\(\int (c+d x)^4 \cot ^3(a+b x) \, dx\) [178]

Optimal result

Integrand size = 16, antiderivative size = 302 \[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=-\frac {2 i d (c+d x)^3}{b^2}-\frac {(c+d x)^4}{2 b}+\frac {i (c+d x)^5}{5 d}-\frac {2 d (c+d x)^3 \cot (a+b x)}{b^2}-\frac {(c+d x)^4 \cot ^2(a+b x)}{2 b}+\frac {6 d^2 (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {(c+d x)^4 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}+\frac {3 d^4 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^5}-\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}+\frac {3 d^4 \operatorname {PolyLog}\left (5,e^{2 i (a+b x)}\right )}{2 b^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3801, 3798, 2221, 2611, 2320, 6724, 32, 6744} \[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=\frac {3 d^4 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^5}+\frac {3 d^4 \operatorname {PolyLog}\left (5,e^{2 i (a+b x)}\right )}{2 b^5}-\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^4}-\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}-\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^3}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac {2 d (c+d x)^3 \cot (a+b x)}{b^2}-\frac {(c+d x)^4 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {(c+d x)^4 \cot ^2(a+b x)}{2 b}-\frac {2 i d (c+d x)^3}{b^2}-\frac {(c+d x)^4}{2 b}+\frac {i (c+d x)^5}{5 d} \]

[In]

Int[(c + d*x)^4*Cot[a + b*x]^3,x]

[Out]

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

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1632\) vs. \(2(302)=604\).

Time = 7.09 (sec) , antiderivative size = 1632, normalized size of antiderivative = 5.40 \[ \int (c+d x)^4 \cot ^3(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 ) \cot (a)-\frac {(c+d x)^4 \csc ^2(a+b x)}{2 b}+\frac {c^2 d^2 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^3}-\frac {d^4 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^5}+\frac {c d^3 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^4}+\frac {d^4 e^{i a} \csc (a) \left (2 b^5 e^{-2 i a} x^5+5 i b^4 \left (1-e^{-2 i a}\right ) x^4 \log \left (1-e^{-i (a+b x)}\right )+5 i b^4 \left (1-e^{-2 i a}\right ) x^4 \log \left (1+e^{-i (a+b x)}\right )-20 b^3 \left (1-e^{-2 i a}\right ) x^3 \operatorname {PolyLog}\left (2,-e^{-i (a+b x)}\right )-20 b^3 \left (1-e^{-2 i a}\right ) x^3 \operatorname {PolyLog}\left (2,e^{-i (a+b x)}\right )+60 i b^2 \left (1-e^{-2 i a}\right ) x^2 \operatorname {PolyLog}\left (3,-e^{-i (a+b x)}\right )+60 i b^2 \left (1-e^{-2 i a}\right ) x^2 \operatorname {PolyLog}\left (3,e^{-i (a+b x)}\right )+120 b \left (1-e^{-2 i a}\right ) x \operatorname {PolyLog}\left (4,-e^{-i (a+b x)}\right )+120 b \left (1-e^{-2 i a}\right ) x \operatorname {PolyLog}\left (4,e^{-i (a+b x)}\right )-120 i \left (1-e^{-2 i a}\right ) \operatorname {PolyLog}\left (5,-e^{-i (a+b x)}\right )-120 i \left (1-e^{-2 i a}\right ) \operatorname {PolyLog}\left (5,e^{-i (a+b x)}\right )\right )}{10 b^5}-\frac {c^4 \csc (a) (-b x \cos (a)+\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a))}{b \left (\cos ^2(a)+\sin ^2(a)\right )}+\frac {6 c^2 d^2 \csc (a) (-b x \cos (a)+\log (\cos (b x) \sin (a)+\cos (a) \sin (b x)) \sin (a))}{b^3 \left (\cos ^2(a)+\sin ^2(a)\right )}+\frac {2 \csc (a) \csc (a+b x) \left (c^3 d \sin (b x)+3 c^2 d^2 x \sin (b x)+3 c d^3 x^2 \sin (b x)+d^4 x^3 \sin (b x)\right )}{b^2}+\frac {2 c^3 d \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^2 \sqrt {\sec ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}}-\frac {6 c d^3 \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^4 \sqrt {\sec ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}} \]

[In]

Integrate[(c + d*x)^4*Cot[a + b*x]^3,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)*Cot[a]) - ((c + d*x)^4*Csc[a + b*x]^2)/(
2*b) + (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))*Poly
Log[3, -E^((-I)*(a + b*x))] + (6*I)*(1 - E^((-2*I)*a))*PolyLog[3, E^((-I)*(a + b*x))]))/b^3 - (d^4*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^5 + (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*Poly
Log[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*PolyLog[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)) + (6*c^2*d^2*Csc[a]*(-(b*x*Cos
[a]) + Log[Cos[b*x]*Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) + (2*Csc[a]*Csc[a + b*x]*(c
^3*d*Sin[b*x] + 3*c^2*d^2*x*Sin[b*x] + 3*c*d^3*x^2*Sin[b*x] + d^4*x^3*Sin[b*x]))/b^2 + (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 + ArcTa
n[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 + ArcTa
n[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)]) - (6*c*d^3*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^4*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. 1876 vs. \(2 (273 ) = 546\).

