\(\int \frac {a+b \text {arccosh}(c x)}{x^3 (d+e x^2)^3} \, dx\) [510]

Optimal result

Integrand size = 21, antiderivative size = 834 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^3 x}+\frac {b c e^2 x \left (1-c^2 x^2\right )}{8 d^3 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2}-\frac {e (a+b \text {arccosh}(c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \text {arccosh}(c x))}{d^3 \left (d+e x^2\right )}-\frac {3 e (a+b \text {arccosh}(c x))^2}{b d^4}+\frac {b c e \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^4}+\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4} \]

[Out]

1/2*(-a-b*arccosh(c*x))/d^3/x^2-1/4*e*(a+b*arccosh(c*x))/d^2/(e*x^2+d)^2-e*(a+b*arccosh(c*x))/d^3/(e*x^2+d)-3*
e*(a+b*arccosh(c*x))^2/b/d^4-3*e*(a+b*arccosh(c*x))*ln(1+1/(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d^4+3/2*e*(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^4+3/2*e*(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^4+3/2*e*(a+b*a
rccosh(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^4+3/2*e*(a+b*ar
ccosh(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^4+3/2*b*e*polylo
g(2,-1/(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d^4+3/2*b*e*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^4+3/2*b*e*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^4+3/2*b*e*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^4+3/2*b*e*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^4+1/
8*b*c*e^2*x*(-c^2*x^2+1)/d^3/(c^2*d+e)/(e*x^2+d)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/2*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/
2)/d^3/x+1/8*b*c*e*(2*c^2*d+e)*arctanh(x*(c^2*d+e)^(1/2)/d^(1/2)/(c^2*x^2-1)^(1/2))*(c^2*x^2-1)^(1/2)/d^(7/2)/
(c^2*d+e)^(3/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+b*c*e*arctanh(x*(c^2*d+e)^(1/2)/d^(1/2)/(c^2*x^2-1)^(1/2))*(c^2*x^
2-1)^(1/2)/d^(7/2)/(c^2*d+e)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 834, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5959, 5883, 97, 5882, 3799, 2221, 2317, 2438, 5957, 533, 390, 385, 214, 5962, 5681} \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\frac {b c x \left (1-c^2 x^2\right ) e^2}{8 d^3 \left (d c^2+e\right ) \sqrt {c x-1} \sqrt {c x+1} \left (e x^2+d\right )}-\frac {3 (a+b \text {arccosh}(c x))^2 e}{b d^4}-\frac {(a+b \text {arccosh}(c x)) e}{d^3 \left (e x^2+d\right )}-\frac {(a+b \text {arccosh}(c x)) e}{4 d^2 \left (e x^2+d\right )^2}+\frac {b c \left (2 d c^2+e\right ) \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {\sqrt {d c^2+e} x}{\sqrt {d} \sqrt {c^2 x^2-1}}\right ) e}{8 d^{7/2} \left (d c^2+e\right )^{3/2} \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {\sqrt {d c^2+e} x}{\sqrt {d} \sqrt {c^2 x^2-1}}\right ) e}{d^{7/2} \sqrt {d c^2+e} \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right ) e}{d^4}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (\frac {e^{\text {arccosh}(c x)} \sqrt {e}}{c \sqrt {-d}-\sqrt {-d c^2-e}}+1\right ) e}{2 d^4}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 (a+b \text {arccosh}(c x)) \log \left (\frac {e^{\text {arccosh}(c x)} \sqrt {e}}{\sqrt {-d} c+\sqrt {-d c^2-e}}+1\right ) e}{2 d^4}+\frac {3 b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right ) e}{2 d^4}+\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) e}{2 d^4}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 d^3 x} \]

[In]

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

[Out]

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*d^3*x) + (b*c*e^2*x*(1 - c^2*x^2))/(8*d^3*(c^2*d + e)*Sqrt[-1 + c*x]*Sqr
t[1 + c*x]*(d + e*x^2)) - (a + b*ArcCosh[c*x])/(2*d^3*x^2) - (e*(a + b*ArcCosh[c*x]))/(4*d^2*(d + e*x^2)^2) -
(e*(a + b*ArcCosh[c*x]))/(d^3*(d + e*x^2)) - (3*e*(a + b*ArcCosh[c*x])^2)/(b*d^4) + (b*c*e*Sqrt[-1 + c^2*x^2]*
ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(d^(7/2)*Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*
x]) + (b*c*e*(2*c^2*d + e)*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(8*d^
(7/2)*(c^2*d + e)^(3/2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*e*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])]
)/d^4 + (3*e*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*d^4)
+ (3*e*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*d^4) + (3
*e*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*d^4) + (3*e*(a
+ b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*d^4) + (3*b*e*PolyL
og[2, -E^(-2*ArcCosh[c*x])])/(2*d^4) + (3*b*e*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d
) - e]))])/(2*d^4) + (3*b*e*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*d^4) +
(3*b*e*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(2*d^4) + (3*b*e*PolyLog[2,
(Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*d^4)

