\(\int (c+d x)^4 \cot ^2(a+b x) \, dx\) [105]

Optimal result

Integrand size = 16, antiderivative size = 155 \[ \int (c+d x)^4 \cot ^2(a+b x) \, dx=-\frac {i (c+d x)^4}{b}-\frac {(c+d x)^5}{5 d}-\frac {(c+d x)^4 \cot (a+b x)}{b}+\frac {4 d (c+d x)^3 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {6 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^3}+\frac {6 d^3 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^4}+\frac {3 i d^4 \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^5} \] Output:

-I*(d*x+c)^4/b-1/5*(d*x+c)^5/d-(d*x+c)^4*cot(b*x+a)/b+4*d*(d*x+c)^3*ln(1-e 
xp(2*I*(b*x+a)))/b^2-6*I*d^2*(d*x+c)^2*polylog(2,exp(2*I*(b*x+a)))/b^3+6*d 
^3*(d*x+c)*polylog(3,exp(2*I*(b*x+a)))/b^4+3*I*d^4*polylog(4,exp(2*I*(b*x+ 
a)))/b^5
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(598\) vs. \(2(155)=310\).

Time = 5.31 (sec) , antiderivative size = 598, normalized size of antiderivative = 3.86 \[ \int (c+d x)^4 \cot ^2(a+b x) \, dx=-\frac {1}{5} x \left (5 c^4+10 c^3 d x+10 c^2 d^2 x^2+5 c d^3 x^3+d^4 x^4\right )-\frac {4 c^3 d (b x \cot (a)-\log (\sin (a+b x)))}{b^2}+\frac {6 c^2 d^2 \left (i b x (\pi -2 \arctan (\tan (a)))+\pi \log \left (1+e^{-2 i b x}\right )+2 (b x+\arctan (\tan (a))) \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )-\pi \log (\cos (b x))-2 \arctan (\tan (a)) \log (\sin (b x+\arctan (\tan (a))))-i \operatorname {PolyLog}\left (2,e^{2 i (b x+\arctan (\tan (a)))}\right )-b^2 e^{i \arctan (\tan (a))} x^2 \cot (a) \sqrt {\sec ^2(a)}\right )}{b^3}+\frac {4 c d^3 e^{-i a} (i+\cot (a)) \left (i b^3 x^3-b^3 x^3 \cot (a)+3 b^2 x^2 \log \left (1-e^{-i (a+b x)}\right )+3 b^2 x^2 \log \left (1+e^{-i (a+b x)}\right )+6 i b x \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )+6 i b x \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+6 \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )\right ) \sin (a)}{b^4}+\frac {d^4 e^{-i a} (i+\cot (a)) \left (i b^4 x^4-b^4 x^4 \cot (a)+4 b^3 x^3 \log \left (1-e^{-i (a+b x)}\right )+4 b^3 x^3 \log \left (1+e^{-i (a+b x)}\right )+12 i b^2 x^2 \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )+12 i b^2 x^2 \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )+24 b x \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+24 b x \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )-24 i \operatorname {PolyLog}\left (4,-e^{-i (a+b x)}\right )-24 i \operatorname {PolyLog}\left (4,e^{-i (a+b x)}\right )\right ) \sin (a)}{b^5}+\frac {(c+d x)^4 \csc (a) \csc (a+b x) \sin (b x)}{b} \] Input:

Integrate[(c + d*x)^4*Cot[a + b*x]^2,x]
 

Output:

