\(\int \frac {\text {arccosh}(a x)^3}{(c-a^2 c x^2)^{3/2}} \, dx\) [250]

Optimal result

Integrand size = 22, antiderivative size = 241 \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \operatorname {PolyLog}\left (3,e^{2 \text {arccosh}(a x)}\right )}{2 a c \sqrt {c-a^2 c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5899, 5913, 3797, 2221, 2611, 2320, 6724} \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=-\frac {3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {a x-1} \sqrt {a x+1} \operatorname {PolyLog}\left (3,e^{2 \text {arccosh}(a x)}\right )}{2 a c \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}} \]

[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 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 3797

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))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5899

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcCosh
[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Dist[b*c*(n/d)*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])], Int[x
*((a + b*ArcCosh[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ
[n, 0]

Rule 5913

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 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*} \text {integral}& = \frac {x \text {arccosh}(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 \text {arccosh}(a x)^2}{1-a^2 x^2} \, dx}{c \sqrt {c-a^2 c x^2}} \\ & = \frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}-\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int x^2 \coth (x) \, dx,x,\text {arccosh}(a x)\right )}{a c \sqrt {c-a^2 c x^2}} \\ & = \frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{a c \sqrt {c-a^2 c x^2}}+\frac {\left (6 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {e^{2 x} x^2}{1-e^{2 x}} \, dx,x,\text {arccosh}(a x)\right )}{a c \sqrt {c-a^2 c x^2}} \\ & = \frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {\left (6 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int x \log \left (1-e^{2 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c \sqrt {c-a^2 c x^2}} \\ & = \frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c \sqrt {c-a^2 c x^2}} \\ & = \frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 \text {arccosh}(a x)}\right )}{2 a c \sqrt {c-a^2 c x^2}} \\ & = \frac {x \text {arccosh}(a x)^3}{c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )}{a c \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \operatorname {PolyLog}\left (3,e^{2 \text {arccosh}(a x)}\right )}{2 a c \sqrt {c-a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.67 \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {-1+a x} \sqrt {1+a x} \left (\text {arccosh}(a x)^3+\frac {a x \text {arccosh}(a x)^3}{\sqrt {-1+a x} \sqrt {1+a x}}-3 \text {arccosh}(a x)^2 \log \left (1-e^{\text {arccosh}(a x)}\right )-3 \text {arccosh}(a x)^2 \log \left (1+e^{\text {arccosh}(a x)}\right )-6 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )-6 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )+6 \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )\right )}{a c \sqrt {c-a^2 c x^2}} \]

[In]

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

[Out]

(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(ArcCosh[a*x]^3 + (a*x*ArcCosh[a*x]^3)/(Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - 3*ArcCos
h[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^Arc
Cosh[a*x]] - 6*ArcCosh[a*x]*PolyLog[2, E^ArcCosh[a*x]] + 6*PolyLog[3, -E^ArcCosh[a*x]] + 6*PolyLog[3, E^ArcCos
h[a*x]]))/(a*c*Sqrt[c - a^2*c*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(547\) vs. \(2(252)=504\).

Time = 1.14 (sec) , antiderivative size = 548, normalized size of antiderivative = 2.27

method	result	size
default	\(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-\sqrt {a x -1}\, \sqrt {a x +1}+a x \right ) \operatorname {arccosh}\left (a x \right )^{3}}{c^{2} a \left (a^{2} x^{2}-1\right )}-\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right )^{3}}{c^{2} a \left (a^{2} x^{2}-1\right )}+\frac {3 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}+\frac {6 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}-\frac {6 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {polylog}\left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}+\frac {3 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}+\frac {6 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}-\frac {6 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {polylog}\left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{2} a \left (a^{2} x^{2}-1\right )}\)	\(548\)

[In]

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

[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))

Fricas [F]

\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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