\(\int x^2 \cot (a+b x) \, dx\) [2]

Optimal result

Integrand size = 10, antiderivative size = 74 \[ \int x^2 \cot (a+b x) \, dx=-\frac {i x^3}{3}+\frac {x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {i x \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}+\frac {\operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3798, 2221, 2611, 2320, 6724} \[ \int x^2 \cot (a+b x) \, dx=\frac {\operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}-\frac {i x \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}+\frac {x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {i x^3}{3} \]

[In]

Int[x^2*Cot[a + b*x],x]

[Out]

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

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 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 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 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]

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

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.84 \[ \int x^2 \cot (a+b x) \, dx=\frac {i b^3 x^3+3 b^2 x^2 \log \left (1-e^{-i (a+b x)}\right )+3 b^2 x^2 \log \left (1+e^{-i (a+b x)}\right )+6 i b x \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )+6 i b x \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+6 \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )}{3 b^3} \]

[In]

Integrate[x^2*Cot[a + b*x],x]

[Out]

(I*b^3*x^3 + 3*b^2*x^2*Log[1 - E^((-I)*(a + b*x))] + 3*b^2*x^2*Log[1 + E^((-I)*(a + b*x))] + (6*I)*b*x*PolyLog
[2, -E^((-I)*(a + b*x))] + (6*I)*b*x*PolyLog[2, E^((-I)*(a + b*x))] + 6*PolyLog[3, -E^((-I)*(a + b*x))] + 6*Po
lyLog[3, E^((-I)*(a + b*x))])/(3*b^3)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (62 ) = 124\).

Time = 0.35 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.68

method	result	size
risch	\(-\frac {i x^{3}}{3}+\frac {4 i a^{3}}{3 b^{3}}+\frac {2 i a^{2} x}{b^{2}}+\frac {\ln \left (1+{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b}-\frac {2 i \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {2 \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {\ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b}-\frac {a^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {2 i \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {2 \operatorname {polylog}\left (3, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}}\)	\(198\)

[In]

int(x^2*cot(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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. 244 vs. \(2 (59) = 118\).

Time = 0.29 (sec) , antiderivative size = 244, normalized size of antiderivative = 3.30 \[ \int x^2 \cot (a+b x) \, dx=\frac {-2 i \, b x {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + 2 i \, b x {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + 2 \, a^{2} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) + 2 \, a^{2} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) - \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) + 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right )}{4 \, b^{3}} \]

[In]

integrate(x^2*cot(b*x+a),x, algorithm="fricas")

[Out]

1/4*(-2*I*b*x*dilog(cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a)) + 2*I*b*x*dilog(cos(2*b*x + 2*a) - I*sin(2*b*x + 2*
a)) + 2*a^2*log(-1/2*cos(2*b*x + 2*a) + 1/2*I*sin(2*b*x + 2*a) + 1/2) + 2*a^2*log(-1/2*cos(2*b*x + 2*a) - 1/2*
I*sin(2*b*x + 2*a) + 1/2) + 2*(b^2*x^2 - a^2)*log(-cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a) + 1) + 2*(b^2*x^2 - a
^2)*log(-cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a) + 1) + polylog(3, cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a)) + poly
log(3, cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a)))/b^3

Sympy [F]

\[ \int x^2 \cot (a+b x) \, dx=\int x^{2} \cot {\left (a + b x \right )}\, dx \]

[In]

integrate(x**2*cot(b*x+a),x)

[Out]

Integral(x**2*cot(a + b*x), 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. 257 vs. \(2 (59) = 118\).

Time = 0.32 (sec) , antiderivative size = 257, normalized size of antiderivative = 3.47 \[ \int x^2 \cot (a+b x) \, dx=-\frac {2 i \, {\left (b x + a\right )}^{3} - 6 i \, {\left (b x + a\right )}^{2} a + 12 i \, b x {\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) + 12 i \, b x {\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) - 6 \, a^{2} \log \left (\sin \left (b x + a\right )\right ) + 6 \, {\left (-i \, {\left (b x + a\right )}^{2} + 2 i \, {\left (b x + a\right )} a\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) + 6 \, {\left (i \, {\left (b x + a\right )}^{2} - 2 i \, {\left (b x + a\right )} a\right )} \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) - 3 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) - 3 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right ) - 12 \, {\rm Li}_{3}(-e^{\left (i \, b x + i \, a\right )}) - 12 \, {\rm Li}_{3}(e^{\left (i \, b x + i \, a\right )})}{6 \, b^{3}} \]

[In]

integrate(x^2*cot(b*x+a),x, algorithm="maxima")

[Out]

-1/6*(2*I*(b*x + a)^3 - 6*I*(b*x + a)^2*a + 12*I*b*x*dilog(-e^(I*b*x + I*a)) + 12*I*b*x*dilog(e^(I*b*x + I*a))
- 6*a^2*log(sin(b*x + a)) + 6*(-I*(b*x + a)^2 + 2*I*(b*x + a)*a)*arctan2(sin(b*x + a), cos(b*x + a) + 1) + 6*
(I*(b*x + a)^2 - 2*I*(b*x + a)*a)*arctan2(sin(b*x + a), -cos(b*x + a) + 1) - 3*((b*x + a)^2 - 2*(b*x + a)*a)*l
og(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) - 3*((b*x + a)^2 - 2*(b*x + a)*a)*log(cos(b*x + a)^2
+ sin(b*x + a)^2 - 2*cos(b*x + a) + 1) - 12*polylog(3, -e^(I*b*x + I*a)) - 12*polylog(3, e^(I*b*x + I*a)))/b^3

Giac [F]

\[ \int x^2 \cot (a+b x) \, dx=\int { x^{2} \cot \left (b x + a\right ) \,d x } \]

[In]

integrate(x^2*cot(b*x+a),x, algorithm="giac")

[Out]

integrate(x^2*cot(b*x + a), x)

Mupad [F(-1)]

Timed out. \[ \int x^2 \cot (a+b x) \, dx=\int x^2\,\mathrm {cot}\left (a+b\,x\right ) \,d x \]

[In]

int(x^2*cot(a + b*x),x)

[Out]

int(x^2*cot(a + b*x), x)