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

Optimal. Leaf size=144 \[ -\frac{6 i b^2 \text{PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c}+\frac{6 i b^2 \text{PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c}+\frac{6 b^3 \text{PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )}{c}-\frac{6 b^3 \text{PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c}+x \left (a+b \csc ^{-1}(c x)\right )^3+\frac{6 b \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2}{c} \]

[Out]

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

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Rubi [A]  time = 0.112179, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5217, 4410, 4183, 2531, 2282, 6589} \[ -\frac{6 i b^2 \text{PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c}+\frac{6 i b^2 \text{PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c}+\frac{6 b^3 \text{PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )}{c}-\frac{6 b^3 \text{PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c}+x \left (a+b \csc ^{-1}(c x)\right )^3+\frac{6 b \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2}{c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsc[c*x])^3,x]

[Out]

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

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 4410

Int[Cot[(a_.) + (b_.)*(x_)]^(p_.)*Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp
[((c + d*x)^m*Csc[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /; Fr
eeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

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 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int (a+b x)^3 \cot (x) \csc (x) \, dx,x,\csc ^{-1}(c x)\right )}{c}\\ &=x \left (a+b \csc ^{-1}(c x)\right )^3-\frac{(3 b) \operatorname{Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\csc ^{-1}(c x)\right )}{c}\\ &=x \left (a+b \csc ^{-1}(c x)\right )^3+\frac{6 b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c}+\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c}-\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c}\\ &=x \left (a+b \csc ^{-1}(c x)\right )^3+\frac{6 b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c}-\frac{6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{c}+\frac{6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c}+\frac{\left (6 i b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c}-\frac{\left (6 i b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c}\\ &=x \left (a+b \csc ^{-1}(c x)\right )^3+\frac{6 b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c}-\frac{6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{c}+\frac{6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c}+\frac{\left (6 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{c}-\frac{\left (6 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{c}\\ &=x \left (a+b \csc ^{-1}(c x)\right )^3+\frac{6 b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c}-\frac{6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{c}+\frac{6 i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c}+\frac{6 b^3 \text{Li}_3\left (-e^{i \csc ^{-1}(c x)}\right )}{c}-\frac{6 b^3 \text{Li}_3\left (e^{i \csc ^{-1}(c x)}\right )}{c}\\ \end{align*}

Mathematica [A]  time = 0.293682, size = 265, normalized size = 1.84 \[ \frac{-6 i b^2 \text{PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )+6 i b^2 \text{PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )+6 b^3 \text{PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )-6 b^3 \text{PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )+3 a^2 b \log \left (c x \left (\sqrt{1-\frac{1}{c^2 x^2}}+1\right )\right )+3 a^2 b c x \csc ^{-1}(c x)+a^3 c x+3 a b^2 c x \csc ^{-1}(c x)^2-6 a b^2 \csc ^{-1}(c x) \log \left (1-e^{i \csc ^{-1}(c x)}\right )+6 a b^2 \csc ^{-1}(c x) \log \left (1+e^{i \csc ^{-1}(c x)}\right )+b^3 c x \csc ^{-1}(c x)^3-3 b^3 \csc ^{-1}(c x)^2 \log \left (1-e^{i \csc ^{-1}(c x)}\right )+3 b^3 \csc ^{-1}(c x)^2 \log \left (1+e^{i \csc ^{-1}(c x)}\right )}{c} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCsc[c*x])^3,x]

[Out]

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

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Maple [B]  time = 0.319, size = 437, normalized size = 3. \begin{align*} x{b}^{3} \left ({\rm arccsc} \left (cx\right ) \right ) ^{3}+3\,xa{b}^{2} \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}-3\,{\frac{{b}^{3} \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}}{c}\ln \left ( 1-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+3\,{\frac{{b}^{3} \left ({\rm arccsc} \left (cx\right ) \right ) ^{2}}{c}\ln \left ( 1+{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+{\frac{6\,i{b}^{3}{\rm arccsc} \left (cx\right )}{c}{\it polylog} \left ( 2,{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }-{\frac{6\,i{b}^{3}{\rm arccsc} \left (cx\right )}{c}{\it polylog} \left ( 2,{\frac{-i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+3\,x{a}^{2}b{\rm arccsc} \left (cx\right )-6\,{\frac{a{b}^{2}{\rm arccsc} \left (cx\right )}{c}\ln \left ( 1-{\frac{i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+6\,{\frac{a{b}^{2}{\rm arccsc} \left (cx\right )}{c}\ln \left ( 1+{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+{\frac{6\,ia{b}^{2}}{c}{\it polylog} \left ( 2,{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }-{\frac{6\,ia{b}^{2}}{c}{\it polylog} \left ( 2,{\frac{-i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+x{a}^{3}-6\,{\frac{{b}^{3}}{c}{\it polylog} \left ( 3,{\frac{i}{cx}}+\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+6\,{\frac{{b}^{3}}{c}{\it polylog} \left ( 3,{\frac{-i}{cx}}-\sqrt{1-{\frac{1}{{c}^{2}{x}^{2}}}} \right ) }+3\,{\frac{{a}^{2}b}{c}\ln \left ( cx+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))^3,x)

