\(\int \frac {\text {csch}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx\) [26]

Optimal result

Integrand size = 19, antiderivative size = 61 \[ \int \frac {\text {csch}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\frac {\text {csch}^{-1}(a+b x)^2}{2 d}-\frac {\text {csch}^{-1}(a+b x) \log \left (1-e^{2 \text {csch}^{-1}(a+b x)}\right )}{d}-\frac {\operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(a+b x)}\right )}{2 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {\text {csch}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\frac {\text {csch}^{-1}(a+b x)^2-2 \text {csch}^{-1}(a+b x) \log \left (1-e^{2 \text {csch}^{-1}(a+b x)}\right )-\operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(a+b x)}\right )}{2 d} \] Input:

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

Output:

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

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.53 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.20, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {6874, 27, 6836, 6190, 3042, 26, 4199, 25, 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.

\(\displaystyle \int \frac {\text {csch}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx\)

\(\Big \downarrow \) 6874

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6836

\(\displaystyle -\frac {\int (a+b x) \text {arcsinh}\left (\frac {1}{a+b x}\right )d\frac {1}{a+b x}}{d}\)

\(\Big \downarrow \) 6190

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int -i \text {arcsinh}\left (\frac {1}{a+b x}\right ) \tan \left (i \text {arcsinh}\left (\frac {1}{a+b x}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (\frac {1}{a+b x}\right )}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {i \int \text {arcsinh}\left (\frac {1}{a+b x}\right ) \tan \left (i \text {arcsinh}\left (\frac {1}{a+b x}\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (\frac {1}{a+b x}\right )}{d}\)

\(\Big \downarrow \) 4199

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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 4199

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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 
))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In 
tegerQ[4*k] && IGtQ[m, 0]
 

rule 6190

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

rule 6836

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> -Subst[Int[(a + 
b*ArcSinh[x/c])/x, x], x, 1/x] /; FreeQ[{a, b, c}, x]
 

rule 6874

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

Maple [F]

\[\int \frac {\operatorname {arccsch}\left (b x +a \right )}{\frac {a d}{b}+d x}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\text {csch}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\int { \frac {\operatorname {arcsch}\left (b x + a\right )}{d x + \frac {a d}{b}} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {\text {csch}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\frac {b \int \frac {\operatorname {acsch}{\left (a + b x \right )}}{a + b x}\, dx}{d} \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\text {csch}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\int { \frac {\operatorname {arcsch}\left (b x + a\right )}{d x + \frac {a d}{b}} \,d x } \] Input:

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

Output:

-1/4*(2*log(b^2*x^2 + 2*a*b*x + a^2 + 1)*log(b*x + a) + dilog(-b^2*x^2 - 2 
*a*b*x - a^2))/d - 1/2*(log(b*x + a)^2 - 2*log(b*x + a)*log(sqrt(b^2*x^2 + 
 2*a*b*x + a^2 + 1) + 1))/d + integrate((b^2*x + a*b)*log(b*x + a)/(b^2*d* 
x^2 + 2*a*b*d*x + a^2*d + (b^2*d*x^2 + 2*a*b*d*x + a^2*d + d)*sqrt(b^2*x^2 
 + 2*a*b*x + a^2 + 1) + d), x)
 

Giac [F]

\[ \int \frac {\text {csch}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\int { \frac {\operatorname {arcsch}\left (b x + a\right )}{d x + \frac {a d}{b}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {csch}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\int \frac {\mathrm {asinh}\left (\frac {1}{a+b\,x}\right )}{d\,x+\frac {a\,d}{b}} \,d x \] Input:

int(asinh(1/(a + b*x))/(d*x + (a*d)/b),x)
 

Output:

int(asinh(1/(a + b*x))/(d*x + (a*d)/b), x)
 

Reduce [F]

\[ \int \frac {\text {csch}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx=\frac {\left (\int \frac {\mathit {acsch} \left (b x +a \right )}{b x +a}d x \right ) b}{d} \] Input:

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

Output:

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