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

Optimal result

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

[Out]

-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-exp(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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3801, 3798, 2221, 2611, 6744, 2320, 6724, 32} \[ \int (c+d x)^4 \cot ^2(a+b x) \, dx=\frac {3 i d^4 \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^5}+\frac {6 d^3 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^4}-\frac {6 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^3}+\frac {4 d (c+d x)^3 \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {(c+d x)^4 \cot (a+b x)}{b}-\frac {i (c+d x)^4}{b}-\frac {(c+d x)^5}{5 d} \]

[In]

Int[(c + d*x)^4*Cot[a + b*x]^2,x]

[Out]

((-I)*(c + d*x)^4)/b - (c + d*x)^5/(5*d) - ((c + d*x)^4*Cot[a + b*x])/b + (4*d*(c + d*x)^3*Log[1 - E^((2*I)*(a
+ b*x))])/b^2 - ((6*I)*d^2*(c + d*x)^2*PolyLog[2, E^((2*I)*(a + b*x))])/b^3 + (6*d^3*(c + d*x)*PolyLog[3, E^(
(2*I)*(a + b*x))])/b^4 + ((3*I)*d^4*PolyLog[4, E^((2*I)*(a + b*x))])/b^5

Rule 32

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

Rule 2221

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]
- Dist[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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{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 2611

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] + Dist[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 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3801

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

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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 6744

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^4 \cot (a+b x)}{b}+\frac {(4 d) \int (c+d x)^3 \cot (a+b x) \, dx}{b}-\int (c+d x)^4 \, 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 {(8 i d) \int \frac {e^{2 i (a+b x)} (c+d x)^3}{1-e^{2 i (a+b x)}} \, dx}{b} \\ & = -\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 {\left (12 d^2\right ) \int (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right ) \, dx}{b^2} \\ & = -\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 {\left (12 i d^3\right ) \int (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right ) \, dx}{b^3} \\ & = -\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 {\left (6 d^4\right ) \int \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right ) \, dx}{b^4} \\ & = -\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 {\left (3 i d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{b^5} \\ & = -\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} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(833\) vs. \(2(155)=310\).

Time = 6.65 (sec) , antiderivative size = 833, normalized size of antiderivative = 5.37 \[ \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 {2 c d^3 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 )}{b^4}-\frac {d^4 e^{i a} \csc (a) \left (b^4 e^{-2 i a} x^4+2 i b^3 \left (1-e^{-2 i a}\right ) x^3 \log \left (1-e^{-i (a+b x)}\right )+2 i b^3 \left (1-e^{-2 i a}\right ) x^3 \log \left (1+e^{-i (a+b x)}\right )-6 b^2 \left (1-e^{-2 i a}\right ) x^2 \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )-6 b^2 \left (1-e^{-2 i a}\right ) x^2 \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )+12 i b \left (1-e^{-2 i a}\right ) x \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+12 i b \left (1-e^{-2 i a}\right ) x \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )+12 \left (1-e^{-2 i a}\right ) \operatorname {PolyLog}\left (4,-e^{-i (a+b x)}\right )+12 \left (1-e^{-2 i a}\right ) \operatorname {PolyLog}\left (4,e^{-i (a+b x)}\right )\right )}{b^5}+\frac {4 c^3 d \csc (a) (-b x \cos (a)+\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a))}{b^2 \left (\cos ^2(a)+\sin ^2(a)\right )}+\frac {\csc (a) \csc (a+b x) \left (c^4 \sin (b x)+4 c^3 d x \sin (b x)+6 c^2 d^2 x^2 \sin (b x)+4 c d^3 x^3 \sin (b x)+d^4 x^4 \sin (b x)\right )}{b}-\frac {6 c^2 d^2 \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^3 \sqrt {\sec ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}} \]

[In]

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

[Out]

-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)) - (2*c*d^3*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^4 - (d^4*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*
PolyLog[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))]))/b^5 + (4*c^3*d*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]
]*Sin[a]))/(b^2*(Cos[a]^2 + Sin[a]^2)) + (Csc[a]*Csc[a + b*x]*(c^4*Sin[b*x] + 4*c^3*d*x*Sin[b*x] + 6*c^2*d^2*x
^2*Sin[b*x] + 4*c*d^3*x^3*Sin[b*x] + d^4*x^4*Sin[b*x]))/b - (6*c^2*d^2*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^3*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])

