3.1.32 \(\int (c+d x)^4 \cot (a+b x) \, dx\) [32]

3.1.32.1 Optimal result

Integrand size = 14, antiderivative size = 151 \[ \int (c+d x)^4 \cot (a+b x) \, dx=-\frac {i (c+d x)^5}{5 d}+\frac {(c+d x)^4 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}+\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}+\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}-\frac {3 d^4 \operatorname {PolyLog}\left (5,e^{2 i (a+b x)}\right )}{2 b^5} \]

output

-1/5*I*(d*x+c)^5/d+(d*x+c)^4*ln(1-exp(2*I*(b*x+a)))/b-2*I*d*(d*x+c)^3*poly 
log(2,exp(2*I*(b*x+a)))/b^2+3*d^2*(d*x+c)^2*polylog(3,exp(2*I*(b*x+a)))/b^ 
3+3*I*d^3*(d*x+c)*polylog(4,exp(2*I*(b*x+a)))/b^4-3/2*d^4*polylog(5,exp(2* 
I*(b*x+a)))/b^5
 

3.1.32.2 Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(799\) vs. \(2(151)=302\).

Time = 6.27 (sec) , antiderivative size = 799, normalized size of antiderivative = 5.29 \[ \int (c+d x)^4 \cot (a+b x) \, dx=\frac {2 i c^3 d \pi x}{b}+2 i c^2 d^2 x^3+i c d^3 x^4+\frac {1}{5} i d^4 x^5-\frac {4 i c^3 d x \arctan (\tan (a))}{b}+2 c^3 d x^2 \cot (a)+\frac {2 c^3 d \pi \log \left (1+e^{-2 i b x}\right )}{b^2}+\frac {6 c^2 d^2 x^2 \log \left (1-e^{-i (a+b x)}\right )}{b}+\frac {4 c d^3 x^3 \log \left (1-e^{-i (a+b x)}\right )}{b}+\frac {d^4 x^4 \log \left (1-e^{-i (a+b x)}\right )}{b}+\frac {6 c^2 d^2 x^2 \log \left (1+e^{-i (a+b x)}\right )}{b}+\frac {4 c d^3 x^3 \log \left (1+e^{-i (a+b x)}\right )}{b}+\frac {d^4 x^4 \log \left (1+e^{-i (a+b x)}\right )}{b}+\frac {4 c^3 d x \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )}{b}+\frac {4 c^3 d \arctan (\tan (a)) \log \left (1-e^{2 i (b x+\arctan (\tan (a)))}\right )}{b^2}-\frac {2 c^3 d \pi \log (\cos (b x))}{b^2}+\frac {c^4 \log (\sin (a+b x))}{b}-\frac {4 c^3 d \arctan (\tan (a)) \log (\sin (b x+\arctan (\tan (a))))}{b^2}+\frac {4 i d^2 x \left (3 c^2+3 c d x+d^2 x^2\right ) \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )}{b^2}+\frac {4 i d^2 x \left (3 c^2+3 c d x+d^2 x^2\right ) \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )}{b^2}-\frac {2 i c^3 d \operatorname {PolyLog}\left (2,e^{2 i (b x+\arctan (\tan (a)))}\right )}{b^2}+\frac {12 c^2 d^2 \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )}{b^3}+\frac {24 c d^3 x \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )}{b^3}+\frac {12 d^4 x^2 \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )}{b^3}+\frac {12 c^2 d^2 \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )}{b^3}+\frac {24 c d^3 x \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )}{b^3}+\frac {12 d^4 x^2 \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )}{b^3}-\frac {24 i c d^3 \operatorname {PolyLog}\left (4,-e^{-i (a+b x)}\right )}{b^4}-\frac {24 i d^4 x \operatorname {PolyLog}\left (4,-e^{-i (a+b x)}\right )}{b^4}-\frac {24 i c d^3 \operatorname {PolyLog}\left (4,e^{-i (a+b x)}\right )}{b^4}-\frac {24 i d^4 x \operatorname {PolyLog}\left (4,e^{-i (a+b x)}\right )}{b^4}-\frac {24 d^4 \operatorname {PolyLog}\left (5,-e^{-i (a+b x)}\right )}{b^5}-\frac {24 d^4 \operatorname {PolyLog}\left (5,e^{-i (a+b x)}\right )}{b^5}-2 c^3 d e^{i \arctan (\tan (a))} x^2 \cot (a) \sqrt {\sec ^2(a)} \]

