3.39 \(\int \frac{\cosh ^{-1}(a x)^4}{x^2} \, dx\)

Optimal. Leaf size=150 \[ -12 i a \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )+12 i a \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+24 i a \cosh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )-24 i a \cosh ^{-1}(a x) \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )-24 i a \text{PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )+24 i a \text{PolyLog}\left (4,i e^{\cosh ^{-1}(a x)}\right )-\frac{\cosh ^{-1}(a x)^4}{x}+8 a \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right ) \]

[Out]

-(ArcCosh[a*x]^4/x) + 8*a*ArcCosh[a*x]^3*ArcTan[E^ArcCosh[a*x]] - (12*I)*a*ArcCosh[a*x]^2*PolyLog[2, (-I)*E^Ar
cCosh[a*x]] + (12*I)*a*ArcCosh[a*x]^2*PolyLog[2, I*E^ArcCosh[a*x]] + (24*I)*a*ArcCosh[a*x]*PolyLog[3, (-I)*E^A
rcCosh[a*x]] - (24*I)*a*ArcCosh[a*x]*PolyLog[3, I*E^ArcCosh[a*x]] - (24*I)*a*PolyLog[4, (-I)*E^ArcCosh[a*x]] +
(24*I)*a*PolyLog[4, I*E^ArcCosh[a*x]]

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Rubi [A]  time = 0.334837, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5662, 5761, 4180, 2531, 6609, 2282, 6589} \[ -12 i a \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )+12 i a \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+24 i a \cosh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )-24 i a \cosh ^{-1}(a x) \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )-24 i a \text{PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )+24 i a \text{PolyLog}\left (4,i e^{\cosh ^{-1}(a x)}\right )-\frac{\cosh ^{-1}(a x)^4}{x}+8 a \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^4/x^2,x]

[Out]

-(ArcCosh[a*x]^4/x) + 8*a*ArcCosh[a*x]^3*ArcTan[E^ArcCosh[a*x]] - (12*I)*a*ArcCosh[a*x]^2*PolyLog[2, (-I)*E^Ar
cCosh[a*x]] + (12*I)*a*ArcCosh[a*x]^2*PolyLog[2, I*E^ArcCosh[a*x]] + (24*I)*a*ArcCosh[a*x]*PolyLog[3, (-I)*E^A
rcCosh[a*x]] - (24*I)*a*ArcCosh[a*x]*PolyLog[3, I*E^ArcCosh[a*x]] - (24*I)*a*PolyLog[4, (-I)*E^ArcCosh[a*x]] +
(24*I)*a*PolyLog[4, I*E^ArcCosh[a*x]]

Rule 5662

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))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5761

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]
), x_Symbol] :> Dist[1/(c^(m + 1)*Sqrt[-(d1*d2)]), Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /
; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[n, 0] && GtQ[d1, 0] &&
LtQ[d2, 0] && IntegerQ[m]

Rule 4180

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

Rule 2531

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 6609

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

Rule 2282

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 6589

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*} \int \frac{\cosh ^{-1}(a x)^4}{x^2} \, dx &=-\frac{\cosh ^{-1}(a x)^4}{x}+(4 a) \int \frac{\cosh ^{-1}(a x)^3}{x \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{\cosh ^{-1}(a x)^4}{x}+(4 a) \operatorname{Subst}\left (\int x^3 \text{sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac{\cosh ^{-1}(a x)^4}{x}+8 a \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-(12 i a) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )+(12 i a) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac{\cosh ^{-1}(a x)^4}{x}+8 a \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-12 i a \cosh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+12 i a \cosh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+(24 i a) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )-(24 i a) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac{\cosh ^{-1}(a x)^4}{x}+8 a \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-12 i a \cosh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+12 i a \cosh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+24 i a \cosh ^{-1}(a x) \text{Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )-24 i a \cosh ^{-1}(a x) \text{Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )-(24 i a) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )+(24 i a) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac{\cosh ^{-1}(a x)^4}{x}+8 a \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-12 i a \cosh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+12 i a \cosh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+24 i a \cosh ^{-1}(a x) \text{Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )-24 i a \cosh ^{-1}(a x) \text{Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )-(24 i a) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )+(24 i a) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )\\ &=-\frac{\cosh ^{-1}(a x)^4}{x}+8 a \cosh ^{-1}(a x)^3 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-12 i a \cosh ^{-1}(a x)^2 \text{Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+12 i a \cosh ^{-1}(a x)^2 \text{Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+24 i a \cosh ^{-1}(a x) \text{Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )-24 i a \cosh ^{-1}(a x) \text{Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )-24 i a \text{Li}_4\left (-i e^{\cosh ^{-1}(a x)}\right )+24 i a \text{Li}_4\left (i e^{\cosh ^{-1}(a x)}\right )\\ \end{align*}

