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

Optimal. Leaf size=91 \[ i b \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )-\frac{1}{2} b^2 \text{PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right )+\frac{i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2 \]

[Out]

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

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Rubi [A]  time = 0.123399, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5223, 3717, 2190, 2531, 2282, 6589} \[ i b \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )-\frac{1}{2} b^2 \text{PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right )+\frac{i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsc[c*x])^2/x,x]

[Out]

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

Rule 5223

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b*
x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcCsc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G
tQ[n, 0] || LtQ[m, -1])

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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \csc ^{-1}(c x)\right )^2}{x} \, dx &=-\operatorname{Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac{i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}+2 i \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)^2}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac{i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\left (a+b \csc ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+(2 b) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac{i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\left (a+b \csc ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+i b \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\left (i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac{i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\left (a+b \csc ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+i b \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\frac{1}{2} b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )\\ &=\frac{i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\left (a+b \csc ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+i b \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\frac{1}{2} b^2 \text{Li}_3\left (e^{2 i \csc ^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.14246, size = 137, normalized size = 1.51 \[ i a b \left (\csc ^{-1}(c x)^2+\text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right )+\frac{1}{24} i b^2 \left (-24 \csc ^{-1}(c x) \text{PolyLog}\left (2,e^{-2 i \csc ^{-1}(c x)}\right )+12 i \text{PolyLog}\left (3,e^{-2 i \csc ^{-1}(c x)}\right )-8 \csc ^{-1}(c x)^3+24 i \csc ^{-1}(c x)^2 \log \left (1-e^{-2 i \csc ^{-1}(c x)}\right )+\pi ^3\right )+a^2 \log (c x)-2 a b \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCsc[c*x])^2/x,x]

[Out]

-2*a*b*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] + a^2*Log[c*x] + I*a*b*(ArcCsc[c*x]^2 + PolyLog[2, E^((2*I)*
ArcCsc[c*x])]) + (I/24)*b^2*(Pi^3 - 8*ArcCsc[c*x]^3 + (24*I)*ArcCsc[c*x]^2*Log[1 - E^((-2*I)*ArcCsc[c*x])] - 2
4*ArcCsc[c*x]*PolyLog[2, E^((-2*I)*ArcCsc[c*x])] + (12*I)*PolyLog[3, E^((-2*I)*ArcCsc[c*x])])

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Maple [B]  time = 0.295, size = 361, normalized size = 4. \begin{align*}{a}^{2}\ln \left ( cx \right ) +{\frac{i}{3}}{b}^{2} \left ({\rm arccsc} \left (cx\right ) \right ) ^{3}-{b}^{2} \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}\ln \left ( 1-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) +2\,i{b}^{2}{\rm arccsc} \left (cx\right ){\it polylog} \left ( 2,{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) -2\,{b}^{2}{\it polylog} \left ( 3,{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) -{b}^{2} \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}\ln \left ( 1+{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) +2\,i{b}^{2}{\rm arccsc} \left (cx\right ){\it polylog} \left ( 2,{\frac{-i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) -2\,{b}^{2}{\it polylog} \left ( 3,{\frac{-i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) +iab \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}+2\,iab{\it polylog} \left ( 2,{\frac{-i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) -2\,ab{\rm arccsc} \left (cx\right )\ln \left ( 1-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) -2\,ab{\rm arccsc} \left (cx\right )\ln \left ( 1+{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) +2\,iab{\it polylog} \left ( 2,{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*ln(c*x)+1/3*I*b^2*arccsc(c*x)^3-b^2*arccsc(c*x)^2*ln(1-I/c/x-(1-1/c^2/x^2)^(1/2))+2*I*b^2*arccsc(c*x)*poly
log(2,I/c/x+(1-1/c^2/x^2)^(1/2))-2*b^2*polylog(3,I/c/x+(1-1/c^2/x^2)^(1/2))-b^2*arccsc(c*x)^2*ln(1+I/c/x+(1-1/
c^2/x^2)^(1/2))+2*I*b^2*arccsc(c*x)*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))-2*b^2*polylog(3,-I/c/x-(1-1/c^2/x^2)
^(1/2))+I*a*b*arccsc(c*x)^2+2*I*a*b*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))-2*a*b*arccsc(c*x)*ln(1-I/c/x-(1-1/c^
2/x^2)^(1/2))-2*a*b*arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))+2*I*a*b*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, b^{2} c^{2}{\left (\frac{\log \left (c x + 1\right )}{c^{2}} + \frac{\log \left (c x - 1\right )}{c^{2}}\right )} \log \left (c\right )^{2} + b^{2} c^{2} \int \frac{x^{2} \log \left (c^{2} x^{2}\right )}{c^{2} x^{3} - x}\,{d x} \log \left (c\right ) - 2 \, b^{2} c^{2} \int \frac{x^{2} \log \left (x\right )}{c^{2} x^{3} - x}\,{d x} \log \left (c\right ) + 2 \, b^{2} c^{2} \int \frac{x^{2} \log \left (c^{2} x^{2}\right ) \log \left (x\right )}{c^{2} x^{3} - x}\,{d x} - b^{2} c^{2} \int \frac{x^{2} \log \left (x\right )^{2}}{c^{2} x^{3} - x}\,{d x} + 2 \, a b c^{2} \int \frac{x^{2} \arctan \left (\frac{1}{\sqrt{c x + 1} \sqrt{c x - 1}}\right )}{c^{2} x^{3} - x}\,{d x} + \frac{1}{2} \, b^{2}{\left (\log \left (c x + 1\right ) + \log \left (c x - 1\right ) - 2 \, \log \left (x\right )\right )} \log \left (c\right )^{2} + b^{2} \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right )^{2} \log \left (x\right ) - \frac{1}{4} \, b^{2} \log \left (c^{2} x^{2}\right )^{2} \log \left (x\right ) - b^{2} \int \frac{\log \left (c^{2} x^{2}\right )}{c^{2} x^{3} - x}\,{d x} \log \left (c\right ) + 2 \, b^{2} \int \frac{\log \left (x\right )}{c^{2} x^{3} - x}\,{d x} \log \left (c\right ) + 2 \, b^{2} \int \frac{\sqrt{c x + 1} \sqrt{c x - 1} \arctan \left (\frac{1}{\sqrt{c x + 1} \sqrt{c x - 1}}\right ) \log \left (x\right )}{c^{2} x^{3} - x}\,{d x} - 2 \, b^{2} \int \frac{\log \left (c^{2} x^{2}\right ) \log \left (x\right )}{c^{2} x^{3} - x}\,{d x} + b^{2} \int \frac{\log \left (x\right )^{2}}{c^{2} x^{3} - x}\,{d x} - 2 \, a b \int \frac{\arctan \left (\frac{1}{\sqrt{c x + 1} \sqrt{c x - 1}}\right )}{c^{2} x^{3} - x}\,{d x} + a^{2} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acsc(c*x))**2/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccsc(c*x) + a)^2/x, x)