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

Optimal. Leaf size=254 \[ -\frac{2 b \text{PolyLog}\left (2,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}+\frac{2 b \text{PolyLog}\left (2,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}-\frac{2 i b \csc ^{-1}(a+b x) \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}+\frac{2 i b \csc ^{-1}(a+b x) \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}-\frac{b \csc ^{-1}(a+b x)^2}{a}-\frac{\csc ^{-1}(a+b x)^2}{x} \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.415623, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {5259, 4427, 4191, 3323, 2264, 2190, 2279, 2391} \[ -\frac{2 b \text{PolyLog}\left (2,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}+\frac{2 b \text{PolyLog}\left (2,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}-\frac{2 i b \csc ^{-1}(a+b x) \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}+\frac{2 i b \csc ^{-1}(a+b x) \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}-\frac{b \csc ^{-1}(a+b x)^2}{a}-\frac{\csc ^{-1}(a+b x)^2}{x} \]

Antiderivative was successfully verified.

[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 5259

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]

Rule 4427

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 4191

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 3323

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 2264

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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]

Rubi steps

\begin{align*} \int \frac{\csc ^{-1}(a+b x)^2}{x^2} \, dx &=-\left (b \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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 \text{Li}_2\left (-\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{Li}_2\left (-\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]  time = 3.09822, size = 802, normalized size = 3.16 \[ -\frac{b \left (\frac{(a+b x) \csc ^{-1}(a+b x)^2}{b x}+\frac{2 \pi \tan ^{-1}\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 \cos ^{-1}\left (\frac{1}{a}\right ) \tanh ^{-1}\left (\frac{(a+1) \cot \left (\frac{1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )}{\sqrt{a^2-1}}\right )+\left (\pi -2 \csc ^{-1}(a+b x)\right ) \tanh ^{-1}\left (\frac{(a-1) \tan \left (\frac{1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )}{\sqrt{a^2-1}}\right )+\left (\cos ^{-1}\left (\frac{1}{a}\right )+2 i \left (\tanh ^{-1}\left (\frac{(a-1) \tan \left (\frac{1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )}{\sqrt{a^2-1}}\right )-\tanh ^{-1}\left (\frac{(a+1) \cot \left (\frac{1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )}{\sqrt{a^2-1}}\right )\right )\right ) \log \left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{a^2-1} e^{-\frac{1}{2} i \csc ^{-1}(a+b x)}}{\sqrt{a} \sqrt{-\frac{b x}{a+b x}}}\right )+\left (\cos ^{-1}\left (\frac{1}{a}\right )+2 i \tanh ^{-1}\left (\frac{(a+1) \cot \left (\frac{1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )}{\sqrt{a^2-1}}\right )-2 i \tanh ^{-1}\left (\frac{(a-1) \tan \left (\frac{1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )}{\sqrt{a^2-1}}\right )\right ) \log \left (\frac{\left (\frac{1}{2}-\frac{i}{2}\right ) \sqrt{a^2-1} e^{\frac{1}{2} i \csc ^{-1}(a+b x)}}{\sqrt{a} \sqrt{-\frac{b x}{a+b x}}}\right )-\left (\cos ^{-1}\left (\frac{1}{a}\right )-2 i \tanh ^{-1}\left (\frac{(a+1) \cot \left (\frac{1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )}{\sqrt{a^2-1}}\right )\right ) \log \left (\frac{(a-1) \left (i a+\sqrt{a^2-1}+i\right ) \left (\cot \left (\frac{1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )-i\right )}{a \left (a+\sqrt{a^2-1} \cot \left (\frac{1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )-1\right )}\right )-\left (\cos ^{-1}\left (\frac{1}{a}\right )+2 i \tanh ^{-1}\left (\frac{(a+1) \cot \left (\frac{1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )}{\sqrt{a^2-1}}\right )\right ) \log \left (\frac{(a-1) \left (-i a+\sqrt{a^2-1}-i\right ) \left (\cot \left (\frac{1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )+i\right )}{a \left (a+\sqrt{a^2-1} \cot \left (\frac{1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )-1\right )}\right )+i \left (\text{PolyLog}\left (2,\frac{\left (i \sqrt{a^2-1}+1\right ) \left (-a+\sqrt{a^2-1} \cot \left (\frac{1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )+1\right )}{a \left (a+\sqrt{a^2-1} \cot \left (\frac{1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )-1\right )}\right )-\text{PolyLog}\left (2,\frac{\left (1-i \sqrt{a^2-1}\right ) \left (-a+\sqrt{a^2-1} \cot \left (\frac{1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )+1\right )}{a \left (a+\sqrt{a^2-1} \cot \left (\frac{1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )-1\right )}\right )\right )\right )}{\sqrt{a^2-1}}\right )}{a} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.493, size = 307, normalized size = 1.2 \begin{align*} -{\frac{b \left ({\rm arccsc} \left (bx+a\right ) \right ) ^{2}}{a}}-{\frac{ \left ({\rm arccsc} \left (bx+a\right ) \right ) ^{2}}{x}}+2\,{\frac{b{\rm arccsc} \left (bx+a\right )}{a\sqrt{{a}^{2}-1}}\ln \left ({\frac{1}{i-\sqrt{{a}^{2}-1}} \left ( -a \left ({\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) +i-\sqrt{{a}^{2}-1} \right ) } \right ) }-2\,{\frac{b{\rm arccsc} \left (bx+a\right )}{a\sqrt{{a}^{2}-1}}\ln \left ({\frac{1}{i+\sqrt{{a}^{2}-1}} \left ( -a \left ({\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) +\sqrt{{a}^{2}-1}+i \right ) } \right ) }+{\frac{2\,ib}{a}{\it dilog} \left ({ \left ( -a \left ({\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) +\sqrt{{a}^{2}-1}+i \right ) \left ( i+\sqrt{{a}^{2}-1} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}-1}}}}-{\frac{2\,ib}{a}{\it dilog} \left ({ \left ( -a \left ({\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) +i-\sqrt{{a}^{2}-1} \right ) \left ( i-\sqrt{{a}^{2}-1} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arccsc}\left (b x + a\right )^{2}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccsc(b*x + a)^2/x^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acsc}^{2}{\left (a + b x \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acsc(a + b*x)**2/x**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccsc}\left (b x + a\right )^{2}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccsc(b*x + a)^2/x^2, x)