Mathematica [B]  time = 0.648351, size = 478, normalized size = 3.19 \[ a \left (-12 i \cosh ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )+12 \pi \cosh ^{-1}(a x) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )-24 i \cosh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(a x)}\right )+24 i \cosh ^{-1}(a x) \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )+3 i \left (\pi -2 i \cosh ^{-1}(a x)\right )^2 \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )+3 i \pi ^2 \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+12 \pi \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(a x)}\right )-12 \pi \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )-24 i \text{PolyLog}\left (4,-i e^{-\cosh ^{-1}(a x)}\right )-24 i \text{PolyLog}\left (4,-i e^{\cosh ^{-1}(a x)}\right )-\frac{\cosh ^{-1}(a x)^4}{a x}+i \cosh ^{-1}(a x)^4-2 \pi \cosh ^{-1}(a x)^3-\frac{3}{2} i \pi ^2 \cosh ^{-1}(a x)^2+\frac{1}{2} \pi ^3 \cosh ^{-1}(a x)+4 i \cosh ^{-1}(a x)^3 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-4 i \cosh ^{-1}(a x)^3 \log \left (1+i e^{\cosh ^{-1}(a x)}\right )-6 \pi \cosh ^{-1}(a x)^2 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+6 \pi \cosh ^{-1}(a x)^2 \log \left (1-i e^{\cosh ^{-1}(a x)}\right )-3 i \pi ^2 \cosh ^{-1}(a x) \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+3 i \pi ^2 \cosh ^{-1}(a x) \log \left (1-i e^{\cosh ^{-1}(a x)}\right )+\frac{1}{2} \pi ^3 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-\frac{1}{2} \pi ^3 \log \left (1+i e^{\cosh ^{-1}(a x)}\right )+\frac{1}{2} \pi ^3 \log \left (\tan \left (\frac{1}{4} \left (\pi +2 i \cosh ^{-1}(a x)\right )\right )\right )-\frac{7 i \pi ^4}{16}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[a*x]^4/x^2,x]

[Out]

a*(((-7*I)/16)*Pi^4 + (Pi^3*ArcCosh[a*x])/2 - ((3*I)/2)*Pi^2*ArcCosh[a*x]^2 - 2*Pi*ArcCosh[a*x]^3 + I*ArcCosh[
a*x]^4 - ArcCosh[a*x]^4/(a*x) + (Pi^3*Log[1 + I/E^ArcCosh[a*x]])/2 - (3*I)*Pi^2*ArcCosh[a*x]*Log[1 + I/E^ArcCo
sh[a*x]] - 6*Pi*ArcCosh[a*x]^2*Log[1 + I/E^ArcCosh[a*x]] + (4*I)*ArcCosh[a*x]^3*Log[1 + I/E^ArcCosh[a*x]] + (3
*I)*Pi^2*ArcCosh[a*x]*Log[1 - I*E^ArcCosh[a*x]] + 6*Pi*ArcCosh[a*x]^2*Log[1 - I*E^ArcCosh[a*x]] - (Pi^3*Log[1
+ I*E^ArcCosh[a*x]])/2 - (4*I)*ArcCosh[a*x]^3*Log[1 + I*E^ArcCosh[a*x]] + (Pi^3*Log[Tan[(Pi + (2*I)*ArcCosh[a*
x])/4]])/2 + (3*I)*(Pi - (2*I)*ArcCosh[a*x])^2*PolyLog[2, (-I)/E^ArcCosh[a*x]] - (12*I)*ArcCosh[a*x]^2*PolyLog
[2, (-I)*E^ArcCosh[a*x]] + (3*I)*Pi^2*PolyLog[2, I*E^ArcCosh[a*x]] + 12*Pi*ArcCosh[a*x]*PolyLog[2, I*E^ArcCosh
[a*x]] + 12*Pi*PolyLog[3, (-I)/E^ArcCosh[a*x]] - (24*I)*ArcCosh[a*x]*PolyLog[3, (-I)/E^ArcCosh[a*x]] + (24*I)*
ArcCosh[a*x]*PolyLog[3, (-I)*E^ArcCosh[a*x]] - 12*Pi*PolyLog[3, I*E^ArcCosh[a*x]] - (24*I)*PolyLog[4, (-I)/E^A
rcCosh[a*x]] - (24*I)*PolyLog[4, (-I)*E^ArcCosh[a*x]])

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{4}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)^4/x^2,x)

[Out]

int(arccosh(a*x)^4/x^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{4}}{x} + \int \frac{4 \,{\left (a^{3} x^{2} + \sqrt{a x + 1} \sqrt{a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{3}}{a^{3} x^{4} - a x^{2} +{\left (a^{2} x^{3} - x\right )} \sqrt{a x + 1} \sqrt{a x - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^4/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arcosh}\left (a x\right )^{4}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^4/x^2,x, algorithm="fricas")

[Out]

integral(arccosh(a*x)^4/x^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acosh}^{4}{\left (a x \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**4/x**2,x)

[Out]

Integral(acosh(a*x)**4/x**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcosh}\left (a x\right )^{4}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^4/x^2,x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^4/x^2, x)