-1/5*(x*(5*c^4 + 10*c^3*d*x + 10*c^2*d^2*x^2 + 5*c*d^3*x^3 + d^4*x^4)) - ( 
4*c^3*d*(b*x*Cot[a] - Log[Sin[a + b*x]]))/b^2 + (6*c^2*d^2*(I*b*x*(Pi - 2* 
ArcTan[Tan[a]]) + Pi*Log[1 + E^((-2*I)*b*x)] + 2*(b*x + ArcTan[Tan[a]])*Lo 
g[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[T 
an[a]]))] - b^2*E^(I*ArcTan[Tan[a]])*x^2*Cot[a]*Sqrt[Sec[a]^2]))/b^3 + (4* 
c*d^3*(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*Poly 
Log[3, -E^((-I)*(a + b*x))] + 6*PolyLog[3, E^((-I)*(a + b*x))])*Sin[a])/(b 
^4*E^(I*a)) + (d^4*(I + Cot[a])*(I*b^4*x^4 - b^4*x^4*Cot[a] + 4*b^3*x^3*Lo 
g[1 - E^((-I)*(a + b*x))] + 4*b^3*x^3*Log[1 + E^((-I)*(a + b*x))] + (12*I) 
*b^2*x^2*PolyLog[2, -E^((-I)*(a + b*x))] + (12*I)*b^2*x^2*PolyLog[2, E^((- 
I)*(a + b*x))] + 24*b*x*PolyLog[3, -E^((-I)*(a + b*x))] + 24*b*x*PolyLog[3 
, E^((-I)*(a + b*x))] - (24*I)*PolyLog[4, -E^((-I)*(a + b*x))] - (24*I)*Po 
lyLog[4, E^((-I)*(a + b*x))])*Sin[a])/(b^5*E^(I*a)) + ((c + d*x)^4*Csc[a]* 
Csc[a + b*x]*Sin[b*x])/b
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.35, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 4203, 17, 25, 3042, 25, 4202, 2620, 3011, 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.

\(\displaystyle \int (c+d x)^4 \cot ^2(a+b x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (c+d x)^4 \tan \left (a+b x+\frac {\pi }{2}\right )^2dx\)

\(\Big \downarrow \) 4203

\(\displaystyle -\frac {4 d \int -(c+d x)^3 \cot (a+b x)dx}{b}-\int (c+d x)^4dx-\frac {(c+d x)^4 \cot (a+b x)}{b}\)

\(\Big \downarrow \) 17

\(\displaystyle -\frac {4 d \int -(c+d x)^3 \cot (a+b x)dx}{b}-\frac {(c+d x)^4 \cot (a+b x)}{b}-\frac {(c+d x)^5}{5 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {4 d \int (c+d x)^3 \cot (a+b x)dx}{b}-\frac {(c+d x)^4 \cot (a+b x)}{b}-\frac {(c+d x)^5}{5 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {4 d \int -(c+d x)^3 \tan \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x)^4 \cot (a+b x)}{b}-\frac {(c+d x)^5}{5 d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {4 d \int (c+d x)^3 \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )dx}{b}-\frac {(c+d x)^4 \cot (a+b x)}{b}-\frac {(c+d x)^5}{5 d}\)

\(\Big \downarrow \) 4202

\(\displaystyle -\frac {4 d \left (\frac {i (c+d x)^4}{4 d}-2 i \int \frac {e^{i (2 a+2 b x+\pi )} (c+d x)^3}{1+e^{i (2 a+2 b x+\pi )}}dx\right )}{b}-\frac {(c+d x)^4 \cot (a+b x)}{b}-\frac {(c+d x)^5}{5 d}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {4 d \left (\frac {i (c+d x)^4}{4 d}-2 i \left (\frac {3 i d \int (c+d x)^2 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )\right )}{b}-\frac {(c+d x)^4 \cot (a+b x)}{b}-\frac {(c+d x)^5}{5 d}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {4 d \left (\frac {i (c+d x)^4}{4 d}-2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )\right )}{b}-\frac {(c+d x)^4 \cot (a+b x)}{b}-\frac {(c+d x)^5}{5 d}\)

\(\Big \downarrow \) 7163

\(\displaystyle -\frac {4 d \left (\frac {i (c+d x)^4}{4 d}-2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i d \left (\frac {i d \int \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )dx}{2 b}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )\right )}{b}-\frac {(c+d x)^4 \cot (a+b x)}{b}-\frac {(c+d x)^5}{5 d}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {4 d \left (\frac {i (c+d x)^4}{4 d}-2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i d \left (\frac {d \int e^{-i (2 a+2 b x+\pi )} \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )de^{i (2 a+2 b x+\pi )}}{4 b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )\right )}{b}-\frac {(c+d x)^4 \cot (a+b x)}{b}-\frac {(c+d x)^5}{5 d}\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {4 d \left (\frac {i (c+d x)^4}{4 d}-2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}-\frac {i d \left (\frac {d \operatorname {PolyLog}\left (4,-e^{i (2 a+2 b x+\pi )}\right )}{4 b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{i (2 a+2 b x+\pi )}\right )}{2 b}\right )\right )}{b}-\frac {(c+d x)^4 \cot (a+b x)}{b}-\frac {(c+d x)^5}{5 d}\)

