3.1.8 \(\int \frac {(d-c^2 d x^2) (a+b \text {arccosh}(c x))}{x^3} \, dx\) [8]

3.1.8.1 Optimal result

Integrand size = 23, antiderivative size = 135 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {1}{2} b c^2 d \text {arccosh}(c x)-\frac {d \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))}{2 x^2}-\frac {c^2 d (a+b \text {arccosh}(c x))^2}{2 b}-c^2 d (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )+\frac {1}{2} b c^2 d \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right ) \]

output

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

3.1.8.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.79 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=-\frac {a d}{2 x^2}+\frac {b c d \sqrt {-1+c x} \sqrt {1+c x}}{2 x}-\frac {b d \text {arccosh}(c x)}{2 x^2}-a c^2 d \log (x)-\frac {1}{2} b c^2 d \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right ) \]

input

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

output

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

3.1.8.3 Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.68 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6335, 108, 27, 43, 6297, 25, 3042, 26, 4201, 2620, 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.

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

\(\Big \downarrow \) 6335

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 43

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

\(\Big \downarrow \) 6297

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

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

output

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

3.1.8.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 43

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a 
*d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
 

rule 108

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

rule 2620

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] - Si 
mp[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 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 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x 
_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I   Int[ 
(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6297

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b 
 Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a 
, b, c}, x] && IGtQ[n, 0]
 

rule 6335

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

3.1.8.4 Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.95

method	result	size
parts	\(-\frac {d a}{2 x^{2}}-d a \,c^{2} \ln \left (x \right )-d b \,c^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\right )\)	\(128\)
derivativedivides	\(c^{2} \left (-d a \left (\ln \left (c x \right )+\frac {1}{2 c^{2} x^{2}}\right )-d b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\right )\right )\)	\(130\)
default	\(c^{2} \left (-d a \left (\ln \left (c x \right )+\frac {1}{2 c^{2} x^{2}}\right )-d b \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}+\operatorname {arccosh}\left (c x \right )}{2 c^{2} x^{2}}+\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\right )\right )\)	\(130\)

input

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

output

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

3.1.8.5 Fricas [F]

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

input

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

output

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

3.1.8.6 Sympy [F]

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

input

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

output

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

3.1.8.7 Maxima [F]

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

input

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

output

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

3.1.8.8 Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x^3} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

3.1.8.9 Mupad [F(-1)]

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

input

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

output

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