3.513 \(\int \frac{a+b \cosh ^{-1}(c x)}{(d+e x^2)^3} \, dx\)

Optimal. Leaf size=1234 \[ \text{result too large to display} \]

[Out]

-(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x)) - (b*c*Sqrt[-1 + c*x]*S
qrt[1 + c*x])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (a + b*ArcCosh[c*x])/(16*(-d)^(3/2)*Sqrt[e]
*(Sqrt[-d] - Sqrt[e]*x)^2) - (3*(a + b*ArcCosh[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcCosh
[c*x])/(16*(-d)^(3/2)*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)^2) + (3*(a + b*ArcCosh[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] +
Sqrt[e]*x)) - (b*c^3*ArcTanh[(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 +
c*x])])/(8*d*(c*Sqrt[-d] - Sqrt[e])^(3/2)*(c*Sqrt[-d] + Sqrt[e])^(3/2)*Sqrt[e]) + (3*b*c*ArcTanh[(Sqrt[c*Sqrt
[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(8*d^2*Sqrt[c*Sqrt[-d] - Sqrt[e]]
*Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[e]) + (b*c^3*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[
-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(8*d*(c*Sqrt[-d] - Sqrt[e])^(3/2)*(c*Sqrt[-d] + Sqrt[e])^(3/2)*Sqrt[e]) - (3*
b*c*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(8*d^2*Sq
rt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[e]) + (3*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcC
osh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[
e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(16*(-d)^(5/2)*Sqrt[e]) + (3*(a + b*ArcCosh[c*x])*Log[1
- (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcCosh[c*
x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*b*PolyLo
g[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(16*(-d)^(5/2)*Sqrt[e]) + (3*b*PolyLog[2,
(Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*b*PolyLog[2, -((Sqr
t[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(16*(-d)^(5/2)*Sqrt[e]) + (3*b*PolyLog[2, (Sqrt[e]*E
^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(16*(-d)^(5/2)*Sqrt[e])

________________________________________________________________________________________

Rubi [A]  time = 1.46034, antiderivative size = 1234, normalized size of antiderivative = 1., number of steps used = 34, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {5707, 5802, 96, 93, 208, 5800, 5562, 2190, 2279, 2391} \[ -\frac{b \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x+1}}{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x-1}}\right ) c^3}{8 d \left (c \sqrt{-d}-\sqrt{e}\right )^{3/2} \left (\sqrt{-d} c+\sqrt{e}\right )^{3/2} \sqrt{e}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x+1}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x-1}}\right ) c^3}{8 d \left (c \sqrt{-d}-\sqrt{e}\right )^{3/2} \left (\sqrt{-d} c+\sqrt{e}\right )^{3/2} \sqrt{e}}+\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x+1}}{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x-1}}\right ) c}{8 d^2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{e}}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x+1}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x-1}}\right ) c}{8 d^2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{e}}-\frac{b \sqrt{c x-1} \sqrt{c x+1} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{b \sqrt{c x-1} \sqrt{c x+1} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\sqrt{e} x+\sqrt{-d}\right )}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{e} x+\sqrt{-d}\right )}-\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )^2}+\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{e} x+\sqrt{-d}\right )^2}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{c \sqrt{-d}-\sqrt{-d c^2-e}}+1\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d} c+\sqrt{-d c^2-e}}+1\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 b \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 b \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right )}{16 (-d)^{5/2} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x)) - (b*c*Sqrt[-1 + c*x]*S
qrt[1 + c*x])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (a + b*ArcCosh[c*x])/(16*(-d)^(3/2)*Sqrt[e]
*(Sqrt[-d] - Sqrt[e]*x)^2) - (3*(a + b*ArcCosh[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcCosh
[c*x])/(16*(-d)^(3/2)*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)^2) + (3*(a + b*ArcCosh[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] +
Sqrt[e]*x)) - (b*c^3*ArcTanh[(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 +
c*x])])/(8*d*(c*Sqrt[-d] - Sqrt[e])^(3/2)*(c*Sqrt[-d] + Sqrt[e])^(3/2)*Sqrt[e]) + (3*b*c*ArcTanh[(Sqrt[c*Sqrt
[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(8*d^2*Sqrt[c*Sqrt[-d] - Sqrt[e]]
*Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[e]) + (b*c^3*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[
-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(8*d*(c*Sqrt[-d] - Sqrt[e])^(3/2)*(c*Sqrt[-d] + Sqrt[e])^(3/2)*Sqrt[e]) - (3*
b*c*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(8*d^2*Sq
rt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[e]) + (3*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcC
osh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[
e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(16*(-d)^(5/2)*Sqrt[e]) + (3*(a + b*ArcCosh[c*x])*Log[1
- (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcCosh[c*
x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*b*PolyLo
g[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(16*(-d)^(5/2)*Sqrt[e]) + (3*b*PolyLog[2,
(Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*b*PolyLog[2, -((Sqr
t[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(16*(-d)^(5/2)*Sqrt[e]) + (3*b*PolyLog[2, (Sqrt[e]*E
^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(16*(-d)^(5/2)*Sqrt[e])