Rule 97

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] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0
] && NeQ[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 385

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

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a
*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 533

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_)*((a2_) + (b2_.)*(x_)^(non2_.))^(
p_), x_Symbol] :> Dist[(a1 + b1*x^(n/2))^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*a2 + b1*b2*x^n)^FracPa
rt[p]), Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[
non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(EqQ[n, 2] && IGtQ[q, 0])

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 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m
+ 1)/(d*(m + 1))), x] + Dist[2*I, Int[(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

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 5882

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Tanh[-a/b + x/b], x],
x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 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 5957

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

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]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \text {arccosh}(c x)}{d^3 x^3}-\frac {3 e (a+b \text {arccosh}(c x))}{d^4 x}+\frac {e^2 x (a+b \text {arccosh}(c x))}{d^2 \left (d+e x^2\right )^3}+\frac {2 e^2 x (a+b \text {arccosh}(c x))}{d^3 \left (d+e x^2\right )^2}+\frac {3 e^2 x (a+b \text {arccosh}(c x))}{d^4 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {a+b \text {arccosh}(c x)}{x^3} \, dx}{d^3}-\frac {(3 e) \int \frac {a+b \text {arccosh}(c x)}{x} \, dx}{d^4}+\frac {\left (3 e^2\right ) \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx}{d^4}+\frac {\left (2 e^2\right ) \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx}{d^3}+\frac {e^2 \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx}{d^2} \\ & = -\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2}-\frac {e (a+b \text {arccosh}(c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \text {arccosh}(c x))}{d^3 \left (d+e x^2\right )}+\frac {(b c) \int \frac {1}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 d^3}+\frac {(3 e) \text {Subst}\left (\int x \tanh \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b d^4}+\frac {(b c e) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )} \, dx}{d^3}+\frac {(b c e) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )^2} \, dx}{4 d^2}+\frac {\left (3 e^2\right ) \int \left (-\frac {a+b \text {arccosh}(c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arccosh}(c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{d^4} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^3 x}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2}-\frac {e (a+b \text {arccosh}(c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \text {arccosh}(c x))}{d^3 \left (d+e x^2\right )}-\frac {3 e (a+b \text {arccosh}(c x))^2}{2 b d^4}+\frac {(6 e) \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x}{1+e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arccosh}(c x)\right )}{b d^4}-\frac {\left (3 e^{3/2}\right ) \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 d^4}+\frac {\left (3 e^{3/2}\right ) \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 d^4}+\frac {\left (b c e \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )} \, dx}{d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c e \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 d^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^3 x}+\frac {b c e^2 x \left (1-c^2 x^2\right )}{8 d^3 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2}-\frac {e (a+b \text {arccosh}(c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \text {arccosh}(c x))}{d^3 \left (d+e x^2\right )}-\frac {3 e (a+b \text {arccosh}(c x))^2}{2 b d^4}-\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^4}+\frac {(3 e) \text {Subst}\left (\int \log \left (1+e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{d^4}-\frac {\left (3 e^{3/2}\right ) \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}-\sqrt {e} \cosh (x)} \, dx,x,\text {arccosh}(c x)\right )}{2 d^4}+\frac {\left (3 e^{3/2}\right ) \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}+\sqrt {e} \cosh (x)} \, dx,x,\text {arccosh}(c x)\right )}{2 d^4}+\frac {\left (b c e \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{d-\left (c^2 d+e\right ) x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )} \, dx}{8 d^3 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^3 x}+\frac {b c e^2 x \left (1-c^2 x^2\right )}{8 d^3 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2}-\frac {e (a+b \text {arccosh}(c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \text {arccosh}(c x))}{d^3 \left (d+e x^2\right )}-\frac {3 e (a+b \text {arccosh}(c x))^2}{b d^4}+\frac {b c e \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^4}-\frac {(3 b e) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{2 d^4}-\frac {\left (3 e^{3/2}\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d-e}-\sqrt {e} e^x} \, dx,x,\text {arccosh}(c x)\right )}{2 d^4}-\frac {\left (3 e^{3/2}\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d-e}-\sqrt {e} e^x} \, dx,x,\text {arccosh}(c x)\right )}{2 d^4}+\frac {\left (3 