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

Optimal. Leaf size=324 \[ -2 i \csc ^{-1}(a+b x) \text{PolyLog}\left (2,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )-2 i \csc ^{-1}(a+b x) \text{PolyLog}\left (2,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )+2 \text{PolyLog}\left (3,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+2 \text{PolyLog}\left (3,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )+i \csc ^{-1}(a+b x) \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{2 i \csc ^{-1}(a+b x)}\right )+\csc ^{-1}(a+b x)^2 \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+\csc ^{-1}(a+b x)^2 \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )-\csc ^{-1}(a+b x)^2 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right ) \]

[Out]

ArcCsc[a + b*x]^2*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])] + ArcCsc[a + b*x]^2*Log[1 + (I*a*E^
(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] - ArcCsc[a + b*x]^2*Log[1 - E^((2*I)*ArcCsc[a + b*x])] - (2*I)*ArcCs
c[a + b*x]*PolyLog[2, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])] - (2*I)*ArcCsc[a + b*x]*PolyLog[2, (
(-I)*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] + I*ArcCsc[a + b*x]*PolyLog[2, E^((2*I)*ArcCsc[a + b*x])] +
2*PolyLog[3, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])] + 2*PolyLog[3, ((-I)*a*E^(I*ArcCsc[a + b*x])
)/(1 + Sqrt[1 - a^2])] - PolyLog[3, E^((2*I)*ArcCsc[a + b*x])]/2

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Rubi [A]  time = 0.493471, antiderivative size = 324, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {5259, 4552, 4529, 3717, 2190, 2531, 2282, 6589, 4519} \[ -2 i \csc ^{-1}(a+b x) \text{PolyLog}\left (2,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )-2 i \csc ^{-1}(a+b x) \text{PolyLog}\left (2,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )+2 \text{PolyLog}\left (3,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+2 \text{PolyLog}\left (3,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )+i \csc ^{-1}(a+b x) \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{2 i \csc ^{-1}(a+b x)}\right )+\csc ^{-1}(a+b x)^2 \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+\csc ^{-1}(a+b x)^2 \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )-\csc ^{-1}(a+b x)^2 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCsc[a + b*x]^2/x,x]

[Out]

ArcCsc[a + b*x]^2*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])] + ArcCsc[a + b*x]^2*Log[1 + (I*a*E^
(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] - ArcCsc[a + b*x]^2*Log[1 - E^((2*I)*ArcCsc[a + b*x])] - (2*I)*ArcCs
c[a + b*x]*PolyLog[2, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])] - (2*I)*ArcCsc[a + b*x]*PolyLog[2, (
(-I)*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] + I*ArcCsc[a + b*x]*PolyLog[2, E^((2*I)*ArcCsc[a + b*x])] +
2*PolyLog[3, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])] + 2*PolyLog[3, ((-I)*a*E^(I*ArcCsc[a + b*x])
)/(1 + Sqrt[1 - a^2])] - PolyLog[3, E^((2*I)*ArcCsc[a + b*x])]/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 4552

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 4529

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 3717

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 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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 4519

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]

