3.28 \(\int \frac{(a+b \csc ^{-1}(c x))^3}{x} \, dx\)

Optimal. Leaf size=124 \[ -\frac{3}{2} b^2 \text{PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )+\frac{3}{2} i b \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2-\frac{3}{4} i b^3 \text{PolyLog}\left (4,e^{2 i \csc ^{-1}(c x)}\right )+\frac{i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^3 \]

[Out]

((I/4)*(a + b*ArcCsc[c*x])^4)/b - (a + b*ArcCsc[c*x])^3*Log[1 - E^((2*I)*ArcCsc[c*x])] + ((3*I)/2)*b*(a + b*Ar
cCsc[c*x])^2*PolyLog[2, E^((2*I)*ArcCsc[c*x])] - (3*b^2*(a + b*ArcCsc[c*x])*PolyLog[3, E^((2*I)*ArcCsc[c*x])])
/2 - ((3*I)/4)*b^3*PolyLog[4, E^((2*I)*ArcCsc[c*x])]

________________________________________________________________________________________

Rubi [A]  time = 0.143158, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5223, 3717, 2190, 2531, 6609, 2282, 6589} \[ -\frac{3}{2} b^2 \text{PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )+\frac{3}{2} i b \text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2-\frac{3}{4} i b^3 \text{PolyLog}\left (4,e^{2 i \csc ^{-1}(c x)}\right )+\frac{i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^3 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsc[c*x])^3/x,x]

[Out]

((I/4)*(a + b*ArcCsc[c*x])^4)/b - (a + b*ArcCsc[c*x])^3*Log[1 - E^((2*I)*ArcCsc[c*x])] + ((3*I)/2)*b*(a + b*Ar
cCsc[c*x])^2*PolyLog[2, E^((2*I)*ArcCsc[c*x])] - (3*b^2*(a + b*ArcCsc[c*x])*PolyLog[3, E^((2*I)*ArcCsc[c*x])])
/2 - ((3*I)/4)*b^3*PolyLog[4, E^((2*I)*ArcCsc[c*x])]

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 6609

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, 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 )^3}{x} \, dx &=-\operatorname{Subst}\left (\int (a+b x)^3 \cot (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac{i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}+2 i \operatorname{Subst}\left (\int \frac{e^{2 i x} (a+b x)^3}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac{i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+(3 b) \operatorname{Subst}\left (\int (a+b x)^2 \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 )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac{3}{2} i b \left (a+b \csc ^{-1}(c x)\right )^2 \text{Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\left (3 i b^2\right ) \operatorname{Subst}\left (\int (a+b x) \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 )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac{3}{2} i b \left (a+b \csc ^{-1}(c x)\right )^2 \text{Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\frac{3}{2} b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_3\left (e^{2 i \csc ^{-1}(c x)}\right )+\frac{1}{2} \left (3 b^3\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac{i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac{3}{2} i b \left (a+b \csc ^{-1}(c x)\right )^2 \text{Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\frac{3}{2} b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_3\left (e^{2 i \csc ^{-1}(c x)}\right )-\frac{1}{4} \left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )\\ &=\frac{i \left (a+b \csc ^{-1}(c x)\right )^4}{4 b}-\left (a+b \csc ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac{3}{2} i b \left (a+b \csc ^{-1}(c x)\right )^2 \text{Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\frac{3}{2} b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_3\left (e^{2 i \csc ^{-1}(c x)}\right )-\frac{3}{4} i b^3 \text{Li}_4\left (e^{2 i \csc ^{-1}(c x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.213212, size = 242, normalized size = 1.95 \[ \frac{3}{2} i a^2 b \left (\text{PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )+\csc ^{-1}(c x) \left (\csc ^{-1}(c x)+2 i \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )\right )\right )+\frac{1}{8} i a b^2 \left (-24 \csc ^{-1}(c x) \text{PolyLog}\left (2,e^{-2 i \csc ^{-1}(c x)}\right )+12 i \text{PolyLog}\left (3,e^{-2 i \csc ^{-1}(c x)}\right )-8 \csc ^{-1}(c x)^3+24 i \csc ^{-1}(c x)^2 \log \left (1-e^{-2 i \csc ^{-1}(c x)}\right )+\pi ^3\right )+\frac{1}{64} i b^3 \left (-96 \csc ^{-1}(c x)^2 \text{PolyLog}\left (2,e^{-2 i \csc ^{-1}(c x)}\right )+96 i \csc ^{-1}(c x) \text{PolyLog}\left (3,e^{-2 i \csc ^{-1}(c x)}\right )+48 \text{PolyLog}\left (4,e^{-2 i \csc ^{-1}(c x)}\right )-16 \csc ^{-1}(c x)^4+64 i \csc ^{-1}(c x)^3 \log \left (1-e^{-2 i \csc ^{-1}(c x)}\right )+\pi ^4\right )+a^3 \log (c x) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCsc[c*x])^3/x,x]

