\(\int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx\) [164]

Optimal result

Integrand size = 22, antiderivative size = 307 \[ \int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx=-\frac {3 c d^3 x}{2 b^3}-\frac {3 d^4 x^2}{4 b^3}+\frac {(c+d x)^4}{4 b}-\frac {i (c+d x)^5}{5 d}+\frac {(c+d x)^4 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}+\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}+\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}-\frac {3 d^4 \operatorname {PolyLog}\left (5,e^{2 i (a+b x)}\right )}{2 b^5}+\frac {3 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{2 b^4}-\frac {d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b^2}-\frac {3 d^4 \sin ^2(a+b x)}{4 b^5}+\frac {3 d^2 (c+d x)^2 \sin ^2(a+b x)}{2 b^3}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b} \]

[Out]

-3/2*c*d^3*x/b^3-3/4*d^4*x^2/b^3+1/4*(d*x+c)^4/b-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*polylog(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)*polylo
g(4,exp(2*I*(b*x+a)))/b^4-3/2*d^4*polylog(5,exp(2*I*(b*x+a)))/b^5+3/2*d^3*(d*x+c)*cos(b*x+a)*sin(b*x+a)/b^4-d*
(d*x+c)^3*cos(b*x+a)*sin(b*x+a)/b^2-3/4*d^4*sin(b*x+a)^2/b^5+3/2*d^2*(d*x+c)^2*sin(b*x+a)^2/b^3-1/2*(d*x+c)^4*
sin(b*x+a)^2/b

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4493, 4489, 3392, 32, 3391, 3798, 2221, 2611, 6744, 2320, 6724} \[ \int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx=-\frac {3 d^4 \operatorname {PolyLog}\left (5,e^{2 i (a+b x)}\right )}{2 b^5}-\frac {3 d^4 \sin ^2(a+b x)}{4 b^5}+\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}+\frac {3 d^3 (c+d x) \sin (a+b x) \cos (a+b x)}{2 b^4}+\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x)^2 \sin ^2(a+b x)}{2 b^3}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac {d (c+d x)^3 \sin (a+b x) \cos (a+b x)}{b^2}+\frac {(c+d x)^4 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac {3 c d^3 x}{2 b^3}-\frac {3 d^4 x^2}{4 b^3}+\frac {(c+d x)^4}{4 b}-\frac {i (c+d x)^5}{5 d} \]

[In]

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

[Out]

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

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 3391

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

Rule 3392

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

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 4489

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] - Dist[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 4493

