\(\int (c+d x)^2 \cot ^3(a+b x) \, dx\) [180]

Optimal result

Integrand size = 16, antiderivative size = 161 \[ \int (c+d x)^2 \cot ^3(a+b x) \, dx=-\frac {(c+d x)^2}{2 b}+\frac {i (c+d x)^3}{3 d}-\frac {d (c+d x) \cot (a+b x)}{b^2}-\frac {(c+d x)^2 \cot ^2(a+b x)}{2 b}-\frac {(c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b}+\frac {d^2 \log (\sin (a+b x))}{b^3}+\frac {i d (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac {d^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3} \] Output:

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(548\) vs. \(2(161)=322\).

Time = 6.49 (sec) , antiderivative size = 548, normalized size of antiderivative = 3.40 \[ \int (c+d x)^2 \cot ^3(a+b x) \, dx=-\frac {1}{3} x \left (3 c^2+3 c d x+d^2 x^2\right ) \cot (a)-\frac {(c+d x)^2 \csc ^2(a+b x)}{2 b}+\frac {d^2 e^{i a} \csc (a) \left (2 b^3 e^{-2 i a} x^3+3 i b^2 \left (1-e^{-2 i a}\right ) x^2 \log \left (1-e^{-i (a+b x)}\right )+3 i b^2 \left (1-e^{-2 i a}\right ) x^2 \log \left (1+e^{-i (a+b x)}\right )-6 b \left (1-e^{-2 i a}\right ) x \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )-6 b \left (1-e^{-2 i a}\right ) x \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )+6 i \left (1-e^{-2 i a}\right ) \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+6 i \left (1-e^{-2 i a}\right ) \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )\right )}{6 b^3}-\frac {c^2 \csc (a) (-b x \cos (a)+\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a))}{b \left (\cos ^2(a)+\sin ^2(a)\right )}+\frac {d^2 \csc (a) (-b x \cos (a)+\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a))}{b^3 \left (\cos ^2(a)+\sin ^2(a)\right )}+\frac {\csc (a) \csc (a+b x) \left (c d \sin (b x)+d^2 x \sin (b x)\right )}{b^2}+\frac {c d \csc (a) \sec (a) \left (b^2 e^{i \arctan (\tan (a))} x^2+\frac {\left (i b x (-\pi +2 \arctan (\tan (a)))-\pi \log \left (1+e^{-2 i b x}\right )-2 (b x+\arctan (\tan (a))) \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )+\pi \log (\cos (b x))+2 \arctan (\tan (a)) \log (\sin (b x+\arctan (\tan (a))))+i \operatorname {PolyLog}\left (2,e^{2 i (b x+\arctan (\tan (a)))}\right )\right ) \tan (a)}{\sqrt {1+\tan ^2(a)}}\right )}{b^2 \sqrt {\sec ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}} \] Input:

Integrate[(c + d*x)^2*Cot[a + b*x]^3,x]
 

Output:

-1/3*(x*(3*c^2 + 3*c*d*x + d^2*x^2)*Cot[a]) - ((c + d*x)^2*Csc[a + b*x]^2) 
/(2*b) + (d^2*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^( 
(-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x))] + (3*I)*b^2*(1 - E^((-2*I)*a))* 
x^2*Log[1 + E^((-I)*(a + b*x))] - 6*b*(1 - E^((-2*I)*a))*x*PolyLog[2, -E^( 
(-I)*(a + b*x))] - 6*b*(1 - E^((-2*I)*a))*x*PolyLog[2, E^((-I)*(a + b*x))] 
 + (6*I)*(1 - E^((-2*I)*a))*PolyLog[3, -E^((-I)*(a + b*x))] + (6*I)*(1 - E 
^((-2*I)*a))*PolyLog[3, E^((-I)*(a + b*x))]))/(6*b^3) - (c^2*Csc[a]*(-(b*x 
*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b*(Cos[a]^2 + 
Sin[a]^2)) + (d^2*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin 
[b*x]]*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) + (Csc[a]*Csc[a + b*x]*(c*d*Si 
n[b*x] + d^2*x*Sin[b*x]))/b^2 + (c*d*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]))/(b^2*Sqr 
t[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.22, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.062, Rules used = {3042, 25, 4203, 25, 3042, 25, 4202, 2620, 3011, 2720, 4203, 17, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4203

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 4203

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3956

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

\(\Big \downarrow \) 7143

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

Input:

Int[(c + d*x)^2*Cot[a + b*x]^3,x]
 

Output:

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

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, 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. 643 vs. \(2 (145 ) = 290\).

