\(\int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx\) [32]

Optimal result

Integrand size = 12, antiderivative size = 254 \[ \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx=-\frac {b \csc ^{-1}(a+b x)^2}{a}-\frac {\csc ^{-1}(a+b x)^2}{x}-\frac {2 i b \csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {2 i b \csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {2 b \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {2 b \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5367, 4512, 4276, 3404, 2296, 2221, 2317, 2438} \[ \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx=-\frac {2 b \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {2 b \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}-\frac {2 i b \csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {2 i b \csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )}{a \sqrt {1-a^2}}-\frac {b \csc ^{-1}(a+b x)^2}{a}-\frac {\csc ^{-1}(a+b x)^2}{x} \]

[In]

Int[ArcCsc[a + b*x]^2/x^2,x]

[Out]

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

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 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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 3404

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

Rule 4276

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

Rule 4512

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

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}& = -\left (b \text {Subst}\left (\int \frac {x^2 \cot (x) \csc (x)}{(-a+\csc (x))^2} \, dx,x,\csc ^{-1}(a+b x)\right )\right ) \\ & = -\frac {\csc ^{-1}(a+b x)^2}{x}+(2 b) \text {Subst}\left (\int \frac {x}{-a+\csc (x)} \, dx,x,\csc ^{-1}(a+b x)\right ) \\ & = -\frac {\csc ^{-1}(a+b x)^2}{x}+(2 b) \text {Subst}\left (\int \left (-\frac {x}{a}+\frac {x}{a (1-a \sin (x))}\right ) \, dx,x,\csc ^{-1}(a+b x)\right ) \\ & = -\frac {b \csc ^{-1}(a+b x)^2}{a}-\frac {\csc ^{-1}(a+b x)^2}{x}+\frac {(2 b) \text {Subst}\left (\int \frac {x}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right )}{a} \\ & = -\frac {b \csc ^{-1}(a+b x)^2}{a}-\frac {\csc ^{-1}(a+b x)^2}{x}+\frac {(4 b) \text {Subst}\left (\int \frac {e^{i x} x}{-i a+2 e^{i x}+i a e^{2 i x}} \, dx,x,\csc ^{-1}(a+b x)\right )}{a} \\ & = -\frac {b \csc ^{-1}(a+b x)^2}{a}-\frac {\csc ^{-1}(a+b x)^2}{x}+\frac {(4 i b) \text {Subst}\left (\int \frac {e^{i x} x}{2-2 \sqrt {1-a^2}+2 i a e^{i x}} \, dx,x,\csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}-\frac {(4 i b) \text {Subst}\left (\int \frac {e^{i x} x}{2+2 \sqrt {1-a^2}+2 i a e^{i x}} \, dx,x,\csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}} \\ & = -\frac {b \csc ^{-1}(a+b x)^2}{a}-\frac {\csc ^{-1}(a+b x)^2}{x}-\frac {2 i b \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 \sqrt {1-a^2}}+\frac {2 i b \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 \sqrt {1-a^2}}+\frac {(2 i b) \text {Subst}\left (\int \log \left (1+\frac {2 i a e^{i x}}{2-2 \sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{a \sqrt {1-a^2}}-\frac {(2 i b) \text {Subst}\left (\int \log \left (1+\frac {2 i a e^{i x}}{2+2 \sqrt {1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{a \sqrt {1-a^2}} \\ & = -\frac {b \csc ^{-1}(a+b x)^2}{a}-\frac {\csc ^{-1}(a+b x)^2}{x}-\frac {2 i b \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 \sqrt {1-a^2}}+\frac {2 i b \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 \sqrt {1-a^2}}+\frac {(2 b) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 i a x}{2-2 \sqrt {1-a^2}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{a \sqrt {1-a^2}}-\frac {(2 b) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 i a x}{2+2 \sqrt {1-a^2}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{a \sqrt {1-a^2}} \\ & = -\frac {b \csc ^{-1}(a+b x)^2}{a}-\frac {\csc ^{-1}(a+b x)^2}{x}-\frac {2 i b \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 \sqrt {1-a^2}}+\frac {2 i b \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 \sqrt {1-a^2}}-\frac {2 b \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {2 b \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(802\) vs. \(2(254)=508\).

