3.6.12 \(\int \frac {x^2 (a+b \cosh ^{-1}(c x))}{(d+e x^2)^3} \, dx\) [512]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 2.27, antiderivative size = 1234, normalized size of antiderivative = 1.00, number of steps used = 62, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {5959, 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 \left (c \sqrt {-d}-\sqrt {e}\right )^{3/2} \left (\sqrt {-d} c+\sqrt {e}\right )^{3/2} e^{3/2}}-\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 \left (c \sqrt {-d}-\sqrt {e}\right )^{3/2} \left (\sqrt {-d} c+\sqrt {e}\right )^{3/2} e^{3/2}}+\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}{8 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{3/2}}-\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}{8 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{3/2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} c}{16 \sqrt {-d} e \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 \sqrt {-d} e \left (d c^2+e\right ) \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {a+b \cosh ^{-1}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {a+b \cosh ^{-1}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {e} x+\sqrt {-d}\right )^2}-\frac {\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)^{3/2} e^{3/2}}+\frac {\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)^{3/2} e^{3/2}}-\frac {\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)^{3/2} e^{3/2}}+\frac {\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)^{3/2} e^{3/2}}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{16 (-d)^{3/2} e^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=\int \left (-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{e \left (d+e x^2\right )^3}+\frac {a+b \cosh ^{-1}(c x)}{e \left (d+e x^2\right )^2}\right ) \, dx\\ &=\frac {\int \frac {a+b \cosh ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx}{e}-\frac {d \int \frac {a+b \cosh ^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx}{e}\\ &=\frac {\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}-\frac {d \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}{e}\\ &=\frac {3 \int \frac {a+b \cosh ^{-1}(c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx}{16 d}+\frac {3 \int \frac {a+b \cosh ^{-1}(c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx}{16 d}-\frac {\int \frac {a+b \cosh ^{-1}(c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx}{4 d}-\frac {\int \frac {a+b \cosh ^{-1}(c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx}{4 d}+\frac {3 \int \frac {a+b \cosh ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{8 d}-\frac {\int \frac {a+b \cosh ^{-1}(c x)}{-d e-e^2 x^2} \, dx}{2 d}-\frac {\sqrt {e} \int \frac {a+b \cosh ^{-1}(c x)}{\left (\sqrt {-d} \sqrt {e}-e x\right )^3} \, dx}{8 \sqrt {-d}}-\frac {\sqrt {e} \int \frac {a+b \cosh ^{-1}(c x)}{\left (\sqrt {-d} \sqrt {e}+e x\right )^3} \, dx}{8 \sqrt {-d}}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \cosh ^{-1}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {3 \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}-\frac {\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}-\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 e}+\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 e}+\frac {(b c) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (\sqrt {-d} \sqrt {e}-e x\right )} \, dx}{4 d e}-\frac {(b c) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (\sqrt {-d} \sqrt {e}+e x\right )} \, dx}{4 d e}+\frac {(b c) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (\sqrt {-d} \sqrt {e}-e x\right )^2} \, dx}{16 \sqrt {-d} \sqrt {e}}-\frac {(b c) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (\sqrt {-d} \sqrt {e}+e x\right )^2} \, dx}{16 \sqrt {-d} \sqrt {e}}\\ &=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 \sqrt {-d} e \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 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \cosh ^{-1}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \cosh ^{-1}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \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)^{3/2} e}-\frac {3 \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{16 (-d)^{3/2} e}+\frac {\int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 (-d)^{3/2} e}+\frac {\int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 (-d)^{3/2} e}-\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 e}+\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 e}+\frac {(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 )}{2 d e}-\frac {(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 )}{2 d e}-\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 e \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 e \left (c^2 d+e\right )}\\ &=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 \sqrt {-d} e \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 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \cosh ^{-1}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \cosh ^{-1}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2}}-\frac {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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2}}-\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)^{3/2} 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)^{3/2} e}+\frac {\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)^{3/2} e}+\frac {\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)^{3/2} e}-\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 e \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 e \left (c^2 d+e\right )}\\ &=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 \sqrt {-d} e \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 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \cosh ^{-1}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \cosh ^{-1}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2}}-\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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2} \left (c^2 d+e\right )}-\frac {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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2}}+\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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2} \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)^{3/2} e}-\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)^{3/2} e}-\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)^{3/2} e}-\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)^{3/2} e}+\frac {\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)^{3/2} e}+\frac {\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)^{3/2} e}+\frac {\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)^{3/2} e}+\frac {\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)^{3/2} e}\\ &=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 \sqrt {-d} e \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 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \cosh ^{-1}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \cosh ^{-1}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2}}-\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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2} \left (c^2 d+e\right )}-\frac {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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2}}+\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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2} \left (c^2 d+e\right )}-\frac {\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)^{3/2} e^{3/2}}+\frac {\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)^{3/2} e^{3/2}}-\frac {\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)^{3/2} e^{3/2}}+\frac {\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)^{3/2} e^{3/2}}-\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)^{3/2} e^{3/2}}+\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)^{3/2} e^{3/2}}-\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)^{3/2} e^{3/2}}+\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)^{3/2} e^{3/2}}+\frac {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 )}{4 (-d)^{3/2} e^{3/2}}-\frac {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 )}{4 (-d)^{3/2} e^{3/2}}+\frac {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 )}{4 (-d)^{3/2} e^{3/2}}-\frac {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 )}{4 (-d)^{3/2} e^{3/2}}\\ &=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 \sqrt {-d} e \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 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \cosh ^{-1}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \cosh ^{-1}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2}}-\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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2} \left (c^2 d+e\right )}-\frac {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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2}}+\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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2} \left (c^2 d+e\right )}-\frac {\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)^{3/2} e^{3/2}}+\frac {\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)^{3/2} e^{3/2}}-\frac {\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)^{3/2} e^{3/2}}+\frac {\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)^{3/2} e^{3/2}}-\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)^{3/2} e^{3/2}}+\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)^{3/2} e^{3/2}}-\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)^{3/2} e^{3/2}}+\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)^{3/2} e^{3/2}}+\frac {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 )}{4 (-d)^{3/2} e^{3/2}}-\frac {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 )}{4 (-d)^{3/2} e^{3/2}}+\frac {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 )}{4 (-d)^{3/2} e^{3/2}}-\frac {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 )}{4 (-d)^{3/2} e^{3/2}}\\ &=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 \sqrt {-d} e \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 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \cosh ^{-1}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \cosh ^{-1}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2}}-\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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2} \left (c^2 d+e\right )}-\frac {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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2}}+\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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2} \left (c^2 d+e\right )}-\frac {\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)^{3/2} e^{3/2}}+\frac {\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)^{3/2} e^{3/2}}-\frac {\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)^{3/2} e^{3/2}}+\frac {\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)^{3/2} e^{3/2}}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{3/2} e^{3/2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 6.04, size = 1193, normalized size = 0.97 \begin {gather*} -\frac {a x}{4 e \left (d+e x^2\right )^2}+\frac {a x}{8 d e \left (d+e x^2\right )}+\frac {a \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} e^{3/2}}+b \left (\frac {\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}}}{16 d e^{3/2}}-\frac {-\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}}}{16 d e^{3/2}}-\frac {i \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 )}{16 \sqrt {d} e}+\frac {i \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 )}{16 \sqrt {d} e}+\frac {i \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 )}{32 d^{3/2} e^{3/2}}-\frac {i \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 )}{32 d^{3/2} e^{3/2}}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/4*(a*x)/(e*(d + e*x^2)^2) + (a*x)/(8*d*e*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2))
+ b*((ArcCosh[c*x]/((-I)*Sqrt[d] + Sqrt[e]*x) + (c*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d) - e]
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e])/(16*d*e^(
3/2)) - (-(ArcCosh[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2*d) - e
]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) - e])/(16*d*e^
(3/2)) - ((I/16)*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCosh[c*x]/(Sq
rt[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))))/(Sqrt[d]*e) + ((I/16)*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCos
h[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))))/(Sqrt[d]*e) + ((I/32)*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*S
qrt[d] - Sqrt[-(c^2*d) - e])] + Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*Pol
yLog[2, (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*
x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))]))/(d^(3/2)*e^(3/2)) - ((I/32)*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1
+ (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqr
t[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))
] + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])]))/(d^(3/2)*e^(3/2)))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 60.57, size = 2292, normalized size = 1.86

