\(\int x^3 \cot ^3(a+b x) \, dx\) [11]

Optimal result

Integrand size = 12, antiderivative size = 202 \[ \int x^3 \cot ^3(a+b x) \, dx=-\frac {3 i x^2}{2 b^2}-\frac {x^3}{2 b}+\frac {i x^4}{4}-\frac {3 x^2 \cot (a+b x)}{2 b^2}-\frac {x^3 \cot ^2(a+b x)}{2 b}+\frac {3 x \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {x^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {3 i \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 i \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{4 b^4} \]

[Out]

-3/2*I*x^2/b^2-1/2*x^3/b+1/4*I*x^4-3/2*x^2*cot(b*x+a)/b^2-1/2*x^3*cot(b*x+a)^2/b+3*x*ln(1-exp(2*I*(b*x+a)))/b^
3-x^3*ln(1-exp(2*I*(b*x+a)))/b-3/2*I*polylog(2,exp(2*I*(b*x+a)))/b^4+3/2*I*x^2*polylog(2,exp(2*I*(b*x+a)))/b^2
-3/2*x*polylog(3,exp(2*I*(b*x+a)))/b^3-3/4*I*polylog(4,exp(2*I*(b*x+a)))/b^4

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {3801, 3798, 2221, 2317, 2438, 30, 2611, 6744, 2320, 6724} \[ \int x^3 \cot ^3(a+b x) \, dx=-\frac {3 i \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 i \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{4 b^4}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 x \log \left (1-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 x^2 \cot (a+b x)}{2 b^2}-\frac {x^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {x^3 \cot ^2(a+b x)}{2 b}-\frac {3 i x^2}{2 b^2}-\frac {x^3}{2 b}+\frac {i x^4}{4} \]

[In]

Int[x^3*Cot[a + b*x]^3,x]

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2317

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 2320

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 2438

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 2611

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 3798

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 3801

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 6724

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]

Rule 6744

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \cot ^2(a+b x)}{2 b}+\frac {3 \int x^2 \cot ^2(a+b x) \, dx}{2 b}-\int x^3 \cot (a+b x) \, dx \\ & = \frac {i x^4}{4}-\frac {3 x^2 \cot (a+b x)}{2 b^2}-\frac {x^3 \cot ^2(a+b x)}{2 b}+2 i \int \frac {e^{2 i (a+b x)} x^3}{1-e^{2 i (a+b x)}} \, dx+\frac {3 \int x \cot (a+b x) \, dx}{b^2}-\frac {3 \int x^2 \, dx}{2 b} \\ & = -\frac {3 i x^2}{2 b^2}-\frac {x^3}{2 b}+\frac {i x^4}{4}-\frac {3 x^2 \cot (a+b x)}{2 b^2}-\frac {x^3 \cot ^2(a+b x)}{2 b}-\frac {x^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {(6 i) \int \frac {e^{2 i (a+b x)} x}{1-e^{2 i (a+b x)}} \, dx}{b^2}+\frac {3 \int x^2 \log \left (1-e^{2 i (a+b x)}\right ) \, dx}{b} \\ & = -\frac {3 i x^2}{2 b^2}-\frac {x^3}{2 b}+\frac {i x^4}{4}-\frac {3 x^2 \cot (a+b x)}{2 b^2}-\frac {x^3 \cot ^2(a+b x)}{2 b}+\frac {3 x \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {x^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 \int \log \left (1-e^{2 i (a+b x)}\right ) \, dx}{b^3}-\frac {(3 i) \int x \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right ) \, dx}{b^2} \\ & = -\frac {3 i x^2}{2 b^2}-\frac {x^3}{2 b}+\frac {i x^4}{4}-\frac {3 x^2 \cot (a+b x)}{2 b^2}-\frac {x^3 \cot ^2(a+b x)}{2 b}+\frac {3 x \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {x^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}+\frac {(3 i) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 \int \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right ) \, dx}{2 b^3} \\ & = -\frac {3 i x^2}{2 b^2}-\frac {x^3}{2 b}+\frac {i x^4}{4}-\frac {3 x^2 \cot (a+b x)}{2 b^2}-\frac {x^3 \cot ^2(a+b x)}{2 b}+\frac {3 x \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {x^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {3 i \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}-\frac {(3 i) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{4 b^4} \\ & = -\frac {3 i x^2}{2 b^2}-\frac {x^3}{2 b}+\frac {i x^4}{4}-\frac {3 x^2 \cot (a+b x)}{2 b^2}-\frac {x^3 \cot ^2(a+b x)}{2 b}+\frac {3 x \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {x^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {3 i \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^4}+\frac {3 i x^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 i \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{4 b^4} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(491\) vs. \(2(202)=404\).

