3.1.27 \(\int \frac {\cosh ^{-1}(a x)^3}{x} \, dx\) [27]

Optimal. Leaf size=87 \[ -\frac {1}{4} \cosh ^{-1}(a x)^4+\cosh ^{-1}(a x)^3 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{2} \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} \cosh ^{-1}(a x) \text {PolyLog}\left (3,-e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{4} \text {PolyLog}\left (4,-e^{2 \cosh ^{-1}(a x)}\right ) \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5882, 3799, 2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {3}{2} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} \cosh ^{-1}(a x) \text {Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{4} \text {Li}_4\left (-e^{2 \cosh ^{-1}(a x)}\right )-\frac {1}{4} \cosh ^{-1}(a x)^4+\cosh ^{-1}(a x)^3 \log \left (e^{2 \cosh ^{-1}(a x)}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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]

Rule 6744

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]

Rubi steps

\begin {align*} \int \frac {\cosh ^{-1}(a x)^3}{x} \, dx &=\text {Subst}\left (\int x^3 \tanh (x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {1}{4} \cosh ^{-1}(a x)^4+2 \text {Subst}\left (\int \frac {e^{2 x} x^3}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {1}{4} \cosh ^{-1}(a x)^4+\cosh ^{-1}(a x)^3 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-3 \text {Subst}\left (\int x^2 \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {1}{4} \cosh ^{-1}(a x)^4+\cosh ^{-1}(a x)^3 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{2} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )-3 \text {Subst}\left (\int x \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {1}{4} \cosh ^{-1}(a x)^4+\cosh ^{-1}(a x)^3 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{2} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} \cosh ^{-1}(a x) \text {Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{2} \text {Subst}\left (\int \text {Li}_3\left (-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {1}{4} \cosh ^{-1}(a x)^4+\cosh ^{-1}(a x)^3 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{2} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} \cosh ^{-1}(a x) \text {Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{4} \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )\\ &=-\frac {1}{4} \cosh ^{-1}(a x)^4+\cosh ^{-1}(a x)^3 \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{2} \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} \cosh ^{-1}(a x) \text {Li}_3\left (-e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{4} \text {Li}_4\left (-e^{2 \cosh ^{-1}(a x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 82, normalized size = 0.94 \begin {gather*} \frac {1}{4} \left (\cosh ^{-1}(a x)^4+4 \cosh ^{-1}(a x)^3 \log \left (1+e^{-2 \cosh ^{-1}(a x)}\right )-6 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(a x)}\right )-6 \cosh ^{-1}(a x) \text {PolyLog}\left (3,-e^{-2 \cosh ^{-1}(a x)}\right )-3 \text {PolyLog}\left (4,-e^{-2 \cosh ^{-1}(a x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 2.57, size = 132, normalized size = 1.52

method	result	size
derivativedivides	\(-\frac {\mathrm {arccosh}\left (a x \right )^{4}}{4}+\mathrm {arccosh}\left (a x \right )^{3} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \polylog \left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}-\frac {3 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}+\frac {3 \polylog \left (4, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{4}\)	\(132\)
default	\(-\frac {\mathrm {arccosh}\left (a x \right )^{4}}{4}+\mathrm {arccosh}\left (a x \right )^{3} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \polylog \left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}-\frac {3 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (3, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}+\frac {3 \polylog \left (4, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{4}\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccosh(a*x)^3/x, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccosh(a*x)^3/x, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(a*x)**3/x, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccosh(a*x)^3/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acosh(a*x)^3/x, x)

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