\(\int \frac {x^5 (a+b \text {arccosh}(c x))}{(d+e x^2)^3} \, dx\) [506]

Optimal result

Integrand size = 21, antiderivative size = 737 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b c d x \left (1-c^2 x^2\right )}{8 e^2 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {d^2 (a+b \text {arccosh}(c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \text {arccosh}(c x))}{e^3 \left (d+e x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b e^3}-\frac {b c \sqrt {d} \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{e^3 \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c \sqrt {d} \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^3} \]

[Out]

-1/4*d^2*(a+b*arccosh(c*x))/e^3/(e*x^2+d)^2+d*(a+b*arccosh(c*x))/e^3/(e*x^2+d)-1/2*(a+b*arccosh(c*x))^2/b/e^3+
1/2*(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^3+1/2
*(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^3+1/2*(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^3+1/2*(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^3+1/2*b*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)))/e^3+1/2*b*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^3+1/2*b*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^3+1/2*b*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^3+1/8*b*c*d*x*(-c^2*x^2+1)/e^2/(c^2*d+e)/(e*x^2+d)/(c*x-1)^(1/2)/(c*x+1)
^(1/2)+1/8*b*c*(2*c^2*d+e)*arctanh(x*(c^2*d+e)^(1/2)/d^(1/2)/(c^2*x^2-1)^(1/2))*d^(1/2)*(c^2*x^2-1)^(1/2)/e^3/
(c^2*d+e)^(3/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-b*c*arctanh(x*(c^2*d+e)^(1/2)/d^(1/2)/(c^2*x^2-1)^(1/2))*d^(1/2)*(
c^2*x^2-1)^(1/2)/e^3/(c^2*d+e)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 737, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {5959, 5957, 533, 390, 385, 214, 5962, 5681, 2221, 2317, 2438} \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 e^3}+\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 e^3}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 e^3}+\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 e^3}-\frac {d^2 (a+b \text {arccosh}(c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \text {arccosh}(c x))}{e^3 \left (d+e x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^3}+\frac {b c \sqrt {d} \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right ) \text {arctanh}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{8 e^3 \sqrt {c x-1} \sqrt {c x+1} \left (c^2 d+e\right )^{3/2}}-\frac {b c \sqrt {d} \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{e^3 \sqrt {c x-1} \sqrt {c x+1} \sqrt {c^2 d+e}}+\frac {b c d x \left (1-c^2 x^2\right )}{8 e^2 \sqrt {c x-1} \sqrt {c x+1} \left (c^2 d+e\right ) \left (d+e x^2\right )} \]

[In]

Int[(x^5*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3,x]

[Out]

(b*c*d*x*(1 - c^2*x^2))/(8*e^2*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2)) - (d^2*(a + b*ArcCosh[c*x
]))/(4*e^3*(d + e*x^2)^2) + (d*(a + b*ArcCosh[c*x]))/(e^3*(d + e*x^2)) - (a + b*ArcCosh[c*x])^2/(2*b*e^3) - (b
*c*Sqrt[d]*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(e^3*Sqrt[c^2*d + e]*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*Sqrt[d]*(2*c^2*d + e)*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqr
t[d]*Sqrt[-1 + c^2*x^2])])/(8*e^3*(c^2*d + e)^(3/2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((a + b*ArcCosh[c*x])*Log[
1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^3) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt
[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^3) + ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^Arc
Cosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e^3) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x]
)/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e^3) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(
c^2*d) - e]))])/(2*e^3) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e^3) +
(b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(2*e^3) + (b*PolyLog[2, (Sqrt[e
]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e^3)

Rule 214

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

Rule 385

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

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a
*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 533

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_)*((a2_) + (b2_.)*(x_)^(non2_.))^(
p_), x_Symbol] :> Dist[(a1 + b1*x^(n/2))^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*a2 + b1*b2*x^n)^FracPa
rt[p]), Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[
non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(EqQ[n, 2] && IGtQ[q, 0])

