3.6.13 \(\int \frac {a+b \cosh ^{-1}(c x)}{(d+e x^2)^3} \, dx\) [513]

Optimal. Leaf size=1234 \[ -\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 {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 \left (c \sqrt {-d}-\sqrt {e}\right )^{3/2} \left (c \sqrt {-d}+\sqrt {e}\right )^{3/2} \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 {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 \left (c \sqrt {-d}-\sqrt {e}\right )^{3/2} \left (c \sqrt {-d}+\sqrt {e}\right )^{3/2} \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 \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 {PolyLog}\left (2,-\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 {PolyLog}\left (2,\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 {PolyLog}\left (2,-\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 {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{5/2} \sqrt {e}} \]

[Out]

3/16*(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)^(
5/2)/e^(1/2)-3/16*(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)^(5/2)/e^(1/2)+3/16*(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)^(5/2)/e^(1/2)-3/16*(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)^(5/2)/e^(1/2)-3/16*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)^(5/2)/e^(1/2)+3/16*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)^(5/2)/e^(1/2)-3/16*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)^(5/2)/e^(1/2)+3/16*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)^(5/2)/e^(1/2)-1/8*b*c^3*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))/d/(c*(-d)^(1/2)-e^(1/2))^(3/2)/e^(1/2)/(c*(-d)^(1/2)+e^(1
/2))^(3/2)+1/8*b*c^3*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))/d/(c*(-d)^(1/2)-e^(1/2))^(3/2)/e^(1/2)/(c*(-d)^(1/2)+e^(1/2))^(3/2)+1/16*(-a-b*arccosh(c*x))/(-d)^(3/2)/
e^(1/2)/((-d)^(1/2)-x*e^(1/2))^2-3/16*(a+b*arccosh(c*x))/d^2/e^(1/2)/((-d)^(1/2)-x*e^(1/2))+1/16*(a+b*arccosh(
c*x))/(-d)^(3/2)/e^(1/2)/((-d)^(1/2)+x*e^(1/2))^2+3/16*(a+b*arccosh(c*x))/d^2/e^(1/2)/((-d)^(1/2)+x*e^(1/2))-1
/16*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(-d)^(3/2)/(c^2*d+e)/((-d)^(1/2)-x*e^(1/2))-1/16*b*c*(c*x-1)^(1/2)*(c*x+1)
^(1/2)/(-d)^(3/2)/(c^2*d+e)/((-d)^(1/2)+x*e^(1/2))+3/8*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))/d^2/e^(1/2)/(c*(-d)^(1/2)-e^(1/2))^(1/2)/(c*(-d)^(1/2)+e^(1/2))^(1
/2)-3/8*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))/d^2
/e^(1/2)/(c*(-d)^(1/2)-e^(1/2))^(1/2)/(c*(-d)^(1/2)+e^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.16, antiderivative size = 1234, normalized size of antiderivative = 1.00, 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 = {5909, 5963, 98, 95, 214, 5962, 5681, 2221, 2317, 2438} \begin {gather*} -\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 {Li}_2\left (-\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 {Li}_2\left (\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 {Li}_2\left (-\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 {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{16 (-d)^{5/2} \sqrt {e}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((-d)^(3/2)*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x)) - (b*c*Sqrt[-1 + c*x]
*Sqrt[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*ArcCo
sh[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*Sq
rt[-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*Sqr
t[-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]) + (3*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^Ar
cCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcCosh[c*x])*Log[1 + (Sqr
t[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*Poly
Log[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, -((S
qrt[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 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 98

