\(\int \frac {\text {arccosh}(a x)^2}{x^5} \, dx\) [21]

Optimal result

Integrand size = 10, antiderivative size = 95 \[ \int \frac {\text {arccosh}(a x)^2}{x^5} \, dx=\frac {a^2}{12 x^2}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{6 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{3 x}-\frac {\text {arccosh}(a x)^2}{4 x^4}-\frac {1}{3} a^4 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5883, 5933, 5918, 29, 30} \[ \int \frac {\text {arccosh}(a x)^2}{x^5} \, dx=-\frac {1}{3} a^4 \log (x)+\frac {a^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{3 x}+\frac {a^2}{12 x^2}-\frac {\text {arccosh}(a x)^2}{4 x^4}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{6 x^3} \]

[In]

Int[ArcCosh[a*x]^2/x^5,x]

[Out]

a^2/(12*x^2) + (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(6*x^3) + (a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCo
sh[a*x])/(3*x) - ArcCosh[a*x]^2/(4*x^4) - (a^4*Log[x])/3

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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]

Rule 5918

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d1*d
2*f*(m + 1))), x] + Dist[b*c*(n/(f*(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p],
Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a,
b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] &&
NeQ[p, -1]

Rule 5933

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d1*d
2*f*(m + 1))), x] + (Dist[c^2*((m + 2*p + 3)/(f^2*(m + 1))), Int[(f*x)^(m + 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a
+ b*ArcCosh[c*x])^n, x], x] + Dist[b*c*(n/(f*(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1
+ c*x)^p], Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /;
FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && ILtQ[m, -1]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.73 \[ \int \frac {\text {arccosh}(a x)^2}{x^5} \, dx=\frac {a^2 x^2+2 a x \sqrt {-1+a x} \sqrt {1+a x} \left (1+2 a^2 x^2\right ) \text {arccosh}(a x)-3 \text {arccosh}(a x)^2-4 a^4 x^4 \log (x)}{12 x^4} \]

[In]

Integrate[ArcCosh[a*x]^2/x^5,x]

[Out]

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.29

method	result	size
derivativedivides	\(a^{4} \left (\frac {2 \,\operatorname {arccosh}\left (a x \right )}{3}-\frac {-4 a^{3} x^{3} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+4 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )-2 a x \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+3 \operatorname {arccosh}\left (a x \right )^{2}-a^{2} x^{2}}{12 a^{4} x^{4}}-\frac {\ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{3}\right )\)	\(123\)
default	\(a^{4} \left (\frac {2 \,\operatorname {arccosh}\left (a x \right )}{3}-\frac {-4 a^{3} x^{3} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+4 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )-2 a x \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+3 \operatorname {arccosh}\left (a x \right )^{2}-a^{2} x^{2}}{12 a^{4} x^{4}}-\frac {\ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{3}\right )\)	\(123\)

[In]

int(arccosh(a*x)^2/x^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \frac {\text {arccosh}(a x)^2}{x^5} \, dx=-\frac {4 \, a^{4} x^{4} \log \left (x\right ) - a^{2} x^{2} - 2 \, {\left (2 \, a^{3} x^{3} + a x\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + 3 \, \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2}}{12 \, x^{4}} \]

[In]

integrate(arccosh(a*x)^2/x^5,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x^5} \, dx=\int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{x^{5}}\, dx \]

[In]

integrate(acosh(a*x)**2/x**5,x)

[Out]

Integral(acosh(a*x)**2/x**5, x)

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.76 \[ \int \frac {\text {arccosh}(a x)^2}{x^5} \, dx=-\frac {1}{12} \, {\left (4 \, a^{2} \log \left (x\right ) - \frac {1}{x^{2}}\right )} a^{2} + \frac {1}{6} \, {\left (\frac {2 \, \sqrt {a^{2} x^{2} - 1} a^{2}}{x} + \frac {\sqrt {a^{2} x^{2} - 1}}{x^{3}}\right )} a \operatorname {arcosh}\left (a x\right ) - \frac {\operatorname {arcosh}\left (a x\right )^{2}}{4 \, x^{4}} \]

[In]

integrate(arccosh(a*x)^2/x^5,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.55 \[ \int \frac {\text {arccosh}(a x)^2}{x^5} \, dx=-\frac {1}{12} \, {\left (2 \, a^{3} \log \left (x^{2}\right ) - 4 \, a^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) - \frac {8 \, {\left (3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )} a^{2} {\left | a \right |} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{3}} - \frac {2 \, a^{3} x^{2} + a}{x^{2}}\right )} a - \frac {\log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2}}{4 \, x^{4}} \]

[In]

integrate(arccosh(a*x)^2/x^5,x, algorithm="giac")

[Out]

-1/12*(2*a^3*log(x^2) - 4*a^3*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1))) - 8*(3*(x*abs(a) - sqrt(a^2*x^2 - 1))^2
+ 1)*a^2*abs(a)*log(a*x + sqrt(a^2*x^2 - 1))/((x*abs(a) - sqrt(a^2*x^2 - 1))^2 + 1)^3 - (2*a^3*x^2 + a)/x^2)*a
- 1/4*log(a*x + sqrt(a^2*x^2 - 1))^2/x^4

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^2}{x^5} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{x^5} \,d x \]

[In]

int(acosh(a*x)^2/x^5,x)

[Out]

int(acosh(a*x)^2/x^5, x)