Time = 2.49 (sec) , antiderivative size = 802, normalized size of antiderivative = 3.16 \[ \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx=-\frac {b \left (\frac {(a+b x) \csc ^{-1}(a+b x)^2}{b x}+\frac {2 \pi \arctan \left (\frac {a-\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )}{\sqrt {1-a^2}}+\frac {2 \left (-2 \arccos \left (\frac {1}{a}\right ) \text {arctanh}\left (\frac {(1+a) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )}{\sqrt {-1+a^2}}\right )+\left (\pi -2 \csc ^{-1}(a+b x)\right ) \text {arctanh}\left (\frac {(-1+a) \tan \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )}{\sqrt {-1+a^2}}\right )+\left (\arccos \left (\frac {1}{a}\right )+2 i \left (-\text {arctanh}\left (\frac {(1+a) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )}{\sqrt {-1+a^2}}\right )+\text {arctanh}\left (\frac {(-1+a) \tan \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )}{\sqrt {-1+a^2}}\right )\right )\right ) \log \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-1+a^2} e^{-\frac {1}{2} i \csc ^{-1}(a+b x)}}{\sqrt {a} \sqrt {-\frac {b x}{a+b x}}}\right )+\left (\arccos \left (\frac {1}{a}\right )+2 i \text {arctanh}\left (\frac {(1+a) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )}{\sqrt {-1+a^2}}\right )-2 i \text {arctanh}\left (\frac {(-1+a) \tan \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )}{\sqrt {-1+a^2}}\right )\right ) \log \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {-1+a^2} e^{\frac {1}{2} i \csc ^{-1}(a+b x)}}{\sqrt {a} \sqrt {-\frac {b x}{a+b x}}}\right )-\left (\arccos \left (\frac {1}{a}\right )-2 i \text {arctanh}\left (\frac {(1+a) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )}{\sqrt {-1+a^2}}\right )\right ) \log \left (\frac {(-1+a) \left (i+i a+\sqrt {-1+a^2}\right ) \left (-i+\cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )\right )}{a \left (-1+a+\sqrt {-1+a^2} \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )\right )}\right )-\left (\arccos \left (\frac {1}{a}\right )+2 i \text {arctanh}\left (\frac {(1+a) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )}{\sqrt {-1+a^2}}\right )\right ) \log \left (\frac {(-1+a) \left (-i-i a+\sqrt {-1+a^2}\right ) \left (i+\cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )\right )}{a \left (-1+a+\sqrt {-1+a^2} \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )\right )}\right )+i \left (-\operatorname {PolyLog}\left (2,\frac {\left (1-i \sqrt {-1+a^2}\right ) \left (1-a+\sqrt {-1+a^2} \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )\right )}{a \left (-1+a+\sqrt {-1+a^2} \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )\right )}\right )+\operatorname {PolyLog}\left (2,\frac {\left (1+i \sqrt {-1+a^2}\right ) \left (1-a+\sqrt {-1+a^2} \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )\right )}{a \left (-1+a+\sqrt {-1+a^2} \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(a+b x)\right )\right )\right )}\right )\right )\right )}{\sqrt {-1+a^2}}\right )}{a} \]

[In]

Integrate[ArcCsc[a + b*x]^2/x^2,x]

[Out]

