3.38 \(\int \frac{\sqrt{d+c^2 d x^2} (a+b \sinh ^{-1}(c x))}{(f+g x)^2} \, dx\)

Optimal. Leaf size=781 \[ -\frac{b c^2 f \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}\right )}{g^2 \sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}+\frac{b c^2 f \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}\right )}{g^2 \sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}-\frac{\sqrt{c^2 d x^2+d} \left (g-c^2 f x\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt{c^2 x^2+1} \left (c^2 f^2+g^2\right ) (f+g x)^2}+\frac{\sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}+\frac{a c^3 f^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{g^2 \sqrt{c^2 x^2+1} \left (c^2 f^2+g^2\right )}+\frac{a c^2 f \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (\frac{g-c^2 f x}{\sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}\right )}{g^2 \sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}-\frac{a \sqrt{c^2 d x^2+d}}{g (f+g x)}+\frac{b c^3 f^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2}{2 g^2 \sqrt{c^2 x^2+1} \left (c^2 f^2+g^2\right )}-\frac{b c^2 f \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right )}{g^2 \sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}+\frac{b c^2 f \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right )}{g^2 \sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}+\frac{b c \sqrt{c^2 d x^2+d} \log (f+g x)}{g^2 \sqrt{c^2 x^2+1}}-\frac{b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{g (f+g x)} \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 2.54735, antiderivative size = 781, normalized size of antiderivative = 1., number of steps used = 35, number of rules used = 22, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.733, Rules used = {5835, 5823, 37, 5813, 12, 1651, 844, 215, 725, 206, 5859, 5857, 5675, 5831, 3324, 3322, 2264, 2190, 2279, 2391, 2668, 31} \[ -\frac{b c^2 f \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}\right )}{g^2 \sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}+\frac{b c^2 f \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}\right )}{g^2 \sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}-\frac{\sqrt{c^2 d x^2+d} \left (g-c^2 f x\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt{c^2 x^2+1} \left (c^2 f^2+g^2\right ) (f+g x)^2}+\frac{\sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}+\frac{a c^3 f^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{g^2 \sqrt{c^2 x^2+1} \left (c^2 f^2+g^2\right )}+\frac{a c^2 f \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (\frac{g-c^2 f x}{\sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}\right )}{g^2 \sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}-\frac{a \sqrt{c^2 d x^2+d}}{g (f+g x)}+\frac{b c^3 f^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2}{2 g^2 \sqrt{c^2 x^2+1} \left (c^2 f^2+g^2\right )}-\frac{b c^2 f \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right )}{g^2 \sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}+\frac{b c^2 f \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right )}{g^2 \sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}+\frac{b c \sqrt{c^2 d x^2+d} \log (f+g x)}{g^2 \sqrt{c^2 x^2+1}}-\frac{b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{g (f+g x)} \]

Antiderivative was successfully verified.

[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 5835

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol]
:> Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 + c^2*x^2)^FracPart[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 5823

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 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 5813

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 12

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

Rule 1651

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 844

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 215

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 725

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 206

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

Rule 5859

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]

