\(\int \frac {\text {arccosh}(a x)}{x^6} \, dx\) [11]

Optimal result

Integrand size = 8, antiderivative size = 93 \[ \int \frac {\text {arccosh}(a x)}{x^6} \, dx=\frac {a \sqrt {-1+a x} \sqrt {1+a x}}{20 x^4}+\frac {3 a^3 \sqrt {-1+a x} \sqrt {1+a x}}{40 x^2}-\frac {\text {arccosh}(a x)}{5 x^5}+\frac {3}{40} a^5 \arctan \left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \]

[Out]

-1/5*arccosh(a*x)/x^5+3/40*a^5*arctan((a*x-1)^(1/2)*(a*x+1)^(1/2))+1/20*a*(a*x-1)^(1/2)*(a*x+1)^(1/2)/x^4+3/40
*a^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/x^2

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5883, 105, 12, 94, 211} \[ \int \frac {\text {arccosh}(a x)}{x^6} \, dx=\frac {3}{40} a^5 \arctan \left (\sqrt {a x-1} \sqrt {a x+1}\right )+\frac {3 a^3 \sqrt {a x-1} \sqrt {a x+1}}{40 x^2}-\frac {\text {arccosh}(a x)}{5 x^5}+\frac {a \sqrt {a x-1} \sqrt {a x+1}}{20 x^4} \]

[In]

Int[ArcCosh[a*x]/x^6,x]

[Out]

(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(20*x^4) + (3*a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(40*x^2) - ArcCosh[a*x]/(5*x^
5) + (3*a^5*ArcTan[Sqrt[-1 + a*x]*Sqrt[1 + a*x]])/40

Rule 12

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

Rule 94

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 105

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)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 211

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

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arccosh}(a x)}{5 x^5}+\frac {1}{5} a \int \frac {1}{x^5 \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {a \sqrt {-1+a x} \sqrt {1+a x}}{20 x^4}-\frac {\text {arccosh}(a x)}{5 x^5}+\frac {1}{20} a \int \frac {3 a^2}{x^3 \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {a \sqrt {-1+a x} \sqrt {1+a x}}{20 x^4}-\frac {\text {arccosh}(a x)}{5 x^5}+\frac {1}{20} \left (3 a^3\right ) \int \frac {1}{x^3 \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {a \sqrt {-1+a x} \sqrt {1+a x}}{20 x^4}+\frac {3 a^3 \sqrt {-1+a x} \sqrt {1+a x}}{40 x^2}-\frac {\text {arccosh}(a x)}{5 x^5}+\frac {1}{40} \left (3 a^3\right ) \int \frac {a^2}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {a \sqrt {-1+a x} \sqrt {1+a x}}{20 x^4}+\frac {3 a^3 \sqrt {-1+a x} \sqrt {1+a x}}{40 x^2}-\frac {\text {arccosh}(a x)}{5 x^5}+\frac {1}{40} \left (3 a^5\right ) \int \frac {1}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {a \sqrt {-1+a x} \sqrt {1+a x}}{20 x^4}+\frac {3 a^3 \sqrt {-1+a x} \sqrt {1+a x}}{40 x^2}-\frac {\text {arccosh}(a x)}{5 x^5}+\frac {1}{40} \left (3 a^6\right ) \text {Subst}\left (\int \frac {1}{a+a x^2} \, dx,x,\sqrt {-1+a x} \sqrt {1+a x}\right ) \\ & = \frac {a \sqrt {-1+a x} \sqrt {1+a x}}{20 x^4}+\frac {3 a^3 \sqrt {-1+a x} \sqrt {1+a x}}{40 x^2}-\frac {\text {arccosh}(a x)}{5 x^5}+\frac {3}{40} a^5 \arctan \left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.12 \[ \int \frac {\text {arccosh}(a x)}{x^6} \, dx=-\frac {2 a x+a^3 x^3-3 a^5 x^5+8 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)-3 a^5 x^5 \sqrt {-1+a^2 x^2} \arctan \left (\sqrt {-1+a^2 x^2}\right )}{40 x^5 \sqrt {-1+a x} \sqrt {1+a x}} \]

[In]

Integrate[ArcCosh[a*x]/x^6,x]

[Out]

