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

Optimal. Leaf size=307 \[ -\frac {3 d^4 \text {Li}_5\left (e^{2 i (a+b x)}\right )}{2 b^5}-\frac {3 d^4 \sin ^2(a+b x)}{4 b^5}+\frac {3 i d^3 (c+d x) \text {Li}_4\left (e^{2 i (a+b x)}\right )}{b^4}+\frac {3 d^3 (c+d x) \sin (a+b x) \cos (a+b x)}{2 b^4}+\frac {3 d^2 (c+d x)^2 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x)^2 \sin ^2(a+b x)}{2 b^3}-\frac {2 i d (c+d x)^3 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}-\frac {d (c+d x)^3 \sin (a+b x) \cos (a+b x)}{b^2}+\frac {(c+d x)^4 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac {3 c d^3 x}{2 b^3}-\frac {3 d^4 x^2}{4 b^3}+\frac {(c+d x)^4}{4 b}-\frac {i (c+d x)^5}{5 d} \]

[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

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Rubi [A]  time = 0.34, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4408, 4404, 3311, 32, 3310, 3717, 2190, 2531, 6609, 2282, 6589} \[ \frac {3 d^2 (c+d x)^2 \text {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}+\frac {3 i d^3 (c+d x) \text {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}-\frac {2 i d (c+d x)^3 \text {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^4 \text {PolyLog}\left (5,e^{2 i (a+b x)}\right )}{2 b^5}+\frac {3 d^2 (c+d x)^2 \sin ^2(a+b x)}{2 b^3}+\frac {3 d^3 (c+d x) \sin (a+b x) \cos (a+b x)}{2 b^4}-\frac {d (c+d x)^3 \sin (a+b x) \cos (a+b x)}{b^2}-\frac {3 d^4 \sin ^2(a+b x)}{4 b^5}+\frac {(c+d x)^4 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {(c+d x)^4 \sin ^2(a+b x)}{2 b}-\frac {3 c d^3 x}{2 b^3}-\frac {3 d^4 x^2}{4 b^3}+\frac {(c+d x)^4}{4 b}-\frac {i (c+d x)^5}{5 d} \]

Antiderivative was successfully verified.

[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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2282

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 2531

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 3310

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 3311

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 3717

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 4404

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 4408

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 6589

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 6609

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*} \int (c+d x)^4 \cos ^2(a+b x) \cot (a+b x) \, dx &=\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 \text {Li}_2\left (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 \text {Li}_2\left (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 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}+\frac {3 d^2 (c+d x)^2 \text {Li}_3\left (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) \text {Li}_3\left (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 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}+\frac {3 d^2 (c+d x)^2 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {3 i d^3 (c+d x) \text {Li}_4\left (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 \text {Li}_4\left (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 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}+\frac {3 d^2 (c+d x)^2 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {3 i d^3 (c+d x) \text {Li}_4\left (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 ) \operatorname {Subst}\left (\int \frac {\text {Li}_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 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}+\frac {3 d^2 (c+d x)^2 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {3 i d^3 (c+d x) \text {Li}_4\left (e^{2 i (a+b x)}\right )}{b^4}-\frac {3 d^4 \text {Li}_5\left (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*}

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Mathematica [B]  time = 6.52, size = 2828, normalized size = 9.21 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[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*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, -E^
((-I)*(a + b*x))] - I*PolyLog[3, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b*x*PolyLog[2, E^
((-I)*(a + b*x))] - I*PolyLog[3, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/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*L
og[1 + E^((-I)*(a + b*x))] - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[2, -E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLo
g[3, -E^((-I)*(a + b*x))] - 2*PolyLog[4, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (6*(-1 + E^((2*I)*a))*(b^2*x^2*P
olyLog[2, E^((-I)*(a + b*x))] - (2*I)*b*x*PolyLog[3, E^((-I)*(a + b*x))] - 2*PolyLog[4, E^((-I)*(a + b*x))]))/
E^((2*I)*a)))/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*(-1 + E^((2*I)*a))*(b^3
*x^3*PolyLog[2, -E^((-I)*(a + b*x))] - (3*I)*b^2*x^2*PolyLog[3, -E^((-I)*(a + b*x))] - 6*b*x*PolyLog[4, -E^((-
I)*(a + b*x))] + (6*I)*PolyLog[5, -E^((-I)*(a + b*x))]))/E^((2*I)*a) - (20*(-1 + E^((2*I)*a))*(b^3*x^3*PolyLog
[2, E^((-I)*(a + b*x))] - (3*I)*b^2*x^2*PolyLog[3, E^((-I)*(a + b*x))] - 6*b*x*PolyLog[4, E^((-I)*(a + b*x))]
+ (6*I)*PolyLog[5, E^((-I)*(a + b*x))]))/E^((2*I)*a)))/(10*b^5) + (c^4*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*Si
n[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] + 1
20*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*Si
n[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*Si
n[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)])

