\(\int (c+d x)^3 \cos ^2(a+b x) \cot (a+b x) \, dx\) [165]

Optimal result

Integrand size = 22, antiderivative size = 246 \[ \int (c+d x)^3 \cos ^2(a+b x) \cot (a+b x) \, dx=-\frac {3 d^3 x}{8 b^3}+\frac {(c+d x)^3}{4 b}-\frac {i (c+d x)^4}{4 d}+\frac {(c+d x)^3 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{2 b^2}+\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{2 b^3}+\frac {3 i d^3 \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{4 b^4}+\frac {3 d^3 \cos (a+b x) \sin (a+b x)}{8 b^4}-\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{4 b^2}+\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b} \] Output:

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1956\) vs. \(2(246)=492\).

Time = 6.24 (sec) , antiderivative size = 1956, normalized size of antiderivative = 7.95 \[ \int (c+d x)^3 \cos ^2(a+b x) \cot (a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*(c*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))]))/b^3 - (d^3*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*Poly 
Log[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))*PolyLog[4, E^((-I)*(a + b*x))]))/(4*b^4) 
+ (c^3*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)) + Csc[a]*(Cos[2*a + 2*b*x]/(64*b^4) - ((I/6 
4)*Sin[2*a + 2*b*x])/b^4)*(32*b^4*c^3*x*Cos[a + 2*b*x] + 48*b^4*c^2*d*x^2* 
Cos[a + 2*b*x] + 32*b^4*c*d^2*x^3*Cos[a + 2*b*x] + 8*b^4*d^3*x^4*Cos[a + 2 
*b*x] + 32*b^4*c^3*x*Cos[3*a + 2*b*x] + 48*b^4*c^2*d*x^2*Cos[3*a + 2*b*x] 
+ 32*b^4*c*d^2*x^3*Cos[3*a + 2*b*x] + 8*b^4*d^3*x^4*Cos[3*a + 2*b*x] + (4* 
I)*b^3*c^3*Cos[3*a + 4*b*x] - 6*b^2*c^2*d*Cos[3*a + 4*b*x] - (6*I)*b*c*...
 

Rubi [A] (verified)

Time = 1.42 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.21, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {4908, 3042, 25, 4202, 2620, 3011, 4904, 3042, 3792, 17, 3042, 3115, 24, 7163, 2720, 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.

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

\(\Big \downarrow \) 4908

\(\displaystyle \int (c+d x)^3 \cot (a+b x)dx-\int (c+d x)^3 \cos (a+b x) \sin (a+b x)dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4202

\(\displaystyle 2 i \int \frac {e^{i (2 a+2 b x+\pi )} (c+d x)^3}{1+e^{i (2 a+2 b x+\pi )}}dx-\int (c+d x)^3 \cos (a+b x) \sin (a+b x)dx-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 2620

\(\displaystyle 2 i \left (\frac {3 i d \int (c+d x)^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\int (c+d x)^3 \cos (a+b x) \sin (a+b x)dx-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 4904

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3792

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

\(\Big \downarrow \) 17

\(\displaystyle \frac {3 d \left (-\frac {d^2 \int \sin ^2(a+b x)dx}{2 b^2}+\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}+2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 d \left (-\frac {d^2 \int \sin (a+b x)^2dx}{2 b^2}+\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}+2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {3 d \left (-\frac {d^2 \left (\frac {\int 1dx}{2}-\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}+\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}+2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 24

\(\displaystyle 2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {3 d \left (\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {d^2 \left (\frac {x}{2}-\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 7163

\(\displaystyle 2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i d \left (\frac {i d \int \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {3 d \left (\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {d^2 \left (\frac {x}{2}-\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 2720

\(\displaystyle 2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i d \left (\frac {d \int e^{-i (2 a+2 b x+\pi )} \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )de^{i (2 a+2 b x+\pi )}}{4 b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )+\frac {3 d \left (\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {d^2 \left (\frac {x}{2}-\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {3 d \left (\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {d^2 \left (\frac {x}{2}-\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}+2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i d \left (\frac {d \operatorname {PolyLog}\left (4,-e^{i (2 a+2 b x+\pi )}\right )}{4 b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {i (c+d x)^4}{4 d}\)

