3.6.10 \(\int \frac {a+b \cosh ^{-1}(c x)}{x^3 (d+e x^2)^3} \, dx\) [510]

Optimal. Leaf size=834 \[ \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 \cosh ^{-1}(c x)}{2 d^3 x^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}-\frac {3 e \left (a+b \cosh ^{-1}(c x)\right )^2}{b d^4}+\frac {b c e \sqrt {-1+c^2 x^2} \tanh ^{-1}\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} \tanh ^{-1}\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 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )}{d^4}+\frac {3 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )}{2 d^4}+\frac {3 b e \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(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]
time = 0.93, antiderivative size = 834, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 15, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5959, 5883, 97, 5882, 3799, 2221, 2317, 2438, 5957, 533, 390, 385, 214, 5962, 5681} \begin {gather*} \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 \left (a+b \cosh ^{-1}(c x)\right )^2 e}{b d^4}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) e}{d^3 \left (e x^2+d\right )}-\frac {\left (a+b \cosh ^{-1}(c x)\right ) 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} \tanh ^{-1}\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} \tanh ^{-1}\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 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right ) e}{d^4}+\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 ) e}{2 d^4}+\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 ) e}{2 d^4}+\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 ) e}{2 d^4}+\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 ) e}{2 d^4}+\frac {3 b \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right ) e}{2 d^4}+\frac {3 b \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 b \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 b \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) e}{2 d^4}+\frac {3 b \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right ) e}{2 d^4}-\frac {a+b \cosh ^{-1}(c x)}{2 d^3 x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 d^3 x} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {a+b \cosh ^{-1}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx &=\int \left (\frac {a+b \cosh ^{-1}(c x)}{d^3 x^3}-\frac {3 e \left (a+b \cosh ^{-1}(c x)\right )}{d^4 x}+\frac {e^2 x \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (d+e x^2\right )^3}+\frac {2 e^2 x \left (a+b \cosh ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )^2}+\frac {3 e^2 x \left (a+b \cosh ^{-1}(c x)\right )}{d^4 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {a+b \cosh ^{-1}(c x)}{x^3} \, dx}{d^3}-\frac {(3 e) \int \frac {a+b \cosh ^{-1}(c x)}{x} \, dx}{d^4}+\frac {\left (3 e^2\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d+e x^2} \, dx}{d^4}+\frac {\left (2 e^2\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx}{d^3}+\frac {e^2 \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx}{d^2}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{2 d^3 x^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{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 (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{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 \cosh ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \cosh ^{-1}(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 \cosh ^{-1}(c x)}{2 d^3 x^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}+\frac {3 e \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b d^4}-\frac {(6 e) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{d^4}-\frac {\left (3 e^{3/2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 d^4}+\frac {\left (3 e^{3/2}\right ) \int \frac {a+b \cosh ^{-1}(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 \cosh ^{-1}(c x)}{2 d^3 x^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}+\frac {3 e \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b d^4}-\frac {3 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d^4}+\frac {(3 b e) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(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,\cosh ^{-1}(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,\cosh ^{-1}(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 \cosh ^{-1}(c x)}{2 d^3 x^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}+\frac {b c e \sqrt {-1+c^2 x^2} \tanh ^{-1}\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 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d^4}+\frac {(3 b e) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(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,\cosh ^{-1}(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,\cosh ^{-1}(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,\cosh ^{-1}(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,\cosh ^{-1}(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 \cosh ^{-1}(c x)}{2 d^3 x^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}+\frac {b c e \sqrt {-1+c^2 x^2} \tanh ^{-1}\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} \tanh ^{-1}\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 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}-\frac {3 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d^4}-\frac {3 b e \text {Li}_2\left (-e^{2 \cosh ^{-1}(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,\cosh ^{-1}(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,\cosh ^{-1}(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,\cosh ^{-1}(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,\cosh ^{-1}(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 \cosh ^{-1}(c x)}{2 d^3 x^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}+\frac {b c e \sqrt {-1+c^2 x^2} \tanh ^{-1}\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} \tanh ^{-1}\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 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}-\frac {3 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d^4}-\frac {3 b e \text {Li}_2\left (-e^{2 \cosh ^{-1}(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^{\cosh ^{-1}(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^{\cosh ^{-1}(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^{\cosh ^{-1}(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^{\cosh ^{-1}(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 \cosh ^{-1}(c x)}{2 d^3 x^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}+\frac {b c e \sqrt {-1+c^2 x^2} \tanh ^{-1}\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} \tanh ^{-1}\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 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}-\frac {3 e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d^4}+\frac {3 b e \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}+\frac {3 b e \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^4}-\frac {3 b e \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{2 d^4}\\ \end {align*}

