] + I*(PolyLog[2, ((I*c*f + Sqrt[-(c^2*f^2) - g^2])*(I*c*f + g - I*Sqrt[-(c^2*f^2) - g^2]*Co
t[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))]
- PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) - g^2])*(-(c*f) + I*g + Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[
c*x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))])))/(Sqrt[-(c^2*f^2) -
g^2]*Sqrt[1 + c^2*x^2])))/(2*g^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1700\) vs. \(2(745)=1490\).

Time = 0.82 (sec) , antiderivative size = 1701, normalized size of antiderivative = 2.18

method	result	size
default	\(\text {Expression too large to display}\)	\(1701\)
parts	\(\text {Expression too large to display}\)	\(1701\)

[In]

int((a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(g*x+f)^2,x,method=_RETURNVERBOSE)

[Out]

a/g^2*(-1/d/(c^2*f^2+g^2)*g^2/(x+f/g)*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(3/2)-c^2*f*g/
(c^2*f^2+g^2)*(((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2)-c^2*d*f/g*ln((-c^2*d*f/g+c^2*d*
(x+f/g))/(c^2*d)^(1/2)+((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2))/(c^2*d)^(1/2)-d*(c^2*f
^2+g^2)/g^2/(d*(c^2*f^2+g^2)/g^2)^(1/2)*ln((2*d*(c^2*f^2+g^2)/g^2-2*c^2*d*f/g*(x+f/g)+2*(d*(c^2*f^2+g^2)/g^2)^
(1/2)*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2))/(x+f/g)))+2*c^2/(c^2*f^2+g^2)*g^2*(1/4*
(2*c^2*d*(x+f/g)-2*c^2*d*f/g)/c^2/d*((x+f/g)^2*c^2*d-2*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2)+1/8*(4*c^2
*d^2*(c^2*f^2+g^2)/g^2-4*c^4*d^2*f^2/g^2)/c^2/d*ln((-c^2*d*f/g+c^2*d*(x+f/g))/(c^2*d)^(1/2)+((x+f/g)^2*c^2*d-2
*c^2*d*f/g*(x+f/g)+d*(c^2*f^2+g^2)/g^2)^(1/2))/(c^2*d)^(1/2)))+1/2*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*a
rcsinh(c*x)^2*c/g^2+b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/(c^2*x^2+1)/g^2/(g*x+f)*x^3*c^4*f-b*(d*(c^2*x^2+1))^(
1/2)*arcsinh(c*x)/g^2/(g*x+f)*x*c^2*f-b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/(c^2*x^2+1)/g/(g*x+f)*x^2*c^2+b*(d*
(c^2*x^2+1))^(1/2)*arcsinh(c*x)/(c^2*x^2+1)^(1/2)/g/(g*x+f)*x*c+b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/(c^2*x^2+
1)/g^2/(g*x+f)*x*c^2*f+b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)/(c^2*x^2+1)^(1/2)/g^2/(g*x+f)*c*f-b*(d*(c^2*x^2+1)
)^(1/2)*arcsinh(c*x)/(c^2*x^2+1)/g/(g*x+f)-b*(d*(c^2*x^2+1))^(1/2)*c^2/(c^2*x^2+1)^(1/2)/g^2/(c^2*f^2+g^2)^(1/
2)*arcsinh(c*x)*ln((-(c*x+(c^2*x^2+1)^(1/2))*g-c*f+(c^2*f^2+g^2)^(1/2))/(-c*f+(c^2*f^2+g^2)^(1/2)))*f+b*(d*(c^
2*x^2+1))^(1/2)*c^2/(c^2*x^2+1)^(1/2)/g^2/(c^2*f^2+g^2)^(1/2)*arcsinh(c*x)*ln(((c*x+(c^2*x^2+1)^(1/2))*g+c*f+(
c^2*f^2+g^2)^(1/2))/(c*f+(c^2*f^2+g^2)^(1/2)))*f+b*(d*(c^2*x^2+1))^(1/2)*c^3/(c^2*x^2+1)^(1/2)/g^2/(c^2*f^2+g^
2)*ln((c*x+(c^2*x^2+1)^(1/2))^2*g+2*c*f*(c*x+(c^2*x^2+1)^(1/2))-g)*f^2-2*b*(d*(c^2*x^2+1))^(1/2)*c^3/(c^2*x^2+
1)^(1/2)/g^2/(c^2*f^2+g^2)*ln(c*x+(c^2*x^2+1)^(1/2))*f^2-b*(d*(c^2*x^2+1))^(1/2)*c^2/(c^2*x^2+1)^(1/2)/g^2/(c^
2*f^2+g^2)^(1/2)*dilog((-(c*x+(c^2*x^2+1)^(1/2))*g-c*f+(c^2*f^2+g^2)^(1/2))/(-c*f+(c^2*f^2+g^2)^(1/2)))*f+b*(d
*(c^2*x^2+1))^(1/2)*c^2/(c^2*x^2+1)^(1/2)/g^2/(c^2*f^2+g^2)^(1/2)*dilog(((c*x+(c^2*x^2+1)^(1/2))*g+c*f+(c^2*f^
2+g^2)^(1/2))/(c*f+(c^2*f^2+g^2)^(1/2)))*f+b*(d*(c^2*x^2+1))^(1/2)*c/(c^2*x^2+1)^(1/2)/(c^2*f^2+g^2)*ln((c*x+(
c^2*x^2+1)^(1/2))^2*g+2*c*f*(c*x+(c^2*x^2+1)^(1/2))-g)-2*b*(d*(c^2*x^2+1))^(1/2)*c/(c^2*x^2+1)^(1/2)/(c^2*f^2+
g^2)*ln(c*x+(c^2*x^2+1)^(1/2))

Fricas [F]

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

[In]

integrate((a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(g*x+f)^2,x, algorithm="fricas")

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)/(g^2*x^2 + 2*f*g*x + f^2), x)

Sympy [F]

\[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{(f+g x)^2} \, dx=\int \frac {\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (f + g x\right )^{2}}\, dx \]

[In]

integrate((a+b*asinh(c*x))*(c**2*d*x**2+d)**(1/2)/(g*x+f)**2,x)

[Out]

Integral(sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))/(f + g*x)**2, x)

Maxima [F]

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

[In]

integrate((a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(g*x+f)^2,x, algorithm="maxima")

[Out]

-(c^2*d*f*arcsinh(c*f*x/(g*abs(x + f/g)) - 1/(c*abs(x + f/g)))/(sqrt(c^2*d*f^2/g^2 + d)*g^3) - c*sqrt(d)*arcsi
nh(c*x)/g^2 + sqrt(c^2*d*x^2 + d)/(g^2*x + f*g))*a + b*integrate(sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 +
1))/(g^2*x^2 + 2*f*g*x + f^2), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{(f+g x)^2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(g*x+f)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

int(((a + b*asinh(c*x))*(d + c^2*d*x^2)^(1/2))/(f + g*x)^2,x)

[Out]

int(((a + b*asinh(c*x))*(d + c^2*d*x^2)^(1/2))/(f + g*x)^2, x)