\(\int \frac {\text {arccosh}(a x)^3}{c-a^2 c x^2} \, dx\) [243]

Optimal result

Integrand size = 20, antiderivative size = 144 \[ \int \frac {\text {arccosh}(a x)^3}{c-a^2 c x^2} \, dx=\frac {2 \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c}+\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{a c}-\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{a c}-\frac {6 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{a c}+\frac {6 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{a c}+\frac {6 \operatorname {PolyLog}\left (4,-e^{\text {arccosh}(a x)}\right )}{a c}-\frac {6 \operatorname {PolyLog}\left (4,e^{\text {arccosh}(a x)}\right )}{a c} \]

[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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5903, 4267, 2611, 6744, 2320, 6724} \[ \int \frac {\text {arccosh}(a x)^3}{c-a^2 c x^2} \, dx=\frac {2 \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c}+\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{a c}-\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{a c}-\frac {6 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{a c}+\frac {6 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{a c}+\frac {6 \operatorname {PolyLog}\left (4,-e^{\text {arccosh}(a x)}\right )}{a c}-\frac {6 \operatorname {PolyLog}\left (4,e^{\text {arccosh}(a x)}\right )}{a c} \]

[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 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 4267

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 5903

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 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*} \text {integral}& = -\frac {\text {Subst}\left (\int x^3 \text {csch}(x) \, dx,x,\text {arccosh}(a x)\right )}{a c} \\ & = \frac {2 \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c}+\frac {3 \text {Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c}-\frac {3 \text {Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c} \\ & = \frac {2 \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c}+\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{a c}-\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{a c}-\frac {6 \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c}+\frac {6 \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c} \\ & = \frac {2 \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c}+\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{a c}-\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{a c}-\frac {6 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{a c}+\frac {6 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{a c}+\frac {6 \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c}-\frac {6 \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c} \\ & = \frac {2 \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c}+\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{a c}-\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{a c}-\frac {6 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{a c}+\frac {6 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{a c}+\frac {6 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{a c}-\frac {6 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{a c} \\ & = \frac {2 \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )}{a c}+\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )}{a c}-\frac {3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )}{a c}-\frac {6 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )}{a c}+\frac {6 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )}{a c}+\frac {6 \operatorname {PolyLog}\left (4,-e^{\text {arccosh}(a x)}\right )}{a c}-\frac {6 \operatorname {PolyLog}\left (4,e^{\text {arccosh}(a x)}\right )}{a c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.90 \[ \int \frac {\text {arccosh}(a x)^3}{c-a^2 c x^2} \, dx=\frac {-\text {arccosh}(a x)^3 \log \left (1-e^{\text {arccosh}(a x)}\right )+\text {arccosh}(a x)^3 \log \left (1+e^{\text {arccosh}(a x)}\right )+3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )-3 \text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )-6 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )+6 \text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )+6 \operatorname {PolyLog}\left (4,-e^{\text {arccosh}(a x)}\right )-6 \operatorname {PolyLog}\left (4,e^{\text {arccosh}(a x)}\right )}{a c} \]

[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)

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.76

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {arccosh}\left (a x \right )^{3} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}+\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}-\frac {6 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}+\frac {6 \operatorname {polylog}\left (4, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}-\frac {\operatorname {arccosh}\left (a x \right )^{3} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}-\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}+\frac {6 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}-\frac {6 \operatorname {polylog}\left (4, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}}{a}\)	\(253\)
default	\(\frac {\frac {\operatorname {arccosh}\left (a x \right )^{3} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}+\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}-\frac {6 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}+\frac {6 \operatorname {polylog}\left (4, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}-\frac {\operatorname {arccosh}\left (a x \right )^{3} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}-\frac {3 \operatorname {arccosh}\left (a x \right )^{2} \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}+\frac {6 \,\operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}-\frac {6 \operatorname {polylog}\left (4, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c}}{a}\)	\(253\)

[In]

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

[Out]

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

Fricas [F]

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

[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)

Sympy [F]

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

[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

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[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)