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]
/; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 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}& = \int (c+d x)^4 \cot (a+b x) \, dx-\int (c+d x)^4 \cos (a+b x) \sin (a+b x) \, dx \\ & = -\frac {i (c+d x)^5}{5 d}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-2 i \int \frac {e^{2 i (a+b x)} (c+d x)^4}{1-e^{2 i (a+b x)}} \, dx+\frac {(2 d) \int (c+d x)^3 \sin ^2(a+b x) \, dx}{b} \\ & = -\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 {d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b^2}+\frac {3 d^2 (c+d x)^2 \sin ^2(a+b x)}{2 b^3}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}+\frac {d \int (c+d x)^3 \, dx}{b}-\frac {(4 d) \int (c+d x)^3 \log \left (1-e^{2 i (a+b x)}\right ) \, dx}{b}-\frac {\left (3 d^3\right ) \int (c+d x) \sin ^2(a+b x) \, dx}{b^3} \\ & = \frac {(c+d x)^4}{4 b}-\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^3 (c+d x) \cos (a+b x) \sin (a+b x)}{2 b^4}-\frac {d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b^2}-\frac {3 d^4 \sin ^2(a+b x)}{4 b^5}+\frac {3 d^2 (c+d x)^2 \sin ^2(a+b x)}{2 b^3}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}+\frac {\left (6 i d^2\right ) \int (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right ) \, dx}{b^2}-\frac {\left (3 d^3\right ) \int (c+d x) \, dx}{2 b^3} \\ & = -\frac {3 c d^3 x}{2 b^3}-\frac {3 d^4 x^2}{4 b^3}+\frac {(c+d x)^4}{4 b}-\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 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{2 b^4}-\frac {d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b^2}-\frac {3 d^4 \sin ^2(a+b x)}{4 b^5}+\frac {3 d^2 (c+d x)^2 \sin ^2(a+b x)}{2 b^3}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac {\left (6 d^3\right ) \int (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right ) \, dx}{b^3} \\ & = -\frac {3 c d^3 x}{2 b^3}-\frac {3 d^4 x^2}{4 b^3}+\frac {(c+d x)^4}{4 b}-\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^3 (c+d x) \cos (a+b x) \sin (a+b x)}{2 b^4}-\frac {d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b^2}-\frac {3 d^4 \sin ^2(a+b x)}{4 b^5}+\frac {3 d^2 (c+d x)^2 \sin ^2(a+b x)}{2 b^3}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac {\left (3 i d^4\right ) \int \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right ) \, dx}{b^4} \\ & = -\frac {3 c d^3 x}{2 b^3}-\frac {3 d^4 x^2}{4 b^3}+\frac {(c+d x)^4}{4 b}-\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^3 (c+d x) \cos (a+b x) \sin (a+b x)}{2 b^4}-\frac {d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b^2}-\frac {3 d^4 \sin ^2(a+b x)}{4 b^5}+\frac {3 d^2 (c+d x)^2 \sin ^2(a+b x)}{2 b^3}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac {\left (3 d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(4,x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^5} \\ & = -\frac {3 c d^3 x}{2 b^3}-\frac {3 d^4 x^2}{4 b^3}+\frac {(c+d x)^4}{4 b}-\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}+\frac {3 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{2 b^4}-\frac {d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b^2}-\frac {3 d^4 \sin ^2(a+b x)}{4 b^5}+\frac {3 d^2 (c+d x)^2 \sin ^2(a+b x)}{2 b^3}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2918\) vs. \(2(307)=614\).

Time = 6.45 (sec) , antiderivative size = 2918, normalized size of antiderivative = 9.50 \[ \int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx=\text {Result too large to show} \]

[In]

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

[Out]

