\(\int \frac {x}{\text {arccosh}(a x)^3} \, dx\) [61]

Optimal result

Integrand size = 8, antiderivative size = 68 \[ \int \frac {x}{\text {arccosh}(a x)^3} \, dx=-\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {1}{2 a^2 \text {arccosh}(a x)}-\frac {x^2}{\text {arccosh}(a x)}+\frac {\text {Shi}(2 \text {arccosh}(a x))}{a^2} \]

[Out]

1/2/a^2/arccosh(a*x)-x^2/arccosh(a*x)+Shi(2*arccosh(a*x))/a^2-1/2*x*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)
^2

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5886, 5951, 5887, 5556, 12, 3379, 5893} \[ \int \frac {x}{\text {arccosh}(a x)^3} \, dx=\frac {\text {Shi}(2 \text {arccosh}(a x))}{a^2}+\frac {1}{2 a^2 \text {arccosh}(a x)}-\frac {x^2}{\text {arccosh}(a x)}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2} \]

[In]

Int[x/ArcCosh[a*x]^3,x]

[Out]

-1/2*(x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x]^2) + 1/(2*a^2*ArcCosh[a*x]) - x^2/ArcCosh[a*x] + SinhInt
egral[2*ArcCosh[a*x]]/a^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 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5886

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((
a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Dist[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcCosh
[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Dist[m/(b*c*(n + 1)), Int[x^(m - 1)*((a + b*ArcCosh[c
*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 5887

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Cosh
[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

Rule 5951

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 +
e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Dist[f*(m/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]
]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b,
c, d1, e1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}-\frac {\int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2} \, dx}{2 a}+a \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2} \, dx \\ & = -\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {1}{2 a^2 \text {arccosh}(a x)}-\frac {x^2}{\text {arccosh}(a x)}+2 \int \frac {x}{\text {arccosh}(a x)} \, dx \\ & = -\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {1}{2 a^2 \text {arccosh}(a x)}-\frac {x^2}{\text {arccosh}(a x)}+\frac {2 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{a^2} \\ & = -\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {1}{2 a^2 \text {arccosh}(a x)}-\frac {x^2}{\text {arccosh}(a x)}+\frac {2 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\text {arccosh}(a x)\right )}{a^2} \\ & = -\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {1}{2 a^2 \text {arccosh}(a x)}-\frac {x^2}{\text {arccosh}(a x)}+\frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{a^2} \\ & = -\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {1}{2 a^2 \text {arccosh}(a x)}-\frac {x^2}{\text {arccosh}(a x)}+\frac {\text {Shi}(2 \text {arccosh}(a x))}{a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.99 \[ \int \frac {x}{\text {arccosh}(a x)^3} \, dx=-\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}+\frac {1-2 a^2 x^2}{2 a^2 \text {arccosh}(a x)}+\frac {\text {Shi}(2 \text {arccosh}(a x))}{a^2} \]

[In]

Integrate[x/ArcCosh[a*x]^3,x]

[Out]

-1/2*(x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x]^2) + (1 - 2*a^2*x^2)/(2*a^2*ArcCosh[a*x]) + SinhIntegral
[2*ArcCosh[a*x]]/a^2

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.63

method	result	size
derivativedivides	\(\frac {-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{4 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{2 \,\operatorname {arccosh}\left (a x \right )}+\operatorname {Shi}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{a^{2}}\)	\(43\)
default	\(\frac {-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{4 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {\cosh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{2 \,\operatorname {arccosh}\left (a x \right )}+\operatorname {Shi}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{a^{2}}\)	\(43\)

[In]

int(x/arccosh(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/a^2*(-1/4/arccosh(a*x)^2*sinh(2*arccosh(a*x))-1/2/arccosh(a*x)*cosh(2*arccosh(a*x))+Shi(2*arccosh(a*x)))

Fricas [F]

\[ \int \frac {x}{\text {arccosh}(a x)^3} \, dx=\int { \frac {x}{\operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]

[In]

integrate(x/arccosh(a*x)^3,x, algorithm="fricas")

[Out]

integral(x/arccosh(a*x)^3, x)

Sympy [F]

\[ \int \frac {x}{\text {arccosh}(a x)^3} \, dx=\int \frac {x}{\operatorname {acosh}^{3}{\left (a x \right )}}\, dx \]

[In]

integrate(x/acosh(a*x)**3,x)

[Out]

Integral(x/acosh(a*x)**3, x)

Maxima [F]

\[ \int \frac {x}{\text {arccosh}(a x)^3} \, dx=\int { \frac {x}{\operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]

[In]

integrate(x/arccosh(a*x)^3,x, algorithm="maxima")

[Out]

-1/2*(a^8*x^8 - 3*a^6*x^6 + 3*a^4*x^4 + (a^5*x^5 - a^3*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) - a^2*x^2 + (3*a^6
*x^6 - 5*a^4*x^4 + 2*a^2*x^2)*(a*x + 1)*(a*x - 1) + (3*a^7*x^7 - 7*a^5*x^5 + 5*a^3*x^3 - a*x)*sqrt(a*x + 1)*sq
rt(a*x - 1) + (2*a^8*x^8 - 6*a^6*x^6 + 6*a^4*x^4 + 2*(a^5*x^5 - a^3*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) - 2*a
^2*x^2 + (6*a^6*x^6 - 10*a^4*x^4 + 5*a^2*x^2 - 1)*(a*x + 1)*(a*x - 1) + (6*a^7*x^7 - 14*a^5*x^5 + 11*a^3*x^3 -
3*a*x)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1)))/((a^8*x^6 + (a*x + 1)^(3/2)*(a*x
- 1)^(3/2)*a^5*x^3 - 3*a^6*x^4 + 3*a^4*x^2 + 3*(a^6*x^4 - a^4*x^2)*(a*x + 1)*(a*x - 1) + 3*(a^7*x^5 - 2*a^5*x^
3 + a^3*x)*sqrt(a*x + 1)*sqrt(a*x - 1) - a^2)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2) + integrate(1/2*(4*a^9
*x^9 + 4*(a*x + 1)^2*(a*x - 1)^2*a^5*x^5 - 16*a^7*x^7 + 24*a^5*x^5 - 16*a^3*x^3 + (16*a^6*x^6 - 16*a^4*x^4 + 3
)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 24*(a^7*x^7 - 2*a^5*x^5 + a^3*x^3)*(a*x + 1)*(a*x - 1) + (16*a^8*x^8 - 48*
a^6*x^6 + 48*a^4*x^4 - 19*a^2*x^2 + 3)*sqrt(a*x + 1)*sqrt(a*x - 1) + 4*a*x)/((a^9*x^8 + (a*x + 1)^2*(a*x - 1)^
2*a^5*x^4 - 4*a^7*x^6 + 6*a^5*x^4 - 4*a^3*x^2 + 4*(a^6*x^5 - a^4*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 6*(a^7
*x^6 - 2*a^5*x^4 + a^3*x^2)*(a*x + 1)*(a*x - 1) + 4*(a^8*x^7 - 3*a^6*x^5 + 3*a^4*x^3 - a^2*x)*sqrt(a*x + 1)*sq
rt(a*x - 1) + a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))), x)

Giac [F]

\[ \int \frac {x}{\text {arccosh}(a x)^3} \, dx=\int { \frac {x}{\operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]

[In]

integrate(x/arccosh(a*x)^3,x, algorithm="giac")

[Out]

integrate(x/arccosh(a*x)^3, x)

Mupad [F(-1)]

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

[In]

int(x/acosh(a*x)^3,x)

[Out]

int(x/acosh(a*x)^3, x)