Time = 0.37 (sec) , antiderivative size = 644, normalized size of antiderivative = 4.00

method	result	size
risch	\(\frac {2 c d a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{2}}-\frac {4 c d a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {2 b \,d^{2} x^{2} {\mathrm e}^{2 i \left (b x +a \right )}+4 b c d x \,{\mathrm e}^{2 i \left (b x +a \right )}+2 b \,c^{2} {\mathrm e}^{2 i \left (b x +a \right )}-2 i d^{2} x \,{\mathrm e}^{2 i \left (b x +a \right )}-2 i c d \,{\mathrm e}^{2 i \left (b x +a \right )}+2 i d^{2} x +2 i d c}{b^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}+\frac {4 i c d a x}{b}-\frac {2 i d^{2} a^{2} x}{b^{2}}+\frac {2 i d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {2 i d^{2} \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+i d c \,x^{2}+\frac {2 d^{2} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {d^{2} a^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}}-\frac {d^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b}-\frac {2 d^{2} \operatorname {polylog}\left (3, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {d^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}}+\frac {d^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{3}}+\frac {i d^{2} x^{3}}{3}-\frac {2 c d \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b}-\frac {2 d c \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b}-\frac {i c^{3}}{3 d}+\frac {2 c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b}-\frac {c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b}-\frac {c^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b}-\frac {2 d^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {d^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a^{2}}{b^{3}}-\frac {d^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{2}}{b}-\frac {4 i d^{2} a^{3}}{3 b^{3}}+\frac {2 i d c \,a^{2}}{b^{2}}-\frac {2 d c \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{2}}+\frac {2 i d c \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {2 i d c \operatorname {polylog}\left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}-i c^{2} x -\frac {2 d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}\)	\(644\)

Input:

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

Output:

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

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. 657 vs. \(2 (142) = 284\).

Time = 0.09 (sec) , antiderivative size = 657, normalized size of antiderivative = 4.08 \[ \int (c+d x)^2 \cot ^3(a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*(4*b^2*d^2*x^2 + 8*b^2*c*d*x + 4*b^2*c^2 - 2*(I*b*d^2*x + I*b*c*d + (- 
I*b*d^2*x - I*b*c*d)*cos(2*b*x + 2*a))*dilog(cos(2*b*x + 2*a) + I*sin(2*b* 
x + 2*a)) - 2*(-I*b*d^2*x - I*b*c*d + (I*b*d^2*x + I*b*c*d)*cos(2*b*x + 2* 
a))*dilog(cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a)) + 2*(b^2*c^2 - 2*a*b*c*d 
+ (a^2 - 1)*d^2 - (b^2*c^2 - 2*a*b*c*d + (a^2 - 1)*d^2)*cos(2*b*x + 2*a))* 
log(-1/2*cos(2*b*x + 2*a) + 1/2*I*sin(2*b*x + 2*a) + 1/2) + 2*(b^2*c^2 - 2 
*a*b*c*d + (a^2 - 1)*d^2 - (b^2*c^2 - 2*a*b*c*d + (a^2 - 1)*d^2)*cos(2*b*x 
 + 2*a))*log(-1/2*cos(2*b*x + 2*a) - 1/2*I*sin(2*b*x + 2*a) + 1/2) + 2*(b^ 
2*d^2*x^2 + 2*b^2*c*d*x + 2*a*b*c*d - a^2*d^2 - (b^2*d^2*x^2 + 2*b^2*c*d*x 
 + 2*a*b*c*d - a^2*d^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*d^2*x^2 + 2*b^2*c*d*x + 2*a*b*c*d - a^2*d^2 - (b^ 
2*d^2*x^2 + 2*b^2*c*d*x + 2*a*b*c*d - a^2*d^2)*cos(2*b*x + 2*a))*log(-cos( 
2*b*x + 2*a) - I*sin(2*b*x + 2*a) + 1) - (d^2*cos(2*b*x + 2*a) - d^2)*poly 
log(3, cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a)) - (d^2*cos(2*b*x + 2*a) - d^ 
2)*polylog(3, cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a)) + 4*(b*d^2*x + b*c*d) 
*sin(2*b*x + 2*a))/(b^3*cos(2*b*x + 2*a) - b^3)
 

Sympy [F]

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

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

Output:

Integral((c + d*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. 1966 vs. \(2 (142) = 284\).

