3.1.22 \(\int \frac {\csc ^{-1}(a+b x)}{x} \, dx\) [22]

3.1.22.1 Optimal result

Integrand size = 10, antiderivative size = 210 \[ \int \frac {\csc ^{-1}(a+b x)}{x} \, dx=\csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-i \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-i \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right ) \]

output

-arccsc(b*x+a)*ln(1-(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))^2)+arccsc(b*x+a)*ln( 
1+I*a*(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/(1-(-a^2+1)^(1/2)))+arccsc(b*x+a)* 
ln(1+I*a*(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/(1+(-a^2+1)^(1/2)))+1/2*I*polyl 
og(2,(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))^2)-I*polylog(2,-I*a*(I/(b*x+a)+(1-1 
/(b*x+a)^2)^(1/2))/(1-(-a^2+1)^(1/2)))-I*polylog(2,-I*a*(I/(b*x+a)+(1-1/(b 
*x+a)^2)^(1/2))/(1+(-a^2+1)^(1/2)))
 

3.1.22.2 Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.79 \[ \int \frac {\csc ^{-1}(a+b x)}{x} \, dx=\frac {1}{8} \left (i \left (\pi -2 \csc ^{-1}(a+b x)\right )^2-32 i \arcsin \left (\frac {\sqrt {\frac {-1+a}{a}}}{\sqrt {2}}\right ) \arctan \left (\frac {(1+a) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )}{\sqrt {1-a^2}}\right )-4 \left (\pi -2 \csc ^{-1}(a+b x)+4 \arcsin \left (\frac {\sqrt {\frac {-1+a}{a}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {i \left (-1+\sqrt {1-a^2}\right ) e^{-i \csc ^{-1}(a+b x)}}{a}\right )-4 \left (\pi -2 \csc ^{-1}(a+b x)-4 \arcsin \left (\frac {\sqrt {\frac {-1+a}{a}}}{\sqrt {2}}\right )\right ) \log \left (1-\frac {i \left (1+\sqrt {1-a^2}\right ) e^{-i \csc ^{-1}(a+b x)}}{a}\right )-8 \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )+4 \left (\pi -2 \csc ^{-1}(a+b x)\right ) \log \left (\frac {b x}{a+b x}\right )+8 \csc ^{-1}(a+b x) \log \left (\frac {b x}{a+b x}\right )+8 i \left (\operatorname {PolyLog}\left (2,-\frac {i \left (-1+\sqrt {1-a^2}\right ) e^{-i \csc ^{-1}(a+b x)}}{a}\right )+\operatorname {PolyLog}\left (2,\frac {i \left (1+\sqrt {1-a^2}\right ) e^{-i \csc ^{-1}(a+b x)}}{a}\right )\right )+4 i \left (\csc ^{-1}(a+b x)^2+\operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )\right )\right ) \]

input

Integrate[ArcCsc[a + b*x]/x,x]
 

output

(I*(Pi - 2*ArcCsc[a + b*x])^2 - (32*I)*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*Ar 
cTan[((1 + a)*Cot[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[1 - a^2]] - 4*(Pi - 2* 
ArcCsc[a + b*x] + 4*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]])*Log[1 + (I*(-1 + Sqr 
t[1 - a^2]))/(a*E^(I*ArcCsc[a + b*x]))] - 4*(Pi - 2*ArcCsc[a + b*x] - 4*Ar 
cSin[Sqrt[(-1 + a)/a]/Sqrt[2]])*Log[1 - (I*(1 + Sqrt[1 - a^2]))/(a*E^(I*Ar 
cCsc[a + b*x]))] - 8*ArcCsc[a + b*x]*Log[1 - E^((2*I)*ArcCsc[a + b*x])] + 
4*(Pi - 2*ArcCsc[a + b*x])*Log[(b*x)/(a + b*x)] + 8*ArcCsc[a + b*x]*Log[(b 
*x)/(a + b*x)] + (8*I)*(PolyLog[2, ((-I)*(-1 + Sqrt[1 - a^2]))/(a*E^(I*Arc 
Csc[a + b*x]))] + PolyLog[2, (I*(1 + Sqrt[1 - a^2]))/(a*E^(I*ArcCsc[a + b* 
x]))]) + (4*I)*(ArcCsc[a + b*x]^2 + PolyLog[2, E^((2*I)*ArcCsc[a + b*x])]) 
)/8
 

3.1.22.3 Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.27, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.600, Rules used = {5782, 25, 5063, 5040, 25, 3042, 25, 4200, 25, 2620, 2715, 2838, 5030, 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.

