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

Optimal. Leaf size=627 \[ -\frac{b (-d)^{3/2} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^{5/2}}+\frac{b (-d)^{3/2} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^{5/2}}-\frac{b (-d)^{3/2} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^{5/2}}+\frac{b (-d)^{3/2} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^{5/2}}+\frac{(-d)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^{5/2}}-\frac{(-d)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{2 e^{5/2}}+\frac{(-d)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^{5/2}}-\frac{(-d)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{2 e^{5/2}}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}-\frac{a d x}{e^2}-\frac{2 b \sqrt{c x-1} \sqrt{c x+1}}{9 c^3 e}+\frac{b d \sqrt{c x-1} \sqrt{c x+1}}{c e^2}-\frac{b d x \cosh ^{-1}(c x)}{e^2}-\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1}}{9 c e} \]

[Out]

-((a*d*x)/e^2) + (b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*e^2) - (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c^3*e) - (
b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c*e) - (b*d*x*ArcCosh[c*x])/e^2 + (x^3*(a + b*ArcCosh[c*x]))/(3*e) + ((
-d)^(3/2)*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^(5/2)
) - ((-d)^(3/2)*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e
^(5/2)) + ((-d)^(3/2)*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])]
)/(2*e^(5/2)) - ((-d)^(3/2)*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d)
- e])])/(2*e^(5/2)) - (b*(-d)^(3/2)*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])
/(2*e^(5/2)) + (b*(-d)^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^(5/2
)) - (b*(-d)^(3/2)*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(2*e^(5/2)) + (b
*(-d)^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 1.05156, antiderivative size = 627, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5792, 5654, 74, 5662, 100, 12, 5707, 5800, 5562, 2190, 2279, 2391} \[ -\frac{b (-d)^{3/2} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^{5/2}}+\frac{b (-d)^{3/2} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^{5/2}}-\frac{b (-d)^{3/2} \text{PolyLog}\left (2,-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^{5/2}}+\frac{b (-d)^{3/2} \text{PolyLog}\left (2,\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^{5/2}}+\frac{(-d)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right )}{2 e^{5/2}}-\frac{(-d)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right )}{2 e^{5/2}}+\frac{(-d)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right )}{2 e^{5/2}}-\frac{(-d)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right )}{2 e^{5/2}}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}-\frac{a d x}{e^2}-\frac{2 b \sqrt{c x-1} \sqrt{c x+1}}{9 c^3 e}+\frac{b d \sqrt{c x-1} \sqrt{c x+1}}{c e^2}-\frac{b d x \cosh ^{-1}(c x)}{e^2}-\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1}}{9 c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*d*x)/e^2) + (b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*e^2) - (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c^3*e) - (
b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c*e) - (b*d*x*ArcCosh[c*x])/e^2 + (x^3*(a + b*ArcCosh[c*x]))/(3*e) + ((
-d)^(3/2)*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^(5/2)
) - ((-d)^(3/2)*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e
^(5/2)) + ((-d)^(3/2)*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])]
)/(2*e^(5/2)) - ((-d)^(3/2)*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d)
- e])])/(2*e^(5/2)) - (b*(-d)^(3/2)*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])
/(2*e^(5/2)) + (b*(-d)^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^(5/2
)) - (b*(-d)^(3/2)*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(2*e^(5/2)) + (b
*(-d)^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e^(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 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 74

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

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 100

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))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 12

