\(\int \frac {\text {arccosh}(a+b x)}{\frac {a d}{b}+d x} \, dx\) [184]

Optimal result

Integrand size = 19, antiderivative size = 60 \[ \int \frac {\text {arccosh}(a+b x)}{\frac {a d}{b}+d x} \, dx=-\frac {\text {arccosh}(a+b x)^2}{2 d}+\frac {\text {arccosh}(a+b x) \log \left (1+e^{2 \text {arccosh}(a+b x)}\right )}{d}+\frac {\operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a+b x)}\right )}{2 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.88 \[ \int \frac {\text {arccosh}(a+b x)}{\frac {a d}{b}+d x} \, dx=\frac {\text {arccosh}(a+b x) \left (\text {arccosh}(a+b x)+2 \log \left (1+e^{-2 \text {arccosh}(a+b x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(a+b x)}\right )}{2 d} \] Input:

Integrate[ArcCosh[a + b*x]/((a*d)/b + d*x),x]
 

Output:

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

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.45 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {6411, 27, 6297, 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.

\(\displaystyle \int \frac {\text {arccosh}(a+b x)}{\frac {a d}{b}+d x} \, dx\)

\(\Big \downarrow \) 6411

\(\displaystyle \frac {\int \frac {b \text {arccosh}(a+b x)}{d (a+b x)}d(a+b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\text {arccosh}(a+b x)}{a+b x}d(a+b x)}{d}\)

\(\Big \downarrow \) 6297

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int -i \text {arccosh}(a+b x) \tan (i \text {arccosh}(a+b x))d\text {arccosh}(a+b x)}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \int \text {arccosh}(a+b x) \tan (i \text {arccosh}(a+b x))d\text {arccosh}(a+b x)}{d}\)

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[ArcCosh[a + b*x]/((a*d)/b + d*x),x]
 

Output:

((-I)*((-1/2*I)*ArcCosh[a + b*x]^2 + (2*I)*((ArcCosh[a + b*x]*Log[1 + E^(2 
*ArcCosh[a + b*x])])/2 + PolyLog[2, -a - b*x]/4)))/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 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 6411

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

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.53

method	result	size
derivativedivides	\(\frac {-\frac {b \operatorname {arccosh}\left (b x +a \right )^{2}}{2 d}+\frac {b \,\operatorname {arccosh}\left (b x +a \right ) \ln \left (1+\left (b x +a +\sqrt {b x +a -1}\, \sqrt {b x +a +1}\right )^{2}\right )}{d}+\frac {b \operatorname {polylog}\left (2, -\left (b x +a +\sqrt {b x +a -1}\, \sqrt {b x +a +1}\right )^{2}\right )}{2 d}}{b}\)	\(92\)
default	\(\frac {-\frac {b \operatorname {arccosh}\left (b x +a \right )^{2}}{2 d}+\frac {b \,\operatorname {arccosh}\left (b x +a \right ) \ln \left (1+\left (b x +a +\sqrt {b x +a -1}\, \sqrt {b x +a +1}\right )^{2}\right )}{d}+\frac {b \operatorname {polylog}\left (2, -\left (b x +a +\sqrt {b x +a -1}\, \sqrt {b x +a +1}\right )^{2}\right )}{2 d}}{b}\)	\(92\)

Input:

int(arccosh(b*x+a)/(a*d/b+d*x),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

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

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

Output:

integral(b*arccosh(b*x + a)/(b*d*x + a*d), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(arccosh(b*x + a)/(d*x + a*d/b), x)
 

Giac [F]

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

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

Output:

integrate(arccosh(b*x + a)/(d*x + a*d/b), x)
 

Mupad [F(-1)]

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

int(acosh(a + b*x)/(d*x + (a*d)/b),x)
 

Output:

int(acosh(a + b*x)/(d*x + (a*d)/b), x)
 

Reduce [F]

\[ \int \frac {\text {arccosh}(a+b x)}{\frac {a d}{b}+d x} \, dx=\frac {\left (\int \frac {\mathit {acosh} \left (b x +a \right )}{b x +a}d x \right ) b}{d} \] Input:

int(acosh(b*x+a)/(a*d/b+d*x),x)
 

Output:

(int(acosh(a + b*x)/(a + b*x),x)*b)/d