\(\int \frac {x^4 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx\) [489]

Optimal result

Integrand size = 21, antiderivative size = 627 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=-\frac {a d x}{e^2}+\frac {b d \sqrt {-1+c x} \sqrt {1+c x}}{c e^2}-\frac {2 b \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3 e}-\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c e}-\frac {b d x \text {arccosh}(c x)}{e^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 e}+\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}+\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}-\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}+\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}-\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}+\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}} \]

[Out]

-a*d*x/e^2-b*d*x*arccosh(c*x)/e^2+1/3*x^3*(a+b*arccosh(c*x))/e+1/2*(-d)^(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)))/e^(5/2)-1/2*(-d)^(3/2)*(a+b*arccosh(c*x))*l
n(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)))/e^(5/2)+1/2*(-d)^(3/2)*(a+b*arc
cosh(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)))/e^(5/2)-1/2*(-d)^(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)))/e^(5/2)
-1/2*b*(-d)^(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)))/e^(5/2
)+1/2*b*(-d)^(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)))/e^(5/2
)-1/2*b*(-d)^(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)))/e^(5/
2)+1/2*b*(-d)^(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)))/e^(5/
2)+b*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/e^2-2/9*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3/e-1/9*b*x^2*(c*x-1)^(1/2)*(c*x+
1)^(1/2)/c/e

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 627, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5959, 5879, 75, 5883, 102, 12, 5909, 5962, 5681, 2221, 2317, 2438} \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 e^{5/2}}+\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 e^{5/2}}-\frac {(-d)^{3/2} (a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 e^{5/2}}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 e}-\frac {a d x}{e^2}-\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^{5/2}}+\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^{5/2}}-\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^{5/2}}+\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^{5/2}}-\frac {b d x \text {arccosh}(c x)}{e^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1}}{9 c^3 e}+\frac {b d \sqrt {c x-1} \sqrt {c x+1}}{c e^2}-\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1}}{9 c e} \]

[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 12

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

Rule 75

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 102

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 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 5879

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 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 {d (a+b \text {arccosh}(c x))}{e^2}+\frac {x^2 (a+b \text {arccosh}(c x))}{e}+\frac {d^2 (a+b \text {arccosh}(c x))}{e^2 \left (d+e x^2\right )}\right ) \, dx \\ & = -\frac {d \int (a+b \text {arccosh}(c x)) \, dx}{e^2}+\frac {d^2 \int \frac {a+b \text {arccosh}(c x)}{d+e x^2} \, dx}{e^2}+\frac {\int x^2 (a+b \text {arccosh}(c x)) \, dx}{e} \\ & = -\frac {a d x}{e^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 e}-\frac {(b d) \int \text {arccosh}(c x) \, dx}{e^2}+\frac {d^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}{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 \text {arccosh}(c x)}{e^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 e}-\frac {(-d)^{3/2} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 e^2}-\frac {(-d)^{3/2} \int \frac {a+b \text {arccosh}(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 \text {arccosh}(c x)}{e^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 e}-\frac {(-d)^{3/2} \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 e^2}-\frac {(-d)^{3/2} \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 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 \text {arccosh}(c x)}{e^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 e}-\frac {(-d)^{3/2} \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 e^2}-\frac {(-d)^{3/2} \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 e^2}-\frac {(-d)^{3/2} \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 e^2}-\frac {(-d)^{3/2} \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 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 \text {arccosh}(c x)}{e^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 e}+\frac {(-d)^{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 e^{5/2}}-\frac {(-d)^{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 e^{5/2}}+\frac {(-d)^{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 e^{5/2}}-\frac {(-d)^{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 e^{5/2}}-\frac {\left (b (-d)^{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 e^{5/2}}+\frac {\left (b (-d)^{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 e^{5/2}}-\frac {\left (b (-d)^{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 e^{5/2}}+\frac {\left (b (-d)^{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 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 \text {arccosh}(c x)}{e^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 e}+\frac {(-d)^{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 e^{5/2}}-\frac {(-d)^{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 e^{5/2}}+\frac {(-d)^{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 e^{5/2}}-\frac {(-d)^{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 e^{5/2}}-\frac {\left (b (-d)^{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 e^{5/2}}+\frac {\left (b (-d)^{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 e^{5/2}}-\frac {\left (b (-d)^{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 e^{5/2}}+\frac {\left (b (-d)^{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 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 \text {arccosh}(c x)}{e^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{3 e}+\frac {(-d)^{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 e^{5/2}}-\frac {(-d)^{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 e^{5/2}}+\frac {(-d)^{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 e^{5/2}}-\frac {(-d)^{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 e^{5/2}}-\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}+\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}-\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}}+\frac {b (-d)^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{5/2}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.86 (sec) , antiderivative size = 524, normalized size of antiderivative = 0.84 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=-\frac {a d x}{e^2}+\frac {a x^3}{3 e}+\frac {a d^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{e^{5/2}}+\frac {b \left (\frac {4 d \sqrt {e} \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)-c x \text {arccosh}(c x)\right )}{c}-\frac {4 e^{3/2} \left (\sqrt {-1+c x} \sqrt {1+c x} \left (2+c^2 x^2\right )-3 c^3 x^3 \text {arccosh}(c x)\right )}{9 c^3}-i d^{3/2} \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \left (\log \left (1+\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )+i d^{3/2} \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \left (\log \left (1+\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {i \sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )}{4 e^{5/2}} \]