Time = 0.30 (sec) , antiderivative size = 1966, normalized size of antiderivative = 12.21 \[ \int (c+d x)^2 \cot ^3(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(c^2*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2)) - 2*a*c*d*(1/sin(b*x + 
a)^2 + log(sin(b*x + a)^2))/b + a^2*d^2*(1/sin(b*x + a)^2 + log(sin(b*x + 
a)^2))/b^2 - 2*(2*(b*x + a)^3*d^2 + 6*(b*c*d - a*d^2)*(b*x + a)^2 + 12*b*c 
*d - 12*a*d^2 - 6*((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - d^2 + ( 
(b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - d^2)*cos(4*b*x + 4*a) - 2* 
((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - d^2)*cos(2*b*x + 2*a) - ( 
-I*(b*x + a)^2*d^2 + 2*(-I*b*c*d + I*a*d^2)*(b*x + a) + I*d^2)*sin(4*b*x + 
 4*a) - 2*(I*(b*x + a)^2*d^2 + 2*(I*b*c*d - I*a*d^2)*(b*x + a) - I*d^2)*si 
n(2*b*x + 2*a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) + 6*(d^2*cos(4*b*x 
 + 4*a) - 2*d^2*cos(2*b*x + 2*a) + I*d^2*sin(4*b*x + 4*a) - 2*I*d^2*sin(2* 
b*x + 2*a) + d^2)*arctan2(sin(b*x + a), cos(b*x + a) - 1) + 6*((b*x + a)^2 
*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) + ((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)* 
(b*x + a))*cos(4*b*x + 4*a) - 2*((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x 
+ a))*cos(2*b*x + 2*a) + (I*(b*x + a)^2*d^2 + 2*(I*b*c*d - I*a*d^2)*(b*x + 
 a))*sin(4*b*x + 4*a) + 2*(-I*(b*x + a)^2*d^2 + 2*(-I*b*c*d + I*a*d^2)*(b* 
x + a))*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), -cos(b*x + a) + 1) + 2*((b 
*x + a)^3*d^2 + 3*(b*c*d - a*d^2)*(b*x + a)^2 - 6*(b*x + a)*d^2)*cos(4*b*x 
 + 4*a) - 4*((b*x + a)^3*d^2 + 3*(b*c*d - (a - I)*d^2)*(b*x + a)^2 + 3*b*c 
*d - 3*a*d^2 - 3*(-2*I*b*c*d + (2*I*a + 1)*d^2)*(b*x + a))*cos(2*b*x + 2*a 
) + 12*(b*c*d + (b*x + a)*d^2 - a*d^2 + (b*c*d + (b*x + a)*d^2 - a*d^2)...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c+d x)^2 \cot ^3(a+b x) \, dx=\frac {4 \left (\int \cot \left (b x +a \right )^{3} x^{2}d x \right ) \sin \left (b x +a \right )^{2} b \,d^{2}+8 \left (\int \cot \left (b x +a \right )^{3} x d x \right ) \sin \left (b x +a \right )^{2} b c d +4 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1\right ) \sin \left (b x +a \right )^{2} c^{2}-4 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sin \left (b x +a \right )^{2} c^{2}+\sin \left (b x +a \right )^{2} c^{2}-2 c^{2}}{4 \sin \left (b x +a \right )^{2} b} \] Input:

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

Output:

(4*int(cot(a + b*x)**3*x**2,x)*sin(a + b*x)**2*b*d**2 + 8*int(cot(a + b*x) 
**3*x,x)*sin(a + b*x)**2*b*c*d + 4*log(tan((a + b*x)/2)**2 + 1)*sin(a + b* 
x)**2*c**2 - 4*log(tan((a + b*x)/2))*sin(a + b*x)**2*c**2 + sin(a + b*x)** 
2*c**2 - 2*c**2)/(4*sin(a + b*x)**2*b)