Rule 2221

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

Rule 2317

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

Rule 2438

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

Rule 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 5957

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

Rule 5959

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

Rule 5962

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 x (a+b \text {arccosh}(c x))}{e^2 \left (d+e x^2\right )^3}-\frac {2 d x (a+b \text {arccosh}(c x))}{e^2 \left (d+e x^2\right )^2}+\frac {x (a+b \text {arccosh}(c x))}{e^2 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx}{e^2}-\frac {(2 d) \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx}{e^2}+\frac {d^2 \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx}{e^2} \\ & = -\frac {d^2 (a+b \text {arccosh}(c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \text {arccosh}(c x))}{e^3 \left (d+e x^2\right )}-\frac {(b c d) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )} \, dx}{e^3}+\frac {\left (b c d^2\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )^2} \, dx}{4 e^3}+\frac {\int \left (-\frac {a+b \text {arccosh}(c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arccosh}(c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{e^2} \\ & = -\frac {d^2 (a+b \text {arccosh}(c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \text {arccosh}(c x))}{e^3 \left (d+e x^2\right )}-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 e^{5/2}}+\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 e^{5/2}}-\frac {\left (b c d \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )} \, dx}{e^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 e^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c d x \left (1-c^2 x^2\right )}{8 e^2 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {d^2 (a+b \text {arccosh}(c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \text {arccosh}(c x))}{e^3 \left (d+e x^2\right )}-\frac {\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^{5/2}}+\frac {\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^{5/2}}-\frac {\left (b c d \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{d-\left (c^2 d+e\right ) x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{e^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )} \, dx}{8 e^3 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c d x \left (1-c^2 x^2\right )}{8 e^2 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {d^2 (a+b \text {arccosh}(c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \text {arccosh}(c x))}{e^3 \left (d+e x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b e^3}-\frac {b c \sqrt {d} \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{e^3 \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\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^{5/2}}-\frac {\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^{5/2}}+\frac {\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^{5/2}}+\frac {\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^{5/2}}+\frac {\left (b c d \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{d-\left (c^2 d+e\right ) x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {b c d x \left (1-c^2 x^2\right )}{8 e^2 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {d^2 (a+b \text {arccosh}(c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \text {arccosh}(c x))}{e^3 \left (d+e x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b e^3}-\frac {b c \sqrt {d} \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{e^3 \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c \sqrt {d} \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(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^3}+\frac {(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^3}+\frac {(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^3}+\frac {(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^3}-\frac {b \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^3}-\frac {b \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^3}-\frac {b \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^3}-\frac {b \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^3} \\ & = \frac {b c d x \left (1-c^2 x^2\right )}{8 e^2 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {d^2 (a+b \text {arccosh}(c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \text {arccosh}(c x))}{e^3 \left (d+e x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b e^3}-\frac {b c \sqrt {d} \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{e^3 \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c \sqrt {d} \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(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^3}+\frac {(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^3}+\frac {(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^3}+\frac {(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^3}-\frac {b \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^3}-\frac {b \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^3}-\frac {b \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^3}-\frac {b \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^3} \\ & = \frac {b c d x \left (1-c^2 x^2\right )}{8 e^2 \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {d^2 (a+b \text {arccosh}(c x))}{4 e^3 \left (d+e x^2\right )^2}+\frac {d (a+b \text {arccosh}(c x))}{e^3 \left (d+e x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b e^3}-\frac {b c \sqrt {d} \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{e^3 \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c \sqrt {d} \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{8 e^3 \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(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^3}+\frac {(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^3}+\frac {(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^3}+\frac {(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^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^3}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^3} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.63 (sec) , antiderivative size = 1097, normalized size of antiderivative = 1.49 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\frac {-\frac {4 a d^2}{\left (d+e x^2\right )^2}+\frac {16 a d}{d+e x^2}+8 a \log \left (d+e x^2\right )+b \left (-\frac {c d \sqrt {e} \sqrt {-1+c x} \sqrt {1+c x}}{\left (c^2 d+e\right ) \left (-i \sqrt {d}+\sqrt {e} x\right )}-\frac {c d \sqrt {e} \sqrt {-1+c x} \sqrt {1+c x}}{\left (c^2 d+e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}+\frac {7 \sqrt {d} \text {arccosh}(c x)}{\sqrt {d}-i \sqrt {e} x}-\frac {d \text {arccosh}(c x)}{\left (\sqrt {d}+i \sqrt {e} x\right )^2}+\frac {7 \sqrt {d} \text {arccosh}(c x)}{\sqrt {d}+i \sqrt {e} x}+\frac {d \text {arccosh}(c x)}{\left (i \sqrt {d}+\sqrt {e} x\right )^2}-8 \text {arccosh}(c x)^2+8 \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+8 \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+8 \text {arccosh}(c x) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+8 \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )-\frac {7 i c \sqrt {d} \log \left (\frac {2 e \left (i \sqrt {e}+c^2 \sqrt {d} x-i \sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}+\frac {7 i c \sqrt {d} \log \left (\frac {2 e \left (-\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}-\frac {c^3 d^{3/2} \log \left (\frac {e \sqrt {c^2 d+e} \left (-i \sqrt {e}-c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c^3 \left (d+i \sqrt {d} \sqrt {e} x\right )}\right )}{\left (c^2 d+e\right )^{3/2}}+\frac {c^3 d^{3/2} \log \left (\frac {e \sqrt {c^2 d+e} \left (-i \sqrt {e}+c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c^3 \left (d-i \sqrt {d} \sqrt {e} x\right )}\right )}{\left (c^2 d+e\right )^{3/2}}+8 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+8 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+8 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+8 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )}{16 e^3} \]

