\(\int \frac {a+b \text {arccosh}(c x)}{x^4 (d+e x^2)} \, dx\) [497]

Optimal result

Integrand size = 21, antiderivative size = 624 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}+\frac {b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}-\frac {b c e \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}-\frac {b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}-\frac {b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}} \]

[Out]

1/3*(-a-b*arccosh(c*x))/d/x^3+e*(a+b*arccosh(c*x))/d^2/x+1/6*b*c^3*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))/d-b*c*e
*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))/d^2+1/2*e^(3/2)*(a+b*arccosh(c*x))*ln(1-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))
*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/(-d)^(5/2)-1/2*e^(3/2)*(a+b*arccosh(c*x))*ln(1+(c*x+(c*x-1)^(1/2)*(c
*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/(-d)^(5/2)+1/2*e^(3/2)*(a+b*arccosh(c*x))*ln(1-(c*x+(c*x
-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/(-d)^(5/2)-1/2*e^(3/2)*(a+b*arccosh(c*x))*ln
(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/(-d)^(5/2)-1/2*b*e^(3/2)*polylog
(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/(-d)^(5/2)+1/2*b*e^(3/2)*polylo
g(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/(-d)^(5/2)-1/2*b*e^(3/2)*polylo
g(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/(-d)^(5/2)+1/2*b*e^(3/2)*polyl
og(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/(-d)^(5/2)+1/6*b*c*(c*x-1)^(1/
2)*(c*x+1)^(1/2)/d/x^2

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5959, 5883, 105, 12, 94, 211, 5909, 5962, 5681, 2221, 2317, 2438} \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 (-d)^{5/2}}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}-\frac {b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 (-d)^{5/2}}-\frac {b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 (-d)^{5/2}}+\frac {b c^3 \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )}{6 d}-\frac {b c e \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 d x^2} \]

[In]

Int[(a + b*ArcCosh[c*x])/(x^4*(d + e*x^2)),x]

[Out]

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*d*x^2) - (a + b*ArcCosh[c*x])/(3*d*x^3) + (e*(a + b*ArcCosh[c*x]))/(d^2*
x) + (b*c^3*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(6*d) - (b*c*e*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/d^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])])/(2*(-d)^(5/2
)) - (e^(3/2)*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*(-d
)^(5/2)) + (e^(3/2)*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/
(2*(-d)^(5/2)) - (e^(3/2)*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) -
e])])/(2*(-d)^(5/2)) - (b*e^(3/2)*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(
2*(-d)^(5/2)) + (b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*(-d)^(5/
2)) - (b*e^(3/2)*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(2*(-d)^(5/2)) + (
b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*(-d)^(5/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 94

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

Rule 105

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[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*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 211

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

Rule 2221

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

Rule 2317

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

Rule 2438

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

Rule 5681

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :
> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 - b^2, 2] + b*E^(c + d
*x))), x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 - b^2, 2] + b*E^(c + d*x))), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 - b^2, 0]

Rule 5883

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

Rule 5909

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p
] && (p > 0 || IGtQ[n, 0])

Rule 5959

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int
[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]

Rule 5962

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Subst[Int[(a + b*x)^n*(Sinh[x
]/(c*d + e*Cosh[x])), x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \text {arccosh}(c x)}{d x^4}-\frac {e (a+b \text {arccosh}(c x))}{d^2 x^2}+\frac {e^2 (a+b \text {arccosh}(c x))}{d^2 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {a+b \text {arccosh}(c x)}{x^4} \, dx}{d}-\frac {e \int \frac {a+b \text {arccosh}(c x)}{x^2} \, dx}{d^2}+\frac {e^2 \int \frac {a+b \text {arccosh}(c x)}{d+e x^2} \, dx}{d^2} \\ & = -\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}+\frac {(b c) \int \frac {1}{x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 d}-\frac {(b c e) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{d^2}+\frac {e^2 \int \left (\frac {\sqrt {-d} (a+b \text {arccosh}(c x))}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} (a+b \text {arccosh}(c x))}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{d^2} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}+\frac {(b c) \int \frac {c^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{6 d}-\frac {\left (b c^2 e\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}-\frac {e^2 \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 (-d)^{5/2}}-\frac {e^2 \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 (-d)^{5/2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}-\frac {b c e \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {\left (b c^3\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{6 d}-\frac {e^2 \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)^{5/2}}-\frac {e^2 \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)^{5/2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}-\frac {b c e \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {\left (b c^4\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}-\frac {e^2 \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)^{5/2}}-\frac {e^2 \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)^{5/2}}-\frac {e^2 \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)^{5/2}}-\frac {e^2 \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)^{5/2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}+\frac {b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}-\frac {b c e \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {e^{3/2} (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)^{5/2}}-\frac {e^{3/2} (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)^{5/2}}+\frac {e^{3/2} (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)^{5/2}}-\frac {e^{3/2} (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)^{5/2}}-\frac {\left (b e^{3/2}\right ) \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)^{5/2}}+\frac {\left (b e^{3/2}\right ) \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)^{5/2}}-\frac {\left (b e^{3/2}\right ) \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)^{5/2}}+\frac {\left (b e^{3/2}\right ) \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)^{5/2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}+\frac {b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}-\frac {b c e \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {e^{3/2} (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)^{5/2}}-\frac {e^{3/2} (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)^{5/2}}+\frac {e^{3/2} (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)^{5/2}}-\frac {e^{3/2} (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)^{5/2}}-\frac {\left (b e^{3/2}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 (-d)^{5/2}}+\frac {\left (b e^{3/2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 (-d)^{5/2}}-\frac {\left (b e^{3/2}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 (-d)^{5/2}}+\frac {\left (b e^{3/2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 (-d)^{5/2}} \\ & = \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \text {arccosh}(c x)}{3 d x^3}+\frac {e (a+b \text {arccosh}(c x))}{d^2 x}+\frac {b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}-\frac {b c e \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {e^{3/2} (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)^{5/2}}-\frac {e^{3/2} (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)^{5/2}}+\frac {e^{3/2} (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)^{5/2}}-\frac {e^{3/2} (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)^{5/2}}-\frac {b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}-\frac {b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}}+\frac {b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 (-d)^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.20 (sec) , antiderivative size = 657, normalized size of antiderivative = 1.05 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\frac {1}{6} \left (-\frac {2 a}{d x^3}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{d x^2}-\frac {2 b \text {arccosh}(c x)}{d x^3}+\frac {6 e (a+b \text {arccosh}(c x))}{d^2 x}+\frac {b c^3 \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{d \sqrt {-1+c x} \sqrt {1+c x}}-\frac {6 b c e \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}+\frac {3 e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}+\frac {3 e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}-\frac {3 e^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}+\frac {3 b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}-\frac {3 b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}-\frac {3 b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}+\frac {3 b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{(-d)^{5/2}}\right ) \]