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

\(\Big \downarrow \) 5782

\(\displaystyle -\int \frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b x}d\csc ^{-1}(a+b x)\)

\(\Big \downarrow \) 25

\(\displaystyle \int -\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b x}d\csc ^{-1}(a+b x)\)

\(\Big \downarrow \) 5063

\(\displaystyle \int \frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{\frac {a}{a+b x}-1}d\csc ^{-1}(a+b x)\)

\(\Big \downarrow \) 5040

\(\displaystyle a \int -\frac {\sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{1-\frac {a}{a+b x}}d\csc ^{-1}(a+b x)-\int (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)d\csc ^{-1}(a+b x)\)

\(\Big \downarrow \) 25

\(\displaystyle -\int (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)d\csc ^{-1}(a+b x)-a \int \frac {\sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{1-\frac {a}{a+b x}}d\csc ^{-1}(a+b x)\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

\(\displaystyle -2 i \int -\frac {e^{2 i \csc ^{-1}(a+b x)} \csc ^{-1}(a+b x)}{1-e^{2 i \csc ^{-1}(a+b x)}}d\csc ^{-1}(a+b x)-a \int \frac {\sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{1-\frac {a}{a+b x}}d\csc ^{-1}(a+b x)+\frac {1}{2} i \csc ^{-1}(a+b x)^2\)

\(\Big \downarrow \) 25

\(\displaystyle 2 i \int \frac {e^{2 i \csc ^{-1}(a+b x)} \csc ^{-1}(a+b x)}{1-e^{2 i \csc ^{-1}(a+b x)}}d\csc ^{-1}(a+b x)-a \int \frac {\sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{1-\frac {a}{a+b x}}d\csc ^{-1}(a+b x)+\frac {1}{2} i \csc ^{-1}(a+b x)^2\)

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 5030

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle -a \left (-\frac {i \int e^{-i \csc ^{-1}(a+b x)} \log \left (\frac {i e^{i \csc ^{-1}(a+b x)} a}{1-\sqrt {1-a^2}}+1\right )de^{i \csc ^{-1}(a+b x)}}{a}-\frac {i \int e^{-i \csc ^{-1}(a+b x)} \log \left (\frac {i e^{i \csc ^{-1}(a+b x)} a}{\sqrt {1-a^2}+1}+1\right )de^{i \csc ^{-1}(a+b x)}}{a}-\frac {\csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a}-\frac {\csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a}+\frac {i \csc ^{-1}(a+b x)^2}{2 a}\right )+2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )+\frac {1}{2} i \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )\right )+\frac {1}{2} i \csc ^{-1}(a+b x)^2\)

\(\Big \downarrow \) 2838

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

input

Int[ArcCsc[a + b*x]/x,x]
 

output

(I/2)*ArcCsc[a + b*x]^2 - a*(((I/2)*ArcCsc[a + b*x]^2)/a - (ArcCsc[a + b*x 
]*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])])/a - (ArcCsc[a 
+ b*x]*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])])/a + (I*Po 
lyLog[2, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])])/a + (I*PolyL 
og[2, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])])/a) + (2*I)*((I/ 
2)*ArcCsc[a + b*x]*Log[1 - E^((2*I)*ArcCsc[a + b*x])] + PolyLog[2, E^((2*I 
)*ArcCsc[a + b*x])]/4)
 

3.1.22.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] 
, x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
 

rule 5030

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[ 
(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1 
))), x] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] - I*b*E^( 
I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] 
 - I*b*E^(I*(c + d*x)))), x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] 
 && PosQ[a^2 - b^2]
 

rule 5040

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

rule 5063

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.)*(G_)[(c_.) + 
 (d_.)*(x_)]^(p_.))/(Csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> In 
t[(e + f*x)^m*Sin[c + d*x]*F[c + d*x]^n*(G[c + d*x]^p/(b + a*Sin[c + d*x])) 
, x] /; FreeQ[{a, b, c, d, e, f}, x] && TrigQ[F] && TrigQ[G] && IntegersQ[m 
, n, p]
 

rule 5782

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

3.1.22.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 606 vs. \(2 (269 ) = 538\).