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 = 1.65 (sec) , antiderivative size = 921, normalized size of antiderivative = 5.94

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

[In]

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

[Out]

24*I*d^4*polylog(4,exp(I*(b*x+a)))/b^5+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^3-4*d^4/b^5*a^3*ln(exp(I*(b*x+a))-1)+8*d^4/b^5*a^3*ln(exp(I*(b*x+a)))+4*d/b^2*
c^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)-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)+24*I*d^4/b^5*polylog(4,-exp(I*(b*x+a)))-6*I*d^4/b^5*a^
4-2*I*d^4/b*x^4+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)))+24*d^4/b^4*poly
log(3,exp(I*(b*x+a)))*x+24*d^4/b^4*polylog(3,-exp(I*(b*x+a)))*x-d^3*c*x^4-2*d^2*c^2*x^3-2*d*c^3*x^2-c^4*x+12*d
^2/b^2*c^2*ln(exp(I*(b*x+a))+1)*x+24*d^2/b^3*c^2*a*ln(exp(I*(b*x+a)))-12*d^2/b^3*c^2*a*ln(exp(I*(b*x+a))-1)+12
*d^3/b^4*c*a^2*ln(exp(I*(b*x+a))-1)-24*d^3/b^4*c*a^2*ln(exp(I*(b*x+a)))-8*I*d^3/b*c*x^3+16*I*d^3/b^4*c*a^3-12*
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,-ex
p(I*(b*x+a)))-8*I*d^4/b^4*a^3*x-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+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-24*I*d^3/b^3*c*polylog(2,-exp(I*(b*x+
a)))*x-24*I*d^2/b^2*c^2*x*a+12*d^3/b^2*c*ln(1-exp(I*(b*x+a)))*x^2+12*d^3/b^2*c*ln(exp(I*(b*x+a))+1)*x^2+12*d^2
/b^2*c^2*ln(1-exp(I*(b*x+a)))*x-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-1/
5*d^4*x^5-1/5/d*c^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 = 0.28 (sec) , antiderivative size = 856, normalized size of antiderivative = 5.52 \[ \int (c+d x)^4 \cot ^2(a+b x) \, dx=-\frac {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} {\rm polylog}\left (4, \cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) \sin \left (2 \, b x + 2 \, a\right ) + 15 i \, d^{4} {\rm polylog}\left (4, \cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) \sin \left (2 \, b x + 2 \, a\right ) + 30 \, {\left (i \, b^{2} d^{4} x^{2} + 2 i \, b^{2} c d^{3} x + i \, b^{2} c^{2} d^{2}\right )} {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) \sin \left (2 \, b x + 2 \, a\right ) + 30 \, {\left (-i \, b^{2} d^{4} x^{2} - 2 i \, b^{2} c d^{3} x - i \, b^{2} c^{2} d^{2}\right )} {\rm Li}_2\left (\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) \sin \left (2 \, b x + 2 \, a\right ) - 20 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\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 ) \sin \left (2 \, b x + 2 \, a\right ) - 20 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\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 ) \sin \left (2 \, b x + 2 \, a\right ) - 20 \, {\left (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}\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) \sin \left (2 \, b x + 2 \, a\right ) - 20 \, {\left (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}\right )} \log \left (-\cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) \sin \left (2 \, b x + 2 \, a\right ) - 30 \, {\left (b d^{4} x + b c d^{3}\right )} {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) + i \, \sin \left (2 \, b x + 2 \, a\right )\right ) \sin \left (2 \, b x + 2 \, a\right ) - 30 \, {\left (b d^{4} x + b c d^{3}\right )} {\rm polylog}\left (3, \cos \left (2 \, b x + 2 \, a\right ) - i \, \sin \left (2 \, b x + 2 \, a\right )\right ) \sin \left (2 \, b x + 2 \, a\right ) + 10 \, {\left (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}\right )} \cos \left (2 \, b x + 2 \, a\right ) + 2 \, {\left (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\right )} \sin \left (2 \, b x + 2 \, a\right )}{10 \, b^{5} \sin \left (2 \, b x + 2 \, a\right )} \]

[In]

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

[Out]

-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)*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) - 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)*polylog(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^5*sin(2*b*x + 2*a))

Sympy [F]

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

[In]

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

[Out]

