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

Optimal. Leaf size=210 \[ -i \text{PolyLog}\left (2,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )-i \text{PolyLog}\left (2,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )+\frac{1}{2} i \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )+\csc ^{-1}(a+b x) \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+\csc ^{-1}(a+b x) \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )-\csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right ) \]

[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])]

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Rubi [A]  time = 0.30023, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {5259, 4552, 4529, 3717, 2190, 2279, 2391, 4519} \[ -i \text{PolyLog}\left (2,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )-i \text{PolyLog}\left (2,-\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )+\frac{1}{2} i \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )+\csc ^{-1}(a+b x) \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+\csc ^{-1}(a+b x) \log \left (1+\frac{i a e^{i \csc ^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )-\csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right ) \]

Antiderivative was successfully verified.

[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 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 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]

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

Mathematica [A]  time = 0.411638, size = 375, normalized size = 1.79 \[ \frac{1}{8} \left (8 i \left (\text{PolyLog}\left (2,-\frac{i \left (\sqrt{1-a^2}-1\right ) e^{-i \csc ^{-1}(a+b x)}}{a}\right )+\text{PolyLog}\left (2,\frac{i \left (\sqrt{1-a^2}+1\right ) e^{-i \csc ^{-1}(a+b x)}}{a}\right )\right )+4 i \left (\csc ^{-1}(a+b x)^2+\text{PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )\right )-4 \log \left (1+\frac{i \left (\sqrt{1-a^2}-1\right ) e^{-i \csc ^{-1}(a+b x)}}{a}\right ) \left (-2 \csc ^{-1}(a+b x)+4 \sin ^{-1}\left (\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right )+\pi \right )-4 \log \left (1-\frac{i \left (\sqrt{1-a^2}+1\right ) e^{-i \csc ^{-1}(a+b x)}}{a}\right ) \left (-2 \csc ^{-1}(a+b x)-4 \sin ^{-1}\left (\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right )+\pi \right )-32 i \sin ^{-1}\left (\frac{\sqrt{\frac{a-1}{a}}}{\sqrt{2}}\right ) \tan ^{-1}\left (\frac{(a+1) \cot \left (\frac{1}{4} \left (2 \csc ^{-1}(a+b x)+\pi \right )\right )}{\sqrt{1-a^2}}\right )+i \left (\pi -2 \csc ^{-1}(a+b x)\right )^2+4 \log \left (\frac{b x}{a+b x}\right ) \left (\pi -2 \csc ^{-1}(a+b x)\right )-8 \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )+8 \log \left (\frac{b x}{a+b x}\right ) \csc ^{-1}(a+b x)\right ) \]

Warning: Unable to verify antiderivative.

[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

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Maple [B]  time = 0.439, size = 607, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccsc(b*x + a)/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 )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccsc(b*x + a)/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acsc}{\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)/x,x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccsc(b*x + a)/x, x)