\(\int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx\) [492]

Optimal result

Integrand size = 19, antiderivative size = 449 \[ \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=-\frac {(a+b \text {arccosh}(c x))^2}{2 b e}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e} \]

[Out]

-1/2*(a+b*arccosh(c*x))^2/b/e+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+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+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+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+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+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+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+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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5959, 5962, 5681, 2221, 2317, 2438} \[ \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 e}+\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 e}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 e}+\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 e}-\frac {(a+b \text {arccosh}(c x))^2}{2 b e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e} \]

[In]

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

[Out]

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

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 5959

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

Rule 5962

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

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a+b \text {arccosh}(c x)}{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 \\ & = -\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 \sqrt {e}}+\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {e}} \\ & = -\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 \sqrt {e}}+\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 \sqrt {e}} \\ & = -\frac {(a+b \text {arccosh}(c x))^2}{2 b e}-\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 \sqrt {e}}-\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 \sqrt {e}}+\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 \sqrt {e}}+\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 \sqrt {e}} \\ & = -\frac {(a+b \text {arccosh}(c x))^2}{2 b e}+\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}+\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}+\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}+\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}-\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}-\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}-\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}-\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} \\ & = -\frac {(a+b \text {arccosh}(c x))^2}{2 b e}+\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}+\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}+\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}+\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}-\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}-\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}-\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}-\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} \\ & = -\frac {(a+b \text {arccosh}(c x))^2}{2 b e}+\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}+\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}+\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}+\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}+\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}+\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}+\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}+\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.00 \[ \int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx=-\frac {b \text {arccosh}(c x)^2}{2 e}+\frac {b \text {arccosh}(c x) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \text {arccosh}(c x) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \text {arccosh}(c x) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {a \log \left (d+e x^2\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 0.88 (sec) , antiderivative size = 1987, normalized size of antiderivative = 4.43

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1987\)
default	\(\text {Expression too large to display}\)	\(1987\)
parts	\(\text {Expression too large to display}\)	\(1988\)

[In]

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

[Out]

1/c^2*(1/2*a*c^2/e*ln(c^2*e*x^2+c^2*d)+b*c^2*(-1/2/e*arccosh(c*x)^2+1/2*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+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/(c^2*d+e)*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/(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))-1/4*(d*c^2*(c^2*d+e))^(1/2)/d/c
^2/(c^2*d+e)*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))+(
-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^2/(c^2*d+e)*arccosh(c*x)^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)/e^2/(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))-(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)
+e)/e^3*c^2*d*arccosh(c*x)^2+1/2*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)/e^3*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))*d*c^2+1/2*(d*c^2*(c^2*d+e))^(1/2)/e/(c^2*d+e)*arccosh(c*x
)^2-1/4*(d*c^2*(c^2*d+e))^(1/2)/e/(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))-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/(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)
-(-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^2/(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)-(-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^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*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)/e^2*arcco
sh(c*x)^2+1/4*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+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))+1/4*(d*c^2*(c^2*d+e))^(1/2)/d/c^2/(c^2*d+e)*arccosh(c*x)^2-1/8*(d*c^2*(c^2*d
+e))^(1/2)/d/c^2/(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))+(-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^3/(c^2*d+e)*arcc
osh(c*x)^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^2*d/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))+(2*c^2*d-2*(d*c^
2*(c^2*d+e))^(1/2)+e)/e^3*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))*c
^2*d*arccosh(c*x)-1/2*(d*c^2*(c^2*d+e))^(1/2)/e/(c^2*d+e)*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/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))))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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