[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))

Maple [C] (verified)

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

Time = 45.91 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.58

method	result	size
parts	\(\frac {a \,x^{3}}{3 e}-\frac {a d x}{e^{2}}+\frac {a \,d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e^{2} \sqrt {d e}}-\frac {b \,x^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{9 c e}-\frac {b d x \,\operatorname {arccosh}\left (c x \right )}{e^{2}}+\frac {b d \sqrt {c x -1}\, \sqrt {c x +1}}{c \,e^{2}}+\frac {b c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2 e^{2}}-\frac {b c \,d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{2 e^{2}}+\frac {b \,\operatorname {arccosh}\left (c x \right ) x^{3}}{3 e}-\frac {2 b \sqrt {c x -1}\, \sqrt {c x +1}}{9 c^{3} e}\)	\(364\)
derivativedivides	\(\frac {-\frac {a \,c^{5} d x}{e^{2}}+\frac {a \,c^{5} x^{3}}{3 e}+\frac {a \,c^{5} d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e^{2} \sqrt {d e}}+\frac {b \,c^{4} \sqrt {c x +1}\, \sqrt {c x -1}\, d}{e^{2}}-\frac {b \,c^{5} \operatorname {arccosh}\left (c x \right ) d x}{e^{2}}+\frac {b \,c^{6} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2 e^{2}}-\frac {b \,c^{6} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{2 e^{2}}-\frac {b \,c^{4} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2}}{9 e}+\frac {b \,c^{5} \operatorname {arccosh}\left (c x \right ) x^{3}}{3 e}-\frac {2 b \,c^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{9 e}}{c^{5}}\)	\(387\)
default	\(\frac {-\frac {a \,c^{5} d x}{e^{2}}+\frac {a \,c^{5} x^{3}}{3 e}+\frac {a \,c^{5} d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{e^{2} \sqrt {d e}}+\frac {b \,c^{4} \sqrt {c x +1}\, \sqrt {c x -1}\, d}{e^{2}}-\frac {b \,c^{5} \operatorname {arccosh}\left (c x \right ) d x}{e^{2}}+\frac {b \,c^{6} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2 e^{2}}-\frac {b \,c^{6} d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{2 e^{2}}-\frac {b \,c^{4} \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2}}{9 e}+\frac {b \,c^{5} \operatorname {arccosh}\left (c x \right ) x^{3}}{3 e}-\frac {2 b \,c^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{9 e}}{c^{5}}\)	\(387\)

[In]

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

[Out]

1/3*a*x^3/e-a*d*x/e^2+a*d^2/e^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-1/9*b*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/e-
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*b*c/e^2*d^2*sum(_R1/(_R1^2*e+2*c^2*d+e)*(arcc
osh(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=R
ootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))-1/2*b*c/e^2*d^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))+1/3*b/e*arccosh(c*x)*x^3-2/9*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3/e

Fricas [F]

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

[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)

Sympy [F]

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

[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)

Maxima [F(-2)]

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

[In]

integrate(x^4*(a+b*arccosh(c*x))/(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]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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