3.1.28 \(\int \frac {\cosh ^{-1}(a x)^3}{x^2} \, dx\) [28]

Optimal. Leaf size=104 \[ -\frac {\cosh ^{-1}(a x)^3}{x}+6 a \cosh ^{-1}(a x)^2 \text {ArcTan}\left (e^{\cosh ^{-1}(a x)}\right )-6 i a \cosh ^{-1}(a x) \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )+6 i a \cosh ^{-1}(a x) \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+6 i a \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )-6 i a \text {PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right ) \]

[Out]

-arccosh(a*x)^3/x+6*a*arccosh(a*x)^2*arctan(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))-6*I*a*arccosh(a*x)*polylog(2,-I*(
a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+6*I*a*arccosh(a*x)*polylog(2,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+6*I*a*poly
log(3,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))-6*I*a*polylog(3,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))

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Rubi [A]
time = 0.20, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5883, 5947, 4265, 2611, 2320, 6724} \begin {gather*} 6 a \cosh ^{-1}(a x)^2 \text {ArcTan}\left (e^{\cosh ^{-1}(a x)}\right )-6 i a \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+6 i a \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+6 i a \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )-6 i a \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )-\frac {\cosh ^{-1}(a x)^3}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^3/x^2,x]

[Out]

-(ArcCosh[a*x]^3/x) + 6*a*ArcCosh[a*x]^2*ArcTan[E^ArcCosh[a*x]] - (6*I)*a*ArcCosh[a*x]*PolyLog[2, (-I)*E^ArcCo
sh[a*x]] + (6*I)*a*ArcCosh[a*x]*PolyLog[2, I*E^ArcCosh[a*x]] + (6*I)*a*PolyLog[3, (-I)*E^ArcCosh[a*x]] - (6*I)
*a*PolyLog[3, I*E^ArcCosh[a*x]]

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 4265

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 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 5947

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]
), x_Symbol] :> Dist[(1/c^(m + 1))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], S
ubst[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*d
1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && IntegerQ[m]

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*} \int \frac {\cosh ^{-1}(a x)^3}{x^2} \, dx &=-\frac {\cosh ^{-1}(a x)^3}{x}+(3 a) \int \frac {\cosh ^{-1}(a x)^2}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {\cosh ^{-1}(a x)^3}{x}+(3 a) \text {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {\cosh ^{-1}(a x)^3}{x}+6 a \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-(6 i a) \text {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )+(6 i a) \text {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {\cosh ^{-1}(a x)^3}{x}+6 a \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-6 i a \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+6 i a \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+(6 i a) \text {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )-(6 i a) \text {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {\cosh ^{-1}(a x)^3}{x}+6 a \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-6 i a \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+6 i a \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+(6 i a) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )-(6 i a) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )\\ &=-\frac {\cosh ^{-1}(a x)^3}{x}+6 a \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-6 i a \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+6 i a \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+6 i a \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )-6 i a \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 128, normalized size = 1.23 \begin {gather*} -\frac {\cosh ^{-1}(a x)^3}{x}+3 i a \left (-\cosh ^{-1}(a x)^2 \left (\log \left (1-i e^{-\cosh ^{-1}(a x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(a x)}\right )\right )-2 \cosh ^{-1}(a x) \left (\text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )-\text {PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )\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 )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[a*x]^3/x^2,x]

[Out]

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

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Maple [F]
time = 1.90, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {arccosh}\left (a x \right )^{3}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)^3/x^2,x)

[Out]

int(arccosh(a*x)^3/x^2,x)

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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^2,x, algorithm="maxima")

[Out]

-log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3/x + integrate(3*(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))^2/(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 = 0.00, 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^2,x, algorithm="fricas")

[Out]

integral(arccosh(a*x)^3/x^2, 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^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**3/x**2,x)

[Out]

Integral(acosh(a*x)**3/x**2, 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^2,x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^3/x^2, 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^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)^3/x^2,x)

[Out]

int(acosh(a*x)^3/x^2, x)

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