[In]

Integrate[(x^5*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3,x]

[Out]

((-4*a*d^2)/(d + e*x^2)^2 + (16*a*d)/(d + e*x^2) + 8*a*Log[d + e*x^2] + b*(-((c*d*Sqrt[e]*Sqrt[-1 + c*x]*Sqrt[
1 + c*x])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))) - (c*d*Sqrt[e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*
(I*Sqrt[d] + Sqrt[e]*x)) + (7*Sqrt[d]*ArcCosh[c*x])/(Sqrt[d] - I*Sqrt[e]*x) - (d*ArcCosh[c*x])/(Sqrt[d] + I*Sq
rt[e]*x)^2 + (7*Sqrt[d]*ArcCosh[c*x])/(Sqrt[d] + I*Sqrt[e]*x) + (d*ArcCosh[c*x])/(I*Sqrt[d] + Sqrt[e]*x)^2 - 8
*ArcCosh[c*x]^2 + 8*ArcCosh[c*x]*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + 8*ArcC
osh[c*x]*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 8*ArcCosh[c*x]*Log[1 - (Sqr
t[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 8*ArcCosh[c*x]*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*
c*Sqrt[d] + Sqrt[-(c^2*d) - e])] - ((7*I)*c*Sqrt[d]*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d) - e]
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] + ((7*I)*c
*Sqrt[d]*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^
2*d) - e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] - (c^3*d^(3/2)*Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e] -
c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))])/(c^2*d + e)^(
3/2) + (c^3*d^(3/2)*Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt
[1 + c*x]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))])/(c^2*d + e)^(3/2) + 8*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*S
qrt[d] - Sqrt[-(c^2*d) - e])] + 8*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] +
8*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))] + 8*PolyLog[2, (Sqrt[e]*E^ArcCos
h[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])]))/(16*e^3)

Maple [C] (warning: unable to verify)

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

Time = 5.52 (sec) , antiderivative size = 3551, normalized size of antiderivative = 4.82

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3551\)
default	\(\text {Expression too large to display}\)	\(3551\)
parts	\(\text {Expression too large to display}\)	\(3556\)