Time = 6.81 (sec) , antiderivative size = 491, normalized size of antiderivative = 2.43 \[ \int x^3 \cot ^3(a+b x) \, dx=-\frac {1}{4} x^4 \cot (a)-\frac {x^3 \csc ^2(a+b x)}{2 b}+\frac {e^{i a} \csc (a) \left (b^4 e^{-2 i a} x^4+2 i b^3 \left (1-e^{-2 i a}\right ) x^3 \log \left (1-e^{-i (a+b x)}\right )+2 i b^3 \left (1-e^{-2 i a}\right ) x^3 \log \left (1+e^{-i (a+b x)}\right )-6 b^2 \left (1-e^{-2 i a}\right ) x^2 \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )-6 b^2 \left (1-e^{-2 i a}\right ) x^2 \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )+12 i b \left (1-e^{-2 i a}\right ) x \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+12 i b \left (1-e^{-2 i a}\right ) x \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )+12 \left (1-e^{-2 i a}\right ) \operatorname {PolyLog}\left (4,-e^{-i (a+b x)}\right )+12 \left (1-e^{-2 i a}\right ) \operatorname {PolyLog}\left (4,e^{-i (a+b x)}\right )\right )}{4 b^4}+\frac {3 x^2 \csc (a) \csc (a+b x) \sin (b x)}{2 b^2}-\frac {3 \csc (a) \sec (a) \left (b^2 e^{i \arctan (\tan (a))} x^2+\frac {\left (i b x (-\pi +2 \arctan (\tan (a)))-\pi \log \left (1+e^{-2 i b x}\right )-2 (b x+\arctan (\tan (a))) \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )+\pi \log (\cos (b x))+2 \arctan (\tan (a)) \log (\sin (b x+\arctan (\tan (a))))+i \operatorname {PolyLog}\left (2,e^{2 i (b x+\arctan (\tan (a)))}\right )\right ) \tan (a)}{\sqrt {1+\tan ^2(a)}}\right )}{2 b^4 \sqrt {\sec ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}} \]

[In]

Integrate[x^3*Cot[a + b*x]^3,x]

[Out]

-1/4*(x^4*Cot[a]) - (x^3*Csc[a + b*x]^2)/(2*b) + (E^(I*a)*Csc[a]*((b^4*x^4)/E^((2*I)*a) + (2*I)*b^3*(1 - E^((-
2*I)*a))*x^3*Log[1 - E^((-I)*(a + b*x))] + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 + E^((-I)*(a + b*x))] - 6*b^
2*(1 - E^((-2*I)*a))*x^2*PolyLog[2, -E^((-I)*(a + b*x))] - 6*b^2*(1 - E^((-2*I)*a))*x^2*PolyLog[2, E^((-I)*(a
+ b*x))] + (12*I)*b*(1 - E^((-2*I)*a))*x*PolyLog[3, -E^((-I)*(a + b*x))] + (12*I)*b*(1 - E^((-2*I)*a))*x*PolyL
og[3, E^((-I)*(a + b*x))] + 12*(1 - E^((-2*I)*a))*PolyLog[4, -E^((-I)*(a + b*x))] + 12*(1 - E^((-2*I)*a))*Poly
Log[4, E^((-I)*(a + b*x))]))/(4*b^4) + (3*x^2*Csc[a]*Csc[a + b*x]*Sin[b*x])/(2*b^2) - (3*Csc[a]*Sec[a]*(b^2*E^
(I*ArcTan[Tan[a]])*x^2 + ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a
]])*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a
]]]] + I*PolyLog[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(2*b^4*Sqrt[Sec[a]^2*(Cos[
a]^2 + Sin[a]^2)])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (167 ) = 334\).

