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

Optimal. Leaf size=241 \[ -\frac{3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )}{2 a c \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{c \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{a c \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.348938, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {5713, 5688, 5715, 3716, 2190, 2531, 2282, 6589} \[ -\frac{3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )}{2 a c \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{c \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{a c \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((-d)^IntPart[p]*(
d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5688

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(((d1_) + (e1_.)*(x_))^(3/2)*((d2_) + (e2_.)*(x_))^(3/2)), x_Sym
bol] :> Simp[(x*(a + b*ArcCosh[c*x])^n)/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x] + Dist[(b*c*n*Sqrt[1 + c*x
]*Sqrt[-1 + c*x])/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), Int[(x*(a + b*ArcCosh[c*x])^(n - 1))/(1 - c^2*x^2),
x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0]

Rule 5715

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

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 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]

Rubi steps

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

Mathematica [A]  time = 0.239766, size = 145, normalized size = 0.6 \[ \frac{\frac{\sqrt{a x-1} \sqrt{a x+1} \left (-6 \cosh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )-6 \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )+6 \text{PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )+6 \text{PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )+\cosh ^{-1}(a x)^3-3 \cosh ^{-1}(a x)^2 \log \left (1-e^{\cosh ^{-1}(a x)}\right )-3 \cosh ^{-1}(a x)^2 \log \left (e^{\cosh ^{-1}(a x)}+1\right )\right )}{a}+x \cosh ^{-1}(a x)^3}{c \sqrt{c-a^2 c x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.21, size = 548, normalized size = 2.3 \begin{align*} -{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{a{c}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( -\sqrt{ax-1}\sqrt{ax+1}+ax \right ) }-2\,{\frac{\sqrt{ax-1}\sqrt{ax+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{a{c}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }}+3\,{\frac{\sqrt{ax-1}\sqrt{ax+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\ln \left ( 1-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }}+6\,{\frac{\sqrt{ax-1}\sqrt{ax+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }{\rm arccosh} \left (ax\right ){\it polylog} \left ( 2,ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }}-6\,{\frac{\sqrt{ax-1}\sqrt{ax+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }{\it polylog} \left ( 3,ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }}+3\,{\frac{\sqrt{ax-1}\sqrt{ax+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\ln \left ( 1+ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }}+6\,{\frac{\sqrt{ax-1}\sqrt{ax+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }{\rm arccosh} \left (ax\right ){\it polylog} \left ( 2,-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }}-6\,{\frac{\sqrt{ax-1}\sqrt{ax+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }{\it polylog} \left ( 3,-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} c x^{2} + c} \operatorname{arcosh}\left (a x\right )^{3}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-a^2*c*x^2 + c)*arccosh(a*x)^3/(a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2), 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 )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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