3.243 \(\int \frac {\cosh ^{-1}(a x)^3}{c-a^2 c x^2} \, dx\)

Optimal. Leaf size=144 \[ \frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {6 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {6 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {6 \text {Li}_4\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {6 \text {Li}_4\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {2 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c} \]

[Out]

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

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Rubi [A]  time = 0.13, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5694, 4182, 2531, 6609, 2282, 6589} \[ \frac {3 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {3 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {6 \cosh ^{-1}(a x) \text {PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {6 \cosh ^{-1}(a x) \text {PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {6 \text {PolyLog}\left (4,-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {6 \text {PolyLog}\left (4,e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {2 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^3/(c - a^2*c*x^2),x]

[Out]

(2*ArcCosh[a*x]^3*ArcTanh[E^ArcCosh[a*x]])/(a*c) + (3*ArcCosh[a*x]^2*PolyLog[2, -E^ArcCosh[a*x]])/(a*c) - (3*A
rcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]])/(a*c) - (6*ArcCosh[a*x]*PolyLog[3, -E^ArcCosh[a*x]])/(a*c) + (6*ArcC
osh[a*x]*PolyLog[3, E^ArcCosh[a*x]])/(a*c) + (6*PolyLog[4, -E^ArcCosh[a*x]])/(a*c) - (6*PolyLog[4, E^ArcCosh[a
*x]])/(a*c)

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 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 4182

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

Rule 5694

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Dist[(c*d)^(-1), Subst[Int[
(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

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 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]

Rubi steps

\begin {align*} \int \frac {\cosh ^{-1}(a x)^3}{c-a^2 c x^2} \, dx &=-\frac {\operatorname {Subst}\left (\int x^3 \text {csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}\\ &=\frac {2 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {3 \operatorname {Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}-\frac {3 \operatorname {Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}\\ &=\frac {2 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {6 \operatorname {Subst}\left (\int x \text {Li}_2\left (-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}+\frac {6 \operatorname {Subst}\left (\int x \text {Li}_2\left (e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}\\ &=\frac {2 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {6 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {6 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {6 \operatorname {Subst}\left (\int \text {Li}_3\left (-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}-\frac {6 \operatorname {Subst}\left (\int \text {Li}_3\left (e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c}\\ &=\frac {2 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {6 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {6 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {6 \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {6 \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c}\\ &=\frac {2 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {6 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {6 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c}+\frac {6 \text {Li}_4\left (-e^{\cosh ^{-1}(a x)}\right )}{a c}-\frac {6 \text {Li}_4\left (e^{\cosh ^{-1}(a x)}\right )}{a c}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 129, normalized size = 0.90 \[ \frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )-3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )-6 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )+6 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )+6 \text {Li}_4\left (-e^{\cosh ^{-1}(a x)}\right )-6 \text {Li}_4\left (e^{\cosh ^{-1}(a x)}\right )+\cosh ^{-1}(a x)^3 \left (-\log \left (1-e^{\cosh ^{-1}(a x)}\right )\right )+\cosh ^{-1}(a x)^3 \log \left (e^{\cosh ^{-1}(a x)}+1\right )}{a c} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[a*x]^3/(c - a^2*c*x^2),x]

[Out]

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

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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\operatorname {arcosh}\left (a x\right )^{3}}{a^{2} c x^{2} - c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3/(-a^2*c*x^2+c),x, algorithm="fricas")

[Out]

integral(-arccosh(a*x)^3/(a^2*c*x^2 - c), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\operatorname {arcosh}\left (a x\right )^{3}}{a^{2} c x^{2} - c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3/(-a^2*c*x^2+c),x, algorithm="giac")

[Out]

integrate(-arccosh(a*x)^3/(a^2*c*x^2 - c), x)

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maple [A]  time = 0.09, size = 273, normalized size = 1.90 \[ -\frac {\mathrm {arccosh}\left (a x \right )^{3} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a c}-\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a c}+\frac {6 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a c}-\frac {6 \polylog \left (4, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a c}+\frac {\mathrm {arccosh}\left (a x \right )^{3} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a c}+\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a c}-\frac {6 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a c}+\frac {6 \polylog \left (4, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (\log \left (a x + 1\right ) - \log \left (a x - 1\right )\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{3}}{2 \, a c} - \int \frac {3 \, {\left ({\left (a x \log \left (a x + 1\right ) - a x \log \left (a x - 1\right )\right )} \sqrt {a x + 1} \sqrt {a x - 1} + {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) - {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{2}}{2 \, {\left (a^{3} c x^{3} - a c x + {\left (a^{2} c x^{2} - c\right )} \sqrt {a x + 1} \sqrt {a x - 1}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3/(-a^2*c*x^2+c),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{c-a^2\,c\,x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)^3/(c - a^2*c*x^2),x)

[Out]

int(acosh(a*x)^3/(c - a^2*c*x^2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{a^{2} x^{2} - 1}\, dx}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**3/(-a**2*c*x**2+c),x)

[Out]

-Integral(acosh(a*x)**3/(a**2*x**2 - 1), x)/c

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