Time = 0.22 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.20

method	result	size
risch	\(\frac {3 i \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}+\frac {3 i \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}+\frac {2 i a^{3} x}{b^{3}}-\frac {\ln \left (1+{\mathrm e}^{i \left (b x +a \right )}\right ) x^{3}}{b}-\frac {\ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{3}}{b}-\frac {a^{3} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {2 a^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{4}}+\frac {3 i a^{4}}{2 b^{4}}-\frac {6 i \operatorname {polylog}\left (4, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {6 i \operatorname {polylog}\left (4, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {6 \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}-\frac {6 \operatorname {polylog}\left (3, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}+\frac {i x^{4}}{4}-\frac {3 i \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {3 i x^{2}}{b^{2}}-\frac {3 i a^{2}}{b^{4}}-\frac {3 i \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {6 a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {3 a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{4}}+\frac {3 \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{4}}+\frac {3 \ln \left (1+{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}+\frac {3 \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}+\frac {x^{2} \left (2 x \,{\mathrm e}^{2 i \left (b x +a \right )} b -3 i {\mathrm e}^{2 i \left (b x +a \right )}+3 i\right )}{b^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}-\frac {6 i a x}{b^{3}}\)	\(444\)

[In]

int(x^3*cot(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/4*I*x^4-1/b*ln(1+exp(I*(b*x+a)))*x^3-1/b*ln(1-exp(I*(b*x+a)))*x^3-1/b^4*a^3*ln(1-exp(I*(b*x+a)))+x^2*(2*x*ex
p(2*I*(b*x+a))*b-3*I*exp(2*I*(b*x+a))+3*I)/b^2/(exp(2*I*(b*x+a))-1)^2-2/b^4*a^3*ln(exp(I*(b*x+a)))+1/b^4*a^3*l
n(exp(I*(b*x+a))-1)-6/b^3*polylog(3,-exp(I*(b*x+a)))*x-6/b^3*polylog(3,exp(I*(b*x+a)))*x+3*I/b^2*polylog(2,-ex
p(I*(b*x+a)))*x^2+3*I/b^2*polylog(2,exp(I*(b*x+a)))*x^2+2*I/b^3*a^3*x-6*I/b^3*a*x+6/b^4*a*ln(exp(I*(b*x+a)))-3
/b^4*a*ln(exp(I*(b*x+a))-1)+3/b^4*ln(1-exp(I*(b*x+a)))*a+3/b^3*ln(1+exp(I*(b*x+a)))*x+3/b^3*ln(1-exp(I*(b*x+a)
))*x-3*I/b^4*polylog(2,exp(I*(b*x+a)))+3/2*I/b^4*a^4-6*I/b^4*polylog(4,-exp(I*(b*x+a)))-6*I/b^4*polylog(4,exp(
I*(b*x+a)))-3*I/b^2*x^2-3*I/b^4*a^2-3*I/b^4*polylog(2,-exp(I*(b*x+a)))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 569 vs. \(2 (160) = 320\).

