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

Optimal result

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

[Out]

-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*pol
ylog(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)))

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5367, 4648, 4625, 3798, 2221, 2317, 2438, 4615} \[ \int \frac {\csc ^{-1}(a+b x)}{x} \, dx=-i \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-i \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+\csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )-\csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right ) \]

[In]

Int[ArcCsc[a + b*x]/x,x]

[Out]

ArcCsc[a + b*x]*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])] + ArcCsc[a + b*x]*Log[1 + (I*a*E^(I*A
rcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] - ArcCsc[a + b*x]*Log[1 - E^((2*I)*ArcCsc[a + b*x])] - I*PolyLog[2, ((-I
)*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])] - I*PolyLog[2, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^
2])] + (I/2)*PolyLog[2, E^((2*I)*ArcCsc[a + b*x])]

Rule 2221

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]
- Dist[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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[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 4615

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 4625

Int[(Cot[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbo
l] :> Dist[1/a, Int[(e + f*x)^m*Cot[c + d*x]^n, x], x] - Dist[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 4648

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.)*(G_)[(c_.) + (d_.)*(x_)]^(p_.))/(Csc[(c_.) + (d
_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Int[(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 5367

Int[((a_.) + ArcCsc[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[-(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]

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x \cot (x) \csc (x)}{-a+\csc (x)} \, dx,x,\csc ^{-1}(a+b x)\right ) \\ & = -\text {Subst}\left (\int \frac {x \cot (x)}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right ) \\ & = -\left (a \text {Subst}\left (\int \frac {x \cos (x)}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right )\right )-\text {Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right ) \\ & = 2 i \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(a+b x)\right )-a \text {Subst}\left (\int \frac {e^{i x} x}{1-\sqrt {1-a^2}+i a e^{i x}} \, dx,x,\csc ^{-1}(a+b x)\right )-a \text {Subst}\left (\int \frac {e^{i x} x}{1+\sqrt {1-a^2}+i a e^{i x}} \, dx,x,\csc ^{-1}(a+b x)\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+\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 )-\text {Subst}\left (\int \log \left (1+\frac {i a e^{i x}}{1-\sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )-\text {Subst}\left (\int \log \left (1+\frac {i a e^{i x}}{1+\sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )+\text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\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+\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 )-\frac {1}{2} i \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(a+b x)}\right )+i \text {Subst}\left (\int \frac {\log \left (1+\frac {i a x}{1-\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )+i \text {Subst}\left (\int \frac {\log \left (1+\frac {i a x}{1+\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\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+\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 ) \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[ArcCsc[a + b*x]/x,x]

[Out]

(I*(Pi - 2*ArcCsc[a + b*x])^2 - (32*I)*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]]*ArcTan[((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 + S
qrt[1 - a^2]))/(a*E^(I*ArcCsc[a + b*x]))] - 4*(Pi - 2*ArcCsc[a + b*x] - 4*ArcSin[Sqrt[(-1 + a)/a]/Sqrt[2]])*Lo
g[1 - (I*(1 + Sqrt[1 - a^2]))/(a*E^(I*ArcCsc[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*ArcCsc[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

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

[In]

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

[Out]

-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)*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)))-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)*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^2-1)*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)))

Fricas [F]

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

[In]

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

[Out]

integral(arccsc(b*x + a)/x, x)

Sympy [F]

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

[In]

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

[Out]

Integral(acsc(a + b*x)/x, x)

Maxima [F]

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

[In]

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

[Out]

integrate(arccsc(b*x + a)/x, x)

Giac [F]

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

[In]

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

[Out]

integrate(arccsc(b*x + a)/x, x)

Mupad [F(-1)]

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

[In]

int(asin(1/(a + b*x))/x,x)

[Out]

int(asin(1/(a + b*x))/x, x)