Input:

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

Output:

((-1/4*I)*(c + d*x)^4)/d + (2*I)*(((-1/2*I)*(c + d*x)^3*Log[1 + E^(I*(2*a 
+ Pi + 2*b*x))])/b + (((3*I)/2)*d*(((I/2)*(c + d*x)^2*PolyLog[2, -E^(I*(2* 
a + Pi + 2*b*x))])/b - (I*d*(((-1/2*I)*(c + d*x)*PolyLog[3, -E^(I*(2*a + P 
i + 2*b*x))])/b + (d*PolyLog[4, -E^(I*(2*a + Pi + 2*b*x))])/(4*b^2)))/b))/ 
b) - ((c + d*x)^3*Sin[a + b*x]^2)/(2*b) + (3*d*((c + d*x)^3/(6*d) - ((c + 
d*x)^2*Cos[a + b*x]*Sin[a + b*x])/(2*b) + (d*(c + d*x)*Sin[a + b*x]^2)/(2* 
b^2) - (d^2*(x/2 - (Cos[a + b*x]*Sin[a + b*x])/(2*b)))/(2*b^2)))/(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 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

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 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

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 4904

Int[Cos[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x 
_)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sin[a + b*x]^(n + 1)/(b*(n + 1))) 
, x] - Simp[d*(m/(b*(n + 1)))   Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n + 1), 
 x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
 

rule 4908

Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*Cot[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d 
_.)*(x_))^(m_.), x_Symbol] :> -Int[(c + d*x)^m*Cos[a + b*x]^n*Cot[a + b*x]^ 
(p - 2), x] + Int[(c + d*x)^m*Cos[a + b*x]^(n - 2)*Cot[a + b*x]^p, x] /; Fr 
eeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 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]
 

rule 7163

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] - Simp[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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 907 vs. \(2 (215 ) = 430\).

Time = 1.76 (sec) , antiderivative size = 908, normalized size of antiderivative = 3.69

method	result	size
risch	\(\text {Expression too large to display}\)	\(908\)

Input:

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

Output:

-3/b^3*c*d^2*ln(1-exp(I*(b*x+a)))*a^2-6/b^3*c*d^2*a^2*ln(exp(I*(b*x+a)))-6 
*I/b*d*c^2*a*x+6*I/b^2*c*d^2*a^2*x-6*I/b^2*c*d^2*polylog(2,-exp(I*(b*x+a)) 
)*x-6*I/b^2*c*d^2*polylog(2,exp(I*(b*x+a)))*x-3/16*d*(2*b^2*d^2*x^2+4*b^2* 
c*d*x+2*b^2*c^2-d^2)/b^4*sin(2*b*x+2*a)+I*c^3*x+1/4*I/d*c^4-3/2*I/b^4*d^3* 
a^4+6*I/b^4*d^3*polylog(4,-exp(I*(b*x+a)))-I*d^2*c*x^3-3/2*I*d*c^2*x^2+1/8 
/b^3*(2*b^2*d^3*x^3+6*b^2*c*d^2*x^2+6*b^2*c^2*d*x+2*b^2*c^3-3*d^3*x-3*c*d^ 
2)*cos(2*b*x+2*a)+2/b^4*d^3*a^3*ln(exp(I*(b*x+a)))-1/b^4*d^3*a^3*ln(exp(I* 
(b*x+a))-1)+6/b^3*c*d^2*polylog(3,-exp(I*(b*x+a)))+6/b^3*c*d^2*polylog(3,e 
xp(I*(b*x+a)))+1/b*d^3*ln(1-exp(I*(b*x+a)))*x^3+1/b*d^3*ln(exp(I*(b*x+a))+ 
1)*x^3+6/b^3*d^3*polylog(3,-exp(I*(b*x+a)))*x+6/b^3*d^3*polylog(3,exp(I*(b 
*x+a)))*x+1/b^4*d^3*ln(1-exp(I*(b*x+a)))*a^3+3/b^3*c*d^2*a^2*ln(exp(I*(b*x 
+a))-1)+6/b^2*c^2*d*a*ln(exp(I*(b*x+a)))-3/b^2*c^2*d*a*ln(exp(I*(b*x+a))-1 
)+3/b^2*d*c^2*ln(1-exp(I*(b*x+a)))*a+3/b*c*d^2*ln(exp(I*(b*x+a))+1)*x^2+3/ 
b*c*d^2*ln(1-exp(I*(b*x+a)))*x^2+3/b*d*c^2*ln(exp(I*(b*x+a))+1)*x+3/b*d*c^ 
2*ln(1-exp(I*(b*x+a)))*x-3*I/b^2*d^3*polylog(2,-exp(I*(b*x+a)))*x^2-3*I/b^ 
2*d^3*polylog(2,exp(I*(b*x+a)))*x^2-3*I/b^2*d*c^2*polylog(2,exp(I*(b*x+a)) 
)-3*I/b^2*d*c^2*a^2-3*I/b^2*d*c^2*polylog(2,-exp(I*(b*x+a)))-2*I/b^3*d^3*a 
^3*x+4*I/b^3*c*d^2*a^3-1/4*I*d^3*x^4-2/b*c^3*ln(exp(I*(b*x+a)))+1/b*c^3*ln 
(exp(I*(b*x+a))-1)+1/b*c^3*ln(exp(I*(b*x+a))+1)+6*I*d^3*polylog(4,exp(I*(b 
*x+a)))/b^4
 

