\(\int \frac {\text {arccosh}(a x)^2}{x^3} \, dx\) [19]

Optimal result

Integrand size = 10, antiderivative size = 48 \[ \int \frac {\text {arccosh}(a x)^2}{x^3} \, dx=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{x}-\frac {\text {arccosh}(a x)^2}{2 x^2}-a^2 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5883, 5918, 29} \[ \int \frac {\text {arccosh}(a x)^2}{x^3} \, dx=a^2 (-\log (x))-\frac {\text {arccosh}(a x)^2}{2 x^2}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)}{x} \]

[In]

Int[ArcCosh[a*x]^2/x^3,x]

[Out]

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

Rule 29

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

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]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arccosh}(a x)^2}{x^3} \, dx=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)}{x}-\frac {\text {arccosh}(a x)^2}{2 x^2}-a^2 \log (x) \]

[In]

Integrate[ArcCosh[a*x]^2/x^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.69

method	result	size
derivativedivides	\(a^{2} \left (2 \,\operatorname {arccosh}\left (a x \right )-\frac {\operatorname {arccosh}\left (a x \right ) \left (2 a^{2} x^{2}-2 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +\operatorname {arccosh}\left (a x \right )\right )}{2 a^{2} x^{2}}-\ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )\right )\)	\(81\)
default	\(a^{2} \left (2 \,\operatorname {arccosh}\left (a x \right )-\frac {\operatorname {arccosh}\left (a x \right ) \left (2 a^{2} x^{2}-2 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +\operatorname {arccosh}\left (a x \right )\right )}{2 a^{2} x^{2}}-\ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )\right )\)	\(81\)

[In]

int(arccosh(a*x)^2/x^3,x,method=_RETURNVERBOSE)

[Out]

a^2*(2*arccosh(a*x)-1/2*arccosh(a*x)*(2*a^2*x^2-2*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x+arccosh(a*x))/a^2/x^2-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 = 65, normalized size of antiderivative = 1.35 \[ \int \frac {\text {arccosh}(a x)^2}{x^3} \, dx=-\frac {2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, \sqrt {a^{2} x^{2} - 1} a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2}}{2 \, x^{2}} \]

[In]

integrate(arccosh(a*x)^2/x^3,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x^3} \, dx=\int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{x^{3}}\, dx \]

[In]

integrate(acosh(a*x)**2/x**3,x)

[Out]

Integral(acosh(a*x)**2/x**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.81 \[ \int \frac {\text {arccosh}(a x)^2}{x^3} \, dx=-a^{2} \log \left (x\right ) + \frac {\sqrt {a^{2} x^{2} - 1} a \operatorname {arcosh}\left (a x\right )}{x} - \frac {\operatorname {arcosh}\left (a x\right )^{2}}{2 \, x^{2}} \]

[In]

integrate(arccosh(a*x)^2/x^3,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (42) = 84\).

Time = 0.32 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.04 \[ \int \frac {\text {arccosh}(a x)^2}{x^3} \, dx={\left (a \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) - a \log \left ({\left | x \right |}\right ) + \frac {2 \, {\left | a \right |} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1}\right )} a - \frac {\log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2}}{2 \, x^{2}} \]

[In]

integrate(arccosh(a*x)^2/x^3,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^2}{x^3} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{x^3} \,d x \]

[In]

int(acosh(a*x)^2/x^3,x)

[Out]

int(acosh(a*x)^2/x^3, x)