e^{3/2}\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d-e}+\sqrt {e} e^x} \, dx,x,\text {arccosh}(c x)\right )}{2 d^4}+\frac {\left (3 e^{3/2}\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d-e}+\sqrt {e} e^x} \, dx,x,\text {arccosh}(c x)\right )}{2 d^4}+\frac {\left (b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{d-\left (c^2 d+e\right ) x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{8 d^3 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^3 x}+\frac {b c e^2 x \left (1-c^2 x^2\right )}{8 d^3 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2}-\frac {e (a+b \text {arccosh}(c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \text {arccosh}(c x))}{d^3 \left (d+e x^2\right )}-\frac {3 e (a+b \text {arccosh}(c x))^2}{b d^4}+\frac {b c e \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^4}+\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{2 d^4}-\frac {(3 b e) \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 d^4}-\frac {(3 b e) \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 d^4}-\frac {(3 b e) \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 d^4}-\frac {(3 b e) \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 d^4} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^3 x}+\frac {b c e^2 x \left (1-c^2 x^2\right )}{8 d^3 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2}-\frac {e (a+b \text {arccosh}(c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \text {arccosh}(c x))}{d^3 \left (d+e x^2\right )}-\frac {3 e (a+b \text {arccosh}(c x))^2}{b d^4}+\frac {b c e \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^4}+\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{2 d^4}-\frac {(3 b e) \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^{\text {arccosh}(c x)}\right )}{2 d^4}-\frac {(3 b e) \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^{\text {arccosh}(c x)}\right )}{2 d^4}-\frac {(3 b e) \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^{\text {arccosh}(c x)}\right )}{2 d^4}-\frac {(3 b e) \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^{\text {arccosh}(c x)}\right )}{2 d^4} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d^3 x}+\frac {b c e^2 x \left (1-c^2 x^2\right )}{8 d^3 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {a+b \text {arccosh}(c x)}{2 d^3 x^2}-\frac {e (a+b \text {arccosh}(c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \text {arccosh}(c x))}{d^3 \left (d+e x^2\right )}-\frac {3 e (a+b \text {arccosh}(c x))^2}{b d^4}+\frac {b c e \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{d^{7/2} \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{8 d^{7/2} \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )}{d^4}+\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}\right )}{2 d^4} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.09 (sec) , antiderivative size = 1261, normalized size of antiderivative = 1.51 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx=-\frac {a}{2 d^3 x^2}-\frac {a e}{4 d^2 \left (d+e x^2\right )^2}-\frac {a e}{d^3 \left (d+e x^2\right )}-\frac {3 a e \log (x)}{d^4}+\frac {3 a e \log \left (d+e x^2\right )}{2 d^4}+b \left (\frac {c x \sqrt {-1+c x} \sqrt {1+c x}-\text {arccosh}(c x)}{2 d^3 x^2}+\frac {9 i e \left (\frac {\text {arccosh}(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 )}{16 d^{7/2}}+\frac {9 i e \left (-\frac {\text {arccosh}(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 )}{16 d^{7/2}}-\frac {e^{3/2} \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 {\text {arccosh}(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 d^3}-\frac {e^{3/2} \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 {\text {arccosh}(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 d^3}-\frac {3 e \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right )}{2 d^4}+\frac {3 e \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )}{4 d^4}+\frac {3 e \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+\log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )}{4 d^4}\right ) \]

[In]

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

[Out]

-1/2*a/(d^3*x^2) - (a*e)/(4*d^2*(d + e*x^2)^2) - (a*e)/(d^3*(d + e*x^2)) - (3*a*e*Log[x])/d^4 + (3*a*e*Log[d +
e*x^2])/(2*d^4) + b*((c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - ArcCosh[c*x])/(2*d^3*x^2) + (((9*I)/16)*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]))/d^(7/2) + (((9*I)/16)*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]))/d^(7/2) - (e^(3
/2)*((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)*S
qrt[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))))/(16*d
^3) - (e^(3/2)*((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 + Sq
rt[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))))/
(16*d^3) - (3*e*(ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]) - PolyLog[2, -E^(-2*ArcCosh[c*x]
)]))/(2*d^4) + (3*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]))]))/(4*d^4) + (3*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^Ar
cCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])]))/(4*d^4))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.69 (sec) , antiderivative size = 1455, normalized size of antiderivative = 1.74

method	result	size
parts	\(\text {Expression too large to display}\)	\(1455\)
derivativedivides	\(\text {Expression too large to display}\)	\(1515\)
default	\(\text {Expression too large to display}\)	\(1515\)