[In]

int(x^5*(a+b*arccosh(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)

[Out]

1/c^6*(a*c^6*(-1/4*c^4*d^2/e^3/(c^2*e*x^2+c^2*d)^2+1/2/e^3*ln(c^2*e*x^2+c^2*d)+1/e^3*d*c^2/(c^2*e*x^2+c^2*d))+
b*c^6*((2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)/e^5/(c^2*d+e)*c^4*d^2*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/
(-2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))*arccosh(c*x)-1/4*(d*c^2*(c^2*d+e))^(1/2)/e/(c^2*d+e)^2/d/c^2*arccosh(c
*x)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)-e))-1/2*(d*c^2*(c^2*d+e))^(
1/2)/e^3/(c^2*d+e)^2*d*c^2*arccosh(c*x)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d+2*(d*c^2*(c^2*d+e
))^(1/2)-e))+3/2*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)/e^4/(c^2*d+e)*c^2*d*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(
1/2))^2/(-2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))*arccosh(c*x)-(-2*(d*c^2*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2
*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)*c^4*d^2/e^5/(c^4*d^2+2*c^2*d*e+e^2)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2
/(-2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))*arccosh(c*x)-2*(-2*(d*c^2*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-
(d*c^2*(c^2*d+e))^(1/2)*e)*c^2*d/e^4/(c^4*d^2+2*c^2*d*e+e^2)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^
2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))*arccosh(c*x)-1/4*(-2*(d*c^2*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*c^
2*(c^2*d+e))^(1/2)*e)/c^2/d/e^2/(c^4*d^2+2*c^2*d*e+e^2)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2
*(d*c^2*(c^2*d+e))^(1/2)-e))*arccosh(c*x)+(-2*(d*c^2*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*c^2*(c^2*d+
e))^(1/2)*e)*c^4*d^2/e^5/(c^4*d^2+2*c^2*d*e+e^2)*arccosh(c*x)^2+1/4*(-2*(d*c^2*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^
2+2*c^2*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)/c^2/d/e^2/(c^4*d^2+2*c^2*d*e+e^2)*arccosh(c*x)^2+1/4*(d*c^2*(c^2*d+e))^
(1/2)/e/(c^2*d+e)^2/d/c^2*arccosh(c*x)^2-1/8*(-2*(d*c^2*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*c^2*(c^2
*d+e))^(1/2)*e)/c^2/d/e^2/(c^4*d^2+2*c^2*d*e+e^2)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*
(d*c^2*(c^2*d+e))^(1/2)-e))+1/2*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)/e^5/(c^2*d+e)*c^4*d^2*polylog(2,e*(c*x+(
c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))-1/4*(d*c^2*(c^2*d+e))^(1/2)/e^3/(c^2*d+e
)^2*d*c^2*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)-e))+1/8*d*c^2*(6
*c^4*d^2*arccosh(c*x)+8*arccosh(c*x)*c^4*d*e*x^2-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*d*e*x-(c*x-1)^(1/2)*(c*x+1)^(
1/2)*e^2*c^3*x^3+c^4*d^2+2*c^4*d*e*x^2+c^4*e^2*x^4+6*c^2*d*e*arccosh(c*x)+8*arccosh(c*x)*e^2*c^2*x^2)/e^3/(c^2
*d+e)/(c^2*e*x^2+c^2*d)^2-3/2*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)/e^4/(c^2*d+e)*c^2*d*arccosh(c*x)^2-1/e^2/(
c^2*d+e)*arccosh(c*x)^2+1/2*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)/e^3/(c^2*d+e)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x
+1)^(1/2))^2/(-2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))*arccosh(c*x)+1/2/e^3/(c^2*d+e)*c^2*d*sum((_R1^2*e+4*c^2*d
+2*e)/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*ln((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(
1/2)*(c*x+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))-1/e^3/(c^2*d+e)*c^2*d*arccosh(c*x)^2-5/4*(-