-((c^2*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) - (c*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*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^4 - (d^4*E^(I*a)*Csc[a]*((2*b^5*
x^5)/E^((2*I)*a) + (5*I)*b^4*(1 - E^((-2*I)*a))*x^4*Log[1 - E^((-I)*(a + b*x))] + (5*I)*b^4*(1 - E^((-2*I)*a))
*x^4*Log[1 + E^((-I)*(a + b*x))] - 20*b^3*(1 - E^((-2*I)*a))*x^3*PolyLog[2, -E^((-I)*(a + b*x))] - 20*b^3*(1 -
E^((-2*I)*a))*x^3*PolyLog[2, E^((-I)*(a + b*x))] + (60*I)*b^2*(1 - E^((-2*I)*a))*x^2*PolyLog[3, -E^((-I)*(a +
b*x))] + (60*I)*b^2*(1 - E^((-2*I)*a))*x^2*PolyLog[3, E^((-I)*(a + b*x))] + 120*b*(1 - E^((-2*I)*a))*x*PolyLo
g[4, -E^((-I)*(a + b*x))] + 120*b*(1 - E^((-2*I)*a))*x*PolyLog[4, E^((-I)*(a + b*x))] - (120*I)*(1 - E^((-2*I)
*a))*PolyLog[5, -E^((-I)*(a + b*x))] - (120*I)*(1 - E^((-2*I)*a))*PolyLog[5, E^((-I)*(a + b*x))]))/(10*b^5) +
(c^4*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]/(160*b^5) - ((I/160)*Sin[2*a + 2*b*x])/b^5)*(80*b^5*c^4*x*Cos[a + 2*b*x] + 160*b^5*c^3*d*x
^2*Cos[a + 2*b*x] + 160*b^5*c^2*d^2*x^3*Cos[a + 2*b*x] + 80*b^5*c*d^3*x^4*Cos[a + 2*b*x] + 16*b^5*d^4*x^5*Cos[
a + 2*b*x] + 80*b^5*c^4*x*Cos[3*a + 2*b*x] + 160*b^5*c^3*d*x^2*Cos[3*a + 2*b*x] + 160*b^5*c^2*d^2*x^3*Cos[3*a
+ 2*b*x] + 80*b^5*c*d^3*x^4*Cos[3*a + 2*b*x] + 16*b^5*d^4*x^5*Cos[3*a + 2*b*x] + (10*I)*b^4*c^4*Cos[3*a + 4*b*
x] - 20*b^3*c^3*d*Cos[3*a + 4*b*x] - (30*I)*b^2*c^2*d^2*Cos[3*a + 4*b*x] + 30*b*c*d^3*Cos[3*a + 4*b*x] + (15*I
)*d^4*Cos[3*a + 4*b*x] + (40*I)*b^4*c^3*d*x*Cos[3*a + 4*b*x] - 60*b^3*c^2*d^2*x*Cos[3*a + 4*b*x] - (60*I)*b^2*
c*d^3*x*Cos[3*a + 4*b*x] + 30*b*d^4*x*Cos[3*a + 4*b*x] + (60*I)*b^4*c^2*d^2*x^2*Cos[3*a + 4*b*x] - 60*b^3*c*d^
3*x^2*Cos[3*a + 4*b*x] - (30*I)*b^2*d^4*x^2*Cos[3*a + 4*b*x] + (40*I)*b^4*c*d^3*x^3*Cos[3*a + 4*b*x] - 20*b^3*
d^4*x^3*Cos[3*a + 4*b*x] + (10*I)*b^4*d^4*x^4*Cos[3*a + 4*b*x] - (10*I)*b^4*c^4*Cos[5*a + 4*b*x] + 20*b^3*c^3*
d*Cos[5*a + 4*b*x] + (30*I)*b^2*c^2*d^2*Cos[5*a + 4*b*x] - 30*b*c*d^3*Cos[5*a + 4*b*x] - (15*I)*d^4*Cos[5*a +
4*b*x] - (40*I)*b^4*c^3*d*x*Cos[5*a + 4*b*x] + 60*b^3*c^2*d^2*x*Cos[5*a + 4*b*x] + (60*I)*b^2*c*d^3*x*Cos[5*a
