\(\int \frac {a+b \text {arccosh}(c+d x)}{c e+d e x} \, dx\) [16]

Optimal result

Integrand size = 21, antiderivative size = 81 \[ \int \frac {a+b \text {arccosh}(c+d x)}{c e+d e x} \, dx=-\frac {(a+b \text {arccosh}(c+d x))^2}{2 b d e}+\frac {(a+b \text {arccosh}(c+d x)) \log \left (1+e^{2 \text {arccosh}(c+d x)}\right )}{d e}+\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(c+d x)}\right )}{2 d e} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \text {arccosh}(c+d x)}{c e+d e x} \, dx=\frac {b \text {arccosh}(c+d x)^2+2 b \text {arccosh}(c+d x) \log \left (1+e^{-2 \text {arccosh}(c+d x)}\right )+2 a \log (c+d x)-b \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c+d x)}\right )}{2 d e} \] Input:

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

Output:

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

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.91 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {6411, 27, 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.

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

\(\Big \downarrow \) 6411

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6297

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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 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 = 0.16 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.20

method	result	size
derivativedivides	\(\frac {\frac {a \ln \left (d x +c \right )}{e}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{2}+\operatorname {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e}}{d}\)	\(97\)
default	\(\frac {\frac {a \ln \left (d x +c \right )}{e}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{2}+\operatorname {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e}}{d}\)	\(97\)
parts	\(\frac {a \ln \left (d x +c \right )}{e d}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{2}+\operatorname {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e d}\)	\(99\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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