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

3.3.43.1 Optimal result

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

output

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*poly 
log(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*po 
lylog(4,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))/a/c
 

3.3.43.2 Mathematica [A] (verified)

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

input

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

output

(-(ArcCosh[a*x]^3*Log[1 - E^ArcCosh[a*x]]) + ArcCosh[a*x]^3*Log[1 + E^ArcC 
osh[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)
 

3.3.43.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.60 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.89, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6318, 3042, 26, 4670, 3011, 7163, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6318

\(\displaystyle -\frac {\int \frac {\text {arccosh}(a x)^3}{\sqrt {\frac {a x-1}{a x+1}} (a x+1)}d\text {arccosh}(a x)}{a c}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int i \text {arccosh}(a x)^3 \csc (i \text {arccosh}(a x))d\text {arccosh}(a x)}{a c}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \int \text {arccosh}(a x)^3 \csc (i \text {arccosh}(a x))d\text {arccosh}(a x)}{a c}\)

\(\Big \downarrow \) 4670

\(\displaystyle -\frac {i \left (3 i \int \text {arccosh}(a x)^2 \log \left (1-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-3 i \int \text {arccosh}(a x)^2 \log \left (1+e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{a c}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {i \left (-3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \int \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{a c}\)

\(\Big \downarrow \) 7163

\(\displaystyle -\frac {i \left (-3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )-\int \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )-\int \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{a c}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {i \left (-3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )-\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )-\int e^{-\text {arccosh}(a x)} \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )+2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )\right )}{a c}\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {i \left (2 i \text {arccosh}(a x)^3 \text {arctanh}\left (e^{\text {arccosh}(a x)}\right )-3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(a x)}\right )-\operatorname {PolyLog}\left (4,-e^{\text {arccosh}(a x)}\right )\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(a x)}\right )\right )+3 i \left (2 \left (\text {arccosh}(a x) \operatorname {PolyLog}\left (3,e^{\text {arccosh}(a x)}\right )-\operatorname {PolyLog}\left (4,e^{\text {arccosh}(a x)}\right )\right )-\text {arccosh}(a x)^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(a x)}\right )\right )\right )}{a c}\)

input

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

output

((-I)*((2*I)*ArcCosh[a*x]^3*ArcTanh[E^ArcCosh[a*x]] - (3*I)*(-(ArcCosh[a*x 
]^2*PolyLog[2, -E^ArcCosh[a*x]]) + 2*(ArcCosh[a*x]*PolyLog[3, -E^ArcCosh[a 
*x]] - PolyLog[4, -E^ArcCosh[a*x]])) + (3*I)*(-(ArcCosh[a*x]^2*PolyLog[2, 
E^ArcCosh[a*x]]) + 2*(ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]] - PolyLog[4, 
 E^ArcCosh[a*x]]))))/(a*c)
 

3.3.43.3.1 Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{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 3011

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] + Simp[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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[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 6318

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[-(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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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 7163

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] - Simp[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]
 

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

input

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

output

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)*polylo 
g(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*a 
rccosh(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)))
 

3.3.43.5 Fricas [F]

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

input

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

output

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

3.3.43.6 Sympy [F]

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

input

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

output

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

3.3.43.7 Maxima [F]

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

input

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

output

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)
 

3.3.43.8 Giac [F]

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

input

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

output

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

3.3.43.9 Mupad [F(-1)]

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

input

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

output

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