3.1.42 \(\int \frac {a+b \text {arccosh}(c x)}{x (d-c^2 d x^2)^2} \, dx\) [42]

3.1.42.1 Optimal result

Integrand size = 25, antiderivative size = 116 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=-\frac {b c x}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{2 \text {arccosh}(c x)}\right )}{d^2}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c x)}\right )}{2 d^2}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{2 d^2} \]

output

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

3.1.42.2 Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(244\) vs. \(2(116)=232\).

Time = 0.61 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.10 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=\frac {-b \sqrt {\frac {-1+c x}{1+c x}}+\frac {b \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}+\frac {b c x \sqrt {\frac {-1+c x}{1+c x}}}{1-c x}-\frac {2 a}{-1+c^2 x^2}+\frac {b \text {arccosh}(c x)}{1-c x}+\frac {b \text {arccosh}(c x)}{1+c x}+4 b \text {arccosh}(c x)^2+4 b \text {arccosh}(c x) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-4 b \text {arccosh}(c x) \log \left (1-e^{\text {arccosh}(c x)}\right )-4 b \text {arccosh}(c x) \log \left (1+e^{\text {arccosh}(c x)}\right )+4 a \log (x)-2 a \log \left (1-c^2 x^2\right )-2 b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )-4 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-4 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{4 d^2} \]

input

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

output

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

3.1.42.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6351, 27, 41, 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.

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

\(\Big \downarrow \) 6351

\(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c x)}{d x \left (1-c^2 x^2\right )}dx}{d}+\frac {b c \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 d^2}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )}dx}{d^2}+\frac {b c \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 d^2}+\frac {a+b \text {arccosh}(c x)}{2 d^2 \left (1-c^2 x^2\right )}\)

\(\Big \downarrow \) 41

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

\(\Big \downarrow \) 6331

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

\(\Big \downarrow \) 5984

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

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

output

-1/2*(b*c*x)/(d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a + b*ArcCosh[c*x])/(2* 
d^2*(1 - c^2*x^2)) - ((2*I)*(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^2
 

3.1.42.3.1 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 27

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

rule 41

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> S 
imp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && Eq 
Q[b*c + a*d, 0]
 

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 6351

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/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1 
))   Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[ 
b*c*(n/(2*f*(p + 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, m}, x] && EqQ[c^2*d + e, 0] & 
& GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || 
EqQ[n, 1])
 

3.1.42.4 Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.19

method	result	size
parts	\(\frac {a \left (\frac {1}{4 c x +4}-\frac {\ln \left (c x +1\right )}{2}+\ln \left (x \right )-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, c x -c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )+1}{2 \left (c^{2} x^{2}-1\right )}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}-\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}\)	\(254\)
derivativedivides	\(\frac {a \left (\ln \left (c x \right )+\frac {1}{4 c x +4}-\frac {\ln \left (c x +1\right )}{2}-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, c x -c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )+1}{2 \left (c^{2} x^{2}-1\right )}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}-\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}\)	\(256\)
default	\(\frac {a \left (\ln \left (c x \right )+\frac {1}{4 c x +4}-\frac {\ln \left (c x +1\right )}{2}-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, c x -c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )+1}{2 \left (c^{2} x^{2}-1\right )}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}-\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )-\operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{d^{2}}\)	\(256\)

input

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

output

a/d^2*(1/4/(c*x+1)-1/2*ln(c*x+1)+ln(x)-1/4/(c*x-1)-1/2*ln(c*x-1))+b/d^2*(- 
1/2*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x-c^2*x^2+arccosh(c*x)+1)/(c^2*x^2-1)+a 
rccosh(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))-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)))
 

3.1.42.5 Fricas [F]

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

input

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

output

integral((b*arccosh(c*x) + a)/(c^4*d^2*x^5 - 2*c^2*d^2*x^3 + d^2*x), x)
 

3.1.42.6 Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{5} - 2 c^{2} x^{3} + x}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{5} - 2 c^{2} x^{3} + x}\, dx}{d^{2}} \]

input

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

output

(Integral(a/(c**4*x**5 - 2*c**2*x**3 + x), x) + Integral(b*acosh(c*x)/(c** 
4*x**5 - 2*c**2*x**3 + x), x))/d**2
 

3.1.42.7 Maxima [F]

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

input

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

output

-1/2*a*(1/(c^2*d^2*x^2 - d^2) + log(c*x + 1)/d^2 + log(c*x - 1)/d^2 - 2*lo 
g(x)/d^2) + b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^4*d^2*x^ 
5 - 2*c^2*d^2*x^3 + d^2*x), x)
 

3.1.42.8 Giac [F]

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

input

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

output

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

3.1.42.9 Mupad [F(-1)]

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

input

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

output

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