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fricas [C]  time = 0.66, size = 1453, normalized size = 4.73 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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 - 24*I*b^3*c*d^3*x^2 - 24*I*b^3*c^2*d^2*x - 8*I*b^3*c^3
*d)*dilog(cos(b*x + a) + I*sin(b*x + a)) - (8*I*b^3*d^4*x^3 + 24*I*b^3*c*d^3*x^2 + 24*I*b^3*c^2*d^2*x + 8*I*b^
3*c^3*d)*dilog(cos(b*x + a) - I*sin(b*x + a)) - (8*I*b^3*d^4*x^3 + 24*I*b^3*c*d^3*x^2 + 24*I*b^3*c^2*d^2*x + 8
*I*b^3*c^3*d)*dilog(-cos(b*x + a) + I*sin(b*x + a)) - (-8*I*b^3*d^4*x^3 - 24*I*b^3*c*d^3*x^2 - 24*I*b^3*c^2*d^
2*x - 8*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 + 48*I*b*c*d^3)*polylog(4, cos(b*x + a) + I*sin(b*x + a)) - (-48*I*b*d^4*
x - 48*I*b*c*d^3)*polylog(4, cos(b*x + a) - I*sin(b*x + a)) - (-48*I*b*d^4*x - 48*I*b*c*d^3)*polylog(4, -cos(b
*x + a) + I*sin(b*x + a)) - (48*I*b*d^4*x + 48*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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{4} \cos \left (b x + a\right )^{2} \cot \left (b x + a\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.54, size = 1326, normalized size = 4.32 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.06, size = 1635, normalized size = 5.33 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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 + 40*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 + 160*I*a*b*c*d^3 - 80*I*a^2*d^4)*(b*x + a)^3 + (-80*I*b^3*c^3*d + 240*I*a*b^2*c^2*d^2 - 240*I*a^2*b*c*
d^3 + 80*I*a^3*d^4)*(b*x + a)^2 + (40*I*(b*x + a)^4*d^4 + (160*I*b*c*d^3 - 160*I*a*d^4)*(b*x + a)^3 + (240*I*b
^2*c^2*d^2 - 480*I*a*b*c*d^3 + 240*I*a^2*d^4)*(b*x + a)^2 + (160*I*b^3*c^3*d - 480*I*a*b^2*c^2*d^2 + 480*I*a^2
*b*c*d^3 - 160*I*a^3*d^4)*(b*x + a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) + (-40*I*(b*x + a)^4*d^4 + (-160*
I*b*c*d^3 + 160*I*a*d^4)*(b*x + a)^3 + (-240*I*b^2*c^2*d^2 + 480*I*a*b*c*d^3 - 240*I*a^2*d^4)*(b*x + a)^2 + (-
160*I*b^3*c^3*d + 480*I*a*b^2*c^2*d^2 - 480*I*a^2*b*c*d^3 + 160*I*a^3*d^4)*(b*x + a))*arctan2(sin(b*x + a), -c
os(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 + 480*I*a*b^2
*c^2*d^2 - 480*I*a^2*b*c*d^3 - 160*I*(b*x + a)^3*d^4 + 160*I*a^3*d^4 + (-480*I*b*c*d^3 + 480*I*a*d^4)*(b*x + a
)^2 + (-480*I*b^2*c^2*d^2 + 960*I*a*b*c*d^3 - 480*I*a^2*d^4)*(b*x + a))*dilog(-e^(I*b*x + I*a)) + (-160*I*b^3*
c^3*d + 480*I*a*b^2*c^2*d^2 - 480*I*a^2*b*c*d^3 - 160*I*(b*x + a)^3*d^4 + 160*I*a^3*d^4 + (-480*I*b*c*d^3 + 48
0*I*a*d^4)*(b*x + a)^2 + (-480*I*b^2*c^2*d^2 + 960*I*a*b*c*d^3 - 480*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 + 960*I*(b*x + a)*d^4 - 960*I*a*d^4)*polylog(4, -e^
(I*b*x + I*a)) + (960*I*b*c*d^3 + 960*I*(b*x + a)*d^4 - 960*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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (a+b\,x\right )}^2\,\mathrm {cot}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right )^{4} \cos ^{2}{\left (a + b x \right )} \cot {\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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