[Out]

x*b^3*arccsc(c*x)^3+3*x*a*b^2*arccsc(c*x)^2-3/c*b^3*arccsc(c*x)^2*ln(1-I/c/x-(1-1/c^2/x^2)^(1/2))+3/c*b^3*arcc
sc(c*x)^2*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))+6*I/c*b^3*arccsc(c*x)*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2))-6*I/c*arc
csc(c*x)*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))*b^3+3*x*a^2*b*arccsc(c*x)-6/c*a*b^2*arccsc(c*x)*ln(1-I/c/x-(1-1
/c^2/x^2)^(1/2))+6/c*a*b^2*arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))+6*I/c*a*b^2*polylog(2,I/c/x+(1-1/c^2/x^
2)^(1/2))-6*I/c*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))*a*b^2+x*a^3-6*b^3*polylog(3,I/c/x+(1-1/c^2/x^2)^(1/2))/c
+6*b^3*polylog(3,-I/c/x-(1-1/c^2/x^2)^(1/2))/c+3/c*ln(c*x+c*x*(1-1/c^2/x^2)^(1/2))*a^2*b

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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, algorithm="maxima")

[Out]

-3/2*a*b^2*c^2*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c^3)*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^2 - 1), x)*log(c)^2 + b^3*x*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^3
- 3/4*b^3*x*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)^2 + 12*b^3*c^2*integrate(1/4*x^2*arctan(1/(sq
rt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) - 24*b^3*c^2*integrate(1/4*x^2*arctan(1/(sqr
t(c*x + 1)*sqrt(c*x - 1)))*log(x)/(c^2*x^2 - 1), x)*log(c) + 12*a*b^2*c^2*integrate(1/4*x^2*log(c^2*x^2)/(c^2*
x^2 - 1), x)*log(c) - 24*a*b^2*c^2*integrate(1/4*x^2*log(x)/(c^2*x^2 - 1), x)*log(c) + 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)*log(x)/(c^2*x^2 - 1), 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^2 - 1), 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^2 - 1), x) + 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^2 - 1), x) - 3*a*b^2*c^2*integrate(1/4*x^2*log(c^2*x^2)^2/(c^2*x^2 - 1), x
) + 12*a*b^2*c^2*integrate(1/4*x^2*log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x) - 12*a*b^2*c^2*integrate(1/4*x^2*log(
x)^2/(c^2*x^2 - 1), x) - 3/2*a*b^2*(log(c*x + 1)/c - log(c*x - 1)/c)*log(c)^2 + 12*b^3*integrate(1/4*arctan(1/
(sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^2 - 1), 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^2 - 1), x)*log(c) + 24*b^3*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1))
)*log(x)/(c^2*x^2 - 1), x)*log(c) - 12*a*b^2*integrate(1/4*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) + 24*a*b^2*in
tegrate(1/4*log(x)/(c^2*x^2 - 1), x)*log(c) + a^3*x + 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/(c^2*x^2 - 1), x) - 3*b^3*integrate(1/4*sqrt(c*x + 1)*sqrt(c*x - 1)*log(c^2
*x^2)^2/(c^2*x^2 - 1), x) - 12*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^2 - 1), x) + 12*b^3*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2/(c^2*x^2 - 1), x) - 1
2*a*b^2*integrate(1/4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^2 - 1), x) - 12*b^3*integrate(1/4*arcta
n(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^2 - 1), x) + 3*a*b^2*integrate(1/4*log(c^2*x^2)^2/(c^2*
x^2 - 1), x) - 12*a*b^2*integrate(1/4*log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x) + 12*a*b^2*integrate(1/4*log(x)^2/
(c^2*x^2 - 1), x) + 3/2*(2*c*x*arccsc(c*x) + log(sqrt(-1/(c^2*x^2) + 1) + 1) - log(-sqrt(-1/(c^2*x^2) + 1) + 1
))*a^2*b/c

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (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\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acsc(c*x))**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arccsc(c*x) + a)^3, x)