\(\int x^2 \cot ^3(a+b x) \, dx\) [12]

Optimal result

Integrand size = 12, antiderivative size = 126 \[ \int x^2 \cot ^3(a+b x) \, dx=-\frac {x^2}{2 b}+\frac {i x^3}{3}-\frac {x \cot (a+b x)}{b^2}-\frac {x^2 \cot ^2(a+b x)}{2 b}-\frac {x^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}+\frac {\log (\sin (a+b x))}{b^3}+\frac {i x \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac {\operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3} \] Output:

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

Mathematica [A] (verified)

Time = 3.38 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.75 \[ \int x^2 \cot ^3(a+b x) \, dx=-\frac {6 b x \cot (a)+2 b^3 x^3 \cot (a)+3 b^2 x^2 \csc ^2(a+b x)-6 \log (\sin (a+b x))+2 e^{-i a} (i+\cot (a)) \left (i b^3 x^3-b^3 x^3 \cot (a)+3 b^2 x^2 \log \left (1-e^{-i (a+b x)}\right )+3 b^2 x^2 \log \left (1+e^{-i (a+b x)}\right )+6 i b x \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )+6 i b x \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+6 \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )\right ) \sin (a)-6 b x \csc (a) \csc (a+b x) \sin (b x)}{6 b^3} \] Input:

Integrate[x^2*Cot[a + b*x]^3,x]
 

Output:

-1/6*(6*b*x*Cot[a] + 2*b^3*x^3*Cot[a] + 3*b^2*x^2*Csc[a + b*x]^2 - 6*Log[S 
in[a + b*x]] + (2*(I + Cot[a])*(I*b^3*x^3 - b^3*x^3*Cot[a] + 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*PolyLog[3, E^((-I)*(a + b*x))] 
)*Sin[a])/E^(I*a) - 6*b*x*Csc[a]*Csc[a + b*x]*Sin[b*x])/b^3
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.29, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.417, Rules used = {3042, 25, 4203, 25, 3042, 25, 4202, 2620, 3011, 2720, 4203, 15, 25, 3042, 25, 3956, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 \cot ^3(a+b x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -x^2 \tan \left (a+b x+\frac {\pi }{2}\right )^3dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int x^2 \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )^3dx\)

\(\Big \downarrow \) 4203

\(\displaystyle \int -x^2 \cot (a+b x)dx+\frac {\int x \cot ^2(a+b x)dx}{b}-\frac {x^2 \cot ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\int x^2 \cot (a+b x)dx+\frac {\int x \cot ^2(a+b x)dx}{b}-\frac {x^2 \cot ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\int -x^2 \tan \left (a+b x+\frac {\pi }{2}\right )dx+\frac {\int x \tan \left (a+b x+\frac {\pi }{2}\right )^2dx}{b}-\frac {x^2 \cot ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 25

\(\displaystyle \int x^2 \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )dx+\frac {\int x \tan \left (a+b x+\frac {\pi }{2}\right )^2dx}{b}-\frac {x^2 \cot ^2(a+b x)}{2 b}\)

\(\Big \downarrow \) 4202

\(\displaystyle -2 i \int \frac {e^{i (2 a+2 b x+\pi )} x^2}{1+e^{i (2 a+2 b x+\pi )}}dx+\frac {\int x \tan \left (a+b x+\frac {\pi }{2}\right )^2dx}{b}-\frac {x^2 \cot ^2(a+b x)}{2 b}+\frac {i x^3}{3}\)

\(\Big \downarrow \) 2620

\(\displaystyle -2 i \left (\frac {i \int x \log \left (1+e^{i (2 a+2 b x+\pi )}\right )dx}{b}-\frac {i x^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {\int x \tan \left (a+b x+\frac {\pi }{2}\right )^2dx}{b}-\frac {x^2 \cot ^2(a+b x)}{2 b}+\frac {i x^3}{3}\)

\(\Big \downarrow \) 3011

\(\displaystyle -2 i \left (\frac {i \left (\frac {i x \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i \int \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}\right )}{b}-\frac {i x^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {\int x \tan \left (a+b x+\frac {\pi }{2}\right )^2dx}{b}-\frac {x^2 \cot ^2(a+b x)}{2 b}+\frac {i x^3}{3}\)

\(\Big \downarrow \) 2720

\(\displaystyle -2 i \left (\frac {i \left (\frac {i x \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {\int e^{-i (2 a+2 b x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )de^{i (2 a+2 b x+\pi )}}{4 b^2}\right )}{b}-\frac {i x^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {\int x \tan \left (a+b x+\frac {\pi }{2}\right )^2dx}{b}-\frac {x^2 \cot ^2(a+b x)}{2 b}+\frac {i x^3}{3}\)

\(\Big \downarrow \) 4203

\(\displaystyle -2 i \left (\frac {i \left (\frac {i x \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {\int e^{-i (2 a+2 b x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )de^{i (2 a+2 b x+\pi )}}{4 b^2}\right )}{b}-\frac {i x^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {-\frac {\int -\cot (a+b x)dx}{b}-\int xdx-\frac {x \cot (a+b x)}{b}}{b}-\frac {x^2 \cot ^2(a+b x)}{2 b}+\frac {i x^3}{3}\)