2*(d*c^2*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)/e^3/(c^4*d^2+2*c^2*d*e+e^2)*ln(
1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))*arccosh(c*x)-3/4*(d*c^2*(c^2*d
+e))^(1/2)/e^2/(c^2*d+e)^2*arccosh(c*x)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d+2*(d*c^2*(c^2*d+e
))^(1/2)-e))+1/2*arccosh(c*x)^2/e^3+1/2/e^2/(c^2*d+e)*sum((_R1^2*e+4*c^2*d+2*e)/(_R1^2*e+2*c^2*d+e)*(arccosh(c
*x)*ln((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)),_R1=RootOf
(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))-1/2*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)/e^3/(c^2*d+e)*arccosh(c*x)^2-5/8*(-2*
(d*c^2*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)/e^3/(c^4*d^2+2*c^2*d*e+e^2)*polyl
og(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))+7/8*(d*c^2*(c^2*d+e))^(1/2)
/e^2/(c^2*d+e)^2*arctanh(1/4*(4*c^2*d+2*e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+2*e)/(c^4*d^2+c^2*d*e)^(1/2))+5/
4*(-2*(d*c^2*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)/e^3/(c^4*d^2+2*c^2*d*e+e^2)
*arccosh(c*x)^2+3/4*(d*c^2*(c^2*d+e))^(1/2)/e^2/(c^2*d+e)^2*arccosh(c*x)^2+1/4*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1
/2)+e)/e^3/(c^2*d+e)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))-3
/8*(d*c^2*(c^2*d+e))^(1/2)/e^2/(c^2*d+e)^2*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d+2*(d*c^2*
(c^2*d+e))^(1/2)-e))+1/2*(d*c^2*(c^2*d+e))^(1/2)/e^3/(c^2*d+e)^2*d*c^2*arccosh(c*x)^2-1/8*(d*c^2*(c^2*d+e))^(1
/2)/e/(c^2*d+e)^2/d/c^2*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)-e)
)-(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)/e^5/(c^2*d+e)*c^4*d^2*arccosh(c*x)^2+3/4*(d*c^2*(c^2*d+e))^(1/2)/e^3/(
c^2*d+e)^2*d*c^2*arctanh(1/4*(4*c^2*d+2*e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+2*e)/(c^4*d^2+c^2*d*e)^(1/2))-1/
2*(-2*(d*c^2*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)*c^4*d^2/e^5/(c^4*d^2+2*c^2*
d*e+e^2)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))+2*(-2*(d*c^2*
(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)*c^2*d/e^4/(c^4*d^2+2*c^2*d*e+e^2)*arccos
h(c*x)^2+3/4*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)/e^4/(c^2*d+e)*c^2*d*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^
(1/2))^2/(-2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))-(-2*(d*c^2*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*c^2*
(c^2*d+e))^(1/2)*e)*c^2*d/e^4/(c^4*d^2+2*c^2*d*e+e^2)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*
d-2*(d*c^2*(c^2*d+e))^(1/2)-e))))

Fricas [F]

\[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]

[In]

integrate(x^5*(a+b*arccosh(c*x))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

integral((b*x^5*arccosh(c*x) + a*x^5)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(x**5*(a+b*acosh(c*x))/(e*x**2+d)**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]

[In]

integrate(x^5*(a+b*arccosh(c*x))/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

1/4*a*((4*d*e*x^2 + 3*d^2)/(e^5*x^4 + 2*d*e^4*x^2 + d^2*e^3) + 2*log(e*x^2 + d)/e^3) + b*integrate(x^5*log(c*x
+ sqrt(c*x + 1)*sqrt(c*x - 1))/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^5*(a+b*arccosh(c*x))/(e*x^2+d)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]

[In]

int((x^5*(a + b*acosh(c*x)))/(d + e*x^2)^3,x)

[Out]

int((x^5*(a + b*acosh(c*x)))/(d + e*x^2)^3, x)