Rubi steps

\begin{align*} \int \frac{\csc ^{-1}(a+b x)^2}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2 \cot (x) \csc (x)}{-a+\csc (x)} \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=-\operatorname{Subst}\left (\int \frac{x^2 \cot (x)}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=-\left (a \operatorname{Subst}\left (\int \frac{x^2 \cos (x)}{1-a \sin (x)} \, dx,x,\csc ^{-1}(a+b x)\right )\right )-\operatorname{Subst}\left (\int x^2 \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=2 i \operatorname{Subst}\left (\int \frac{e^{2 i x} x^2}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(a+b x)\right )-a \operatorname{Subst}\left (\int \frac{e^{i x} x^2}{1-\sqrt{1-a^2}+i a e^{i x}} \, dx,x,\csc ^{-1}(a+b x)\right )-a \operatorname{Subst}\left (\int \frac{e^{i x} x^2}{1+\sqrt{1-a^2}+i a e^{i x}} \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=\csc ^{-1}(a+b x)^2 \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+\csc ^{-1}(a+b x)^2 \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )-\csc ^{-1}(a+b x)^2 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-2 \operatorname{Subst}\left (\int x \log \left (1+\frac{i a e^{i x}}{1-\sqrt{1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )-2 \operatorname{Subst}\left (\int x \log \left (1+\frac{i a e^{i x}}{1+\sqrt{1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )+2 \operatorname{Subst}\left (\int x \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=\csc ^{-1}(a+b x)^2 \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+\csc ^{-1}(a+b x)^2 \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )-\csc ^{-1}(a+b x)^2 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-2 i \csc ^{-1}(a+b x) \text{Li}_2\left (-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )-2 i \csc ^{-1}(a+b x) \text{Li}_2\left (-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )+i \csc ^{-1}(a+b x) \text{Li}_2\left (e^{2 i \csc ^{-1}(a+b x)}\right )-i \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )+2 i \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{i a e^{i x}}{1-\sqrt{1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )+2 i \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{i a e^{i x}}{1+\sqrt{1-a^2}}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )\\ &=\csc ^{-1}(a+b x)^2 \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+\csc ^{-1}(a+b x)^2 \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )-\csc ^{-1}(a+b x)^2 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-2 i \csc ^{-1}(a+b x) \text{Li}_2\left (-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )-2 i \csc ^{-1}(a+b x) \text{Li}_2\left (-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )+i \csc ^{-1}(a+b x) \text{Li}_2\left (e^{2 i \csc ^{-1}(a+b x)}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 i \csc ^{-1}(a+b x)}\right )+2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i a x}{-1+\sqrt{1-a^2}}\right )}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )+2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\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)^2 \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+\csc ^{-1}(a+b x)^2 \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )-\csc ^{-1}(a+b x)^2 \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )-2 i \csc ^{-1}(a+b x) \text{Li}_2\left (-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )-2 i \csc ^{-1}(a+b x) \text{Li}_2\left (-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )+i \csc ^{-1}(a+b x) \text{Li}_2\left (e^{2 i \csc ^{-1}(a+b x)}\right )+2 \text{Li}_3\left (-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+2 \text{Li}_3\left (-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )-\frac{1}{2} \text{Li}_3\left (e^{2 i \csc ^{-1}(a+b x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.320781, size = 408, normalized size = 1.26 \[ -2 i \csc ^{-1}(a+b x) \text{PolyLog}\left (2,\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right )-2 i \csc ^{-1}(a+b x) \text{PolyLog}\left (2,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )+2 \text{PolyLog}\left (3,\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right )+2 \text{PolyLog}\left (3,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )-2 i \csc ^{-1}(a+b x) \text{PolyLog}\left (2,e^{-i \csc ^{-1}(a+b x)}\right )+2 i \csc ^{-1}(a+b x) \text{PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )-2 \text{PolyLog}\left (3,e^{-i \csc ^{-1}(a+b x)}\right )-2 \text{PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )+\csc ^{-1}(a+b x)^2 \log \left (1-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right )+\csc ^{-1}(a+b x)^2 \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )-\frac{1}{3} i \csc ^{-1}(a+b x)^3-\csc ^{-1}(a+b x)^2 \log \left (1-e^{-i \csc ^{-1}(a+b x)}\right )-\csc ^{-1}(a+b x)^2 \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )+\frac{i \pi ^3}{6} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCsc[a + b*x]^2/x,x]

[Out]

(I/6)*Pi^3 - (I/3)*ArcCsc[a + b*x]^3 - ArcCsc[a + b*x]^2*Log[1 - E^((-I)*ArcCsc[a + b*x])] - ArcCsc[a + b*x]^2
*Log[1 + E^(I*ArcCsc[a + b*x])] + ArcCsc[a + b*x]^2*Log[1 - (I*a*E^(I*ArcCsc[a + b*x]))/(-1 + Sqrt[1 - a^2])]
+ ArcCsc[a + b*x]^2*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] - (2*I)*ArcCsc[a + b*x]*PolyLog[2
, E^((-I)*ArcCsc[a + b*x])] + (2*I)*ArcCsc[a + b*x]*PolyLog[2, -E^(I*ArcCsc[a + b*x])] - (2*I)*ArcCsc[a + b*x]
*PolyLog[2, (I*a*E^(I*ArcCsc[a + b*x]))/(-1 + Sqrt[1 - a^2])] - (2*I)*ArcCsc[a + b*x]*PolyLog[2, ((-I)*a*E^(I*
ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] - 2*PolyLog[3, E^((-I)*ArcCsc[a + b*x])] - 2*PolyLog[3, -E^(I*ArcCsc[a
+ b*x])] + 2*PolyLog[3, (I*a*E^(I*ArcCsc[a + b*x]))/(-1 + Sqrt[1 - a^2])] + 2*PolyLog[3, ((-I)*a*E^(I*ArcCsc[a
+ b*x]))/(1 + Sqrt[1 - a^2])]

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Maple [F]  time = 0.804, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm arccsc} \left (bx+a\right ) \right ) ^{2}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccsc(b*x + a)^2/x, x)

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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}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccsc(b*x + a)^2/x, 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}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acsc(a + b*x)**2/x, 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}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccsc(b*x + a)^2/x, x)