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 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 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 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)}{\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) \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 )}{8 d^2}-\frac {(3 b c) \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 )}{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 \text {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 \text {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 ) \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 )}{8 d \left (c^2 d+e\right )}-\frac {\left (b c^3\right ) \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 )}{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 \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 )}{16 (-d)^{5/2}}-\frac {3 \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 )}{16 (-d)^{5/2}}-\frac {3 \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 )}{16 (-d)^{5/2}}-\frac {3 \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 )}{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) \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 )}{16 (-d)^{5/2} \sqrt {e}}+\frac {(3 b) \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {(3 b) \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 )}{16 (-d)^{5/2} \sqrt {e}}+\frac {(3 b) \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 )}{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 {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) \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 )}{16 (-d)^{5/2} \sqrt {e}}-\frac {(3 b) \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 )}{16 (-d)^{5/2} \sqrt {e}}+\frac {(3 b) \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 )}{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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 5.63, size = 1162, normalized size = 0.94 \begin {gather*} \frac {\frac {8 a d^{3/2} x}{\left (d+e x^2\right )^2}+\frac {12 a \sqrt {d} x}{d+e x^2}+\frac {12 a \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {6 b \sqrt {d} \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 )}{\sqrt {e}}-\frac {6 b \sqrt {d} \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 )}{\sqrt {e}}+2 i b d \left (\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{\left (c^2 d+e\right ) \left (-i \sqrt {d}+\sqrt {e} x\right )}-\frac {\cosh ^{-1}(c x)}{\sqrt {e} \left (-i \sqrt {d}+\sqrt {e} x\right )^2}+\frac {c^3 \sqrt {d} \left (\log (4)+\log \left (\frac {e \sqrt {c^2 d+e} \left (-i \sqrt {e}-c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c^3 \left (d+i \sqrt {d} \sqrt {e} x\right )}\right )\right )}{\sqrt {e} \left (c^2 d+e\right )^{3/2}}\right )-2 i b d \left (\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{\left (c^2 d+e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}-\frac {\cosh ^{-1}(c x)}{\sqrt {e} \left (i \sqrt {d}+\sqrt {e} x\right )^2}-\frac {c^3 \sqrt {d} \left (\log (4)+\log \left (\frac {e \sqrt {c^2 d+e} \left (-i \sqrt {e}+c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c^3 \left (d-i \sqrt {d} \sqrt {e} x\right )}\right )\right )}{\sqrt {e} \left (c^2 d+e\right )^{3/2}}\right )+\frac {3 i b \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 )}{\sqrt {e}}+\frac {3 i b \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 )}{\sqrt {e}}}{32 d^{5/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((8*a*d^(3/2)*x)/(d + e*x^2)^2 + (12*a*Sqrt[d]*x)/(d + e*x^2) + (12*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e] + (
6*b*Sqrt[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]))/Sq
rt[e] - (6*b*Sqrt[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]))/Sqrt[e] + (2*I)*b*d*((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))) - (2*I)*b*d*((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*S
qrt[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))) + ((3*I)*b*(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]))]))/Sqrt[e] + ((3*I)*b*(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^Ar
cCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])]))/Sqrt[e])/(32*d^(5/2))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 37.67, size = 3149, normalized size = 2.55

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3149\)
default	\(\text {Expression too large to display}\)	\(3149\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(3/8*a*c^3/d^2*x/(c^2*e*x^2+c^2*d)+7/4*b*c^6*((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)^2/e^2+7/4*b*c^6*(-(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)^2/e^2+3/4*b*c^4*(-(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)^2/e-3/8*b*c
^2*(-(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+3/4*b*c^4*((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)^2/e-3/
8*b*c^2*((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+b*c^6*(-(2*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)+e)*e)^(1/2)*a
rctanh((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))/e^3/(c^2*d+e)^2*(
(c^2*d+e)*c^2*d)^(1/2)+1/4*a*c^5*x/d/(c^2*e*x^2+c^2*d)^2-3/16*b*c^2/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))/_R
1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))-1/8*b*c^6/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*(c*x-1)^(1/2)*(c*x+1)^(1/
2)+3/16*b*c^2/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))+5/8*b*c^5/d
/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arccosh(c*x)*e*x+3/8*b*c^5/d^2/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arccosh(c*x)*e^2*x
^3-1/8*b*c^6/d/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*e*x^2+5/8*b*c^7/(c^2*e*x^2+c^2*d)^2/(
c^2*d+e)*arccosh(c*x)*x-5/4*b*c^4*(-(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^2-5/4*b*c^4*((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+5/4*b*c^4*(-(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)^2/d/e^2*((c^2*d+e)*c^2*d)^(1
/2)-b*c^4*(-(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^3*((c^2*d+e)*c^2*d)^(1/2)-5/4*b*c^4*((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)^2/d/e^2*((c^2*d+e)*c^2*d)^(1/2)+b*c^4*((2*c^2*d+2*((c^2*d+e)*c^2*d)^(1/2)+e)*e)^(1/2)*a
rctan((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^3*((c
^2*d+e)*c^2*d)^(1/2)+3/8*b*c^2*(-(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)^2/e*((c^2*d+e)*c^2*d)^(1/2)-3/4*
b*c^2*(-(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^2*((c^2*d+e)*c^2*d)^(1/2)-3/8*b*c^2*((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)^2/e*((c^2*d+e)*c^2*d)^(1/2)+3/4*b*c^2*((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^2
*((c^2*d+e)*c^2*d)^(1/2)+b*c^8*((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))/e^3/(c^2*d+e)^2*d-b*c^6*((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))/e^3/(c^2*d+e)^2*((c^2*d+e)*c^2*d)^(1/2)+b*c^8*(-(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))/e^3/(c^2*d+e)^2*d+3/16*b*c
^4/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))-3/16*b*c^4/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))-b*c^6*((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))/e^3/(c
^2*d+e)-b*c^6*(-(2*c^2*d-2*((c^2*d+e)*c^2*d)^(1...

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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))/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

1/8*a*((3*x^3*e + 5*d*x)/(d^2*x^4*e^2 + 2*d^3*x^2*e + d^4) + 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^3 + 3*d*x^4*e^2 + 3*d^2*x^2*e + d^3), 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))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((b*arccosh(c*x) + a)/(e*x^2 + d)^3, 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 )}{{\left (e\,x^2+d\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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