Time = 0.30 (sec) , antiderivative size = 569, normalized size of antiderivative = 2.82 \[ \int x^3 \cot ^3(a+b x) \, dx=\frac {8 \, b^{3} x^{3} + 12 \, b^{2} x^{2} \sin \left (2 \, b x + 2 \, a\right ) - 6 \, {\left (i \, b^{2} x^{2} + {\left (-i \, b^{2} x^{2} + i\right )} \cos \left (2 \, b x + 2 \, a\right ) - i\right )} {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) - 6 \, {\left (-i \, b^{2} x^{2} + {\left (i \, b^{2} x^{2} - i\right )} \cos \left (2 \, b x + 2 \, a\right ) + i\right )} {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) - 4 \, {\left (a^{3} - {\left (a^{3} - 3 \, a\right )} \cos \left (2 \, b x + 2 \, a\right ) - 3 \, a\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) - 4 \, {\left (a^{3} - {\left (a^{3} - 3 \, a\right )} \cos \left (2 \, b x + 2 \, a\right ) - 3 \, a\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) - \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) + 4 \, {\left (b^{3} x^{3} + a^{3} - 3 \, b x - {\left (b^{3} x^{3} + a^{3} - 3 \, b x - 3 \, a\right )} \cos \left (2 \, b x + 2 \, a\right ) - 3 \, a\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 4 \, {\left (b^{3} x^{3} + a^{3} - 3 \, b x - {\left (b^{3} x^{3} + a^{3} - 3 \, b x - 3 \, a\right )} \cos \left (2 \, b x + 2 \, a\right ) - 3 \, a\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) - 3 \, {\left (i \, \cos \left (2 \, b x + 2 \, a\right ) - i\right )} {\rm polylog}\left (4, \cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) - 3 \, {\left (-i \, \cos \left (2 \, b x + 2 \, a\right ) + i\right )} {\rm polylog}\left (4, \cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) - 6 \, {\left (b x \cos \left (2 \, b x + 2 \, a\right ) - b x\right )} {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) - 6 \, {\left (b x \cos \left (2 \, b x + 2 \, a\right ) - b x\right )} {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right )}{8 \, {\left (b^{4} \cos \left (2 \, b x + 2 \, a\right ) - b^{4}\right )}} \]

[In]

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

[Out]

1/8*(8*b^3*x^3 + 12*b^2*x^2*sin(2*b*x + 2*a) - 6*(I*b^2*x^2 + (-I*b^2*x^2 + I)*cos(2*b*x + 2*a) - I)*dilog(cos
(2*b*x + 2*a) + I*sin(2*b*x + 2*a)) - 6*(-I*b^2*x^2 + (I*b^2*x^2 - I)*cos(2*b*x + 2*a) + I)*dilog(cos(2*b*x +
2*a) - I*sin(2*b*x + 2*a)) - 4*(a^3 - (a^3 - 3*a)*cos(2*b*x + 2*a) - 3*a)*log(-1/2*cos(2*b*x + 2*a) + 1/2*I*si
n(2*b*x + 2*a) + 1/2) - 4*(a^3 - (a^3 - 3*a)*cos(2*b*x + 2*a) - 3*a)*log(-1/2*cos(2*b*x + 2*a) - 1/2*I*sin(2*b
*x + 2*a) + 1/2) + 4*(b^3*x^3 + a^3 - 3*b*x - (b^3*x^3 + a^3 - 3*b*x - 3*a)*cos(2*b*x + 2*a) - 3*a)*log(-cos(2
*b*x + 2*a) + I*sin(2*b*x + 2*a) + 1) + 4*(b^3*x^3 + a^3 - 3*b*x - (b^3*x^3 + a^3 - 3*b*x - 3*a)*cos(2*b*x + 2
*a) - 3*a)*log(-cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a) + 1) - 3*(I*cos(2*b*x + 2*a) - I)*polylog(4, cos(2*b*x +
2*a) + I*sin(2*b*x + 2*a)) - 3*(-I*cos(2*b*x + 2*a) + I)*polylog(4, cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a)) -
6*(b*x*cos(2*b*x + 2*a) - b*x)*polylog(3, cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a)) - 6*(b*x*cos(2*b*x + 2*a) - b
*x)*polylog(3, cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a)))/(b^4*cos(2*b*x + 2*a) - b^4)

Sympy [F]

\[ \int x^3 \cot ^3(a+b x) \, dx=\int x^{3} \cot ^{3}{\left (a + b x \right )}\, dx \]

[In]

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

[Out]

Integral(x**3*cot(a + b*x)**3, x)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1960 vs. \(2 (160) = 320\).