Input:

Int[(c + d*x)^4*Cot[a + b*x]^2,x]
 

Output:

-1/5*(c + d*x)^5/d - ((c + d*x)^4*Cot[a + b*x])/b - (4*d*(((I/4)*(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 + Pi + 2*b*x))])/b + ( 
d*PolyLog[4, -E^(I*(2*a + Pi + 2*b*x))])/(4*b^2)))/b))/b)))/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 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]
 

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

Time = 0.50 (sec) , antiderivative size = 921, normalized size of antiderivative = 5.94

method	result	size
risch	\(\text {Expression too large to display}\)	\(921\)

Input:

int((d*x+c)^4*cot(b*x+a)^2,x,method=_RETURNVERBOSE)
 

Output:

8*d^4/b^5*a^3*ln(exp(I*(b*x+a)))-4*d^4/b^5*a^3*ln(exp(I*(b*x+a))-1)-8*d/b^ 
2*c^3*ln(exp(I*(b*x+a)))+4*d/b^2*c^3*ln(exp(I*(b*x+a))-1)+4*d/b^2*c^3*ln(e 
xp(I*(b*x+a))+1)+4*d^4/b^5*ln(1-exp(I*(b*x+a)))*a^3+4*d^4/b^2*ln(exp(I*(b* 
x+a))+1)*x^3+4*d^4/b^2*ln(1-exp(I*(b*x+a)))*x^3+24*d^4/b^4*polylog(3,-exp( 
I*(b*x+a)))*x+24*d^4/b^4*polylog(3,exp(I*(b*x+a)))*x+24*d^3/b^4*c*polylog( 
3,-exp(I*(b*x+a)))+24*d^3/b^4*c*polylog(3,exp(I*(b*x+a)))-2*I*d^4/b*x^4-6* 
I*d^4/b^5*a^4+24*I*d^4/b^5*polylog(4,-exp(I*(b*x+a)))-12*I*d^4/b^3*polylog 
(2,-exp(I*(b*x+a)))*x^2-12*I*d^4/b^3*polylog(2,exp(I*(b*x+a)))*x^2-12*I*d^ 
2/b*c^2*x^2-12*I*d^2/b^3*c^2*a^2-12*I*d^2/b^3*c^2*polylog(2,-exp(I*(b*x+a) 
))-12*I*d^2/b^3*c^2*polylog(2,exp(I*(b*x+a)))-12*d^2/b^3*c^2*a*ln(exp(I*(b 
*x+a))-1)-1/5*d^4*x^5-1/5/d*c^5-d^3*c*x^4-2*I*(d^4*x^4+4*c*d^3*x^3+6*c^2*d 
^2*x^2+4*c^3*d*x+c^4)/b/(exp(2*I*(b*x+a))-1)+12*d^3/b^2*c*ln(1-exp(I*(b*x+ 
a)))*x^2+12*d^2/b^2*c^2*ln(exp(I*(b*x+a))+1)*x+12*d^2/b^2*c^2*ln(1-exp(I*( 
b*x+a)))*x-24*d^3/b^4*c*a^2*ln(exp(I*(b*x+a)))-12*d^3/b^4*c*ln(1-exp(I*(b* 
x+a)))*a^2+12*d^2/b^3*c^2*ln(1-exp(I*(b*x+a)))*a-8*I*d^3/b*c*x^3+16*I*d^3/ 
b^4*c*a^3-8*I*d^4/b^4*a^3*x-24*I*d^3/b^3*c*polylog(2,exp(I*(b*x+a)))*x-24* 
I*d^2/b^2*c^2*a*x+24*I*d^3/b^3*c*a^2*x-24*I*d^3/b^3*c*polylog(2,-exp(I*(b* 
x+a)))*x-2*d^2*c^2*x^3-2*d*c^3*x^2-c^4*x+12*d^3/b^2*c*ln(exp(I*(b*x+a))+1) 
*x^2+12*d^3/b^4*c*a^2*ln(exp(I*(b*x+a))-1)+24*d^2/b^3*c^2*a*ln(exp(I*(b*x+ 
a)))+24*I*d^4*polylog(4,exp(I*(b*x+a)))/b^5
 

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. 856 vs. \(2 (138) = 276\).