+ 4*b*x] - 30*b*d^4*x*Cos[5*a + 4*b*x] - (60*I)*b^4*c^2*d^2*x^2*Cos[5*a + 4*b*x] + 60*b^3*c*d^3*x^2*Cos[5*a +
4*b*x] + (30*I)*b^2*d^4*x^2*Cos[5*a + 4*b*x] - (40*I)*b^4*c*d^3*x^3*Cos[5*a + 4*b*x] + 20*b^3*d^4*x^3*Cos[5*a
+ 4*b*x] - (10*I)*b^4*d^4*x^4*Cos[5*a + 4*b*x] + 20*b^4*c^4*Sin[a] - (40*I)*b^3*c^3*d*Sin[a] - 60*b^2*c^2*d^2*
Sin[a] + (60*I)*b*c*d^3*Sin[a] + 30*d^4*Sin[a] + 80*b^4*c^3*d*x*Sin[a] - (120*I)*b^3*c^2*d^2*x*Sin[a] - 120*b^
2*c*d^3*x*Sin[a] + (60*I)*b*d^4*x*Sin[a] + 120*b^4*c^2*d^2*x^2*Sin[a] - (120*I)*b^3*c*d^3*x^2*Sin[a] - 60*b^2*
d^4*x^2*Sin[a] + 80*b^4*c*d^3*x^3*Sin[a] - (40*I)*b^3*d^4*x^3*Sin[a] + 20*b^4*d^4*x^4*Sin[a] + (80*I)*b^5*c^4*
x*Sin[a + 2*b*x] + (160*I)*b^5*c^3*d*x^2*Sin[a + 2*b*x] + (160*I)*b^5*c^2*d^2*x^3*Sin[a + 2*b*x] + (80*I)*b^5*
c*d^3*x^4*Sin[a + 2*b*x] + (16*I)*b^5*d^4*x^5*Sin[a + 2*b*x] + (80*I)*b^5*c^4*x*Sin[3*a + 2*b*x] + (160*I)*b^5
*c^3*d*x^2*Sin[3*a + 2*b*x] + (160*I)*b^5*c^2*d^2*x^3*Sin[3*a + 2*b*x] + (80*I)*b^5*c*d^3*x^4*Sin[3*a + 2*b*x]
+ (16*I)*b^5*d^4*x^5*Sin[3*a + 2*b*x] - 10*b^4*c^4*Sin[3*a + 4*b*x] - (20*I)*b^3*c^3*d*Sin[3*a + 4*b*x] + 30*
b^2*c^2*d^2*Sin[3*a + 4*b*x] + (30*I)*b*c*d^3*Sin[3*a + 4*b*x] - 15*d^4*Sin[3*a + 4*b*x] - 40*b^4*c^3*d*x*Sin[
3*a + 4*b*x] - (60*I)*b^3*c^2*d^2*x*Sin[3*a + 4*b*x] + 60*b^2*c*d^3*x*Sin[3*a + 4*b*x] + (30*I)*b*d^4*x*Sin[3*
a + 4*b*x] - 60*b^4*c^2*d^2*x^2*Sin[3*a + 4*b*x] - (60*I)*b^3*c*d^3*x^2*Sin[3*a + 4*b*x] + 30*b^2*d^4*x^2*Sin[
3*a + 4*b*x] - 40*b^4*c*d^3*x^3*Sin[3*a + 4*b*x] - (20*I)*b^3*d^4*x^3*Sin[3*a + 4*b*x] - 10*b^4*d^4*x^4*Sin[3*
a + 4*b*x] + 10*b^4*c^4*Sin[5*a + 4*b*x] + (20*I)*b^3*c^3*d*Sin[5*a + 4*b*x] - 30*b^2*c^2*d^2*Sin[5*a + 4*b*x]
- (30*I)*b*c*d^3*Sin[5*a + 4*b*x] + 15*d^4*Sin[5*a + 4*b*x] + 40*b^4*c^3*d*x*Sin[5*a + 4*b*x] + (60*I)*b^3*c^
2*d^2*x*Sin[5*a + 4*b*x] - 60*b^2*c*d^3*x*Sin[5*a + 4*b*x] - (30*I)*b*d^4*x*Sin[5*a + 4*b*x] + 60*b^4*c^2*d^2*
x^2*Sin[5*a + 4*b*x] + (60*I)*b^3*c*d^3*x^2*Sin[5*a + 4*b*x] - 30*b^2*d^4*x^2*Sin[5*a + 4*b*x] + 40*b^4*c*d^3*
x^3*Sin[5*a + 4*b*x] + (20*I)*b^3*d^4*x^3*Sin[5*a + 4*b*x] + 10*b^4*d^4*x^4*Sin[5*a + 4*b*x]) - (2*c^3*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*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. 1334 vs. \(2 (276 ) = 552\).

