\(\int \frac {1}{\text {arccosh}(a x)^3} \, dx\) [62]

Optimal result

Integrand size = 6, antiderivative size = 55 \[ \int \frac {1}{\text {arccosh}(a x)^3} \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}-\frac {x}{2 \text {arccosh}(a x)}+\frac {\text {Shi}(\text {arccosh}(a x))}{2 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5880, 5951, 5881, 3379} \[ \int \frac {1}{\text {arccosh}(a x)^3} \, dx=\frac {\text {Shi}(\text {arccosh}(a x))}{2 a}-\frac {x}{2 \text {arccosh}(a x)}-\frac {\sqrt {a x-1} \sqrt {a x+1}}{2 a \text {arccosh}(a x)^2} \]

[In]

Int[ArcCosh[a*x]^(-3),x]

[Out]

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

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 5880

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

Rule 5881

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

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\text {arccosh}(a x)^3} \, dx=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{2 a \text {arccosh}(a x)^2}-\frac {x}{2 \text {arccosh}(a x)}+\frac {\text {Shi}(\text {arccosh}(a x))}{2 a} \]

[In]

Integrate[ArcCosh[a*x]^(-3),x]

[Out]

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

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{2 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {a x}{2 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Shi}\left (\operatorname {arccosh}\left (a x \right )\right )}{2}}{a}\)	\(45\)
default	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{2 \operatorname {arccosh}\left (a x \right )^{2}}-\frac {a x}{2 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Shi}\left (\operatorname {arccosh}\left (a x \right )\right )}{2}}{a}\)	\(45\)

[In]

int(1/arccosh(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {1}{\text {arccosh}(a x)^3} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]

[In]

integrate(1/arccosh(a*x)^3,x, algorithm="fricas")

[Out]

integral(arccosh(a*x)^(-3), x)

Sympy [F]

\[ \int \frac {1}{\text {arccosh}(a x)^3} \, dx=\int \frac {1}{\operatorname {acosh}^{3}{\left (a x \right )}}\, dx \]

[In]

integrate(1/acosh(a*x)**3,x)

[Out]

Integral(acosh(a*x)**(-3), x)

Maxima [F]

\[ \int \frac {1}{\text {arccosh}(a x)^3} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]

[In]

integrate(1/arccosh(a*x)^3,x, algorithm="maxima")

[Out]

-1/2*(a^7*x^7 - 3*a^5*x^5 + 3*a^3*x^3 + (a^4*x^4 - a^2*x^2)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + (3*a^5*x^5 - 5*a
^3*x^3 + 2*a*x)*(a*x + 1)*(a*x - 1) + (3*a^6*x^6 - 7*a^4*x^4 + 5*a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1) - a*
x + (a^7*x^7 - 3*a^5*x^5 + 3*a^3*x^3 + (a^4*x^4 - 1)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 3*(a^5*x^5 - a^3*x^3)*(
a*x + 1)*(a*x - 1) + (3*a^6*x^6 - 6*a^4*x^4 + 4*a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1) - a*x)*log(a*x + sqrt
(a*x + 1)*sqrt(a*x - 1)))/((a^7*x^6 + (a*x + 1)^(3/2)*(a*x - 1)^(3/2)*a^4*x^3 - 3*a^5*x^4 + 3*a^3*x^2 + 3*(a^5
*x^4 - a^3*x^2)*(a*x + 1)*(a*x - 1) + 3*(a^6*x^5 - 2*a^4*x^3 + a^2*x)*sqrt(a*x + 1)*sqrt(a*x - 1) - a)*log(a*x
+ sqrt(a*x + 1)*sqrt(a*x - 1))^2) + integrate(1/2*(a^8*x^8 - 4*a^6*x^6 + 6*a^4*x^4 + (a^4*x^4 + 3)*(a*x + 1)^
2*(a*x - 1)^2 + (4*a^5*x^5 - 4*a^3*x^3 + 3*a*x)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) - 4*a^2*x^2 + 3*(2*a^6*x^6 - 4
*a^4*x^4 + a^2*x^2 + 1)*(a*x + 1)*(a*x - 1) + (4*a^7*x^7 - 12*a^5*x^5 + 9*a^3*x^3 - a*x)*sqrt(a*x + 1)*sqrt(a*
x - 1) + 1)/((a^8*x^8 + (a*x + 1)^2*(a*x - 1)^2*a^4*x^4 - 4*a^6*x^6 + 6*a^4*x^4 + 4*(a^5*x^5 - a^3*x^3)*(a*x +
1)^(3/2)*(a*x - 1)^(3/2) - 4*a^2*x^2 + 6*(a^6*x^6 - 2*a^4*x^4 + a^2*x^2)*(a*x + 1)*(a*x - 1) + 4*(a^7*x^7 - 3
*a^5*x^5 + 3*a^3*x^3 - a*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + 1)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))), x)

Giac [F]

\[ \int \frac {1}{\text {arccosh}(a x)^3} \, dx=\int { \frac {1}{\operatorname {arcosh}\left (a x\right )^{3}} \,d x } \]

[In]

integrate(1/arccosh(a*x)^3,x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^(-3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\text {arccosh}(a x)^3} \, dx=\int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^3} \,d x \]

[In]

int(1/acosh(a*x)^3,x)

[Out]

int(1/acosh(a*x)^3, x)