input

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

output

((2*I)*c^3*d*Pi*x)/b + (2*I)*c^2*d^2*x^3 + I*c*d^3*x^4 + (I/5)*d^4*x^5 - ( 
(4*I)*c^3*d*x*ArcTan[Tan[a]])/b + 2*c^3*d*x^2*Cot[a] + (2*c^3*d*Pi*Log[1 + 
 E^((-2*I)*b*x)])/b^2 + (6*c^2*d^2*x^2*Log[1 - E^((-I)*(a + b*x))])/b + (4 
*c*d^3*x^3*Log[1 - E^((-I)*(a + b*x))])/b + (d^4*x^4*Log[1 - E^((-I)*(a + 
b*x))])/b + (6*c^2*d^2*x^2*Log[1 + E^((-I)*(a + b*x))])/b + (4*c*d^3*x^3*L 
og[1 + E^((-I)*(a + b*x))])/b + (d^4*x^4*Log[1 + E^((-I)*(a + b*x))])/b + 
(4*c^3*d*x*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))])/b + (4*c^3*d*ArcTan[ 
Tan[a]]*Log[1 - E^((2*I)*(b*x + ArcTan[Tan[a]]))])/b^2 - (2*c^3*d*Pi*Log[C 
os[b*x]])/b^2 + (c^4*Log[Sin[a + b*x]])/b - (4*c^3*d*ArcTan[Tan[a]]*Log[Si 
n[b*x + ArcTan[Tan[a]]]])/b^2 + ((4*I)*d^2*x*(3*c^2 + 3*c*d*x + d^2*x^2)*P 
olyLog[2, -E^((-I)*(a + b*x))])/b^2 + ((4*I)*d^2*x*(3*c^2 + 3*c*d*x + d^2* 
x^2)*PolyLog[2, E^((-I)*(a + b*x))])/b^2 - ((2*I)*c^3*d*PolyLog[2, E^((2*I 
)*(b*x + ArcTan[Tan[a]]))])/b^2 + (12*c^2*d^2*PolyLog[3, -E^((-I)*(a + b*x 
))])/b^3 + (24*c*d^3*x*PolyLog[3, -E^((-I)*(a + b*x))])/b^3 + (12*d^4*x^2* 
PolyLog[3, -E^((-I)*(a + b*x))])/b^3 + (12*c^2*d^2*PolyLog[3, E^((-I)*(a + 
 b*x))])/b^3 + (24*c*d^3*x*PolyLog[3, E^((-I)*(a + b*x))])/b^3 + (12*d^4*x 
^2*PolyLog[3, E^((-I)*(a + b*x))])/b^3 - ((24*I)*c*d^3*PolyLog[4, -E^((-I) 
*(a + b*x))])/b^4 - ((24*I)*d^4*x*PolyLog[4, -E^((-I)*(a + b*x))])/b^4 - ( 
(24*I)*c*d^3*PolyLog[4, E^((-I)*(a + b*x))])/b^4 - ((24*I)*d^4*x*PolyLog[4 
, E^((-I)*(a + b*x))])/b^4 - (24*d^4*PolyLog[5, -E^((-I)*(a + b*x))])/b...
 

3.1.32.3 Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.42, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {3042, 25, 4202, 2620, 3011, 7163, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

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

output

((-1/5*I)*(c + d*x)^5)/d + (2*I)*(((-1/2*I)*(c + d*x)^4*Log[1 + E^(I*(2*a 
+ Pi + 2*b*x))])/b + ((2*I)*d*(((I/2)*(c + d*x)^3*PolyLog[2, -E^(I*(2*a + 
Pi + 2*b*x))])/b - (((3*I)/2)*d*(((-1/2*I)*(c + d*x)^2*PolyLog[3, -E^(I*(2 
*a + Pi + 2*b*x))])/b + (I*d*(((-1/2*I)*(c + d*x)*PolyLog[4, -E^(I*(2*a + 
Pi + 2*b*x))])/b + (d*PolyLog[5, -E^(I*(2*a + Pi + 2*b*x))])/(4*b^2)))/b)) 
/b))/b)
 

3.1.32.3.1 Defintions of rubi rules used

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 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]
 

3.1.32.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1158 vs. \(2 (134 ) = 268\).

Time = 1.21 (sec) , antiderivative size = 1159, normalized size of antiderivative = 7.68

method	result	size
risch	\(\text {Expression too large to display}\)	\(1159\)

input

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

output

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

3.1.32.5 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. 1204 vs. \(2 (130) = 260\).

Time = 0.31 (sec) , antiderivative size = 1204, normalized size of antiderivative = 7.97 \[ \int (c+d x)^4 \cot (a+b x) \, dx=\text {Too large to display} \]