[Out]

a^3*Log[c*x] + ((3*I)/2)*a^2*b*(ArcCsc[c*x]*(ArcCsc[c*x] + (2*I)*Log[1 - E^((2*I)*ArcCsc[c*x])]) + PolyLog[2,
E^((2*I)*ArcCsc[c*x])]) + (I/8)*a*b^2*(Pi^3 - 8*ArcCsc[c*x]^3 + (24*I)*ArcCsc[c*x]^2*Log[1 - E^((-2*I)*ArcCsc[
c*x])] - 24*ArcCsc[c*x]*PolyLog[2, E^((-2*I)*ArcCsc[c*x])] + (12*I)*PolyLog[3, E^((-2*I)*ArcCsc[c*x])]) + (I/6
4)*b^3*(Pi^4 - 16*ArcCsc[c*x]^4 + (64*I)*ArcCsc[c*x]^3*Log[1 - E^((-2*I)*ArcCsc[c*x])] - 96*ArcCsc[c*x]^2*Poly
Log[2, E^((-2*I)*ArcCsc[c*x])] + (96*I)*ArcCsc[c*x]*PolyLog[3, E^((-2*I)*ArcCsc[c*x])] + 48*PolyLog[4, E^((-2*
I)*ArcCsc[c*x])])

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Maple [B]  time = 0.333, size = 666, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsc(c*x))^3/x,x)

[Out]

a^3*ln(c*x)+3/2*I*a^2*b*arccsc(c*x)^2-b^3*arccsc(c*x)^3*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))+3*I*a^2*b*polylog(2,I/
c/x+(1-1/c^2/x^2)^(1/2))-6*b^3*arccsc(c*x)*polylog(3,-I/c/x-(1-1/c^2/x^2)^(1/2))-6*I*b^3*polylog(4,I/c/x+(1-1/
c^2/x^2)^(1/2))-b^3*arccsc(c*x)^3*ln(1-I/c/x-(1-1/c^2/x^2)^(1/2))+3*I*b^3*arccsc(c*x)^2*polylog(2,I/c/x+(1-1/c
^2/x^2)^(1/2))-6*b^3*arccsc(c*x)*polylog(3,I/c/x+(1-1/c^2/x^2)^(1/2))+3*I*b^3*arccsc(c*x)^2*polylog(2,-I/c/x-(
1-1/c^2/x^2)^(1/2))+1/4*I*b^3*arccsc(c*x)^4+6*I*a*b^2*arccsc(c*x)*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2))-3*a*b^2
*arccsc(c*x)^2*ln(1-I/c/x-(1-1/c^2/x^2)^(1/2))-3*a*b^2*arccsc(c*x)^2*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))+6*I*a*b^2
*arccsc(c*x)*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))-6*a*b^2*polylog(3,I/c/x+(1-1/c^2/x^2)^(1/2))-6*a*b^2*polylo
g(3,-I/c/x-(1-1/c^2/x^2)^(1/2))+3*I*a^2*b*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))-3*a^2*b*arccsc(c*x)*ln(1-I/c/x
-(1-1/c^2/x^2)^(1/2))-3*a^2*b*arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))+I*a*b^2*arccsc(c*x)^3-6*I*b^3*polylo
g(4,-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*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))^3/x,x, algorithm="maxima")