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Mathematica [F]
time = 7.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \cosh ^{-1}(c x)}{x^3 \left (d+e x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 12.18, size = 1938, normalized size = 2.32

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1938\)
default	\(\text {Expression too large to display}\)	\(1938\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^2*(3/4*b/d^3/(c^2*d+e)*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*b/d^3/(c^2*d+e)*e*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-3*b/d^3/(c^2*d+e)*e*dilog(1-I*(c*x+(c*x-1)^
(1/2)*(c*x+1)^(1/2)))+3/4*b/d^3/(c^2*d+e)*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))*e^2+1/2*b*c^3/x/d/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e+7/8*b*c^3*x/d^2/(c^2*e*x^
2+c^2*d)^2/(c^2*d+e)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^2-3*b/d^3/(c^2*d+e)*e*arccosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2
)*(c*x+1)^(1/2)))-3/8*b*c^4*x^4/d^3/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*e^3-9/4*b*c^2/d^2/(c^2*e*x^2+c^2*d)^2/(c^2*d
+e)*arccosh(c*x)*e^2-9/8*b/c^2*((c^2*d+e)*c^2*d)^(1/2)/d^4/(c^2*d+e)^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))-3*b/c^2/d^4*e^2/(c^2*d+e)*arccosh(c*x)*ln(1+I*(c*x+(c*
x-1)^(1/2)*(c*x+1)^(1/2)))-3*b/c^2/d^4*e^2/(c^2*d+e)*arccosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-1/
2*b*c^4/x^2/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*arccosh(c*x)+3/8*b*c^3*x^3/d^3/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*(c*x-1)
^(1/2)*(c*x+1)^(1/2)*e^3-3/2*b*c^4/d^2/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*arccosh(c*x)*e^2*x^2-3/4*b*c^4*x^2/d^2/(c
^2*e*x^2+c^2*d)^2/(c^2*d+e)*e^2-3/2*b*c^2*x^2/d^3/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*arccosh(c*x)*e^3+1/2*b*c^5/x/(
c^2*e*x^2+c^2*d)^2/(c^2*d+e)*(c*x-1)^(1/2)*(c*x+1)^(1/2)-5/4*b*((c^2*d+e)*c^2*d)^(1/2)/d^3/(c^2*d+e)^2*e*arcta
nh(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))-b*c^6/d/(c^2*d+e)/(c^2*e
*x^2+c^2*d)^2*e*x^2-1/2*b*c^6/d^2/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*e^2*x^4-a*e/d^3/(c^2*e*x^2+c^2*d)-1/2*a/d^3/c^
2/x^2+b*c^5/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+1/2*b*c^5/d^2/(c^2*d+e)/(c^2*e*x^2
+c^2*d)^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^2*x^3-9/4*b*c^4/d/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2*e*arccosh(c*x)-3*b/d^3
/(c^2*d+e)*e*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-3/8*b*c^4/d/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*
e-3*b/c^2/d^4*e^2/(c^2*d+e)*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-3*b/c^2/d^4*e^2/(c^2*d+e)*dilog(1-I*(
c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+3/4*b/c^2/d^4*e^3/(c^2*d+e)*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*b/c^2/d^4*e^2/(c^2*d+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))-1/2*b*c^2/x^2/d/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)*e*arccosh(c*x)-3*a/c^2/d^4*e*ln(c*
x)+3/2*a/c^2*e/d^4*ln(c^2*e*x^2+c^2*d)-1/4*a*c^2*e/d^2/(c^2*e*x^2+c^2*d)^2-1/2*b*c^6/(c^2*d+e)/(c^2*e*x^2+c^2*
d)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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