input

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

output

-1/2*(24*d^4*polylog(5, cos(b*x + a) + I*sin(b*x + a)) + 24*d^4*polylog(5, 
 cos(b*x + a) - I*sin(b*x + a)) + 24*d^4*polylog(5, -cos(b*x + a) + I*sin( 
b*x + a)) + 24*d^4*polylog(5, -cos(b*x + a) - I*sin(b*x + a)) + 4*(I*b^3*d 
^4*x^3 + 3*I*b^3*c*d^3*x^2 + 3*I*b^3*c^2*d^2*x + I*b^3*c^3*d)*dilog(cos(b* 
x + a) + I*sin(b*x + a)) + 4*(-I*b^3*d^4*x^3 - 3*I*b^3*c*d^3*x^2 - 3*I*b^3 
*c^2*d^2*x - I*b^3*c^3*d)*dilog(cos(b*x + a) - I*sin(b*x + a)) + 4*(-I*b^3 
*d^4*x^3 - 3*I*b^3*c*d^3*x^2 - 3*I*b^3*c^2*d^2*x - I*b^3*c^3*d)*dilog(-cos 
(b*x + a) + I*sin(b*x + a)) + 4*(I*b^3*d^4*x^3 + 3*I*b^3*c*d^3*x^2 + 3*I*b 
^3*c^2*d^2*x + I*b^3*c^3*d)*dilog(-cos(b*x + a) - I*sin(b*x + a)) - (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)*lo 
g(cos(b*x + a) + I*sin(b*x + a) + 1) - (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)*log(cos(b*x + a) - I*sin(b*x + 
a) + 1) - (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a 
^4*d^4)*log(-1/2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/2) - (b^4*c^4 - 4*a 
*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(-1/2*cos(b*x 
 + a) - 1/2*I*sin(b*x + a) + 1/2) - (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 + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*a^3*b 
*c*d^3 - a^4*d^4)*log(-cos(b*x + a) + I*sin(b*x + a) + 1) - (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 + 4*a*b^3*c^3*d - 6*a 
^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - a^4*d^4)*log(-cos(b*x + a) - I*sin(b*x...
 

3.1.32.6 Sympy [F]

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

input

integrate((d*x+c)**4*cos(b*x+a)*csc(b*x+a),x)
 

output

Integral((c + d*x)**4*cos(a + b*x)*csc(a + b*x), x)
 

3.1.32.7 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. 1281 vs. \(2 (130) = 260\).

Time = 0.41 (sec) , antiderivative size = 1281, normalized size of antiderivative = 8.48 \[ \int (c+d x)^4 \cot (a+b x) \, dx=\text {Too large to display} \]

input

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

output

1/10*(10*c^4*log(sin(b*x + a)) - 40*a*c^3*d*log(sin(b*x + a))/b + 60*a^2*c 
^2*d^2*log(sin(b*x + a))/b^2 - 40*a^3*c*d^3*log(sin(b*x + a))/b^3 + 10*a^4 
*d^4*log(sin(b*x + a))/b^4 + (-2*I*(b*x + a)^5*d^4 - 10*(I*b*c*d^3 - I*a*d 
^4)*(b*x + a)^4 - 240*d^4*polylog(5, -e^(I*b*x + I*a)) - 240*d^4*polylog(5 
, e^(I*b*x + I*a)) - 20*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + 
 a)^3 - 20*(I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + 3*I*a^2*b*c*d^3 - I*a^3*d^4) 
*(b*x + a)^2 - 10*(-I*(b*x + a)^4*d^4 + 4*(-I*b*c*d^3 + I*a*d^4)*(b*x + a) 
^3 + 6*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a)^2 + 4*(-I*b^ 
3*c^3*d + 3*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 + I*a^3*d^4)*(b*x + a))*arct 
an2(sin(b*x + a), cos(b*x + a) + 1) - 10*(I*(b*x + a)^4*d^4 + 4*(I*b*c*d^3 
 - I*a*d^4)*(b*x + a)^3 + 6*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b 
*x + a)^2 + 4*(I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + 3*I*a^2*b*c*d^3 - I*a^3*d 
^4)*(b*x + a))*arctan2(sin(b*x + a), -cos(b*x + a) + 1) - 40*(I*b^3*c^3*d 
- 3*I*a*b^2*c^2*d^2 + 3*I*a^2*b*c*d^3 + I*(b*x + a)^3*d^4 - I*a^3*d^4 + 3* 
(I*b*c*d^3 - I*a*d^4)*(b*x + a)^2 + 3*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a 
^2*d^4)*(b*x + a))*dilog(-e^(I*b*x + I*a)) - 40*(I*b^3*c^3*d - 3*I*a*b^2*c 
^2*d^2 + 3*I*a^2*b*c*d^3 + I*(b*x + a)^3*d^4 - I*a^3*d^4 + 3*(I*b*c*d^3 - 
I*a*d^4)*(b*x + a)^2 + 3*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x 
+ a))*dilog(e^(I*b*x + I*a)) + 5*((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b 
*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^...
 

3.1.32.8 Giac [F]

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

input

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

output

integrate((d*x + c)^4*cos(b*x + a)*csc(b*x + a), x)
 

3.1.32.9 Mupad [F(-1)]

Timed out. \[ \int (c+d x)^4 \cot (a+b x) \, dx=\int \frac {\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^4}{\sin \left (a+b\,x\right )} \,d x \]

input

int((cos(a + b*x)*(c + d*x)^4)/sin(a + b*x),x)
 

output

int((cos(a + b*x)*(c + d*x)^4)/sin(a + b*x), x)