[Out]

-3/2*a*b^2*c^2*(log(c*x + 1)/c^2 + log(c*x - 1)/c^2)*log(c)^2 - 12*b^3*c^2*integrate(1/4*x^2*arctan(1/(sqrt(c*
x + 1)*sqrt(c*x - 1)))/(c^2*x^3 - x), x)*log(c)^2 + 12*b^3*c^2*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(
c*x - 1)))*log(c^2*x^2)/(c^2*x^3 - x), x)*log(c) - 24*b^3*c^2*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c
*x - 1)))*log(x)/(c^2*x^3 - x), x)*log(c) + 12*a*b^2*c^2*integrate(1/4*x^2*log(c^2*x^2)/(c^2*x^3 - x), x)*log(
c) - 24*a*b^2*c^2*integrate(1/4*x^2*log(x)/(c^2*x^3 - x), x)*log(c) + b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x -
1))^3*log(x) - 3/4*b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)^2*log(x) + 24*b^3*c^2*integrate(1/
4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)*log(x)/(c^2*x^3 - x), x) - 12*b^3*c^2*integrate(1/4
*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2/(c^2*x^3 - x), x) + 12*a*b^2*c^2*integrate(1/4*x^2*arcta
n(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^3 - x), x) - 3*a*b^2*c^2*integrate(1/4*x^2*log(c^2*x^2)^2/(c^2*x^3
- x), x) + 12*a*b^2*c^2*integrate(1/4*x^2*log(c^2*x^2)*log(x)/(c^2*x^3 - x), x) - 12*a*b^2*c^2*integrate(1/4*
x^2*log(x)^2/(c^2*x^3 - x), x) + 12*a^2*b*c^2*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x
^3 - x), x) + 3/2*a*b^2*(log(c*x + 1) + log(c*x - 1) - 2*log(x))*log(c)^2 + 12*b^3*integrate(1/4*arctan(1/(sqr
t(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^3 - x), x)*log(c)^2 - 12*b^3*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x
- 1)))*log(c^2*x^2)/(c^2*x^3 - x), x)*log(c) + 24*b^3*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*lo
g(x)/(c^2*x^3 - x), x)*log(c) - 12*a*b^2*integrate(1/4*log(c^2*x^2)/(c^2*x^3 - x), x)*log(c) + 24*a*b^2*integr
ate(1/4*log(x)/(c^2*x^3 - x), x)*log(c) + 12*b^3*integrate(1/4*sqrt(c*x + 1)*sqrt(c*x - 1)*arctan(1/(sqrt(c*x
+ 1)*sqrt(c*x - 1)))^2*log(x)/(c^2*x^3 - x), x) - 3*b^3*integrate(1/4*sqrt(c*x + 1)*sqrt(c*x - 1)*log(c^2*x^2)
^2*log(x)/(c^2*x^3 - x), x) - 24*b^3*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)*log(x)
/(c^2*x^3 - x), x) + 12*b^3*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2/(c^2*x^3 - x), x) -
12*a*b^2*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^3 - x), x) + 3*a*b^2*integrate(1/4*lo
g(c^2*x^2)^2/(c^2*x^3 - x), x) - 12*a*b^2*integrate(1/4*log(c^2*x^2)*log(x)/(c^2*x^3 - x), x) + 12*a*b^2*integ
rate(1/4*log(x)^2/(c^2*x^3 - x), x) - 12*a^2*b*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^3
- x), x) + a^3*log(x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \operatorname{arccsc}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname{arccsc}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname{arccsc}\left (c x\right ) + a^{3}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))^3/x,x, algorithm="fricas")

[Out]

integral((b^3*arccsc(c*x)^3 + 3*a*b^2*arccsc(c*x)^2 + 3*a^2*b*arccsc(c*x) + a^3)/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 )^{3}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsc(c*x))**3/x,x)

[Out]

Integral((a + b*acsc(c*x))**3/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 )}^{3}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))^3/x,x, algorithm="giac")

[Out]

integrate((b*arccsc(c*x) + a)^3/x, x)