-1/40*(2*a*x + a^3*x^3 - 3*a^5*x^5 + 8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x] - 3*a^5*x^5*Sqrt[-1 + a^2*x^2
]*ArcTan[Sqrt[-1 + a^2*x^2]])/(x^5*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.02

method	result	size
parts	\(-\frac {\operatorname {arccosh}\left (a x \right )}{5 x^{5}}-\frac {a \sqrt {a x -1}\, \sqrt {a x +1}\, \left (3 \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{4} x^{4}-3 a^{2} x^{2} \sqrt {a^{2} x^{2}-1}-2 \sqrt {a^{2} x^{2}-1}\right )}{40 \sqrt {a^{2} x^{2}-1}\, x^{4}}\)	\(95\)
derivativedivides	\(a^{5} \left (-\frac {\operatorname {arccosh}\left (a x \right )}{5 a^{5} x^{5}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (3 \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{4} x^{4}-3 a^{2} x^{2} \sqrt {a^{2} x^{2}-1}-2 \sqrt {a^{2} x^{2}-1}\right )}{40 \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}}\right )\)	\(104\)
default	\(a^{5} \left (-\frac {\operatorname {arccosh}\left (a x \right )}{5 a^{5} x^{5}}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (3 \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{4} x^{4}-3 a^{2} x^{2} \sqrt {a^{2} x^{2}-1}-2 \sqrt {a^{2} x^{2}-1}\right )}{40 \sqrt {a^{2} x^{2}-1}\, a^{4} x^{4}}\right )\)	\(104\)

[In]

int(arccosh(a*x)/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/5*arccosh(a*x)/x^5-1/40*a*(a*x-1)^(1/2)*(a*x+1)^(1/2)*(3*arctan(1/(a^2*x^2-1)^(1/2))*a^4*x^4-3*a^2*x^2*(a^2
*x^2-1)^(1/2)-2*(a^2*x^2-1)^(1/2))/(a^2*x^2-1)^(1/2)/x^4

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.09 \[ \int \frac {\text {arccosh}(a x)}{x^6} \, dx=\frac {6 \, a^{5} x^{5} \arctan \left (-a x + \sqrt {a^{2} x^{2} - 1}\right ) + 8 \, x^{5} \log \left (-a x + \sqrt {a^{2} x^{2} - 1}\right ) + 8 \, {\left (x^{5} - 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + {\left (3 \, a^{3} x^{3} + 2 \, a x\right )} \sqrt {a^{2} x^{2} - 1}}{40 \, x^{5}} \]

[In]

integrate(arccosh(a*x)/x^6,x, algorithm="fricas")

[Out]

1/40*(6*a^5*x^5*arctan(-a*x + sqrt(a^2*x^2 - 1)) + 8*x^5*log(-a*x + sqrt(a^2*x^2 - 1)) + 8*(x^5 - 1)*log(a*x +
sqrt(a^2*x^2 - 1)) + (3*a^3*x^3 + 2*a*x)*sqrt(a^2*x^2 - 1))/x^5

Sympy [F]

\[ \int \frac {\text {arccosh}(a x)}{x^6} \, dx=\int \frac {\operatorname {acosh}{\left (a x \right )}}{x^{6}}\, dx \]

[In]

integrate(acosh(a*x)/x**6,x)

[Out]

Integral(acosh(a*x)/x**6, x)

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.68 \[ \int \frac {\text {arccosh}(a x)}{x^6} \, dx=-\frac {1}{40} \, {\left (3 \, a^{4} \arcsin \left (\frac {1}{a {\left | x \right |}}\right ) - \frac {3 \, \sqrt {a^{2} x^{2} - 1} a^{2}}{x^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} - 1}}{x^{4}}\right )} a - \frac {\operatorname {arcosh}\left (a x\right )}{5 \, x^{5}} \]

[In]

integrate(arccosh(a*x)/x^6,x, algorithm="maxima")

[Out]

-1/40*(3*a^4*arcsin(1/(a*abs(x))) - 3*sqrt(a^2*x^2 - 1)*a^2/x^2 - 2*sqrt(a^2*x^2 - 1)/x^4)*a - 1/5*arccosh(a*x
)/x^5

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.91 \[ \int \frac {\text {arccosh}(a x)}{x^6} \, dx=\frac {3 \, a^{6} \arctan \left (\sqrt {a^{2} x^{2} - 1}\right ) + \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} a^{6} + 5 \, \sqrt {a^{2} x^{2} - 1} a^{6}}{a^{4} x^{4}}}{40 \, a} - \frac {\log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{5 \, x^{5}} \]

[In]

integrate(arccosh(a*x)/x^6,x, algorithm="giac")

[Out]

1/40*(3*a^6*arctan(sqrt(a^2*x^2 - 1)) + (3*(a^2*x^2 - 1)^(3/2)*a^6 + 5*sqrt(a^2*x^2 - 1)*a^6)/(a^4*x^4))/a - 1
/5*log(a*x + sqrt(a^2*x^2 - 1))/x^5

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)}{x^6} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )}{x^6} \,d x \]

[In]

int(acosh(a*x)/x^6,x)

[Out]

int(acosh(a*x)/x^6, x)