3.1.6 \(\int \frac {(d-c^2 d x^2) (a+b \text {arccosh}(c x))}{x} \, dx\) [6]

3.1.6.1 Optimal result

Integrand size = 23, antiderivative size = 117 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=\frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{4} b d \text {arccosh}(c x)+\frac {1}{2} d \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))+\frac {d (a+b \text {arccosh}(c x))^2}{2 b}+d (a+b \text {arccosh}(c x)) \log \left (1+e^{-2 \text {arccosh}(c x)}\right )-\frac {1}{2} b d \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right ) \]

output

-1/4*b*d*arccosh(c*x)+1/2*d*(-c^2*x^2+1)*(a+b*arccosh(c*x))+1/2*d*(a+b*arc 
cosh(c*x))^2/b+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*d*polylog(2,-1/(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)+1/4*b*c*d* 
x*(c*x-1)^(1/2)*(c*x+1)^(1/2)
 

3.1.6.2 Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.11 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \text {arccosh}(c x))}{x} \, dx=-\frac {1}{2} a c^2 d x^2+\frac {1}{4} b c d x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{2} b c^2 d x^2 \text {arccosh}(c x)+\frac {1}{2} b d \text {arctanh}\left (\frac {\sqrt {-1+c x}}{\sqrt {1+c x}}\right )+a d \log (x)+\frac {1}{2} b d \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1+e^{-2 \text {arccosh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c x)}\right )\right ) \]

input

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

output

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

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

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.20, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {6334, 40, 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.

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

\(\Big \downarrow \) 6334

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

\(\Big \downarrow \) 40

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

\(\Big \downarrow \) 43

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

\(\Big \downarrow \) 6297

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

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

output

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

3.1.6.3.1 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 40

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

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 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 6334

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

3.1.6.4 Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.12

method	result	size
parts	\(-d a \left (\frac {c^{2} x^{2}}{2}-\ln \left (x \right )\right )-\frac {d b \operatorname {arccosh}\left (c x \right )^{2}}{2}-\frac {d b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {b c d x \sqrt {c x -1}\, \sqrt {c x +1}}{4}+\frac {b d \,\operatorname {arccosh}\left (c x \right )}{4}+d b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\)	\(131\)
derivativedivides	\(-d a \left (\frac {c^{2} x^{2}}{2}-\ln \left (c x \right )\right )-\frac {d b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {b c d x \sqrt {c x -1}\, \sqrt {c x +1}}{4}-\frac {d b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {b d \,\operatorname {arccosh}\left (c x \right )}{4}+d b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\)	\(133\)
default	\(-d a \left (\frac {c^{2} x^{2}}{2}-\ln \left (c x \right )\right )-\frac {d b \operatorname {arccosh}\left (c x \right )^{2}}{2}+\frac {b c d x \sqrt {c x -1}\, \sqrt {c x +1}}{4}-\frac {d b \,\operatorname {arccosh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {b d \,\operatorname {arccosh}\left (c x \right )}{4}+d b \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d b \operatorname {polylog}\left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}\)	\(133\)

input

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

output

-d*a*(1/2*c^2*x^2-ln(x))-1/2*d*b*arccosh(c*x)^2-1/2*d*b*arccosh(c*x)*c^2*x 
^2+1/4*b*c*d*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)+1/4*b*d*arccosh(c*x)+d*b*arccos 
h(c*x)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)+1/2*d*b*polylog(2,-(c*x+( 
c*x-1)^(1/2)*(c*x+1)^(1/2))^2)
 

3.1.6.5 Fricas [F]

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

input

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

3.1.6.6 Sympy [F]

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

input

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

output

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

3.1.6.7 Maxima [F]

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

input

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

output

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

3.1.6.8 Giac [F(-2)]

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

input

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

3.1.6.9 Mupad [F(-1)]

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

input

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

output

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