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

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{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{d+e x^2} \, dx &=\int \left (-\frac{d \left (a+b \cosh ^{-1}(c x)\right )}{e^2}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{e}+\frac{d^2 \left (a+b \cosh ^{-1}(c x)\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=-\frac{d \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{e^2}+\frac{d^2 \int \frac{a+b \cosh ^{-1}(c x)}{d+e x^2} \, dx}{e^2}+\frac{\int x^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{e}\\ &=-\frac{a d x}{e^2}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}-\frac{(b d) \int \cosh ^{-1}(c x) \, dx}{e^2}+\frac{d^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}{e^2}-\frac{(b c) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 e}\\ &=-\frac{a d x}{e^2}-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x}}{9 c e}-\frac{b d x \cosh ^{-1}(c x)}{e^2}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}-\frac{(-d)^{3/2} \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 e^2}-\frac{(-d)^{3/2} \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 e^2}+\frac{(b c d) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{e^2}-\frac{b \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 c e}\\ &=-\frac{a d x}{e^2}+\frac{b d \sqrt{-1+c x} \sqrt{1+c x}}{c e^2}-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x}}{9 c e}-\frac{b d x \cosh ^{-1}(c x)}{e^2}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}-\frac{(-d)^{3/2} \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 e^2}-\frac{(-d)^{3/2} \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 e^2}-\frac{(2 b) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 c e}\\ &=-\frac{a d x}{e^2}+\frac{b d \sqrt{-1+c x} \sqrt{1+c x}}{c e^2}-\frac{2 b \sqrt{-1+c x} \sqrt{1+c x}}{9 c^3 e}-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x}}{9 c e}-\frac{b d x \cosh ^{-1}(c x)}{e^2}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}-\frac{(-d)^{3/2} \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 e^2}-\frac{(-d)^{3/2} \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 e^2}-\frac{(-d)^{3/2} \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 e^2}-\frac{(-d)^{3/2} \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 e^2}\\ &=-\frac{a d x}{e^2}+\frac{b d \sqrt{-1+c x} \sqrt{1+c x}}{c e^2}-\frac{2 b \sqrt{-1+c x} \sqrt{1+c x}}{9 c^3 e}-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x}}{9 c e}-\frac{b d x \cosh ^{-1}(c x)}{e^2}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}+\frac{(-d)^{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 e^{5/2}}-\frac{(-d)^{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 e^{5/2}}+\frac{(-d)^{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 e^{5/2}}-\frac{(-d)^{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 e^{5/2}}-\frac{\left (b (-d)^{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 e^{5/2}}+\frac{\left (b (-d)^{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 e^{5/2}}-\frac{\left (b (-d)^{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 e^{5/2}}+\frac{\left (b (-d)^{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 e^{5/2}}\\ &=-\frac{a d x}{e^2}+\frac{b d \sqrt{-1+c x} \sqrt{1+c x}}{c e^2}-\frac{2 b \sqrt{-1+c x} \sqrt{1+c x}}{9 c^3 e}-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x}}{9 c e}-\frac{b d x \cosh ^{-1}(c x)}{e^2}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}+\frac{(-d)^{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 e^{5/2}}-\frac{(-d)^{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 e^{5/2}}+\frac{(-d)^{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 e^{5/2}}-\frac{(-d)^{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 e^{5/2}}-\frac{\left (b (-d)^{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 e^{5/2}}+\frac{\left (b (-d)^{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 e^{5/2}}-\frac{\left (b (-d)^{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 e^{5/2}}+\frac{\left (b (-d)^{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 e^{5/2}}\\ &=-\frac{a d x}{e^2}+\frac{b d \sqrt{-1+c x} \sqrt{1+c x}}{c e^2}-\frac{2 b \sqrt{-1+c x} \sqrt{1+c x}}{9 c^3 e}-\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x}}{9 c e}-\frac{b d x \cosh ^{-1}(c x)}{e^2}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}+\frac{(-d)^{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 e^{5/2}}-\frac{(-d)^{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 e^{5/2}}+\frac{(-d)^{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 e^{5/2}}-\frac{(-d)^{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 e^{5/2}}-\frac{b (-d)^{3/2} \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 e^{5/2}}+\frac{b (-d)^{3/2} \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-c^2 d-e}}\right )}{2 e^{5/2}}-\frac{b (-d)^{3/2} \text{Li}_2\left (-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 e^{5/2}}+\frac{b (-d)^{3/2} \text{Li}_2\left (\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}+\sqrt{-c^2 d-e}}\right )}{2 e^{5/2}}\\ \end{align*}

Mathematica [C]  time = 1.38052, size = 524, normalized size = 0.84 \[ \frac{b \left (-i d^{3/2} \left (2 \text{PolyLog}\left (2,\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )+2 \text{PolyLog}\left (2,-\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+\cosh ^{-1}(c x) \left (-\cosh ^{-1}(c x)+2 \left (\log \left (1+\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )+\log \left (1+\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )\right )\right )+i d^{3/2} \left (2 \text{PolyLog}\left (2,\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right )+2 \text{PolyLog}\left (2,\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )+\cosh ^{-1}(c x) \left (-\cosh ^{-1}(c x)+2 \left (\log \left (1+\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right )+\log \left (1-\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right )\right )\right )\right )-\frac{4 e^{3/2} \left (\sqrt{c x-1} \sqrt{c x+1} \left (c^2 x^2+2\right )-3 c^3 x^3 \cosh ^{-1}(c x)\right )}{9 c^3}+\frac{4 d \sqrt{e} \left (\sqrt{\frac{c x-1}{c x+1}} (c x+1)-c x \cosh ^{-1}(c x)\right )}{c}\right )}{4 e^{5/2}}+\frac{a d^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{e^{5/2}}-\frac{a d x}{e^2}+\frac{a x^3}{3 e} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((a*d*x)/e^2) + (a*x^3)/(3*e) + (a*d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/e^(5/2) + (b*((4*d*Sqrt[e]*(Sqrt[(-1
+ c*x)/(1 + c*x)]*(1 + c*x) - c*x*ArcCosh[c*x]))/c - (4*e^(3/2)*(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2 + c^2*x^2) -
3*c^3*x^3*ArcCosh[c*x]))/(9*c^3) - I*d^(3/2)*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (I*Sqrt[e]*E^ArcCosh[c*
x])/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*P
olyLog[2, (I*Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, ((-I)*Sqrt[e]*E^ArcCosh[
c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])]) + I*d^(3/2)*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (I*Sqrt[e]*E^ArcCo
sh[c*x])/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])]
)) + 2*PolyLog[2, (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (I*Sqrt[e]*E^ArcCos
h[c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])])))/(4*e^(5/2))

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Maple [C]  time = 3.506, size = 364, normalized size = 0.6 \begin{align*}{\frac{{x}^{3}a}{3\,e}}-{\frac{adx}{{e}^{2}}}+{\frac{a{d}^{2}}{{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{bdx{\rm arccosh} \left (cx\right )}{{e}^{2}}}+{\frac{bd}{c{e}^{2}}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{cb{d}^{2}}{2\,{e}^{2}}\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}}{{{\it \_R1}}^{2}e+2\,{c}^{2}d+e} \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{cb{d}^{2}}{2\,{e}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ( e{{\it \_Z}}^{4}+ \left ( 4\,{c}^{2}d+2\,e \right ){{\it \_Z}}^{2}+e \right ) }{\frac{1}{{\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{b{x}^{2}}{9\,ce}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{b{\rm arccosh} \left (cx\right ){x}^{3}}{3\,e}}-{\frac{2\,b}{9\,{c}^{3}e}\sqrt{cx-1}\sqrt{cx+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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