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. 984 vs. \(2 (211) = 422\).

Time = 0.14 (sec) , antiderivative size = 984, normalized size of antiderivative = 4.00 \[ \int (c+d x)^3 \cos ^2(a+b x) \cot (a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/8*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 - 24*I*d^3*polylog(4, cos(b*x + a) + 
 I*sin(b*x + a)) + 24*I*d^3*polylog(4, cos(b*x + a) - I*sin(b*x + a)) + 24 
*I*d^3*polylog(4, -cos(b*x + a) + I*sin(b*x + a)) - 24*I*d^3*polylog(4, -c 
os(b*x + a) - I*sin(b*x + a)) - 2*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 2*b^3 
*c^3 - 3*b*c*d^2 + 3*(2*b^3*c^2*d - b*d^3)*x)*cos(b*x + a)^2 + 3*(2*b^2*d^ 
3*x^2 + 4*b^2*c*d^2*x + 2*b^2*c^2*d - d^3)*cos(b*x + a)*sin(b*x + a) + 3*( 
2*b^3*c^2*d - b*d^3)*x + 12*(I*b^2*d^3*x^2 + 2*I*b^2*c*d^2*x + I*b^2*c^2*d 
)*dilog(cos(b*x + a) + I*sin(b*x + a)) + 12*(-I*b^2*d^3*x^2 - 2*I*b^2*c*d^ 
2*x - I*b^2*c^2*d)*dilog(cos(b*x + a) - I*sin(b*x + a)) + 12*(-I*b^2*d^3*x 
^2 - 2*I*b^2*c*d^2*x - I*b^2*c^2*d)*dilog(-cos(b*x + a) + I*sin(b*x + a)) 
+ 12*(I*b^2*d^3*x^2 + 2*I*b^2*c*d^2*x + I*b^2*c^2*d)*dilog(-cos(b*x + a) - 
 I*sin(b*x + a)) - 4*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3* 
c^3)*log(cos(b*x + a) + I*sin(b*x + a) + 1) - 4*(b^3*d^3*x^3 + 3*b^3*c*d^2 
*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*log(cos(b*x + a) - I*sin(b*x + a) + 1) - 4 
*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*log(-1/2*cos(b*x + a) 
 + 1/2*I*sin(b*x + a) + 1/2) - 4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 
- a^3*d^3)*log(-1/2*cos(b*x + a) - 1/2*I*sin(b*x + a) + 1/2) - 4*(b^3*d^3* 
x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^ 
3*d^3)*log(-cos(b*x + a) + I*sin(b*x + a) + 1) - 4*(b^3*d^3*x^3 + 3*b^3*c* 
d^2*x^2 + 3*b^3*c^2*d*x + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*log(...
 