Time = 3.24 (sec) , antiderivative size = 1335, normalized size of antiderivative = 4.35

method	result	size
risch	\(\text {Expression too large to display}\)	\(1335\)

[In]

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

[Out]

-1/4/b^4*d*(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)*sin(2*b*x+2*a)+I*c^4*x+1/5*
I/d*c^5-12*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*d*c^3
*x*a+12*I/b^2*d^2*c^2*a^2*x-1/5*I*d^4*x^5-8*I/b^3*d^3*c*a^3*x+12/b^3*d^2*c^2*polylog(3,-exp(I*(b*x+a)))-1/b^5*
d^4*ln(1-exp(I*(b*x+a)))*a^4+12/b^3*d^4*polylog(3,exp(I*(b*x+a)))*x^2+12/b^3*d^4*polylog(3,-exp(I*(b*x+a)))*x^
2+12/b^3*d^2*c^2*polylog(3,exp(I*(b*x+a)))-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*d^4*ln(1-exp(I*(b*x+a)))*x^4+1/b*d^4*ln(exp(I*(b*x+a))+1)*x^4+8/5*I/b^5*d^4*a^5-24*d^4*polylog(5,-exp(I*(
b*x+a)))/b^5-24*d^4*polylog(5,exp(I*(b*x+a)))/b^5-I*d^3*c*x^4-2*I*d^2*c^2*x^3-2*I*d*c^3*x^2+1/8*(2*b^4*d^4*x^4
+8*b^4*c*d^3*x^3+12*b^4*c^2*d^2*x^2+8*b^4*c^3*d*x+2*b^4*c^4-6*b^2*d^4*x^2-12*b^2*c*d^3*x-6*b^2*c^2*d^2+3*d^4)/
b^5*cos(2*b*x+2*a)+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*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+8/b^2*c^3*d*a*ln(exp(I*(b
*x+a)))-4/b^2*c^3*d*a*ln(exp(I*(b*x+a))-1)-4/b^4*c*d^3*a^3*ln(exp(I*(b*x+a))-1)+24*I/b^4*d^4*polylog(4,exp(I*(
b*x+a)))*x+24*I/b^4*d^4*polylog(4,-exp(I*(b*x+a)))*x+2*I/b^4*d^4*a^4*x+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*c^3*a^2-4*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+8*I/b^3*d^2*c^2*a^3-6*I/b^4*d^3*c*a^4-4*I/b^2*d*c^3*polylog(2,exp(I*(b*x+
a)))-4*I/b^2*d*c^3*polylog(2,-exp(I*(b*x+a)))-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)+4/b^2*d*c^3*ln(1-exp(I*(b*x+a)))*a-6/b^3*d^2*c^2*ln(1-exp(I*(b*x+a)))*a^2+6/b*d^2*c^2*ln(1-exp(I*
(b*x+a)))*x^2+6/b*d^2*c^2*ln(exp(I*(b*x+a))+1)*x^2+24/b^3*d^3*c*polylog(3,-exp(I*(b*x+a)))*x+24/b^3*d^3*c*poly
log(3,exp(I*(b*x+a)))*x+8/b^4*c*d^3*a^3*ln(exp(I*(b*x+a)))+4/b*d^3*c*ln(exp(I*(b*x+a))+1)*x^3+4/b^4*d^3*c*ln(1
-exp(I*(b*x+a)))*a^3+4/b*d*c^3*ln(1-exp(I*(b*x+a)))*x+4/b*d*c^3*ln(exp(I*(b*x+a))+1)*x+4/b*d^3*c*ln(1-exp(I*(b
*x+a)))*x^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. 1453 vs. \(2 (272) = 544\).

Time = 0.35 (sec) , antiderivative size = 1453, normalized size of antiderivative = 4.73 \[ \int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/4*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 48*d^4*polylog(5, cos(b*x + a) + I*sin(b*x + a)) + 48*d^4*polylog(5, cos
(b*x + a) - I*sin(b*x + a)) + 48*d^4*polylog(5, -cos(b*x + a) + I*sin(b*x + a)) + 48*d^4*polylog(5, -cos(b*x +
a) - I*sin(b*x + a)) + 3*(2*b^4*c^2*d^2 - b^2*d^4)*x^2 - (2*b^4*d^4*x^4 + 8*b^4*c*d^3*x^3 + 2*b^4*c^4 - 6*b^2
*c^2*d^2 + 3*d^4 + 6*(2*b^4*c^2*d^2 - b^2*d^4)*x^2 + 4*(2*b^4*c^3*d - 3*b^2*c*d^3)*x)*cos(b*x + a)^2 + 2*(2*b^
3*d^4*x^3 + 6*b^3*c*d^3*x^2 + 2*b^3*c^3*d - 3*b*c*d^3 + 3*(2*b^3*c^2*d^2 - b*d^4)*x)*cos(b*x + a)*sin(b*x + a)
+ 2*(2*b^4*c^3*d - 3*b^2*c*d^3)*x + 8*(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)*d
ilog(cos(b*x + a) + I*sin(b*x + a)) + 8*(-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)) + 8*(-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)) + 8*(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)) - 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 + b^4*c^4)*log(cos(b*x + a) + 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 + b^4*c^4)*log(cos(b*x + a) - 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) - 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) - 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) - 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) + 48*(-I*b*d^4*x - I*b*c*d^3)*polylog(4, cos(b*x + a) + I*sin(b*x + a)) + 48*(I*b*d^4*x + I*b*c*d^3)*poly
log(4, cos(b*x + a) - I*sin(b*x + a)) + 48*(I*b*d^4*x + I*b*c*d^3)*polylog(4, -cos(b*x + a) + I*sin(b*x + a))
+ 48*(-I*b*d^4*x - I*b*c*d^3)*polylog(4, -cos(b*x + a) - I*sin(b*x + a)) - 24*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b
^2*c^2*d^2)*polylog(3, cos(b*x + a) + I*sin(b*x + a)) - 24*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*polylog
(3, cos(b*x + a) - I*sin(b*x + a)) - 24*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*polylog(3, -cos(b*x + a) +
I*sin(b*x + a)) - 24*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*polylog(3, -cos(b*x + a) - I*sin(b*x + a)))/
b^5