\(\Big \downarrow \) 15

\(\displaystyle -2 i \left (\frac {i \left (\frac {i x \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {\int e^{-i (2 a+2 b x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )de^{i (2 a+2 b x+\pi )}}{4 b^2}\right )}{b}-\frac {i x^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {-\frac {\int -\cot (a+b x)dx}{b}-\frac {x \cot (a+b x)}{b}-\frac {x^2}{2}}{b}-\frac {x^2 \cot ^2(a+b x)}{2 b}+\frac {i x^3}{3}\)

\(\Big \downarrow \) 25

\(\displaystyle -2 i \left (\frac {i \left (\frac {i x \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {\int e^{-i (2 a+2 b x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )de^{i (2 a+2 b x+\pi )}}{4 b^2}\right )}{b}-\frac {i x^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {\frac {\int \cot (a+b x)dx}{b}-\frac {x \cot (a+b x)}{b}-\frac {x^2}{2}}{b}-\frac {x^2 \cot ^2(a+b x)}{2 b}+\frac {i x^3}{3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -2 i \left (\frac {i \left (\frac {i x \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {\int e^{-i (2 a+2 b x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )de^{i (2 a+2 b x+\pi )}}{4 b^2}\right )}{b}-\frac {i x^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {\frac {\int -\tan \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {x \cot (a+b x)}{b}-\frac {x^2}{2}}{b}-\frac {x^2 \cot ^2(a+b x)}{2 b}+\frac {i x^3}{3}\)

\(\Big \downarrow \) 25

\(\displaystyle -2 i \left (\frac {i \left (\frac {i x \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {\int e^{-i (2 a+2 b x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )de^{i (2 a+2 b x+\pi )}}{4 b^2}\right )}{b}-\frac {i x^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {-\frac {\int \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )dx}{b}-\frac {x \cot (a+b x)}{b}-\frac {x^2}{2}}{b}-\frac {x^2 \cot ^2(a+b x)}{2 b}+\frac {i x^3}{3}\)

\(\Big \downarrow \) 3956

\(\displaystyle -2 i \left (\frac {i \left (\frac {i x \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {\int e^{-i (2 a+2 b x+\pi )} \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )de^{i (2 a+2 b x+\pi )}}{4 b^2}\right )}{b}-\frac {i x^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {\frac {\log (-\sin (a+b x))}{b^2}-\frac {x \cot (a+b x)}{b}-\frac {x^2}{2}}{b}-\frac {x^2 \cot ^2(a+b x)}{2 b}+\frac {i x^3}{3}\)

\(\Big \downarrow \) 7143

\(\displaystyle -2 i \left (\frac {i \left (\frac {i x \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {\operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{4 b^2}\right )}{b}-\frac {i x^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {\frac {\log (-\sin (a+b x))}{b^2}-\frac {x \cot (a+b x)}{b}-\frac {x^2}{2}}{b}-\frac {x^2 \cot ^2(a+b x)}{2 b}+\frac {i x^3}{3}\)

Input:

Int[x^2*Cot[a + b*x]^3,x]
 

Output:

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

Defintions of rubi rules used

rule 15

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

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2620

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] - Si 
mp[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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{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 3011

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] + Simp[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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3956

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d 
*x], x]]/d, x] /; FreeQ[{c, d}, x]
 

rule 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^m*(E^(2*I*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

rule 4203

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

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (110 ) = 220\).

Time = 0.32 (sec) , antiderivative size = 293, normalized size of antiderivative = 2.33

method	result	size
risch	\(-\frac {4 i a^{3}}{3 b^{3}}+\frac {2 x \left (b x \,{\mathrm e}^{2 i \left (b x +a \right )}-i {\mathrm e}^{2 i \left (b x +a \right )}+i\right )}{b^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}-\frac {2 \operatorname {polylog}\left (3, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {2 \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}}-\frac {2 \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{3}}+\frac {a^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {i x^{3}}{3}+\frac {2 i \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}}+\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {\ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b}-\frac {2 i a^{2} x}{b^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{2}}{b}+\frac {2 i \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}\)	\(293\)

Input:

int(x^2*cot(b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

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

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. 425 vs. \(2 (107) = 214\).

