\(\int \frac {\text {arccosh}(a x)}{x^5} \, dx\) [10]

Optimal result

Integrand size = 8, antiderivative size = 66 \[ \int \frac {\text {arccosh}(a x)}{x^5} \, dx=\frac {a \sqrt {-1+a x} \sqrt {1+a x}}{12 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x}}{6 x}-\frac {\text {arccosh}(a x)}{4 x^4} \]

[Out]

-1/4*arccosh(a*x)/x^4+1/12*a*(a*x-1)^(1/2)*(a*x+1)^(1/2)/x^3+1/6*a^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/x

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5883, 105, 12, 97} \[ \int \frac {\text {arccosh}(a x)}{x^5} \, dx=\frac {a^3 \sqrt {a x-1} \sqrt {a x+1}}{6 x}-\frac {\text {arccosh}(a x)}{4 x^4}+\frac {a \sqrt {a x-1} \sqrt {a x+1}}{12 x^3} \]

[In]

Int[ArcCosh[a*x]/x^5,x]

[Out]

(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(12*x^3) + (a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(6*x) - ArcCosh[a*x]/(4*x^4)

Rule 12

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

Rule 97

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] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0
] && NeQ[m, -1]

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 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)}{4 x^4}+\frac {1}{4} a \int \frac {1}{x^4 \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {a \sqrt {-1+a x} \sqrt {1+a x}}{12 x^3}-\frac {\text {arccosh}(a x)}{4 x^4}+\frac {1}{12} a \int \frac {2 a^2}{x^2 \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {a \sqrt {-1+a x} \sqrt {1+a x}}{12 x^3}-\frac {\text {arccosh}(a x)}{4 x^4}+\frac {1}{6} a^3 \int \frac {1}{x^2 \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {a \sqrt {-1+a x} \sqrt {1+a x}}{12 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x}}{6 x}-\frac {\text {arccosh}(a x)}{4 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.68 \[ \int \frac {\text {arccosh}(a x)}{x^5} \, dx=\frac {a x \sqrt {-1+a x} \sqrt {1+a x} \left (1+2 a^2 x^2\right )-3 \text {arccosh}(a x)}{12 x^4} \]

[In]

Integrate[ArcCosh[a*x]/x^5,x]

[Out]

(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(1 + 2*a^2*x^2) - 3*ArcCosh[a*x])/(12*x^4)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.68

method	result	size
parts	\(-\frac {\operatorname {arccosh}\left (a x \right )}{4 x^{4}}+\frac {a \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {csgn}\left (a \right )^{2} \left (2 a^{2} x^{2}+1\right )}{12 x^{3}}\)	\(45\)
derivativedivides	\(a^{4} \left (-\frac {\operatorname {arccosh}\left (a x \right )}{4 a^{4} x^{4}}+\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (2 a^{2} x^{2}+1\right )}{12 a^{3} x^{3}}\right )\)	\(50\)
default	\(a^{4} \left (-\frac {\operatorname {arccosh}\left (a x \right )}{4 a^{4} x^{4}}+\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (2 a^{2} x^{2}+1\right )}{12 a^{3} x^{3}}\right )\)	\(50\)

[In]

int(arccosh(a*x)/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4*arccosh(a*x)/x^4+1/12*a*(a*x-1)^(1/2)*(a*x+1)^(1/2)*csgn(a)^2*(2*a^2*x^2+1)/x^3

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.73 \[ \int \frac {\text {arccosh}(a x)}{x^5} \, dx=\frac {{\left (2 \, a^{3} x^{3} + a x\right )} \sqrt {a^{2} x^{2} - 1} - 3 \, \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{12 \, x^{4}} \]

[In]

integrate(arccosh(a*x)/x^5,x, algorithm="fricas")

[Out]

1/12*((2*a^3*x^3 + a*x)*sqrt(a^2*x^2 - 1) - 3*log(a*x + sqrt(a^2*x^2 - 1)))/x^4

Sympy [F]

\[ \int \frac {\text {arccosh}(a x)}{x^5} \, dx=\int \frac {\operatorname {acosh}{\left (a x \right )}}{x^{5}}\, dx \]

[In]

integrate(acosh(a*x)/x**5,x)

[Out]

Integral(acosh(a*x)/x**5, x)

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.73 \[ \int \frac {\text {arccosh}(a x)}{x^5} \, dx=\frac {1}{12} \, {\left (\frac {2 \, \sqrt {a^{2} x^{2} - 1} a^{2}}{x} + \frac {\sqrt {a^{2} x^{2} - 1}}{x^{3}}\right )} a - \frac {\operatorname {arcosh}\left (a x\right )}{4 \, x^{4}} \]

[In]

integrate(arccosh(a*x)/x^5,x, algorithm="maxima")

[Out]

1/12*(2*sqrt(a^2*x^2 - 1)*a^2/x + sqrt(a^2*x^2 - 1)/x^3)*a - 1/4*arccosh(a*x)/x^4

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.17 \[ \int \frac {\text {arccosh}(a x)}{x^5} \, dx=\frac {{\left (3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )} a^{3} {\left | a \right |}}{3 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{3}} - \frac {\log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{4 \, x^{4}} \]

[In]

integrate(arccosh(a*x)/x^5,x, algorithm="giac")

[Out]

1/3*(3*(x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)*a^3*abs(a)/((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)^3 - 1/4*log(a*x
+ sqrt(a^2*x^2 - 1))/x^4

Mupad [F(-1)]

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

[In]

int(acosh(a*x)/x^5,x)

[Out]

int(acosh(a*x)/x^5, x)