\(\int \frac {\text {arccosh}(a x)^4}{x^3} \, dx\) [40]

Optimal result

Integrand size = 10, antiderivative size = 115 \[ \int \frac {\text {arccosh}(a x)^4}{x^3} \, dx=2 a^2 \text {arccosh}(a x)^3+\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{x}-\frac {\text {arccosh}(a x)^4}{2 x^2}-6 a^2 \text {arccosh}(a x)^2 \log \left (1+e^{2 \text {arccosh}(a x)}\right )-6 a^2 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )+3 a^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5883, 5918, 5882, 3799, 2221, 2611, 2320, 6724} \[ \int \frac {\text {arccosh}(a x)^4}{x^3} \, dx=-6 a^2 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )+3 a^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arccosh}(a x)}\right )+2 a^2 \text {arccosh}(a x)^3-6 a^2 \text {arccosh}(a x)^2 \log \left (e^{2 \text {arccosh}(a x)}+1\right )-\frac {\text {arccosh}(a x)^4}{2 x^2}+\frac {2 a \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{x} \]

[In]

Int[ArcCosh[a*x]^4/x^3,x]

[Out]

2*a^2*ArcCosh[a*x]^3 + (2*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/x - ArcCosh[a*x]^4/(2*x^2) - 6*a^2*Ar
cCosh[a*x]^2*Log[1 + E^(2*ArcCosh[a*x])] - 6*a^2*ArcCosh[a*x]*PolyLog[2, -E^(2*ArcCosh[a*x])] + 3*a^2*PolyLog[
3, -E^(2*ArcCosh[a*x])]

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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

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]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

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

Mathematica [A] (warning: unable to verify)

Time = 0.78 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {\text {arccosh}(a x)^4}{x^3} \, dx=-\frac {\text {arccosh}(a x)^4}{2 x^2}+a^2 \left (2 \text {arccosh}(a x)^2 \left (-\text {arccosh}(a x)+\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)}{a x}-3 \log \left (1+e^{-2 \text {arccosh}(a x)}\right )\right )+6 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(a x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{-2 \text {arccosh}(a x)}\right )\right ) \]

[In]

Integrate[ArcCosh[a*x]^4/x^3,x]

[Out]

-1/2*ArcCosh[a*x]^4/x^2 + a^2*(2*ArcCosh[a*x]^2*(-ArcCosh[a*x] + (Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh
[a*x])/(a*x) - 3*Log[1 + E^(-2*ArcCosh[a*x])]) + 6*ArcCosh[a*x]*PolyLog[2, -E^(-2*ArcCosh[a*x])] + 3*PolyLog[3
, -E^(-2*ArcCosh[a*x])])

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.30

method	result	size
derivativedivides	\(a^{2} \left (-\frac {\operatorname {arccosh}\left (a x \right )^{3} \left (4 a^{2} x^{2}-4 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +\operatorname {arccosh}\left (a x \right )\right )}{2 a^{2} x^{2}}+4 \operatorname {arccosh}\left (a x \right )^{3}-6 \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-6 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+3 \operatorname {polylog}\left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )\right )\)	\(149\)
default	\(a^{2} \left (-\frac {\operatorname {arccosh}\left (a x \right )^{3} \left (4 a^{2} x^{2}-4 \sqrt {a x -1}\, \sqrt {a x +1}\, a x +\operatorname {arccosh}\left (a x \right )\right )}{2 a^{2} x^{2}}+4 \operatorname {arccosh}\left (a x \right )^{3}-6 \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-6 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+3 \operatorname {polylog}\left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )\right )\)	\(149\)

[In]

int(arccosh(a*x)^4/x^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {\text {arccosh}(a x)^4}{x^3} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{4}}{x^{3}} \,d x } \]

[In]

integrate(arccosh(a*x)^4/x^3,x, algorithm="fricas")

[Out]

integral(arccosh(a*x)^4/x^3, x)

Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^4}{x^3} \, dx=\int \frac {\operatorname {acosh}^{4}{\left (a x \right )}}{x^{3}}\, dx \]

[In]

integrate(acosh(a*x)**4/x**3,x)

[Out]

Integral(acosh(a*x)**4/x**3, x)

Maxima [F]

\[ \int \frac {\text {arccosh}(a x)^4}{x^3} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{4}}{x^{3}} \,d x } \]

[In]

integrate(arccosh(a*x)^4/x^3,x, algorithm="maxima")

[Out]

-1/2*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^4/x^2 + integrate(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))^3/(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. \[ \int \frac {\text {arccosh}(a x)^4}{x^3} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(arccosh(a*x)^4/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. \[ \int \frac {\text {arccosh}(a x)^4}{x^3} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^4}{x^3} \,d x \]

[In]

int(acosh(a*x)^4/x^3,x)

[Out]

int(acosh(a*x)^4/x^3, x)