Time = 0.09 (sec) , antiderivative size = 425, normalized size of antiderivative = 3.37 \[ \int x^2 \cot ^3(a+b x) \, dx=\frac {4 \, b^{2} x^{2} + 4 \, b x \sin \left (2 \, b x + 2 \, a\right ) - 2 \, {\left (-i \, b x \cos \left (2 \, b x + 2 \, a\right ) + i \, b x\right )} {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) - 2 \, {\left (i \, b x \cos \left (2 \, b x + 2 \, a\right ) - i \, b x\right )} {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) + 2 \, {\left (a^{2} - {\left (a^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) + 2 \, {\left (a^{2} - {\left (a^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) - \frac {1}{2} i \, \sin \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) + 2 \, {\left (b^{2} x^{2} - a^{2} - {\left (b^{2} x^{2} - a^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 2 \, {\left (b^{2} x^{2} - a^{2} - {\left (b^{2} x^{2} - a^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) - {\left (\cos \left (2 \, b x + 2 \, a\right ) - 1\right )} {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) - {\left (\cos \left (2 \, b x + 2 \, a\right ) - 1\right )} {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right )}{4 \, {\left (b^{3} \cos \left (2 \, b x + 2 \, a\right ) - b^{3}\right )}} \] Input:

integrate(x^2*cot(b*x+a)^3,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int x^2 \cot ^3(a+b x) \, dx=\int x^{2} \cot ^{3}{\left (a + b x \right )}\, dx \] Input:

integrate(x**2*cot(b*x+a)**3,x)
                                                                                    
                                                                                    
 

Output:

Integral(x**2*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. 1208 vs. \(2 (107) = 214\).

Time = 0.19 (sec) , antiderivative size = 1208, normalized size of antiderivative = 9.59 \[ \int x^2 \cot ^3(a+b x) \, dx=\text {Too large to display} \] Input:

integrate(x^2*cot(b*x+a)^3,x, algorithm="maxima")
 

Output:

-1/2*(a^2*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2)) - 2*(2*(b*x + a)^3 - 6* 
(b*x + a)^2*a - 6*((b*x + a)^2 - 2*(b*x + a)*a + ((b*x + a)^2 - 2*(b*x + a 
)*a - 1)*cos(4*b*x + 4*a) - 2*((b*x + a)^2 - 2*(b*x + a)*a - 1)*cos(2*b*x 
+ 2*a) - (-I*(b*x + a)^2 + 2*I*(b*x + a)*a + I)*sin(4*b*x + 4*a) - 2*(I*(b 
*x + a)^2 - 2*I*(b*x + a)*a - I)*sin(2*b*x + 2*a) - 1)*arctan2(sin(b*x + a 
), cos(b*x + a) + 1) + 6*(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)*arctan2(sin(b*x + a), cos(b*x + a) 
- 1) + 6*((b*x + a)^2 - 2*(b*x + a)*a + ((b*x + a)^2 - 2*(b*x + a)*a)*cos( 
4*b*x + 4*a) - 2*((b*x + a)^2 - 2*(b*x + a)*a)*cos(2*b*x + 2*a) + (I*(b*x 
+ a)^2 - 2*I*(b*x + a)*a)*sin(4*b*x + 4*a) + 2*(-I*(b*x + a)^2 + 2*I*(b*x 
+ a)*a)*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), -cos(b*x + a) + 1) + 2*((b 
*x + a)^3 - 3*(b*x + a)^2*a - 6*b*x - 6*a)*cos(4*b*x + 4*a) - 4*((b*x + a) 
^3 - 3*(b*x + a)^2*(a - I) - 3*(b*x + a)*(2*I*a + 1) - 3*a)*cos(2*b*x + 2* 
a) + 12*(b*x*cos(4*b*x + 4*a) - 2*b*x*cos(2*b*x + 2*a) + I*b*x*sin(4*b*x + 
 4*a) - 2*I*b*x*sin(2*b*x + 2*a) + b*x)*dilog(-e^(I*b*x + I*a)) + 12*(b*x* 
cos(4*b*x + 4*a) - 2*b*x*cos(2*b*x + 2*a) + I*b*x*sin(4*b*x + 4*a) - 2*I*b 
*x*sin(2*b*x + 2*a) + b*x)*dilog(e^(I*b*x + I*a)) + 3*(I*(b*x + a)^2 - 2*I 
*(b*x + a)*a + (I*(b*x + a)^2 - 2*I*(b*x + a)*a - I)*cos(4*b*x + 4*a) + 2* 
(-I*(b*x + a)^2 + 2*I*(b*x + a)*a + I)*cos(2*b*x + 2*a) - ((b*x + a)^2 - 2 
*(b*x + a)*a - 1)*sin(4*b*x + 4*a) + 2*((b*x + a)^2 - 2*(b*x + a)*a - 1...
 

Giac [F]

\[ \int x^2 \cot ^3(a+b x) \, dx=\int { x^{2} \cot \left (b x + a\right )^{3} \,d x } \] Input:

integrate(x^2*cot(b*x+a)^3,x, algorithm="giac")
 

Output:

integrate(x^2*cot(b*x + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \cot ^3(a+b x) \, dx=\int x^2\,{\mathrm {cot}\left (a+b\,x\right )}^3 \,d x \] Input:

int(x^2*cot(a + b*x)^3,x)
 

Output:

int(x^2*cot(a + b*x)^3, x)
 

Reduce [F]

\[ \int x^2 \cot ^3(a+b x) \, dx=\int \cot \left (b x +a \right )^{3} x^{2}d x \] Input:

int(x^2*cot(b*x+a)^3,x)
 

Output:

int(cot(a + b*x)**3*x**2,x)