3.502 \(\int \frac {x^4 (a+b \cosh ^{-1}(c x))}{(d+e x^2)^2} \, dx\)

Optimal. Leaf size=839 \[ \frac {x \cosh ^{-1}(c x) b}{e^2}+\frac {c d \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x+1}}{\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x-1}}\right ) b}{2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{5/2}}-\frac {c d \tanh ^{-1}\left (\frac {\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x+1}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x-1}}\right ) b}{2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{5/2}}-\frac {3 \sqrt {-d} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) b}{4 e^{5/2}}+\frac {3 \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) b}{4 e^{5/2}}-\frac {3 \sqrt {-d} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) b}{4 e^{5/2}}+\frac {3 \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) b}{4 e^{5/2}}-\frac {\sqrt {c x-1} \sqrt {c x+1} b}{c e^2}+\frac {a x}{e^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {e} x+\sqrt {-d}\right )}+\frac {3 \sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e^{\cosh ^{-1}(c x)} \sqrt {e}}{c \sqrt {-d}-\sqrt {-d c^2-e}}+1\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e^{\cosh ^{-1}(c x)} \sqrt {e}}{\sqrt {-d} c+\sqrt {-d c^2-e}}+1\right )}{4 e^{5/2}} \]

[Out]

a*x/e^2+b*x*arccosh(c*x)/e^2+3/4*(a+b*arccosh(c*x))*ln(1-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/
2)-(-c^2*d-e)^(1/2)))*(-d)^(1/2)/e^(5/2)-3/4*(a+b*arccosh(c*x))*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)
/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))*(-d)^(1/2)/e^(5/2)+3/4*(a+b*arccosh(c*x))*ln(1-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1
/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))*(-d)^(1/2)/e^(5/2)-3/4*(a+b*arccosh(c*x))*ln(1+(c*x+(c*x-1)^(1/2
)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))*(-d)^(1/2)/e^(5/2)-3/4*b*polylog(2,-(c*x+(c*x-1)^(1/
2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))*(-d)^(1/2)/e^(5/2)+3/4*b*polylog(2,(c*x+(c*x-1)^(1/
2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))*(-d)^(1/2)/e^(5/2)-3/4*b*polylog(2,-(c*x+(c*x-1)^(1
/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))*(-d)^(1/2)/e^(5/2)+3/4*b*polylog(2,(c*x+(c*x-1)^(1
/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))*(-d)^(1/2)/e^(5/2)-1/4*d*(a+b*arccosh(c*x))/e^(5/2
)/((-d)^(1/2)-x*e^(1/2))+1/4*d*(a+b*arccosh(c*x))/e^(5/2)/((-d)^(1/2)+x*e^(1/2))-b*(c*x-1)^(1/2)*(c*x+1)^(1/2)
/c/e^2+1/2*b*c*d*arctanh((c*x+1)^(1/2)*(c*(-d)^(1/2)-e^(1/2))^(1/2)/(c*x-1)^(1/2)/(c*(-d)^(1/2)+e^(1/2))^(1/2)
)/e^(5/2)/(c*(-d)^(1/2)-e^(1/2))^(1/2)/(c*(-d)^(1/2)+e^(1/2))^(1/2)-1/2*b*c*d*arctanh((c*x+1)^(1/2)*(c*(-d)^(1
/2)+e^(1/2))^(1/2)/(c*x-1)^(1/2)/(c*(-d)^(1/2)-e^(1/2))^(1/2))/e^(5/2)/(c*(-d)^(1/2)-e^(1/2))^(1/2)/(c*(-d)^(1
/2)+e^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 2.19, antiderivative size = 839, normalized size of antiderivative = 1.00, number of steps used = 49, number of rules used = 12, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5792, 5654, 74, 5707, 5802, 93, 208, 5800, 5562, 2190, 2279, 2391} \[ \frac {x \cosh ^{-1}(c x) b}{e^2}+\frac {c d \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x+1}}{\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x-1}}\right ) b}{2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{5/2}}-\frac {c d \tanh ^{-1}\left (\frac {\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x+1}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x-1}}\right ) b}{2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{5/2}}-\frac {3 \sqrt {-d} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) b}{4 e^{5/2}}+\frac {3 \sqrt {-d} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) b}{4 e^{5/2}}-\frac {3 \sqrt {-d} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) b}{4 e^{5/2}}+\frac {3 \sqrt {-d} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) b}{4 e^{5/2}}-\frac {\sqrt {c x-1} \sqrt {c x+1} b}{c e^2}+\frac {a x}{e^2}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{4 e^{5/2} \left (\sqrt {e} x+\sqrt {-d}\right )}+\frac {3 \sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e^{\cosh ^{-1}(c x)} \sqrt {e}}{c \sqrt {-d}-\sqrt {-d c^2-e}}+1\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e^{\cosh ^{-1}(c x)} \sqrt {e}}{\sqrt {-d} c+\sqrt {-d c^2-e}}+1\right )}{4 e^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(a + b*ArcCosh[c*x]))/(d + e*x^2)^2,x]

[Out]

(a*x)/e^2 - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*e^2) + (b*x*ArcCosh[c*x])/e^2 - (d*(a + b*ArcCosh[c*x]))/(4*e^
(5/2)*(Sqrt[-d] - Sqrt[e]*x)) + (d*(a + b*ArcCosh[c*x]))/(4*e^(5/2)*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*d*ArcTanh[(
Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(2*Sqrt[c*Sqrt[-d] - S
qrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(5/2)) - (b*c*d*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[
c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(2*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(5/2)) + (3
*Sqrt[-d]*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(4*e^(5/2)
) - (3*Sqrt[-d]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(4*e
^(5/2)) + (3*Sqrt[-d]*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])]
)/(4*e^(5/2)) - (3*Sqrt[-d]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d)
- e])])/(4*e^(5/2)) - (3*b*Sqrt[-d]*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])
/(4*e^(5/2)) + (3*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(4*e^(5/2
)) - (3*b*Sqrt[-d]*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(4*e^(5/2)) + (3
*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(4*e^(5/2))