[In]

Integrate[(a + b*ArcCosh[c*x])/(x^4*(d + e*x^2)),x]

[Out]

((-2*a)/(d*x^3) + (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(d*x^2) - (2*b*ArcCosh[c*x])/(d*x^3) + (6*e*(a + b*ArcCos
h[c*x]))/(d^2*x) + (b*c^3*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (6
*b*c*e*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*e^(3/2)*(a + b*A
rcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(-d)^(5/2) + (3*e^(3/2)*(a +
b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])])/(-d)^(5/2) + (3*e^(3/
2)*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(-d)^(5/2) - (3*e
^(3/2)*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(-d)^(5/2) +
(3*b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(-d)^(5/2) - (3*b*e^(3/2)
*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])])/(-d)^(5/2) - (3*b*e^(3/2)*PolyLog[
2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(-d)^(5/2) + (3*b*e^(3/2)*PolyLog[2, (Sqrt[
e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(-d)^(5/2))/6

Maple [C] (warning: unable to verify)

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

Time = 32.84 (sec) , antiderivative size = 424, normalized size of antiderivative = 0.68

method	result	size
parts	\(a \left (\frac {e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{d^{2} \sqrt {d e}}-\frac {1}{3 d \,x^{3}}+\frac {e}{d^{2} x}\right )+\frac {b \left (8 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{7} d^{2} x^{3}+4 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} d^{2} x -48 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{5} d e \,x^{3}+24 \,\operatorname {arccosh}\left (c x \right ) c^{4} d e \,x^{2}-8 c^{4} d^{2} \operatorname {arccosh}\left (c x \right )+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{4} x^{3} d^{3}}\)	\(424\)
derivativedivides	\(c^{3} \left (-\frac {a}{3 d \,c^{3} x^{3}}+\frac {a e}{c^{3} d^{2} x}+\frac {a \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c^{3} d^{2} \sqrt {d e}}-\frac {b \left (-8 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{7} d^{2} x^{3}-4 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} d^{2} x +48 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{5} d e \,x^{3}+8 c^{4} d^{2} \operatorname {arccosh}\left (c x \right )-24 \,\operatorname {arccosh}\left (c x \right ) c^{4} d e \,x^{2}+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{7} x^{3} d^{3}}\right )\)	\(437\)
default	\(c^{3} \left (-\frac {a}{3 d \,c^{3} x^{3}}+\frac {a e}{c^{3} d^{2} x}+\frac {a \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c^{3} d^{2} \sqrt {d e}}-\frac {b \left (-8 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{7} d^{2} x^{3}-4 \sqrt {c x +1}\, \sqrt {c x -1}\, c^{5} d^{2} x +48 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{5} d e \,x^{3}+8 c^{4} d^{2} \operatorname {arccosh}\left (c x \right )-24 \,\operatorname {arccosh}\left (c x \right ) c^{4} d e \,x^{2}+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e +e \right ) \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{7} x^{3} d^{3}}\right )\)	\(437\)

[In]

int((a+b*arccosh(c*x))/x^4/(e*x^2+d),x,method=_RETURNVERBOSE)

[Out]

a*(e^2/d^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-1/3/d/x^3+e/d^2/x)+1/24*b/c^4*(8*arctan(c*x+(c*x-1)^(1/2)*(c*x+
1)^(1/2))*c^7*d^2*x^3+4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^5*d^2*x-48*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*c^5*d
*e*x^3+24*arccosh(c*x)*c^4*d*e*x^2-8*c^4*d^2*arccosh(c*x)+3*sum((4*_R1^2*c^2*d+_R1^2*e+e)/_R1/(_R1^2*e+2*c^2*d
+e)*(arccosh(c*x)*ln((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R
1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))*e^2*c^3*x^3-3*sum((_R1^2*e+4*c^2*d+e)/_R1/(_R1^2*e+2*c^2*d+e)*(ar
ccosh(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*c^3*x^3)/x^3/d^3

Fricas [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{4}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))/x^4/(e*x^2+d),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{4} \left (d + e x^{2}\right )}\, dx \]

[In]

integrate((a+b*acosh(c*x))/x**4/(e*x**2+d),x)

[Out]

Integral((a + b*acosh(c*x))/(x**4*(d + e*x**2)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*arccosh(c*x))/x^4/(e*x^2+d),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{4}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))/x^4/(e*x^2+d),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d+e x^2\right )} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,\left (e\,x^2+d\right )} \,d x \]

[In]

int((a + b*acosh(c*x))/(x^4*(d + e*x^2)),x)

[Out]

int((a + b*acosh(c*x))/(x^4*(d + e*x^2)), x)