3.30 \(\int \frac{\cosh ^{-1}(a x)^3}{x^4} \, dx\)

Optimal. Leaf size=183 \[ -i a^3 \cosh ^{-1}(a x) \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )+i a^3 \cosh ^{-1}(a x) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+i a^3 \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )-i a^3 \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )-a^3 \tan ^{-1}\left (\sqrt{a x-1} \sqrt{a x+1}\right )+\frac{a^2 \cosh ^{-1}(a x)}{x}+a^3 \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+\frac{a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{2 x^2}-\frac{\cosh ^{-1}(a x)^3}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.5773, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.9, Rules used = {5662, 5748, 5761, 4180, 2531, 2282, 6589, 92, 205} \[ -i a^3 \cosh ^{-1}(a x) \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )+i a^3 \cosh ^{-1}(a x) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+i a^3 \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )-i a^3 \text{PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )-a^3 \tan ^{-1}\left (\sqrt{a x-1} \sqrt{a x+1}\right )+\frac{a^2 \cosh ^{-1}(a x)}{x}+a^3 \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )+\frac{a \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{2 x^2}-\frac{\cosh ^{-1}(a x)^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*ArcCosh[a*x])/x + (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(2*x^2) - ArcCosh[a*x]^3/(3*x^3) + a^3*
ArcCosh[a*x]^2*ArcTan[E^ArcCosh[a*x]] - a^3*ArcTan[Sqrt[-1 + a*x]*Sqrt[1 + a*x]] - I*a^3*ArcCosh[a*x]*PolyLog[
2, (-I)*E^ArcCosh[a*x]] + I*a^3*ArcCosh[a*x]*PolyLog[2, I*E^ArcCosh[a*x]] + I*a^3*PolyLog[3, (-I)*E^ArcCosh[a*
x]] - I*a^3*PolyLog[3, 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 5748

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*
d2*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*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p
])/(f*(m + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(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, 0] && EqQ[e2 +
c*d2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m] && IntegerQ[p + 1/2]

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 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]

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.545972, size = 201, normalized size = 1.1 \[ \frac{1}{6} \left (-3 i a^3 \left (2 \cosh ^{-1}(a x) \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )-2 \cosh ^{-1}(a x) \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,i e^{-\cosh ^{-1}(a x)}\right )+\cosh ^{-1}(a x)^2 \log \left (1-i e^{-\cosh ^{-1}(a x)}\right )-\cosh ^{-1}(a x)^2 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )-4 i \tan ^{-1}\left (\tanh \left (\frac{1}{2} \cosh ^{-1}(a x)\right )\right )\right )+\frac{6 a^2 \cosh ^{-1}(a x)}{x}+\frac{3 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)^2}{x^2}-\frac{2 \cosh ^{-1}(a x)^3}{x^3}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.169, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(arccosh(a*x)^3/x^4,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 )^{3}}{3 \, x^{3}} + \int \frac{{\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 )^{2}}{a^{3} x^{6} - a x^{4} +{\left (a^{2} x^{5} - x^{3}\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)^3/x^4,x, algorithm="maxima")

[Out]

-1/3*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3/x^3 + integrate((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^6 - a*x^4 + (a^2*x^5 - x^3)*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 )^{3}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccosh(a*x)^3/x^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acosh}^{3}{\left (a x \right )}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(a*x)**3/x**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcosh}\left (a x\right )^{3}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccosh(a*x)^3/x^4, x)