3.6.5 \(\int \frac {a+b \cosh ^{-1}(c x)}{x^2 (d+e x^2)^2} \, dx\) [505]

Optimal. Leaf size=846 \[ -\frac {a+b \cosh ^{-1}(c x)}{d^2 x}+\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c \text {ArcTan}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}-\frac {b c \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}}}+\frac {b c \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}}}-\frac {3 \sqrt {e} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{4 (-d)^{5/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.47, antiderivative size = 846, normalized size of antiderivative = 1.00, number of steps used = 49, number of rules used = 13, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {5959, 5883, 94, 211, 5909, 5963, 95, 214, 5962, 5681, 2221, 2317, 2438} \begin {gather*} -\frac {3 \sqrt {e} \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \log \left (\frac {e^{\cosh ^{-1}(c x)} \sqrt {e}}{c \sqrt {-d}-\sqrt {-d c^2-e}}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 (-d)^{5/2}}-\frac {3 \sqrt {e} \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} \log \left (\frac {e^{\cosh ^{-1}(c x)} \sqrt {e}}{\sqrt {-d} c+\sqrt {-d c^2-e}}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 (-d)^{5/2}}-\frac {a+b \cosh ^{-1}(c x)}{d^2 x}+\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (\sqrt {e} x+\sqrt {-d}\right )}+\frac {b c \text {ArcTan}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d^2}-\frac {b c \sqrt {e} \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 )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}}}+\frac {b c \sqrt {e} \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 )}{2 d^2 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}}}+\frac {3 b \sqrt {e} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{4 (-d)^{5/2}}+\frac {3 b \sqrt {e} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{4 (-d)^{5/2}}-\frac {3 b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{4 (-d)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*ArcCosh[c*x])/(d^2*x)) + (Sqrt[e]*(a + b*ArcCosh[c*x]))/(4*d^2*(Sqrt[-d] - Sqrt[e]*x)) - (Sqrt[e]*(a
+ b*ArcCosh[c*x]))/(4*d^2*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/d^2 - (b*c*Sqrt
[e]*ArcTanh[(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(2*d^2*Sq
rt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]) + (b*c*Sqrt[e]*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1
+ c*x])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(2*d^2*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqr
t[e]]) - (3*Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/
(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d)
- e])])/(4*(-d)^(5/2)) - (3*Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-
(c^2*d) - e])])/(4*(-d)^(5/2)) + (3*Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d]
+ Sqrt[-(c^2*d) - e])])/(4*(-d)^(5/2)) + (3*b*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt
[-(c^2*d) - e]))])/(4*(-d)^(5/2)) - (3*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*
d) - e])])/(4*(-d)^(5/2)) + (3*b*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e
]))])/(4*(-d)^(5/2)) - (3*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(4
*(-d)^(5/2))