Sympy [F]

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

[In]

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

[Out]

Integral((c + d*x)**4*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. 1654 vs. \(2 (272) = 544\).

Time = 0.55 (sec) , antiderivative size = 1654, normalized size of antiderivative = 5.39 \[ \int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/40*(20*(sin(b*x + a)^2 - log(sin(b*x + a)^2))*c^4 - 80*(sin(b*x + a)^2 - log(sin(b*x + a)^2))*a*c^3*d/b + 1
20*(sin(b*x + a)^2 - log(sin(b*x + a)^2))*a^2*c^2*d^2/b^2 - 80*(sin(b*x + a)^2 - log(sin(b*x + a)^2))*a^3*c*d^
3/b^3 + 20*(sin(b*x + a)^2 - log(sin(b*x + a)^2))*a^4*d^4/b^4 - (-8*I*(b*x + a)^5*d^4 - 40*(I*b*c*d^3 - I*a*d^
4)*(b*x + a)^4 - 960*d^4*polylog(5, -e^(I*b*x + I*a)) - 960*d^4*polylog(5, e^(I*b*x + I*a)) - 80*(I*b^2*c^2*d^
2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a)^3 - 80*(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 - 40*(-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))*a
rctan2(sin(b*x + a), cos(b*x + a) + 1) - 40*(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) + 5*(2*(b*x + a)^4*d^4 - 6*b^2*c^2*d^2 + 12*a*b*c
*d^3 - 3*(2*a^2 - 1)*d^4 + 8*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(2*b^2*c^2*d^2 - 4*a*b*c*d^3 + (2*a^2 - 1)*d^4)
*(b*x + a)^2 + 4*(2*b^3*c^3*d - 6*a*b^2*c^2*d^2 + 3*(2*a^2 - 1)*b*c*d^3 - (2*a^3 - 3*a)*d^4)*(b*x + a))*cos(2*
b*x + 2*a) - 160*(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))
- 160*(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)) + 20*((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*c
^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x +
a) + 1) + 20*((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*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a))*log(cos(b*x + a)^2 + sin(b*x +
a)^2 - 2*cos(b*x + a) + 1) - 960*(-I*b*c*d^3 - I*(b*x + a)*d^4 + I*a*d^4)*polylog(4, -e^(I*b*x + I*a)) - 960*
(-I*b*c*d^3 - I*(b*x + a)*d^4 + I*a*d^4)*polylog(4, e^(I*b*x + I*a)) + 480*(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))*polylog(3, -e^(I*b*x + I*a)) + 480*(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))*polylog(3, e^(I*b*x + I*a)) - 10*(2*b^3*c^3*
d - 6*a*b^2*c^2*d^2 + 2*(b*x + a)^3*d^4 + 3*(2*a^2 - 1)*b*c*d^3 - (2*a^3 - 3*a)*d^4 + 6*(b*c*d^3 - a*d^4)*(b*x
+ a)^2 + 3*(2*b^2*c^2*d^2 - 4*a*b*c*d^3 + (2*a^2 - 1)*d^4)*(b*x + a))*sin(2*b*x + 2*a))/b^4)/b

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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