Time = 1.87 (sec) , antiderivative size = 607, normalized size of antiderivative = 2.89

method	result	size
derivativedivides	\(-\frac {i a^{2} \operatorname {dilog}\left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}-\frac {i a^{2} \operatorname {dilog}\left (\frac {\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}-i}{-i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}-\frac {\operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}-\frac {\operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}-i}{-i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}+\frac {a^{2} \operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}-i \operatorname {dilog}\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+i \operatorname {dilog}\left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\frac {a^{2} \operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}-i}{-i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}-\operatorname {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\frac {i \operatorname {dilog}\left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}+\frac {i \operatorname {dilog}\left (\frac {\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}-i}{-i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}\)	\(607\)
default	\(-\frac {i a^{2} \operatorname {dilog}\left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}-\frac {i a^{2} \operatorname {dilog}\left (\frac {\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}-i}{-i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}-\frac {\operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}-\frac {\operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}-i}{-i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}+\frac {a^{2} \operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}-i \operatorname {dilog}\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+i \operatorname {dilog}\left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\frac {a^{2} \operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}-i}{-i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}-\operatorname {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+\frac {i \operatorname {dilog}\left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}+\frac {i \operatorname {dilog}\left (\frac {\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}-i}{-i+\sqrt {a^{2}-1}}\right )}{a^{2}-1}\)	\(607\)

input

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

output

-I*a^2/(a^2-1)*dilog((-(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))*a+(a^2-1)^(1/2)+I 
)/(I+(a^2-1)^(1/2)))-I*a^2/(a^2-1)*dilog(((I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2) 
)*a+(a^2-1)^(1/2)-I)/(-I+(a^2-1)^(1/2)))-arccsc(b*x+a)/(a^2-1)*ln((-(I/(b* 
x+a)+(1-1/(b*x+a)^2)^(1/2))*a+(a^2-1)^(1/2)+I)/(I+(a^2-1)^(1/2)))-arccsc(b 
*x+a)/(a^2-1)*ln(((I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))*a+(a^2-1)^(1/2)-I)/(-I 
+(a^2-1)^(1/2)))+a^2*arccsc(b*x+a)/(a^2-1)*ln((-(I/(b*x+a)+(1-1/(b*x+a)^2) 
^(1/2))*a+(a^2-1)^(1/2)+I)/(I+(a^2-1)^(1/2)))-I*dilog(I/(b*x+a)+(1-1/(b*x+ 
a)^2)^(1/2))+I*dilog(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))+a^2*arccsc(b*x+a)/ 
(a^2-1)*ln(((I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))*a+(a^2-1)^(1/2)-I)/(-I+(a^2- 
1)^(1/2)))-arccsc(b*x+a)*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))+I/(a^2-1)*d 
ilog((-(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))*a+(a^2-1)^(1/2)+I)/(I+(a^2-1)^(1/ 
2)))+I/(a^2-1)*dilog(((I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))*a+(a^2-1)^(1/2)-I) 
/(-I+(a^2-1)^(1/2)))
 

3.1.22.5 Fricas [F]

\[ \int \frac {\csc ^{-1}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )}{x} \,d x } \]

input

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

output

integral(arccsc(b*x + a)/x, x)
 

3.1.22.6 Sympy [F]

\[ \int \frac {\csc ^{-1}(a+b x)}{x} \, dx=\int \frac {\operatorname {acsc}{\left (a + b x \right )}}{x}\, dx \]

input

integrate(acsc(b*x+a)/x,x)
 

output

Integral(acsc(a + b*x)/x, x)
 

3.1.22.7 Maxima [F]

\[ \int \frac {\csc ^{-1}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )}{x} \,d x } \]

input

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

output

integrate(arccsc(b*x + a)/x, x)
 

3.1.22.8 Giac [F]

\[ \int \frac {\csc ^{-1}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )}{x} \,d x } \]

input

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

output

integrate(arccsc(b*x + a)/x, x)
 

3.1.22.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^{-1}(a+b x)}{x} \, dx=\int \frac {\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}{x} \,d x \]

input

int(asin(1/(a + b*x))/x,x)
 

output

int(asin(1/(a + b*x))/x, x)