Rule 5707

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p
] && (p > 0 || IGtQ[n, 0])

Rule 5802

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

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

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

Rule 5800

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Subst[Int[((a + b*x)^n*Sinh[x
])/(c*d + e*Cosh[x]), x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 5562

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

Rubi steps

\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx &=\int \left (-\frac{e^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{8 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}-e x\right )^3}-\frac{3 e \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}-e x\right )^2}-\frac{e^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{8 (-d)^{3/2} \left (\sqrt{-d} \sqrt{e}+e x\right )^3}-\frac{3 e \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \left (\sqrt{-d} \sqrt{e}+e x\right )^2}-\frac{3 e \left (a+b \cosh ^{-1}(c x)\right )}{8 d^2 \left (-d e-e^2 x^2\right )}\right ) \, dx\\ &=-\frac{(3 e) \int \frac{a+b \cosh ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{16 d^2}-\frac{(3 e) \int \frac{a+b \cosh ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{16 d^2}-\frac{(3 e) \int \frac{a+b \cosh ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{8 d^2}-\frac{e^{3/2} \int \frac{a+b \cosh ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}-e x\right )^3} \, dx}{8 (-d)^{3/2}}-\frac{e^{3/2} \int \frac{a+b \cosh ^{-1}(c x)}{\left (\sqrt{-d} \sqrt{e}+e x\right )^3} \, dx}{8 (-d)^{3/2}}\\ &=-\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{(3 b c) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (\sqrt{-d} \sqrt{e}-e x\right )} \, dx}{16 d^2}-\frac{(3 b c) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (\sqrt{-d} \sqrt{e}+e x\right )} \, dx}{16 d^2}+\frac{\left (b c \sqrt{e}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{16 (-d)^{3/2}}-\frac{\left (b c \sqrt{e}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{16 (-d)^{3/2}}-\frac{(3 e) \int \left (-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{2 d e \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{8 d^2}\\ &=-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{3 \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{16 (-d)^{5/2}}-\frac{3 \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{16 (-d)^{5/2}}+\frac{(3 b c) \operatorname{Subst}\left (\int \frac{1}{c \sqrt{-d} \sqrt{e}+e-\left (c \sqrt{-d} \sqrt{e}-e\right ) x^2} \, dx,x,\frac{\sqrt{1+c x}}{\sqrt{-1+c x}}\right )}{8 d^2}-\frac{(3 b c) \operatorname{Subst}\left (\int \frac{1}{c \sqrt{-d} \sqrt{e}-e-\left (c \sqrt{-d} \sqrt{e}+e\right ) x^2} \, dx,x,\frac{\sqrt{1+c x}}{\sqrt{-1+c x}}\right )}{8 d^2}+\frac{\left (b c^3\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (\sqrt{-d} \sqrt{e}-e x\right )} \, dx}{16 d \left (c^2 d+e\right )}-\frac{\left (b c^3\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x} \left (\sqrt{-d} \sqrt{e}+e x\right )} \, dx}{16 d \left (c^2 d+e\right )}\\ &=-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{3 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d^2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e}}-\frac{3 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d^2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e}}-\frac{3 \operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{c \sqrt{-d}-\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{16 (-d)^{5/2}}-\frac{3 \operatorname{Subst}\left (\int \frac{(a+b x) \sinh (x)}{c \sqrt{-d}+\sqrt{e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{16 (-d)^{5/2}}+\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{c \sqrt{-d} \sqrt{e}+e-\left (c \sqrt{-d} \sqrt{e}-e\right ) x^2} \, dx,x,\frac{\sqrt{1+c x}}{\sqrt{-1+c x}}\right )}{8 d \left (c^2 d+e\right )}-\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{c \sqrt{-d} \sqrt{e}-e-\left (c \sqrt{-d} \sqrt{e}+e\right ) x^2} \, dx,x,\frac{\sqrt{1+c x}}{\sqrt{-1+c x}}\right )}{8 d \left (c^2 d+e\right )}\\ &=-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{3 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d^2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e}}+\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e} \left (c^2 d+e\right )}-\frac{3 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d^2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e}}-\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e} \left (c^2 d+e\right )}-\frac{3 \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}-\sqrt{-c^2 d-e}-\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 (-d)^{5/2}}-\frac{3 \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}+\sqrt{-c^2 d-e}-\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 (-d)^{5/2}}-\frac{3 \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}-\sqrt{-c^2 d-e}+\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 (-d)^{5/2}}-\frac{3 \operatorname{Subst}\left (\int \frac{e^x (a+b x)}{c \sqrt{-d}+\sqrt{-c^2 d-e}+\sqrt{e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 (-d)^{5/2}}\\ &=-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{3 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d^2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e}}+\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e} \left (c^2 d+e\right )}-\frac{3 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d^2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e}}-\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e} \left (c^2 d+e\right )}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{(3 b) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{(3 b) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{(3 b) \operatorname{Subst}\left (\int \log \left (1-\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{(3 b) \operatorname{Subst}\left (\int \log \left (1+\frac{\sqrt{e} e^x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{16 (-d)^{5/2} \sqrt{e}}\\ &=-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{3 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d^2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e}}+\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e} \left (c^2 d+e\right )}-\frac{3 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d^2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e}}-\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e} \left (c^2 d+e\right )}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{e} x}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{16 (-d)^{5/2} \sqrt{e}}\\ &=-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )^2}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )^2}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{16 d^2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{3 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d^2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e}}+\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e} \left (c^2 d+e\right )}-\frac{3 b c \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d^2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e}}-\frac{b c^3 \tanh ^{-1}\left (\frac{\sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{1+c x}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{-1+c x}}\right )}{8 d \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}} \sqrt{e} \left (c^2 d+e\right )}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 b \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 b \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 b \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 b \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt{e}}\\ \end{align*}