Time = 0.44 (sec) , antiderivative size = 1960, normalized size of antiderivative = 9.70 \[ \int x^3 \cot ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*(a^3*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2)) + 2*((b*x + a)^4 - 4*(b*x + a)^3*a + 6*(b*x + a)^2*a^2 + 12*
a^2 - 4*((b*x + a)^3 - 3*(b*x + a)^2*a + 3*(a^2 - 1)*(b*x + a) + ((b*x + a)^3 - 3*(b*x + a)^2*a + 3*(a^2 - 1)*
(b*x + a) + 3*a)*cos(4*b*x + 4*a) - 2*((b*x + a)^3 - 3*(b*x + a)^2*a + 3*(a^2 - 1)*(b*x + a) + 3*a)*cos(2*b*x
+ 2*a) - (-I*(b*x + a)^3 + 3*I*(b*x + a)^2*a + 3*(-I*a^2 + I)*(b*x + a) - 3*I*a)*sin(4*b*x + 4*a) - 2*(I*(b*x
+ a)^3 - 3*I*(b*x + a)^2*a + 3*(I*a^2 - I)*(b*x + a) + 3*I*a)*sin(2*b*x + 2*a) + 3*a)*arctan2(sin(b*x + a), co
s(b*x + a) + 1) - 12*(a*cos(4*b*x + 4*a) - 2*a*cos(2*b*x + 2*a) + I*a*sin(4*b*x + 4*a) - 2*I*a*sin(2*b*x + 2*a
) + a)*arctan2(sin(b*x + a), cos(b*x + a) - 1) + 4*((b*x + a)^3 - 3*(b*x + a)^2*a + 3*(a^2 - 1)*(b*x + a) + ((
b*x + a)^3 - 3*(b*x + a)^2*a + 3*(a^2 - 1)*(b*x + a))*cos(4*b*x + 4*a) - 2*((b*x + a)^3 - 3*(b*x + a)^2*a + 3*
(a^2 - 1)*(b*x + a))*cos(2*b*x + 2*a) + (I*(b*x + a)^3 - 3*I*(b*x + a)^2*a + 3*(I*a^2 - I)*(b*x + a))*sin(4*b*
x + 4*a) + 2*(-I*(b*x + a)^3 + 3*I*(b*x + a)^2*a + 3*(-I*a^2 + I)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(sin(b*x
+ a), -cos(b*x + a) + 1) + ((b*x + a)^4 - 4*(b*x + a)^3*a + 6*(a^2 - 2)*(b*x + a)^2 + 24*(b*x + a)*a)*cos(4*b
*x + 4*a) - 2*((b*x + a)^4 - 4*(b*x + a)^3*(a - I) + 6*(a^2 - 2*I*a - 1)*(b*x + a)^2 - 12*(-I*a^2 - a)*(b*x +
a) + 6*a^2)*cos(2*b*x + 2*a) + 12*((b*x + a)^2 - 2*(b*x + a)*a + a^2 + ((b*x + a)^2 - 2*(b*x + a)*a + a^2 - 1)
*cos(4*b*x + 4*a) - 2*((b*x + a)^2 - 2*(b*x + a)*a + a^2 - 1)*cos(2*b*x + 2*a) + (I*(b*x + a)^2 - 2*I*(b*x + a
)*a + I*a^2 - I)*sin(4*b*x + 4*a) + 2*(-I*(b*x + a)^2 + 2*I*(b*x + a)*a - I*a^2 + I)*sin(2*b*x + 2*a) - 1)*dil
og(-e^(I*b*x + I*a)) + 12*((b*x + a)^2 - 2*(b*x + a)*a + a^2 + ((b*x + a)^2 - 2*(b*x + a)*a + a^2 - 1)*cos(4*b
*x + 4*a) - 2*((b*x + a)^2 - 2*(b*x + a)*a + a^2 - 1)*cos(2*b*x + 2*a) + (I*(b*x + a)^2 - 2*I*(b*x + a)*a + I*
a^2 - I)*sin(4*b*x + 4*a) + 2*(-I*(b*x + a)^2 + 2*I*(b*x + a)*a - I*a^2 + I)*sin(2*b*x + 2*a) - 1)*dilog(e^(I*