-((b*(((a + b*x)*ArcCsc[a + b*x]^2)/(b*x) + (2*Pi*ArcTan[(a - Tan[ArcCsc[a + b*x]/2])/Sqrt[1 - a^2]])/Sqrt[1 -
a^2] + (2*(-2*ArcCos[a^(-1)]*ArcTanh[((1 + a)*Cot[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]] + (Pi - 2*ArcC
sc[a + b*x])*ArcTanh[((-1 + a)*Tan[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]] + (ArcCos[a^(-1)] + (2*I)*(-Ar
cTanh[((1 + a)*Cot[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]] + ArcTanh[((-1 + a)*Tan[(Pi + 2*ArcCsc[a + b*x
])/4])/Sqrt[-1 + a^2]]))*Log[((1/2 + I/2)*Sqrt[-1 + a^2])/(Sqrt[a]*E^((I/2)*ArcCsc[a + b*x])*Sqrt[-((b*x)/(a +
b*x))])] + (ArcCos[a^(-1)] + (2*I)*ArcTanh[((1 + a)*Cot[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]] - (2*I)*
ArcTanh[((-1 + a)*Tan[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]])*Log[((1/2 - I/2)*Sqrt[-1 + a^2]*E^((I/2)*A
rcCsc[a + b*x]))/(Sqrt[a]*Sqrt[-((b*x)/(a + b*x))])] - (ArcCos[a^(-1)] - (2*I)*ArcTanh[((1 + a)*Cot[(Pi + 2*Ar
cCsc[a + b*x])/4])/Sqrt[-1 + a^2]])*Log[((-1 + a)*(I + I*a + Sqrt[-1 + a^2])*(-I + Cot[(Pi + 2*ArcCsc[a + b*x]
)/4]))/(a*(-1 + a + Sqrt[-1 + a^2]*Cot[(Pi + 2*ArcCsc[a + b*x])/4]))] - (ArcCos[a^(-1)] + (2*I)*ArcTanh[((1 +
a)*Cot[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]])*Log[((-1 + a)*(-I - I*a + Sqrt[-1 + a^2])*(I + Cot[(Pi +
2*ArcCsc[a + b*x])/4]))/(a*(-1 + a + Sqrt[-1 + a^2]*Cot[(Pi + 2*ArcCsc[a + b*x])/4]))] + I*(-PolyLog[2, ((1 -
I*Sqrt[-1 + a^2])*(1 - a + Sqrt[-1 + a^2]*Cot[(Pi + 2*ArcCsc[a + b*x])/4]))/(a*(-1 + a + Sqrt[-1 + a^2]*Cot[(P
i + 2*ArcCsc[a + b*x])/4]))] + PolyLog[2, ((1 + I*Sqrt[-1 + a^2])*(1 - a + Sqrt[-1 + a^2]*Cot[(Pi + 2*ArcCsc[a
+ b*x])/4]))/(a*(-1 + a + Sqrt[-1 + a^2]*Cot[(Pi + 2*ArcCsc[a + b*x])/4]))])))/Sqrt[-1 + a^2]))/a)

Maple [A] (verified)

Time = 1.64 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.19

method	result	size
derivativedivides	\(b \left (-\frac {\left (b x +a \right ) \operatorname {arccsc}\left (b x +a \right )^{2}}{a b x}-\frac {2 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}+\frac {2 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +i-\sqrt {a^{2}-1}}{i-\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}+\frac {2 i \operatorname {dilog}\left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}-\frac {2 i \operatorname {dilog}\left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +i-\sqrt {a^{2}-1}}{i-\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}\right )\)	\(302\)
default	\(b \left (-\frac {\left (b x +a \right ) \operatorname {arccsc}\left (b x +a \right )^{2}}{a b x}-\frac {2 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}+\frac {2 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +i-\sqrt {a^{2}-1}}{i-\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}+\frac {2 i \operatorname {dilog}\left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +\sqrt {a^{2}-1}+i}{i+\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}-\frac {2 i \operatorname {dilog}\left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +i-\sqrt {a^{2}-1}}{i-\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}\right )\)	\(302\)

[In]

int(arccsc(b*x+a)^2/x^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

integrate(arccsc(b*x+a)^2/x^2,x, algorithm="fricas")

[Out]

integral(arccsc(b*x + a)^2/x^2, x)

Sympy [F]

\[ \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx=\int \frac {\operatorname {acsc}^{2}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

integrate(acsc(b*x+a)**2/x**2,x)

[Out]

Integral(acsc(a + b*x)**2/x**2, x)

Maxima [F]

\[ \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

integrate(arccsc(b*x+a)^2/x^2,x, algorithm="maxima")

[Out]

-1/4*(4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^2 + 4*x*integrate((2*sqrt(b*x + a + 1)*sqrt(b*x + a -
1)*b*x*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + (b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2 - 1)*b*x - a)*l
og(b*x + a)^2 + (b^3*x^3 + 2*a*b^2*x^2 + (a^2 - 1)*b*x - (b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2 - 1)*b*x - a)*l
og(b*x + a))*log(b^2*x^2 + 2*a*b*x + a^2))/(b^3*x^5 + 3*a*b^2*x^4 + (3*a^2 - 1)*b*x^3 + (a^3 - a)*x^2), x) - l
og(b^2*x^2 + 2*a*b*x + a^2)^2)/x

Giac [F]

\[ \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx=\int { \frac {\operatorname {arccsc}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

integrate(arccsc(b*x+a)^2/x^2,x, algorithm="giac")

[Out]

integrate(arccsc(b*x + a)^2/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx=\int \frac {{\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^2}{x^2} \,d x \]

[In]

int(asin(1/(a + b*x))^2/x^2,x)

[Out]

int(asin(1/(a + b*x))^2/x^2, x)