Mathematica [C]  time = 6.3874, size = 1184, normalized size = 0.96 \[ \frac{3 a x}{8 d^2 \left (e x^2+d\right )}+\frac{a x}{4 d \left (e x^2+d\right )^2}+\frac{3 a \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{8 d^{5/2} \sqrt{e}}+b \left (\frac{3 \left (\frac{\cosh ^{-1}(c x)}{\sqrt{e} x-i \sqrt{d}}+\frac{c \log \left (\frac{2 e \left (\sqrt{d} x c^2+i \sqrt{e}-i \sqrt{-d c^2-e} \sqrt{c x-1} \sqrt{c x+1}\right )}{c \sqrt{-d c^2-e} \left (i \sqrt{e} x+\sqrt{d}\right )}\right )}{\sqrt{-d c^2-e}}\right )}{16 d^2 \sqrt{e}}-\frac{3 \left (-\frac{\cosh ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}-\frac{c \log \left (\frac{2 e \left (-i \sqrt{d} x c^2-\sqrt{e}+\sqrt{-d c^2-e} \sqrt{c x-1} \sqrt{c x+1}\right )}{c \sqrt{-d c^2-e} \left (\sqrt{e} x+i \sqrt{d}\right )}\right )}{\sqrt{-d c^2-e}}\right )}{16 d^2 \sqrt{e}}+\frac{i \left (\frac{\sqrt{d} \left (\log \left (\frac{e \sqrt{d c^2+e} \left (-\sqrt{d} x c^2-i \sqrt{e}+\sqrt{d c^2+e} \sqrt{c x-1} \sqrt{c x+1}\right )}{c^3 \left (d+i \sqrt{e} x \sqrt{d}\right )}\right )+\log (4)\right ) c^3}{\sqrt{e} \left (d c^2+e\right )^{3/2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} c}{\left (d c^2+e\right ) \left (\sqrt{e} x-i \sqrt{d}\right )}-\frac{\cosh ^{-1}(c x)}{\sqrt{e} \left (\sqrt{e} x-i \sqrt{d}\right )^2}\right )}{16 d^{3/2}}-\frac{i \left (-\frac{\sqrt{d} \left (\log \left (\frac{e \sqrt{d c^2+e} \left (\sqrt{d} x c^2-i \sqrt{e}+\sqrt{d c^2+e} \sqrt{c x-1} \sqrt{c x+1}\right )}{c^3 \left (d-i \sqrt{d} \sqrt{e} x\right )}\right )+\log (4)\right ) c^3}{\sqrt{e} \left (d c^2+e\right )^{3/2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} c}{\left (d c^2+e\right ) \left (\sqrt{e} x+i \sqrt{d}\right )}-\frac{\cosh ^{-1}(c x)}{\sqrt{e} \left (\sqrt{e} x+i \sqrt{d}\right )^2}\right )}{16 d^{3/2}}+\frac{3 i \left (\cosh ^{-1}(c x) \left (2 \left (\log \left (\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{i c \sqrt{d}-\sqrt{-d c^2-e}}+1\right )+\log \left (\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{i \sqrt{d} c+\sqrt{-d c^2-e}}+1\right )\right )-\cosh ^{-1}(c x)\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right )\right )}{32 d^{5/2} \sqrt{e}}-\frac{3 i \left (\cosh ^{-1}(c x) \left (2 \left (\log \left (\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d c^2-e}-i c \sqrt{d}}+1\right )+\log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right )\right )-\cosh ^{-1}(c x)\right )+2 \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right )\right )}{32 d^{5/2} \sqrt{e}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*x)/(4*d*(d + e*x^2)^2) + (3*a*x)/(8*d^2*(d + e*x^2)) + (3*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]
) + b*((3*(ArcCosh[c*x]/((-I)*Sqrt[d] + Sqrt[e]*x) + (c*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d)
- e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]))/(16*
d^2*Sqrt[e]) - (3*(-(ArcCosh[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(
c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]
))/(16*d^2*Sqrt[e]) + ((I/16)*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - Arc
Cosh[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) + (c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e]
- c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*
(c^2*d + e)^(3/2))))/d^(3/2) - ((I/16)*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))
- ArcCosh[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - (c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt
[e] + c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[
e]*(c^2*d + e)^(3/2))))/d^(3/2) + (((3*I)/32)*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x
])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])
) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, -((Sqrt[e]*E^A
rcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))]))/(d^(5/2)*Sqrt[e]) - (((3*I)/32)*(ArcCosh[c*x]*(-ArcCosh[c*
x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + Log[1 - (Sqrt[e]*E^ArcCosh[c
*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(
c^2*d) - e]))] + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])]))/(d^(5/2)*Sqrt[e])
)