Rule 94

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 95

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 211

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

Rule 214

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

Rule 2221

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

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 5681

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 5883

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

Rule 5909

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 5959

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 5962

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 5963

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

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Mathematica [C] Result contains complex when optimal does not.
time = 1.71, size = 821, normalized size = 0.97 \begin {gather*} \frac {-\frac {8 a \sqrt {d}}{x}-\frac {4 a \sqrt {d} e x}{d+e x^2}-12 a \sqrt {e} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+b \left (8 \sqrt {d} \left (-\frac {\cosh ^{-1}(c x)}{x}+\frac {c \sqrt {-1+c^2 x^2} \text {ArcTan}\left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\right )-2 \sqrt {d} \sqrt {e} \left (\frac {\cosh ^{-1}(c x)}{-i \sqrt {d}+\sqrt {e} x}+\frac {c \log \left (\frac {2 e \left (i \sqrt {e}+c^2 \sqrt {d} x-i \sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}\right )+2 \sqrt {d} \sqrt {e} \left (-\frac {\cosh ^{-1}(c x)}{i \sqrt {d}+\sqrt {e} x}-\frac {c \log \left (\frac {2 e \left (-\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}\right )-3 i \sqrt {e} \left (\cosh ^{-1}(c x) \left (-\cosh ^{-1}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+\log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )+2 \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )+3 i \sqrt {e} \left (\cosh ^{-1}(c x) \left (-\cosh ^{-1}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+\log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )+2 \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+2 \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )}{8 d^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*a*Sqrt[d])/x - (4*a*Sqrt[d]*e*x)/(d + e*x^2) - 12*a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*(8*Sqrt[d]*(-
(ArcCosh[c*x]/x) + (c*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - 2*Sqrt[
d]*Sqrt[e]*(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*
Sqrt[d]*Sqrt[e]*(-(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[e]*(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^Ar
cCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sq
rt[-(c^2*d) - e]))]) + (3*I)*Sqrt[e]*(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])])))/(8*d^(5/2))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 89.48, size = 1821, normalized size = 2.15 \[\text {Expression too large to display}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/d^2/x-1/2*a*e/d^2*x*c^2/(c^2*e*x^2+c^2*d)-3/2*a*e/d^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))-3/2*b*x*c^2*arcco
sh(c*x)/d^2/(c^2*e*x^2+c^2*d)*e-b*c^2/x*arccosh(c*x)/d/(c^2*e*x^2+c^2*d)-b*c^5*(-(2*c^2*d-2*((c^2*d+e)*c^2*d)^
(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+e)*c^2*d)^(1/2)-e)*e)^(1/2)
)/(c^2*d+e)/e^2-b*c^3*(-(2*c^2*d-2*((c^2*d+e)*c^2*d)^(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+e)*c^2*d)^(1/2)-e)*e)^(1/2))/d/(c^2*d+e)/e^2*((c^2*d+e)*c^2*d)^(1/2)-b*c^3*(-(2*c^2*
d-2*((c^2*d+e)*c^2*d)^(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+e)*c^
2*d)^(1/2)-e)*e)^(1/2))/d/(c^2*d+e)/e-1/2*c*b*(-(2*c^2*d-2*((c^2*d+e)*c^2*d)^(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+e)*c^2*d)^(1/2)-e)*e)^(1/2))/d^2/(c^2*d+e)/e*((c^2*d+e)*c^2*
d)^(1/2)+b*c^3*(-(2*c^2*d-2*((c^2*d+e)*c^2*d)^(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+e)*c^2*d)^(1/2)-e)*e)^(1/2))/d/e^2+c*b*(-(2*c^2*d-2*((c^2*d+e)*c^2*d)^(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+e)*c^2*d)^(1/2)-e)*e)^(1/2))/d^2/e^2*((c^2*d+e)*
c^2*d)^(1/2)+1/2*c*b*(-(2*c^2*d-2*((c^2*d+e)*c^2*d)^(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+e)*c^2*d)^(1/2)-e)*e)^(1/2))/d^2/e-b*c^5*((2*c^2*d+2*((c^2*d+e)*c^2*d)^(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+e)*c^2*d)^(1/2)+e)*e)^(1/2))/(c^2*d+e)/e^2+
b*c^3*((2*c^2*d+2*((c^2*d+e)*c^2*d)^(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+e)*c^2*d)^(1/2)+e)*e)^(1/2))/d/(c^2*d+e)/e^2*((c^2*d+e)*c^2*d)^(1/2)-b*c^3*((2*c^2*d+2*((c^2*d+e)*c^2*d
)^(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+e)*c^2*d)^(1/2)+e)*e)^(1/2)
)/d/(c^2*d+e)/e+1/2*c*b*((2*c^2*d+2*((c^2*d+e)*c^2*d)^(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+e)*c^2*d)^(1/2)+e)*e)^(1/2))/d^2/(c^2*d+e)/e*((c^2*d+e)*c^2*d)^(1/2)+b*c^3*((2*c^2*d+
2*((c^2*d+e)*c^2*d)^(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+e)*c^2*d)
^(1/2)+e)*e)^(1/2))/d/e^2-c*b*((2*c^2*d+2*((c^2*d+e)*c^2*d)^(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+e)*c^2*d)^(1/2)+e)*e)^(1/2))/d^2/e^2*((c^2*d+e)*c^2*d)^(1/2)+1/2*c*b*((2*c^2*d+
2*((c^2*d+e)*c^2*d)^(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+e)*c^2*d)
^(1/2)+e)*e)^(1/2))/d^2/e+3/16*b/c/d^3*e*sum((_R1^2*e+4*c^2*d+e)/_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))+2*c*b/d^2*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-3/16*b/c/d^3*e*sum((4*_R1^2*c^2*d+_R1^2*
e+e)/_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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*((3*x^2*e + 2*d)/(d^2*x^3*e + d^3*x) + 3*arctan(x*e^(1/2)/sqrt(d))*e^(1/2)/d^(5/2)) + b*integrate(log(c
*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(x^6*e^2 + 2*d*x^4*e + d^2*x^2), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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