3.497 \(\int \frac{a+b \cosh ^{-1}(c x)}{x^4 (d+e x^2)} \, dx\)

Optimal. Leaf size=624 \[ -\frac{b e^{3/2} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 (-d)^{5/2}}+\frac{b e^{3/2} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 (-d)^{5/2}}-\frac{b e^{3/2} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 (-d)^{5/2}}+\frac{b e^{3/2} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 (-d)^{5/2}}+\frac{e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 (-d)^{5/2}}-\frac{e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{2 (-d)^{5/2}}+\frac{e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 (-d)^{5/2}}-\frac{e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{2 (-d)^{5/2}}+\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac{b c^3 \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{6 d}-\frac{b c e \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{d^2}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{6 d x^2} \]

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.981366, antiderivative size = 624, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5792, 5662, 103, 12, 92, 205, 5707, 5800, 5562, 2190, 2279, 2391} \[ -\frac{b e^{3/2} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 (-d)^{5/2}}+\frac{b e^{3/2} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 (-d)^{5/2}}-\frac{b e^{3/2} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 (-d)^{5/2}}+\frac{b e^{3/2} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 (-d)^{5/2}}+\frac{e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 (-d)^{5/2}}-\frac{e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{2 (-d)^{5/2}}+\frac{e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 (-d)^{5/2}}-\frac{e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{2 (-d)^{5/2}}+\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac{b c^3 \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{6 d}-\frac{b c e \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{d^2}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{6 d x^2} \]

Antiderivative was successfully verified.

[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 5792

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 5662

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))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 103

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] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 12

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

Rule 92

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 205

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

Rule 5707

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 5800

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

Rule 5562

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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]

