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

Optimal. Leaf size=140 \[ -\frac{6 i \csc ^{-1}(a+b x) \text{PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac{6 i \csc ^{-1}(a+b x) \text{PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac{6 \text{PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b}-\frac{6 \text{PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac{(a+b x) \csc ^{-1}(a+b x)^3}{b}+\frac{6 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b} \]

[Out]

((a + b*x)*ArcCsc[a + b*x]^3)/b + (6*ArcCsc[a + b*x]^2*ArcTanh[E^(I*ArcCsc[a + b*x])])/b - ((6*I)*ArcCsc[a + b
*x]*PolyLog[2, -E^(I*ArcCsc[a + b*x])])/b + ((6*I)*ArcCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a + b*x])])/b + (6*P
olyLog[3, -E^(I*ArcCsc[a + b*x])])/b - (6*PolyLog[3, E^(I*ArcCsc[a + b*x])])/b

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Rubi [A]  time = 0.101629, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.875, Rules used = {5253, 5217, 3758, 4183, 2531, 2282, 6589} \[ -\frac{6 i \csc ^{-1}(a+b x) \text{PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac{6 i \csc ^{-1}(a+b x) \text{PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac{6 \text{PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b}-\frac{6 \text{PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac{(a+b x) \csc ^{-1}(a+b x)^3}{b}+\frac{6 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[ArcCsc[a + b*x]^3,x]

[Out]

((a + b*x)*ArcCsc[a + b*x]^3)/b + (6*ArcCsc[a + b*x]^2*ArcTanh[E^(I*ArcCsc[a + b*x])])/b - ((6*I)*ArcCsc[a + b
*x]*PolyLog[2, -E^(I*ArcCsc[a + b*x])])/b + ((6*I)*ArcCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a + b*x])])/b + (6*P
olyLog[3, -E^(I*ArcCsc[a + b*x])])/b - (6*PolyLog[3, E^(I*ArcCsc[a + b*x])])/b

Rule 5253

Int[((a_.) + ArcCsc[(c_) + (d_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcCsc[x])^p, x],
x, c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]

Rule 5217

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

Rule 3758

Int[Cot[(a_.) + (b_.)*(x_)^(n_.)]^(q_.)*Csc[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.), x_Symbol] :> -Simp[(x^
(m - n + 1)*Csc[a + b*x^n]^p)/(b*n*p), x] + Dist[(m - n + 1)/(b*n*p), Int[x^(m - n)*Csc[a + b*x^n]^p, x], x] /
; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m, n] && EqQ[q, 1]

Rule 4183

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E^(I*(e + f*
x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, 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 \csc ^{-1}(a+b x)^3 \, dx &=\frac{\operatorname{Subst}\left (\int \csc ^{-1}(x)^3 \, dx,x,a+b x\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int x^3 \cot (x) \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b}\\ &=\frac{(a+b x) \csc ^{-1}(a+b x)^3}{b}-\frac{3 \operatorname{Subst}\left (\int x^2 \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b}\\ &=\frac{(a+b x) \csc ^{-1}(a+b x)^3}{b}+\frac{6 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac{6 \operatorname{Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b}-\frac{6 \operatorname{Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b}\\ &=\frac{(a+b x) \csc ^{-1}(a+b x)^3}{b}+\frac{6 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b}-\frac{6 i \csc ^{-1}(a+b x) \text{Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac{6 i \csc ^{-1}(a+b x) \text{Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac{(6 i) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b}-\frac{(6 i) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b}\\ &=\frac{(a+b x) \csc ^{-1}(a+b x)^3}{b}+\frac{6 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b}-\frac{6 i \csc ^{-1}(a+b x) \text{Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac{6 i \csc ^{-1}(a+b x) \text{Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac{6 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b}-\frac{6 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b}\\ &=\frac{(a+b x) \csc ^{-1}(a+b x)^3}{b}+\frac{6 \csc ^{-1}(a+b x)^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b}-\frac{6 i \csc ^{-1}(a+b x) \text{Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac{6 i \csc ^{-1}(a+b x) \text{Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac{6 \text{Li}_3\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b}-\frac{6 \text{Li}_3\left (e^{i \csc ^{-1}(a+b x)}\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.118717, size = 162, normalized size = 1.16 \[ \frac{-6 i \csc ^{-1}(a+b x) \text{PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )+6 i \csc ^{-1}(a+b x) \text{PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )+6 \text{PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )-6 \text{PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )+a \csc ^{-1}(a+b x)^3+b x \csc ^{-1}(a+b x)^3-3 \csc ^{-1}(a+b x)^2 \log \left (1-e^{i \csc ^{-1}(a+b x)}\right )+3 \csc ^{-1}(a+b x)^2 \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )}{b} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCsc[a + b*x]^3,x]