Rule 74

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

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 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 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 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

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 5792

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

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 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]

Rubi steps

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

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Mathematica [C]  time = 2.41, size = 777, normalized size = 0.93 \[ \frac {\frac {4 a d \sqrt {e} x}{d+e x^2}-12 a \sqrt {d} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+8 a \sqrt {e} x+b \left (-3 i \sqrt {d} \left (2 \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d c^2-e}-i c \sqrt {d}}\right )+2 \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i \sqrt {d} c+\sqrt {-d c^2-e}}\right )+\cosh ^{-1}(c x) \left (-\cosh ^{-1}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{-\sqrt {c^2 (-d)-e}+i c \sqrt {d}}\right )+\log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+i c \sqrt {d}}\right )\right )\right )\right )+3 i \sqrt {d} \left (2 \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i c \sqrt {d}-\sqrt {-d c^2-e}}\right )+2 \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i \sqrt {d} c+\sqrt {-d c^2-e}}\right )+\cosh ^{-1}(c x) \left (-\cosh ^{-1}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}-i c \sqrt {d}}\right )+\log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+i c \sqrt {d}}\right )\right )\right )\right )+2 d \left (\frac {c \log \left (\frac {2 e \left (-i \sqrt {c x-1} \sqrt {c x+1} \sqrt {c^2 (-d)-e}+c^2 \sqrt {d} x+i \sqrt {e}\right )}{c \sqrt {c^2 (-d)-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {c^2 (-d)-e}}+\frac {\cosh ^{-1}(c x)}{\sqrt {e} x-i \sqrt {d}}\right )+2 d \left (\frac {c \log \left (\frac {2 e \left (\sqrt {c x-1} \sqrt {c x+1} \sqrt {c^2 (-d)-e}-i c^2 \sqrt {d} x-\sqrt {e}\right )}{c \sqrt {c^2 (-d)-e} \left (\sqrt {e} x+i \sqrt {d}\right )}\right )}{\sqrt {c^2 (-d)-e}}+\frac {\cosh ^{-1}(c x)}{\sqrt {e} x+i \sqrt {d}}\right )+\frac {8 \sqrt {e} \left (c x \cosh ^{-1}(c x)-\sqrt {\frac {c x-1}{c x+1}} (c x+1)\right )}{c}\right )}{8 e^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*(a + b*ArcCosh[c*x]))/(d + e*x^2)^2,x]

[Out]

(8*a*Sqrt[e]*x + (4*a*d*Sqrt[e]*x)/(d + e*x^2) - 12*a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*((8*Sqrt[e]*(-(S
qrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + c*x*ArcCosh[c*x]))/c + 2*d*(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]) + 2*d*(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]) - (3*I)*Sqrt[d]*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^Arc
Cosh[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, -((Sqr
t[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))]) + (3*I)*Sqrt[d]*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(L
og[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])])))/(8*e^(5/2))

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fricas [F]  time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{4} \operatorname {arcosh}\left (c x\right ) + a x^{4}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccosh(c*x))/(e*x^2+d)^2,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccosh(c*x))/(e*x^2+d)^2,x, algorithm="giac")

[Out]

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

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maple [C]  time = 7.08, size = 1749, normalized size = 2.08 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x/e^2+1/2*c^2*a/e^2*d*x/(c^2*e*x^2+c^2*d)-3/2*a/e^2*d/(d*e)^(1/2)*arctan(x*e/(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))*d^2/e^5-1/2*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/e^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^2/e^5-1/2*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/e^4+3/4*c*b/e^2*d*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))-3/4*c*b/e^2*d*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)
)-b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/e^2+b*x*arccosh(c*x)/e^2+1/2*c^2*b*arccosh(c*x)*d*x/e^2/(c^2*e*x^2+c^2*d)+c^
5*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*d^3*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^5/(c^2*d+e)+c^3*b*(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*
d^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^4/(c^2*d+e
)-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/e^5*(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)*d^3*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
^5/(c^2*d+e)+c^3*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*d^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^4/(c^2*d+e)+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/e^5*(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)*d^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^5/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)+1/2*c*b*
(-(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*d*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^4/(c^2*d+e)*(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)*d^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^5/(c^2*d+e)*(c^2*d*(c^2*d+e))^(1/2)-1/2*c*b*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*d*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^4/(c^2*d+e)*(c^2*d*(c^2*d+e
))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a {\left (\frac {d x}{e^{3} x^{2} + d e^{2}} - \frac {3 \, d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} e^{2}} + \frac {2 \, x}{e^{2}}\right )} + b \int \frac {x^{4} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*arccosh(c*x))/(e*x^2+d)^2,x, algorithm="maxima")

[Out]

1/2*a*(d*x/(e^3*x^2 + d*e^2) - 3*d*arctan(e*x/sqrt(d*e))/(sqrt(d*e)*e^2) + 2*x/e^2) + b*integrate(x^4*log(c*x
+ sqrt(c*x + 1)*sqrt(c*x - 1))/(e^2*x^4 + 2*d*e*x^2 + d^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(a + b*acosh(c*x)))/(d + e*x^2)^2,x)

[Out]

int((x^4*(a + b*acosh(c*x)))/(d + e*x^2)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*acosh(c*x))/(e*x**2+d)**2,x)

[Out]

Integral(x**4*(a + b*acosh(c*x))/(d + e*x**2)**2, x)

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