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2292\)
default	\(\text {Expression too large to display}\)	\(2292\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(-1/8*b*c^9/e/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*arccosh(c*x)*d*x+1/8*a*c^7/(c^2*e*x^2+c^2*d)^2/d*x^3+1/4*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+1/4*b*c^6*(-(2*c^2*d-2*((c^2*d+e)*c^2*d)^(1/2)+e)*e)^(1/2)*ar
ctanh((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+1/
8*a*c^3/d/e/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+1/4*b*c^6*(-(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))/e^3/(c^2*d+e)^2*((c
^2*d+e)*c^2*d)^(1/2)+1/8*b*c^9/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*arccosh(c*x)*x^3-1/8*a*c^7/(c^2*e*x^2+c^2*d)^2/e*
x-1/8*b*c^7/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*arccosh(c*x)*x-1/8*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^2-1/8*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+1/8*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)-1/4*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^3*((c^2*d+e)*c^2
*d)^(1/2)-1/8*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)+1/4*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^3*((c^2*d+e)*c^2*d)^(1/2)+1/8*b*c^8/e/(c^2*e*x^2+c^2*d)^2*d/(c^2*d+e)*(c*x
-1)^(1/2)*(c*x+1)^(1/2)+1/8*b*c^8/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2+1/4*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-1/4*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))/e^3/(c^2*d+e)^2*((c^2*d+e)
*c^2*d)^(1/2)+1/4*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-1/16*b*c^6/e/(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))+1/16*b*c^6/e/(c^2*d+e)*sum(_R1/(_R1^2*e+2*c^2*d+e)*
(arccosh(c*x)*ln((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)),
_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))+1/16*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))-1/16*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))-1/4*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))/e^3/(c^2*d+e)-1/4*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))/e^3/(c
^2*d+e)+1/8*b*c^7*e/(c^2*e*x^2+c^2*d)^2/d/(c^2*d+e)*arccosh(c*x)*x^3)

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

[Out]

1/8*a*(arctan(x*e^(1/2)/sqrt(d))*e^(-3/2)/d^(3/2) + (x^3*e - d*x)/(d*x^4*e^3 + 2*d^2*x^2*e^2 + d^3*e)) + b*int
egrate(x^2*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(x^2*(a+b*arccosh(c*x))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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