Sympy [F]

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

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

Output:

Integral((c + d*x)**3*cos(a + b*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. 979 vs. \(2 (211) = 422\).

Time = 0.20 (sec) , antiderivative size = 979, normalized size of antiderivative = 3.98 \[ \int (c+d x)^3 \cos ^2(a+b x) \cot (a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/16*(8*(sin(b*x + a)^2 - log(sin(b*x + a)^2))*c^3 - 24*(sin(b*x + a)^2 - 
 log(sin(b*x + a)^2))*a*c^2*d/b + 24*(sin(b*x + a)^2 - log(sin(b*x + a)^2) 
)*a^2*c*d^2/b^2 - 8*(sin(b*x + a)^2 - log(sin(b*x + a)^2))*a^3*d^3/b^3 - ( 
-4*I*(b*x + a)^4*d^3 - 16*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^3 + 96*I*d^3*pol 
ylog(4, -e^(I*b*x + I*a)) + 96*I*d^3*polylog(4, e^(I*b*x + I*a)) - 24*(I*b 
^2*c^2*d - 2*I*a*b*c*d^2 + I*a^2*d^3)*(b*x + a)^2 - 16*(-I*(b*x + a)^3*d^3 
 + 3*(-I*b*c*d^2 + I*a*d^3)*(b*x + a)^2 + 3*(-I*b^2*c^2*d + 2*I*a*b*c*d^2 
- I*a^2*d^3)*(b*x + a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) - 16*(I*(b 
*x + a)^3*d^3 + 3*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 + 3*(I*b^2*c^2*d - 2*I 
*a*b*c*d^2 + I*a^2*d^3)*(b*x + a))*arctan2(sin(b*x + a), -cos(b*x + a) + 1 
) + 2*(2*(b*x + a)^3*d^3 - 3*b*c*d^2 + 3*a*d^3 + 6*(b*c*d^2 - a*d^3)*(b*x 
+ a)^2 + 3*(2*b^2*c^2*d - 4*a*b*c*d^2 + (2*a^2 - 1)*d^3)*(b*x + a))*cos(2* 
b*x + 2*a) - 48*(I*b^2*c^2*d - 2*I*a*b*c*d^2 + I*(b*x + a)^2*d^3 + I*a^2*d 
^3 + 2*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*dilog(-e^(I*b*x + I*a)) - 48*(I*b^ 
2*c^2*d - 2*I*a*b*c*d^2 + I*(b*x + a)^2*d^3 + I*a^2*d^3 + 2*(I*b*c*d^2 - I 
*a*d^3)*(b*x + a))*dilog(e^(I*b*x + I*a)) + 8*((b*x + a)^3*d^3 + 3*(b*c*d^ 
2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(b*x + a))* 
log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) + 8*((b*x + a)^3 
*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2* 
d^3)*(b*x + a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + ...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c+d x)^3 \cos ^2(a+b x) \cot (a+b x) \, dx=\frac {2 \left (\int \cos \left (b x +a \right )^{2} \cot \left (b x +a \right ) x^{3}d x \right ) b \,d^{3}+6 \left (\int \cos \left (b x +a \right )^{2} \cot \left (b x +a \right ) x^{2}d x \right ) b c \,d^{2}+6 \left (\int \cos \left (b x +a \right )^{2} \cot \left (b x +a \right ) x d x \right ) b \,c^{2} d -2 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1\right ) c^{3}+2 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) c^{3}-\sin \left (b x +a \right )^{2} c^{3}}{2 b} \] Input:

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

Output:

(2*int(cos(a + b*x)**2*cot(a + b*x)*x**3,x)*b*d**3 + 6*int(cos(a + b*x)**2 
*cot(a + b*x)*x**2,x)*b*c*d**2 + 6*int(cos(a + b*x)**2*cot(a + b*x)*x,x)*b 
*c**2*d - 2*log(tan((a + b*x)/2)**2 + 1)*c**3 + 2*log(tan((a + b*x)/2))*c* 
*3 - sin(a + b*x)**2*c**3)/(2*b)