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 = 0.68 (sec) , antiderivative size = 3242, normalized size of antiderivative = 20.92 \[ \int (c+d x)^4 \cot ^2(a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-((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)*sin(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*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^2*c*d^3/((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)
*b^3) - 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)*sin(2*b*x + 2*a))*a^3*d^4/((cos(2*b*x + 2*a)^2 + s
in(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*b^4) - (-I*(b*x + a)^5*d^4 - 5*(I*b*c*d^3 - I*a*d^4)*(b*x + a)^4 -
10*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a)^3 - 20*((b*x + a)^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x +
a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a) - ((b*x + a)^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2
+ 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*cos(2*b*x + 2*a) + (-I*(b*x + 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))*sin(2*b*x + 2*a))*arctan2(si
n(b*x + a), cos(b*x + a) + 1) + 20*((b*x + a)^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b
*c*d^3 + a^2*d^4)*(b*x + a) - ((b*x + a)^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^
3 + a^2*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (I*(b*x + 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))*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), -cos(b*x + a) + 1) +
(I*(b*x + a)^5*d^4 - 5*(-I*b*c*d^3 + (I*a + 2)*d^4)*(b*x + a)^4 - 10*(-I*b^2*c^2*d^2 + 2*(I*a + 2)*b*c*d^3 + (
-I*a^2 - 4*a)*d^4)*(b*x + a)^3 - 60*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2)*cos(2*b*x + 2*a) + 60*(
b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a) - (b^2*c^2*d^2 - 2*a*b*c
*d^3 + (b*x + a)^2*d^4 + a^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (I*b^2*c^2*d^2 - 2*I*a*b*
c*d^3 + I*(b*x + a)^2*d^4 + I*a^2*d^4 + 2*(I*b*c*d^3 - I*a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*dilog(-e^(I*b*x +
I*a)) + 60*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a) - (b^2*c^2*
d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (I*b^2*c^2*d
^2 - 2*I*a*b*c*d^3 + I*(b*x + a)^2*d^4 + I*a^2*d^4 + 2*(I*b*c*d^3 - I*a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*dilo
g(e^(I*b*x + I*a)) - 10*(-I*(b*x + 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) + (I*(b*x + 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))*cos(2*b*x + 2*a) - ((b*x + a)^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2
+ 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2
*cos(b*x + a) + 1) - 10*(-I*(b*x + 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) + (I*(b*x + 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))*cos(2*b*x + 2*a) - ((b*x + a)^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2
+ 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2
*cos(b*x + a) + 1) + 120*(d^4*cos(2*b*x + 2*a) + I*d^4*sin(2*b*x + 2*a) - d^4)*polylog(4, -e^(I*b*x + I*a)) +
120*(d^4*cos(2*b*x + 2*a) + I*d^4*sin(2*b*x + 2*a) - d^4)*polylog(4, e^(I*b*x + I*a)) - 120*(-I*b*c*d^3 - I*(b
*x + a)*d^4 + I*a*d^4 + (I*b*c*d^3 + I*(b*x + a)*d^4 - I*a*d^4)*cos(2*b*x + 2*a) - (b*c*d^3 + (b*x + a)*d^4 -
a*d^4)*sin(2*b*x + 2*a))*polylog(3, -e^(I*b*x + I*a)) - 120*(-I*b*c*d^3 - I*(b*x + a)*d^4 + I*a*d^4 + (I*b*c*d
^3 + I*(b*x + a)*d^4 - I*a*d^4)*cos(2*b*x + 2*a) - (b*c*d^3 + (b*x + a)*d^4 - a*d^4)*sin(2*b*x + 2*a))*polylog
(3, e^(I*b*x + I*a)) - ((b*x + a)^5*d^4 + 5*(b*c*d^3 - (a - 2*I)*d^4)*(b*x + a)^4 + 10*(b^2*c^2*d^2 - 2*(a - 2
*I)*b*c*d^3 + (a^2 - 4*I*a)*d^4)*(b*x + a)^3 + 60*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a)^2)*sin
(2*b*x + 2*a))/(-5*I*b^4*cos(2*b*x + 2*a) + 5*b^4*sin(2*b*x + 2*a) + 5*I*b^4))/b

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(cot(a + b*x)^2*(c + d*x)^4,x)

[Out]

int(cot(a + b*x)^2*(c + d*x)^4, x)