Time = 0.10 (sec) , antiderivative size = 856, normalized size of antiderivative = 5.52 \[ \int (c+d x)^4 \cot ^2(a+b x) \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^4*cot(b*x+a)^2,x, algorithm="fricas")
 

Output:

-1/10*(10*b^4*d^4*x^4 + 40*b^4*c*d^3*x^3 + 60*b^4*c^2*d^2*x^2 + 40*b^4*c^3 
*d*x + 10*b^4*c^4 - 15*I*d^4*polylog(4, cos(2*b*x + 2*a) + I*sin(2*b*x + 2 
*a))*sin(2*b*x + 2*a) + 15*I*d^4*polylog(4, cos(2*b*x + 2*a) - I*sin(2*b*x 
 + 2*a))*sin(2*b*x + 2*a) + 30*(I*b^2*d^4*x^2 + 2*I*b^2*c*d^3*x + I*b^2*c^ 
2*d^2)*dilog(cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a))*sin(2*b*x + 2*a) + 30* 
(-I*b^2*d^4*x^2 - 2*I*b^2*c*d^3*x - I*b^2*c^2*d^2)*dilog(cos(2*b*x + 2*a) 
- I*sin(2*b*x + 2*a))*sin(2*b*x + 2*a) - 20*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 
 3*a^2*b*c*d^3 - a^3*d^4)*log(-1/2*cos(2*b*x + 2*a) + 1/2*I*sin(2*b*x + 2* 
a) + 1/2)*sin(2*b*x + 2*a) - 20*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d 
^3 - a^3*d^4)*log(-1/2*cos(2*b*x + 2*a) - 1/2*I*sin(2*b*x + 2*a) + 1/2)*si 
n(2*b*x + 2*a) - 20*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*b^3*c^2*d^2*x + 3*a 
*b^2*c^2*d^2 - 3*a^2*b*c*d^3 + a^3*d^4)*log(-cos(2*b*x + 2*a) + I*sin(2*b* 
x + 2*a) + 1)*sin(2*b*x + 2*a) - 20*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*b^3 
*c^2*d^2*x + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3 + a^3*d^4)*log(-cos(2*b*x + 2 
*a) - I*sin(2*b*x + 2*a) + 1)*sin(2*b*x + 2*a) - 30*(b*d^4*x + b*c*d^3)*po 
lylog(3, cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a))*sin(2*b*x + 2*a) - 30*(b*d 
^4*x + b*c*d^3)*polylog(3, cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a))*sin(2*b* 
x + 2*a) + 10*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c 
^3*d*x + b^4*c^4)*cos(2*b*x + 2*a) + 2*(b^5*d^4*x^5 + 5*b^5*c*d^3*x^4 + 10 
*b^5*c^2*d^2*x^3 + 10*b^5*c^3*d*x^2 + 5*b^5*c^4*x)*sin(2*b*x + 2*a))/(b...
 

Sympy [F]

\[ \int (c+d x)^4 \cot ^2(a+b x) \, dx=\int \left (c + d x\right )^{4} \cot ^{2}{\left (a + b x \right )}\, dx \] Input:

integrate((d*x+c)**4*cot(b*x+a)**2,x)
                                                                                    
                                                                                    
 

Output:

Integral((c + d*x)**4*cot(a + b*x)**2, 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. 3242 vs. \(2 (138) = 276\).

Time = 0.45 (sec) , antiderivative size = 3242, normalized size of antiderivative = 20.92 \[ \int (c+d x)^4 \cot ^2(a+b x) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^4*cot(b*x+a)^2,x, algorithm="maxima")
 

Output:

-((b*x + a + 1/tan(b*x + a))*c^4 - 4*(b*x + a + 1/tan(b*x + a))*a*c^3*d/b 
+ 6*(b*x + a + 1/tan(b*x + a))*a^2*c^2*d^2/b^2 - 4*(b*x + a + 1/tan(b*x + 
a))*a^3*c*d^3/b^3 + (b*x + a + 1/tan(b*x + a))*a^4*d^4/b^4 + 2*((b*x + a)^ 
2*cos(2*b*x + 2*a)^2 + (b*x + a)^2*sin(2*b*x + 2*a)^2 - 2*(b*x + a)^2*cos( 
2*b*x + 2*a) + (b*x + a)^2 - (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2* 
cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) 
 + 1) - (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1) 
*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) + 4*(b*x + a)*s 
in(2*b*x + 2*a))*c^3*d/((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2 
*b*x + 2*a) + 1)*b) - 6*((b*x + a)^2*cos(2*b*x + 2*a)^2 + (b*x + a)^2*sin( 
2*b*x + 2*a)^2 - 2*(b*x + a)^2*cos(2*b*x + 2*a) + (b*x + a)^2 - (cos(2*b*x 
 + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^ 
2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) - (cos(2*b*x + 2*a)^2 + sin(2*b*x 
 + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 
2*cos(b*x + a) + 1) + 4*(b*x + a)*sin(2*b*x + 2*a))*a*c^2*d^2/((cos(2*b*x 
+ 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*b^2) + 6*((b*x + a 
)^2*cos(2*b*x + 2*a)^2 + (b*x + a)^2*sin(2*b*x + 2*a)^2 - 2*(b*x + a)^2*co 
s(2*b*x + 2*a) + (b*x + a)^2 - (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 
2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + 
a) + 1) - (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a)...
 

Giac [F]

\[ \int (c+d x)^4 \cot ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{4} \cot \left (b x + a\right )^{2} \,d x } \] Input:

integrate((d*x+c)^4*cot(b*x+a)^2,x, algorithm="giac")
 

Output:

integrate((d*x + c)^4*cot(b*x + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^4 \cot ^2(a+b x) \, dx=\int {\mathrm {cot}\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^4 \,d x \] Input:

int(cot(a + b*x)^2*(c + d*x)^4,x)
 

Output:

int(cot(a + b*x)^2*(c + d*x)^4, x)
 

Reduce [F]

\[ \int (c+d x)^4 \cot ^2(a+b x) \, dx =\text {Too large to display} \] Input:

int((d*x+c)^4*cot(b*x+a)^2,x)
 

Output:

( - 10*cos(a + b*x)*tan((a + b*x)/2)*b*c**4 - 40*cos(a + b*x)*tan((a + b*x 
)/2)*b*c**3*d*x - 30*cos(a + b*x)*tan((a + b*x)/2)*b*c**2*d**2*x**2 - 20*c 
os(a + b*x)*tan((a + b*x)/2)*b*c*d**3*x**3 - 5*cos(a + b*x)*tan((a + b*x)/ 
2)*b*d**4*x**4 + 20*int(x**3/tan((a + b*x)/2),x)*sin(a + b*x)*tan((a + b*x 
)/2)*b*d**4 + 60*int(x**2/tan((a + b*x)/2),x)*sin(a + b*x)*tan((a + b*x)/2 
)*b*c*d**3 + 60*int(x/tan((a + b*x)/2),x)*sin(a + b*x)*tan((a + b*x)/2)*b* 
c**2*d**2 - 20*int(tan((a + b*x)/2)*x**3,x)*sin(a + b*x)*tan((a + b*x)/2)* 
b*d**4 - 60*int(tan((a + b*x)/2)*x**2,x)*sin(a + b*x)*tan((a + b*x)/2)*b*c 
*d**3 - 60*int(tan((a + b*x)/2)*x,x)*sin(a + b*x)*tan((a + b*x)/2)*b*c**2* 
d**2 - 40*log(tan((a + b*x)/2)**2 + 1)*sin(a + b*x)*tan((a + b*x)/2)*c**3* 
d + 40*log(tan((a + b*x)/2))*sin(a + b*x)*tan((a + b*x)/2)*c**3*d - 10*sin 
(a + b*x)*tan((a + b*x)/2)*b**2*c**4*x - 20*sin(a + b*x)*tan((a + b*x)/2)* 
b**2*c**3*d*x**2 - 20*sin(a + b*x)*tan((a + b*x)/2)*b**2*c**2*d**2*x**3 - 
10*sin(a + b*x)*tan((a + b*x)/2)*b**2*c*d**3*x**4 - 2*sin(a + b*x)*tan((a 
+ b*x)/2)*b**2*d**4*x**5 - 30*sin(a + b*x)*b*c**2*d**2*x**2 - 20*sin(a + b 
*x)*b*c*d**3*x**3 - 5*sin(a + b*x)*b*d**4*x**4 + 30*tan((a + b*x)/2)*b*c** 
2*d**2*x**2 + 20*tan((a + b*x)/2)*b*c*d**3*x**3 + 5*tan((a + b*x)/2)*b*d** 
4*x**4)/(10*sin(a + b*x)*tan((a + b*x)/2)*b**2)