Rule 5857

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 5675

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

Rubi steps

\begin{align*} \int \frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{(f+g x)^2} \, dx &=\frac{\sqrt{d+c^2 d x^2} \int \frac{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{(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} \left (a+b \sinh ^{-1}(c x)\right )^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 ) \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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 \left (a+b \sinh ^{-1}(c x)\right )}{\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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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 \left (a+b \sinh ^{-1}(c x)\right )}{(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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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 \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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 \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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 \sinh ^{-1}(c x)}{g^2 \sqrt{1+c^2 x^2}}+\frac{\left (c^2 f^2+g^2\right )^2 \sinh ^{-1}(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 ) \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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{\sinh ^{-1}(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{\sinh ^{-1}(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{\sinh ^{-1}(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} \sinh ^{-1}(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} \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}+\frac{\left (a c^2 f \sqrt{d+c^2 d x^2}\right ) \operatorname{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 ) \operatorname{Subst}\left (\int \frac{x}{c f+g \sinh (x)} \, dx,x,\sinh ^{-1}(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 ) \operatorname{Subst}\left (\int \frac{x}{(c f+g \sinh (x))^2} \, dx,x,\sinh ^{-1}(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} \sinh ^{-1}(c x)}{g (f+g x)}+\frac{a c^3 f^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(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} \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}+\frac{a c^2 f \sqrt{d+c^2 d x^2} \tanh ^{-1}\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 ) \operatorname{Subst}\left (\int \frac{x}{c f+g \sinh (x)} \, dx,x,\sinh ^{-1}(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 ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c e^x f-g+e^{2 x} g} \, dx,x,\sinh ^{-1}(c x)\right )}{g^2 \sqrt{1+c^2 x^2}}+\frac{\left (b c \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{c f+g \sinh (x)} \, dx,x,\sinh ^{-1}(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} \sinh ^{-1}(c x)}{g (f+g x)}+\frac{a c^3 f^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(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} \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}+\frac{a c^2 f \sqrt{d+c^2 d x^2} \tanh ^{-1}\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 ) \operatorname{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 ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c e^x f-g+e^{2 x} g} \, dx,x,\sinh ^{-1}(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 ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c f+2 e^x g-2 \sqrt{c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(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 ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c f+2 e^x g+2 \sqrt{c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(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} \sinh ^{-1}(c x)}{g (f+g x)}+\frac{a c^3 f^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(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} \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}+\frac{a c^2 f \sqrt{d+c^2 d x^2} \tanh ^{-1}\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} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(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} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(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 ) \operatorname{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,\sinh ^{-1}(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 ) \operatorname{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,\sinh ^{-1}(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 ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c f+2 e^x g-2 \sqrt{c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(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 ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c f+2 e^x g+2 \sqrt{c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(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} \sinh ^{-1}(c x)}{g (f+g x)}+\frac{a c^3 f^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(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} \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}+\frac{a c^2 f \sqrt{d+c^2 d x^2} \tanh ^{-1}\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} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(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} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(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 ) \operatorname{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,\sinh ^{-1}(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 ) \operatorname{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,\sinh ^{-1}(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 ) \operatorname{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^{\sinh ^{-1}(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 ) \operatorname{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^{\sinh ^{-1}(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} \sinh ^{-1}(c x)}{g (f+g x)}+\frac{a c^3 f^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(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} \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}+\frac{a c^2 f \sqrt{d+c^2 d x^2} \tanh ^{-1}\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} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(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} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(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} \text{Li}_2\left (-\frac{e^{\sinh ^{-1}(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{Li}_2\left (-\frac{e^{\sinh ^{-1}(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 ) \operatorname{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^{\sinh ^{-1}(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 ) \operatorname{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^{\sinh ^{-1}(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} \sinh ^{-1}(c x)}{g (f+g x)}+\frac{a c^3 f^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(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} \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}+\frac{a c^2 f \sqrt{d+c^2 d x^2} \tanh ^{-1}\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} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(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} \sinh ^{-1}(c x) \log \left (1+\frac{e^{\sinh ^{-1}(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} \text{Li}_2\left (-\frac{e^{\sinh ^{-1}(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{Li}_2\left (-\frac{e^{\sinh ^{-1}(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]  time = 9.76935, size = 1398, normalized size = 1.79 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((a*Sqrt[d*(1 + c^2*x^2)])/(g*(f + g*x))) - (a*c^2*Sqrt[d]*f*Log[f + g*x])/(g^2*Sqrt[c^2*f^2 + g^2]) + (a*c*S
qrt[d]*Log[c*d*x + Sqrt[d]*Sqrt[d*(1 + c^2*x^2)]])/g^2 + (a*c^2*Sqrt[d]*f*Log[d*g - c^2*d*f*x + Sqrt[d]*Sqrt[c
^2*f^2 + g^2]*Sqrt[d*(1 + c^2*x^2)]])/(g^2*Sqrt[c^2*f^2 + g^2]) + (b*c*Sqrt[d*(1 + c^2*x^2)]*((-2*g*ArcSinh[c*
x])/(c*f + c*g*x) + ArcSinh[c*x]^2/Sqrt[1 + c^2*x^2] + ((2*I)*c*f*Pi*ArcTanh[(-g + c*f*Tanh[ArcSinh[c*x]/2])/S
qrt[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)*Tan[(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 + c*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)*ArcSi
nh[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 + c*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)*ArcSi
nh[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*Sqr
t[-(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]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) - g^2])*(-(c*f) + I*g + 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]))])
))/(Sqrt[-(c^2*f^2) - g^2]*Sqrt[1 + c^2*x^2])))/(2*g^2)

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Maple [B]  time = 0.324, size = 1814, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(g*x+f)^2,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{\sqrt{c^{2} d x^{2} + d}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}{g^{2} x^{2} + 2 \, f g x + f^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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