\(\int \frac {\text {arccosh}(a x)^3}{x^3} \, dx\) [29]

Optimal result

Integrand size = 10, antiderivative size = 98 \[ \int \frac {\text {arccosh}(a x)^3}{x^3} \, dx=\frac {3}{2} a^2 \text {arccosh}(a x)^2+\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 x}-\frac {\text {arccosh}(a x)^3}{2 x^2}-3 a^2 \text {arccosh}(a x) \log \left (1+e^{2 \text {arccosh}(a x)}\right )-\frac {3}{2} a^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right ) \]

[Out]

3/2*a^2*arccosh(a*x)^2-1/2*arccosh(a*x)^3/x^2-3*a^2*arccosh(a*x)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)-3/2
*a^2*polylog(2,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)+3/2*a*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/x

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5883, 5918, 5882, 3799, 2221, 2317, 2438} \[ \int \frac {\text {arccosh}(a x)^3}{x^3} \, dx=-\frac {3}{2} a^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )+\frac {3}{2} a^2 \text {arccosh}(a x)^2-3 a^2 \text {arccosh}(a x) \log \left (e^{2 \text {arccosh}(a x)}+1\right )-\frac {\text {arccosh}(a x)^3}{2 x^2}+\frac {3 a \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 x} \]

[In]

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

[Out]

(3*a^2*ArcCosh[a*x]^2)/2 + (3*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(2*x) - ArcCosh[a*x]^3/(2*x^2) -
3*a^2*ArcCosh[a*x]*Log[1 + E^(2*ArcCosh[a*x])] - (3*a^2*PolyLog[2, -E^(2*ArcCosh[a*x])])/2

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m
+ 1)/(d*(m + 1))), x] + Dist[2*I, Int[(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5882

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

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)^3}{2 x^2}+\frac {1}{2} (3 a) \int \frac {\text {arccosh}(a x)^2}{x^2 \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 x}-\frac {\text {arccosh}(a x)^3}{2 x^2}-\left (3 a^2\right ) \int \frac {\text {arccosh}(a x)}{x} \, dx \\ & = \frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 x}-\frac {\text {arccosh}(a x)^3}{2 x^2}-\left (3 a^2\right ) \text {Subst}(\int x \tanh (x) \, dx,x,\text {arccosh}(a x)) \\ & = \frac {3}{2} a^2 \text {arccosh}(a x)^2+\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 x}-\frac {\text {arccosh}(a x)^3}{2 x^2}-\left (6 a^2\right ) \text {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\text {arccosh}(a x)\right ) \\ & = \frac {3}{2} a^2 \text {arccosh}(a x)^2+\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 x}-\frac {\text {arccosh}(a x)^3}{2 x^2}-3 a^2 \text {arccosh}(a x) \log \left (1+e^{2 \text {arccosh}(a x)}\right )+\left (3 a^2\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arccosh}(a x)\right ) \\ & = \frac {3}{2} a^2 \text {arccosh}(a x)^2+\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 x}-\frac {\text {arccosh}(a x)^3}{2 x^2}-3 a^2 \text {arccosh}(a x) \log \left (1+e^{2 \text {arccosh}(a x)}\right )+\frac {1}{2} \left (3 a^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arccosh}(a x)}\right ) \\ & = \frac {3}{2} a^2 \text {arccosh}(a x)^2+\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 x}-\frac {\text {arccosh}(a x)^3}{2 x^2}-3 a^2 \text {arccosh}(a x) \log \left (1+e^{2 \text {arccosh}(a x)}\right )-\frac {3}{2} a^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right ) \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.64 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.94 \[ \int \frac {\text {arccosh}(a x)^3}{x^3} \, dx=\frac {1}{2} \left (-\frac {\text {arccosh}(a x)^3}{x^2}+3 a^2 \left (\text {arccosh}(a x) \left (-\text {arccosh}(a x)+\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)}{a x}-2 \log \left (1+e^{-2 \text {arccosh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(a x)}\right )\right )\right ) \]

[In]

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

[Out]

(-(ArcCosh[a*x]^3/x^2) + 3*a^2*(ArcCosh[a*x]*(-ArcCosh[a*x] + (Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*
x])/(a*x) - 2*Log[1 + E^(-2*ArcCosh[a*x])]) + PolyLog[2, -E^(-2*ArcCosh[a*x])]))/2

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.18

method	result	size
derivativedivides	\(a^{2} \left (-\frac {\operatorname {arccosh}\left (a x \right )^{2} \left (-3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +3 a^{2} x^{2}+\operatorname {arccosh}\left (a x \right )\right )}{2 a^{2} x^{2}}+3 \operatorname {arccosh}\left (a x \right )^{2}-3 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}\right )\)	\(116\)
default	\(a^{2} \left (-\frac {\operatorname {arccosh}\left (a x \right )^{2} \left (-3 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +3 a^{2} x^{2}+\operatorname {arccosh}\left (a x \right )\right )}{2 a^{2} x^{2}}+3 \operatorname {arccosh}\left (a x \right )^{2}-3 \,\operatorname {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}\right )\)	\(116\)

[In]

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

[Out]

a^2*(-1/2*arccosh(a*x)^2*(-3*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x+3*a^2*x^2+arccosh(a*x))/a^2/x^2+3*arccosh(a*x)^2-
3*arccosh(a*x)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)-3/2*polylog(2,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2))

Fricas [F]

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

[In]

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

[Out]

integral(arccosh(a*x)^3/x^3, x)

Sympy [F]

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

[In]

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

[Out]

Integral(acosh(a*x)**3/x**3, x)

Maxima [F]

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

[In]

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

[Out]

-1/2*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3/x^2 + integrate(3/2*(a^3*x^2 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x
- a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2/(a^3*x^5 - a*x^3 + (a^2*x^4 - x^2)*sqrt(a*x + 1)*sqrt(a*x - 1))
, x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\text {arccosh}(a x)^3}{x^3} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

int(acosh(a*x)^3/x^3, x)