[In]

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

[Out]

3/2*a*e/d^4*ln(e*x^2+d)-1/4*a*e/d^2/(e*x^2+d)^2-a*e/d^3/(e*x^2+d)-1/2*a/d^3/x^2-3*a/d^4*e*ln(x)+b*c^2*(-1/8*(-
4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^7*d^3*x-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^7*d^2*e*x^3-4*(c*x-1)^(1/2)*(c*x+1)^(1
/2)*c^7*d*e^2*x^5+4*c^8*d^3*x^2+8*c^8*d^2*e*x^4+4*c^8*d*e^2*x^6+4*c^6*d^3*arccosh(c*x)+18*arccosh(c*x)*c^6*d^2
*e*x^2+12*arccosh(c*x)*c^6*d*e^2*x^4-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^5*d^2*e*x-7*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c
^5*d*e^2*x^3-3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^3*c^5*x^5+3*c^6*d^2*e*x^2+6*c^6*d*e^2*x^4+3*c^6*e^3*x^6+4*c^4*d^2
*e*arccosh(c*x)+18*arccosh(c*x)*c^4*d*e^2*x^2+12*arccosh(c*x)*e^3*c^4*x^4)/c^2/x^2/d^3/(c^2*d+e)/(c^2*e*x^2+c^
2*d)^2-9/8*(d*c^2*(c^2*d+e))^(1/2)/(c^2*d+e)^2/d^4/c^2*e^2*arctanh(1/4*(4*c^2*d+2*e*(c*x+(c*x-1)^(1/2)*(c*x+1)
^(1/2))^2+2*e)/(c^4*d^2+c^2*d*e)^(1/2))-5/4*(d*c^2*(c^2*d+e))^(1/2)/(c^2*d+e)^2/d^3*e*arctanh(1/4*(4*c^2*d+2*e
*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+2*e)/(c^4*d^2+c^2*d*e)^(1/2))+3/4/(c^2*d+e)*e^2/d^4/c^2*sum((_R1^2*e+4*c^
2*d+e)/(_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/(c^2*d+e)*e^2/d^4/c^2*arccosh(c*x)*ln(1+
I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-3/(c^2*d+e)*e^2/d^4/c^2*arccosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1
/2)))-3/(c^2*d+e)*e^2/d^4/c^2*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-3/(c^2*d+e)*e^2/d^4/c^2*dilog(1-I*(
c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+3/4/(c^2*d+e)*e^3/d^4/c^2*sum((_R1^2+1)/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*ln
((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^
4+(4*c^2*d+2*e)*_Z^2+e))+3/4/(c^2*d+e)/d^3*e*sum((_R1^2*e+4*c^2*d+e)/(_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/(c^2*d+e)/d^3*e*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-3/(c^2*d+e)/d^3*e
*arccosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-3/(c^2*d+e)/d^3*e*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)
^(1/2)))-3/(c^2*d+e)/d^3*e*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+3/4/(c^2*d+e)/d^3*e^2*sum((_R1^2+1)/(_
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)))

Fricas [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{3}} \,d x } \]

[In]

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

[Out]

integral((b*arccosh(c*x) + a)/(e^3*x^9 + 3*d*e^2*x^7 + 3*d^2*e*x^5 + d^3*x^3), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{3}} \,d x } \]

[In]

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

[Out]

-1/4*a*((6*e^2*x^4 + 9*d*e*x^2 + 2*d^2)/(d^3*e^2*x^6 + 2*d^4*e*x^4 + d^5*x^2) - 6*e*log(e*x^2 + d)/d^4 + 12*e*
log(x)/d^4) + b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(e^3*x^9 + 3*d*e^2*x^7 + 3*d^2*e*x^5 + d^3*x^
3), x)

Giac [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^3\,{\left (e\,x^2+d\right )}^3} \,d x \]

[In]

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

[Out]

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