[Out]

(a*ArcCsc[a + b*x]^3 + b*x*ArcCsc[a + b*x]^3 - 3*ArcCsc[a + b*x]^2*Log[1 - E^(I*ArcCsc[a + b*x])] + 3*ArcCsc[a
+ b*x]^2*Log[1 + E^(I*ArcCsc[a + b*x])] - (6*I)*ArcCsc[a + b*x]*PolyLog[2, -E^(I*ArcCsc[a + b*x])] + (6*I)*Ar
cCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a + b*x])] + 6*PolyLog[3, -E^(I*ArcCsc[a + b*x])] - 6*PolyLog[3, E^(I*Arc
Csc[a + b*x])])/b

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Maple [A]  time = 0.306, size = 247, normalized size = 1.8 \begin{align*} x \left ({\rm arccsc} \left (bx+a\right ) \right ) ^{3}+{\frac{ \left ({\rm arccsc} \left (bx+a\right ) \right ) ^{3}a}{b}}+3\,{\frac{ \left ({\rm arccsc} \left (bx+a\right ) \right ) ^{2}}{b}\ln \left ( 1+{\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }-{\frac{6\,i{\rm arccsc} \left (bx+a\right )}{b}{\it polylog} \left ( 2,{\frac{-i}{bx+a}}-\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }-3\,{\frac{ \left ({\rm arccsc} \left (bx+a\right ) \right ) ^{2}}{b}\ln \left ( 1-{\frac{i}{bx+a}}-\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }+{\frac{6\,i{\rm arccsc} \left (bx+a\right )}{b}{\it polylog} \left ( 2,{\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }+6\,{\frac{1}{b}{\it polylog} \left ( 3,{\frac{-i}{bx+a}}-\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) }-6\,{\frac{1}{b}{\it polylog} \left ( 3,{\frac{i}{bx+a}}+\sqrt{1- \left ( bx+a \right ) ^{-2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arccsc(b*x+a)^3+1/b*arccsc(b*x+a)^3*a+3/b*arccsc(b*x+a)^2*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))-6*I*arccsc(b
*x+a)*polylog(2,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))/b-3/b*arccsc(b*x+a)^2*ln(1-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))+
6*I*arccsc(b*x+a)*polylog(2,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b+6*polylog(3,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))/b
-6*polylog(3,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^3 - 3/4*x*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))*lo
g(b^2*x^2 + 2*a*b*x + a^2)^2 - integrate(3/4*(4*(b^3*x^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 3*a
*b^2*x^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + a^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))
+ (3*a^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) - arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))*b
*x - a*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))*log(b*x + a)^2 - (4*b*x*arctan2(1, sqrt(b*x + a + 1)*s
qrt(b*x + a - 1))^2 - b*x*log(b^2*x^2 + 2*a*b*x + a^2)^2)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) - 4*(b^3*x^3*arc
tan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 2*a*b^2*x^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + (
a^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) - arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))*b*x + (
b^3*x^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 3*a*b^2*x^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x +
a - 1)) + a^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + (3*a^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x +
a - 1)) - arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))*b*x - a*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a
- 1)))*log(b*x + a))*log(b^2*x^2 + 2*a*b*x + a^2))/(b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2 - 1)*b*x - a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccsc(b*x + a)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{acsc}^{3}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acsc(a + b*x)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{arccsc}\left (b x + a\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccsc(b*x + a)^3, x)