Rubi steps

\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{x^4 \left (d+e x^2\right )} \, dx &=\int \left (\frac{a+b \cosh ^{-1}(c x)}{d x^4}-\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x^2}+\frac{e^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{a+b \cosh ^{-1}(c x)}{x^4} \, dx}{d}-\frac{e \int \frac{a+b \cosh ^{-1}(c x)}{x^2} \, dx}{d^2}+\frac{e^2 \int \frac{a+b \cosh ^{-1}(c x)}{d+e x^2} \, dx}{d^2}\\ &=-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{2 d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \left (a+b \cosh ^{-1}(c x)\right )}{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 \cosh ^{-1}(c x)}{3 d x^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{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 ) \operatorname{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 \cosh ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 (-d)^{5/2}}-\frac{e^2 \int \frac{a+b \cosh ^{-1}(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 \cosh ^{-1}(c x)}{3 d x^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}-\frac{b c e \tan ^{-1}\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 \operatorname{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)^{5/2}}-\frac{e^2 \operatorname{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)^{5/2}}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{6 d x^2}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}-\frac{b c e \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{d^2}+\frac{\left (b c^4\right ) \operatorname{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 \operatorname{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)^{5/2}}-\frac{e^2 \operatorname{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)^{5/2}}-\frac{e^2 \operatorname{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)^{5/2}}-\frac{e^2 \operatorname{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)^{5/2}}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{6 d x^2}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}+\frac{b c^3 \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{6 d}-\frac{b c e \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{d^2}+\frac{e^{3/2} \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)^{5/2}}-\frac{e^{3/2} \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)^{5/2}}+\frac{e^{3/2} \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)^{5/2}}-\frac{e^{3/2} \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)^{5/2}}-\frac{\left (b e^{3/2}\right ) \operatorname{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)^{5/2}}+\frac{\left (b e^{3/2}\right ) \operatorname{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)^{5/2}}-\frac{\left (b e^{3/2}\right ) \operatorname{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)^{5/2}}+\frac{\left (b e^{3/2}\right ) \operatorname{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)^{5/2}}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{6 d x^2}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}+\frac{b c^3 \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{6 d}-\frac{b c e \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{d^2}+\frac{e^{3/2} \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)^{5/2}}-\frac{e^{3/2} \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)^{5/2}}+\frac{e^{3/2} \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)^{5/2}}-\frac{e^{3/2} \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)^{5/2}}-\frac{\left (b e^{3/2}\right ) \operatorname{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)^{5/2}}+\frac{\left (b e^{3/2}\right ) \operatorname{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)^{5/2}}-\frac{\left (b e^{3/2}\right ) \operatorname{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)^{5/2}}+\frac{\left (b e^{3/2}\right ) \operatorname{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)^{5/2}}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x}}{6 d x^2}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac{e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}+\frac{b c^3 \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{6 d}-\frac{b c e \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{d^2}+\frac{e^{3/2} \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)^{5/2}}-\frac{e^{3/2} \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)^{5/2}}+\frac{e^{3/2} \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)^{5/2}}-\frac{e^{3/2} \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)^{5/2}}-\frac{b e^{3/2} \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 (-d)^{5/2}}+\frac{b e^{3/2} \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 (-d)^{5/2}}-\frac{b e^{3/2} \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 (-d)^{5/2}}+\frac{b e^{3/2} \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 (-d)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 1.48439, size = 641, normalized size = 1.03 \[ \frac{1}{6} \left (\frac{3 b e^{3/2} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{(-d)^{5/2}}-\frac{3 b e^{3/2} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-c \sqrt{-d}}\right )}{(-d)^{5/2}}-\frac{3 b e^{3/2} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{(-d)^{5/2}}+\frac{3 b e^{3/2} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{(-d)^{5/2}}-\frac{3 e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{(-d)^{5/2}}+\frac{3 e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-c \sqrt{-d}}+1\right )}{(-d)^{5/2}}+\frac{3 e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{(-d)^{5/2}}-\frac{3 e^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{(-d)^{5/2}}+\frac{6 e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{d x^3}-\frac{6 b c e \sqrt{c^2 x^2-1} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{d^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c \left (c^2 x^2+c^2 x^2 \sqrt{c^2 x^2-1} \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )-1\right )}{d x^2 \sqrt{c x-1} \sqrt{c x+1}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*(a + b*ArcCosh[c*x]))/(d*x^3) + (6*e*(a + b*ArcCosh[c*x]))/(d^2*x) - (6*b*c*e*Sqrt[-1 + c^2*x^2]*ArcTan[S
qrt[-1 + c^2*x^2]])/(d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*(-1 + c^2*x^2 + c^2*x^2*Sqrt[-1 + c^2*x^2]*ArcTa
n[Sqrt[-1 + c^2*x^2]]))/(d*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (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 + (Sq
rt[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, (Sqr
t[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^ArcCo
sh[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]  time = 0.811, size = 410, normalized size = 0.7 \begin{align*}{\frac{a{e}^{2}}{{d}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{a}{3\,d{x}^{3}}}+{\frac{ae}{{d}^{2}x}}+{\frac{bc}{6\,d{x}^{2}}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{b{\rm arccosh} \left (cx\right )e}{{d}^{2}x}}-{\frac{b{\rm arccosh} \left (cx\right )}{3\,d{x}^{3}}}-2\,{\frac{bce\arctan \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }{{d}^{2}}}-{\frac{b{e}^{2}}{8\,c{d}^{3}}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( 4\,{c}^{2}d+2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{{{\it \_R1}}^{2}e+4\,{c}^{2}d+e}{{\it \_R1}\, \left ({{\it \_R1}}^{2}e+2\,{c}^{2}d+e \right ) } \left ({\rm arccosh} \left (cx\right )\ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) } \right ) \right ) }}+{\frac{{c}^{3}b}{3\,d}\arctan \left ( cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{b{e}^{2}}{8\,c{d}^{3}}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( 4\,{c}^{2}d+2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{4\,{{\it \_R1}}^{2}{c}^{2}d+{{\it \_R1}}^{2}e+e}{{\it \_R1}\, \left ({{\it \_R1}}^{2}e+2\,{c}^{2}d+e \right ) } \left ({\rm arccosh} \left (cx\right )\ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) } \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*e^2/d^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))-1/3*a/d/x^3+a/d^2*e/x+1/6*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/x^2+
b*arccosh(c*x)/d^2*e/x-1/3*b*arccosh(c*x)/d/x^3-2*c*b/d^2*e*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-1/8/c*b/d^
3*e^2*sum((_R1^2*e+4*c^2*d+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))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))+1/3*c^3*b/d*arc
tan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+1/8/c*b/d^3*e^2*sum((4*_R1^2*c^2*d+_R1^2*e+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))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{arcosh}\left (c x\right ) + a}{e x^{6} + d x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{acosh}{\left (c x \right )}}{x^{4} \left (d + e x^{2}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)