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Maple [C]  time = 1.283, size = 3128, normalized size = 2.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))/(e*x^2+d)^3,x)

[Out]

3/8*a/d^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+5/8*c^6*b/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arccosh(c*x)*x-c^5*b*(-(
2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e/((-2*c^2*d+2*(c^2*d*
(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^3/(c^2*d+e)-c^5*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan((c*x+
(c*x-1)^(1/2)*(c*x+1)^(1/2))*e/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^3/(c^2*d+e)+3/16*c*b/d^2/(c^
2*d+e)*e*sum(_R1/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*ln((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*
x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))-3/16*c*b/d^2/(c^2*d+e)*e*sum(1/_
R1/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*ln((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2
)*(c*x+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))+7/4*c^5*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)
+e)*e)^(1/2)*arctanh((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/(c^
2*d+e)^2/e^2+7/4*c^5*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)
)*e/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/(c^2*d+e)^2/e^2-1/8*c^5*b/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*(
c*x+1)^(1/2)*(c*x-1)^(1/2)+3/8*c^4*b/d^2/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*arccosh(c*x)*x^3*e^2+5/8*c^4*b/d/(c^2*e
*x^2+c^2*d)^2/(c^2*d+e)*arccosh(c*x)*x*e+3/8*c^6*b*e/d/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arccosh(c*x)*x^3+3/8*c*b*
(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e/((-2*c^2*d+2*(c^2
*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/d^2/(c^2*d+e)^2/e*(c^2*d*(c^2*d+e))^(1/2)-3/4*c*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+
e))^(1/2)+e)*e)^(1/2)*arctanh((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(
1/2))/e^2/d^2/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)+5/4*c^3*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arc
tanh((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/(c^2*d+e)^2/d/e^2*(
c^2*d*(c^2*d+e))^(1/2)-c^3*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh((c*x+(c*x-1)^(1/2)*(c*x+
1)^(1/2))*e/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^3/d/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)-5/4*c^3*
b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e/((2*c^2*d+2*(c^2*
d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/(c^2*d+e)^2/d/e^2*(c^2*d*(c^2*d+e))^(1/2)+c^3*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^
(1/2)+e)*e)^(1/2)*arctan((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/
e^3/d/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)-3/8*c*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan((c*x+(c
*x-1)^(1/2)*(c*x+1)^(1/2))*e/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/d^2/(c^2*d+e)^2/e*(c^2*d*(c^2*d+
e))^(1/2)+3/4*c*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e/(
(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^2/d^2/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)-5/4*c^3*b*(-(2*c^2*d
-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e/((-2*c^2*d+2*(c^2*d*(c^2*d+
e))^(1/2)-e)*e)^(1/2))/e^2/d/(c^2*d+e)-5/4*c^3*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan((c*x+(
c*x-1)^(1/2)*(c*x+1)^(1/2))*e/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^2/d/(c^2*d+e)+c^7*b*((2*c^2*d
+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e/((2*c^2*d+2*(c^2*d*(c^2*d+e)
)^(1/2)+e)*e)^(1/2))/e^3/(c^2*d+e)^2*d+c^7*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh((c*x+(c*
x-1)^(1/2)*(c*x+1)^(1/2))*e/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^3/(c^2*d+e)^2*d+c^5*b*(-(2*c^2
*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e/((-2*c^2*d+2*(c^2*d*(c^2*
d+e))^(1/2)-e)*e)^(1/2))/e^3/(c^2*d+e)^2*(c^2*d*(c^2*d+e))^(1/2)-c^5*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*
e)^(1/2)*arctan((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^3/(c^2*
d+e)^2*(c^2*d*(c^2*d+e))^(1/2)+3/4*c^3*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh((c*x+(c*x-1)
^(1/2)*(c*x+1)^(1/2))*e/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/d/(c^2*d+e)^2/e-3/8*c*b*(-(2*c^2*d-2
*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctanh((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e/((-2*c^2*d+2*(c^2*d*(c^2*d+e)
)^(1/2)-e)*e)^(1/2))/e/d^2/(c^2*d+e)+3/4*c^3*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan((c*x+(c*
x-1)^(1/2)*(c*x+1)^(1/2))*e/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/d/(c^2*d+e)^2/e-3/8*c*b*((2*c^2*d
+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*arctan((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e/((2*c^2*d+2*(c^2*d*(c^2*d+e)
)^(1/2)+e)*e)^(1/2))/e/d^2/(c^2*d+e)-1/8*c^5*b/d/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2
*e-3/16*c^3*b/d/(c^2*d+e)*sum(1/_R1/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*ln((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))
/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))+3/16*c^3*b/d/
(c^2*d+e)*sum(_R1/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*ln((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c
*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))+1/4*c^4*a*x/d/(c^2*e*x^2+c^2*d)
^2+3/8*c^2*a/d^2*x/(c^2*e*x^2+c^2*d)

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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*arccosh(c*x))/(e*x^2+d)^3,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{b \operatorname{arcosh}\left (c x\right ) + a}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))/(e*x**2+d)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccosh(c*x) + a)/(e*x^2 + d)^3, x)