b*x + I*a)) + 2*(I*(b*x + a)^3 - 3*I*(b*x + a)^2*a + 3*(I*a^2 - I)*(b*x + a) + (I*(b*x + a)^3 - 3*I*(b*x + a)^
2*a + 3*(I*a^2 - I)*(b*x + a) + 3*I*a)*cos(4*b*x + 4*a) + 2*(-I*(b*x + a)^3 + 3*I*(b*x + a)^2*a + 3*(-I*a^2 +
I)*(b*x + a) - 3*I*a)*cos(2*b*x + 2*a) - ((b*x + a)^3 - 3*(b*x + a)^2*a + 3*(a^2 - 1)*(b*x + a) + 3*a)*sin(4*b
*x + 4*a) + 2*((b*x + a)^3 - 3*(b*x + a)^2*a + 3*(a^2 - 1)*(b*x + a) + 3*a)*sin(2*b*x + 2*a) + 3*I*a)*log(cos(
b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) + 2*(I*(b*x + a)^3 - 3*I*(b*x + a)^2*a + 3*(I*a^2 - I)*(b*x
+ a) + (I*(b*x + a)^3 - 3*I*(b*x + a)^2*a + 3*(I*a^2 - I)*(b*x + a) + 3*I*a)*cos(4*b*x + 4*a) + 2*(-I*(b*x + a
)^3 + 3*I*(b*x + a)^2*a + 3*(-I*a^2 + I)*(b*x + a) - 3*I*a)*cos(2*b*x + 2*a) - ((b*x + a)^3 - 3*(b*x + a)^2*a
+ 3*(a^2 - 1)*(b*x + a) + 3*a)*sin(4*b*x + 4*a) + 2*((b*x + a)^3 - 3*(b*x + a)^2*a + 3*(a^2 - 1)*(b*x + a) + 3
*a)*sin(2*b*x + 2*a) + 3*I*a)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) - 24*(cos(4*b*x + 4*a)
- 2*cos(2*b*x + 2*a) + I*sin(4*b*x + 4*a) - 2*I*sin(2*b*x + 2*a) + 1)*polylog(4, -e^(I*b*x + I*a)) - 24*(cos(
4*b*x + 4*a) - 2*cos(2*b*x + 2*a) + I*sin(4*b*x + 4*a) - 2*I*sin(2*b*x + 2*a) + 1)*polylog(4, e^(I*b*x + I*a))
+ 24*(I*b*x*cos(4*b*x + 4*a) - 2*I*b*x*cos(2*b*x + 2*a) - b*x*sin(4*b*x + 4*a) + 2*b*x*sin(2*b*x + 2*a) + I*b
*x)*polylog(3, -e^(I*b*x + I*a)) + 24*(I*b*x*cos(4*b*x + 4*a) - 2*I*b*x*cos(2*b*x + 2*a) - b*x*sin(4*b*x + 4*a
) + 2*b*x*sin(2*b*x + 2*a) + I*b*x)*polylog(3, e^(I*b*x + I*a)) - (-I*(b*x + a)^4 + 4*I*(b*x + a)^3*a - 6*(I*a
^2 - 2*I)*(b*x + a)^2 - 24*I*(b*x + a)*a)*sin(4*b*x + 4*a) + 2*(-I*(b*x + a)^4 + 4*(b*x + a)^3*(I*a + 1) + 6*(
-I*a^2 - 2*a + I)*(b*x + a)^2 + 12*(a^2 - I*a)*(b*x + a) - 6*I*a^2)*sin(2*b*x + 2*a))/(-4*I*cos(4*b*x + 4*a) +
8*I*cos(2*b*x + 2*a) + 4*sin(4*b*x + 4*a) - 8*sin(2*b*x + 2*a) - 4*I))/b^4

Giac [F]

\[ \int x^3 \cot ^3(a+b x) \, dx=\int { x^{3} \cot \left (b x + a\right )^{3} \,d x } \]

[In]

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

[Out]

integrate(x^3*cot(b*x + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int x^3 \cot ^3(a+b x) \, dx=\int x^3\,{\mathrm {cot}\left (a+b\,x\right )}^3 \,d x \]

[In]

int(x^3*cot(a + b*x)^3,x)

[Out]

int(x^3*cot(a + b*x)^3, x)