\(\int \frac {a+b \text {arccosh}(c x)}{x^3 (d-c^2 d x^2)} \, dx\) [35]

Optimal result

Integrand size = 25, antiderivative size = 118 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x}-\frac {a+b \text {arccosh}(c x)}{2 d x^2}+\frac {2 c^2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d}+\frac {b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d}-\frac {b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d} \] Output:

1/2*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/x-1/2*(a+b*arccosh(c*x))/d/x^2+2*c^2 
*(a+b*arccosh(c*x))*arctanh((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d+1/2*b*c 
^2*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d-1/2*b*c^2*polylog(2,( 
c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.80 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=-\frac {a-b c x \sqrt {-1+c x} \sqrt {1+c x}+b \text {arccosh}(c x)-2 b c^2 x^2 \text {arccosh}(c x)^2-2 b c^2 x^2 \text {arccosh}(c x) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+2 b c^2 x^2 \text {arccosh}(c x) \log \left (1-e^{\text {arccosh}(c x)}\right )+2 b c^2 x^2 \text {arccosh}(c x) \log \left (1+e^{\text {arccosh}(c x)}\right )-2 a c^2 x^2 \log (x)+a c^2 x^2 \log \left (1-c^2 x^2\right )+b c^2 x^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )+2 b c^2 x^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )+2 b c^2 x^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{2 d x^2} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6347, 27, 106, 6331, 5984, 3042, 26, 4670, 2715, 2838}

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.

\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx\)

\(\Big \downarrow \) 6347

\(\displaystyle c^2 \int \frac {a+b \text {arccosh}(c x)}{d x \left (1-c^2 x^2\right )}dx+\frac {b c \int \frac {1}{x^2 \sqrt {c x-1} \sqrt {c x+1}}dx}{2 d}-\frac {a+b \text {arccosh}(c x)}{2 d x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx}{d}+\frac {b c \int \frac {1}{x^2 \sqrt {c x-1} \sqrt {c x+1}}dx}{2 d}-\frac {a+b \text {arccosh}(c x)}{2 d x^2}\)

\(\Big \downarrow \) 106

\(\displaystyle \frac {c^2 \int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx}{d}-\frac {a+b \text {arccosh}(c x)}{2 d x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 d x}\)

\(\Big \downarrow \) 6331

\(\displaystyle -\frac {c^2 \int \frac {a+b \text {arccosh}(c x)}{c x \sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{d}-\frac {a+b \text {arccosh}(c x)}{2 d x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 d x}\)

\(\Big \downarrow \) 5984

\(\displaystyle -\frac {2 c^2 \int (a+b \text {arccosh}(c x)) \text {csch}(2 \text {arccosh}(c x))d\text {arccosh}(c x)}{d}-\frac {a+b \text {arccosh}(c x)}{2 d x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 d x}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 c^2 \int i (a+b \text {arccosh}(c x)) \csc (2 i \text {arccosh}(c x))d\text {arccosh}(c x)}{d}-\frac {a+b \text {arccosh}(c x)}{2 d x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 d x}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {2 i c^2 \int (a+b \text {arccosh}(c x)) \csc (2 i \text {arccosh}(c x))d\text {arccosh}(c x)}{d}-\frac {a+b \text {arccosh}(c x)}{2 d x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 d x}\)

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {2 i c^2 \left (\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{4} i b \int e^{-2 \text {arccosh}(c x)} \log \left (1+e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}+i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{d}-\frac {a+b \text {arccosh}(c x)}{2 d x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 d x}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {2 i c^2 \left (i \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )}{d}-\frac {a+b \text {arccosh}(c x)}{2 d x^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 d x}\)

Input:

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

Output:

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*d*x) - (a + b*ArcCosh[c*x])/(2*d*x^2 
) - ((2*I)*c^2*(I*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])] + (I/4) 
*b*PolyLog[2, -E^(2*ArcCosh[c*x])] - (I/4)*b*PolyLog[2, E^(2*ArcCosh[c*x]) 
]))/d
 

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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 106

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] /; FreeQ[{a, b, c, d, e, f, m, n, 
p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 1), 0] && NeQ[m, -1]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

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 5984

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n   Int[(c + d*x)^m*Csch[2*a + 2*b*x 
]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
 

rule 6331

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

rule 6347

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

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.06

method	result	size
derivativedivides	\(c^{2} \left (-\frac {a \left (\frac {1}{2 c^{2} x^{2}}-\ln \left (c x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (\frac {-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}\right )\)	\(243\)
default	\(c^{2} \left (-\frac {a \left (\frac {1}{2 c^{2} x^{2}}-\ln \left (c x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (\frac {-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}\right )\)	\(243\)
parts	\(-\frac {a \left (\frac {1}{2 x^{2}}-c^{2} \ln \left (x \right )+\frac {c^{2} \ln \left (c x +1\right )}{2}+\frac {c^{2} \ln \left (c x -1\right )}{2}\right )}{d}-\frac {b \,c^{2} \left (\frac {-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )+\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d}\)	\(246\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(-(b*arccosh(c*x) + a)/(c^2*d*x^5 - d*x^3), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=- \frac {\int \frac {a}{c^{2} x^{5} - x^{3}}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{2} x^{5} - x^{3}}\, dx}{d} \] Input:

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

Output:

-(Integral(a/(c**2*x**5 - x**3), x) + Integral(b*acosh(c*x)/(c**2*x**5 - x 
**3), x))/d
 

Maxima [F]

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

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

Output:

-1/2*(c^2*log(c*x + 1)/d + c^2*log(c*x - 1)/d - 2*c^2*log(x)/d + 1/(d*x^2) 
)*a - b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^2*d*x^5 - d*x^ 
3), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx=\frac {-2 \left (\int \frac {\mathit {acosh} \left (c x \right )}{c^{2} x^{5}-x^{3}}d x \right ) b \,x^{2}-\mathrm {log}\left (c^{2} x -c \right ) a \,c^{2} x^{2}-\mathrm {log}\left (c^{2} x +c \right ) a \,c^{2} x^{2}+2 \,\mathrm {log}\left (x \right ) a \,c^{2} x^{2}-a}{2 d \,x^{2}} \] Input:

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

Output:

( - 2*int(acosh(c*x)/(c**2*x**5 - x**3),x)*b*x**2 - log(c**2*x - c)*a*c**2 
*x**2 - log(c**2*x + c)*a*c**2*x**2 + 2*log(x)*a*c**2*x**2 - a)/(2*d*x**2)