Time = 1.38 (sec) , antiderivative size = 1877, normalized size of antiderivative = 6.22

method	result	size
risch	\(\text {Expression too large to display}\)	\(1877\)

[In]

int((d*x+c)^4*cot(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

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

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. 1751 vs. \(2 (266) = 532\).

Time = 0.30 (sec) , antiderivative size = 1751, normalized size of antiderivative = 5.80 \[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^4*cot(b*x+a)^3,x, algorithm="fricas")

[Out]

1/4*(4*b^4*d^4*x^4 + 16*b^4*c*d^3*x^3 + 24*b^4*c^2*d^2*x^2 + 16*b^4*c^3*d*x + 4*b^4*c^4 - 4*(I*b^3*d^4*x^3 + 3
*I*b^3*c*d^3*x^2 + I*b^3*c^3*d - 3*I*b*c*d^3 + 3*I*(b^3*c^2*d^2 - b*d^4)*x + (-I*b^3*d^4*x^3 - 3*I*b^3*c*d^3*x
^2 - I*b^3*c^3*d + 3*I*b*c*d^3 - 3*I*(b^3*c^2*d^2 - b*d^4)*x)*cos(2*b*x + 2*a))*dilog(cos(2*b*x + 2*a) + I*sin
(2*b*x + 2*a)) - 4*(-I*b^3*d^4*x^3 - 3*I*b^3*c*d^3*x^2 - I*b^3*c^3*d + 3*I*b*c*d^3 - 3*I*(b^3*c^2*d^2 - b*d^4)
*x + (I*b^3*d^4*x^3 + 3*I*b^3*c*d^3*x^2 + I*b^3*c^3*d - 3*I*b*c*d^3 + 3*I*(b^3*c^2*d^2 - b*d^4)*x)*cos(2*b*x +
2*a))*dilog(cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a)) + 2*(b^4*c^4 - 4*a*b^3*c^3*d + 6*(a^2 - 1)*b^2*c^2*d^2 - 4
*(a^3 - 3*a)*b*c*d^3 + (a^4 - 6*a^2)*d^4 - (b^4*c^4 - 4*a*b^3*c^3*d + 6*(a^2 - 1)*b^2*c^2*d^2 - 4*(a^3 - 3*a)*
b*c*d^3 + (a^4 - 6*a^2)*d^4)*cos(2*b*x + 2*a))*log(-1/2*cos(2*b*x + 2*a) + 1/2*I*sin(2*b*x + 2*a) + 1/2) + 2*(
b^4*c^4 - 4*a*b^3*c^3*d + 6*(a^2 - 1)*b^2*c^2*d^2 - 4*(a^3 - 3*a)*b*c*d^3 + (a^4 - 6*a^2)*d^4 - (b^4*c^4 - 4*a
*b^3*c^3*d + 6*(a^2 - 1)*b^2*c^2*d^2 - 4*(a^3 - 3*a)*b*c*d^3 + (a^4 - 6*a^2)*d^4)*cos(2*b*x + 2*a))*log(-1/2*c
os(2*b*x + 2*a) - 1/2*I*sin(2*b*x + 2*a) + 1/2) + 2*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 4*a*b^3*c^3*d - 6*a^2*b^2
*c^2*d^2 + 4*(a^3 - 3*a)*b*c*d^3 - (a^4 - 6*a^2)*d^4 + 6*(b^4*c^2*d^2 - b^2*d^4)*x^2 + 4*(b^4*c^3*d - 3*b^2*c*
d^3)*x - (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*(a^3 - 3*a)*b*c*d^3 - (a^4 - 6
*a^2)*d^4 + 6*(b^4*c^2*d^2 - b^2*d^4)*x^2 + 4*(b^4*c^3*d - 3*b^2*c*d^3)*x)*cos(2*b*x + 2*a))*log(-cos(2*b*x +
2*a) + I*sin(2*b*x + 2*a) + 1) + 2*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*(a^3
- 3*a)*b*c*d^3 - (a^4 - 6*a^2)*d^4 + 6*(b^4*c^2*d^2 - b^2*d^4)*x^2 + 4*(b^4*c^3*d - 3*b^2*c*d^3)*x - (b^4*d^4
*x^4 + 4*b^4*c*d^3*x^3 + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*(a^3 - 3*a)*b*c*d^3 - (a^4 - 6*a^2)*d^4 + 6*(b^
4*c^2*d^2 - b^2*d^4)*x^2 + 4*(b^4*c^3*d - 3*b^2*c*d^3)*x)*cos(2*b*x + 2*a))*log(-cos(2*b*x + 2*a) - I*sin(2*b*
x + 2*a) + 1) + 3*(d^4*cos(2*b*x + 2*a) - d^4)*polylog(5, cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a)) + 3*(d^4*cos(
2*b*x + 2*a) - d^4)*polylog(5, cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a)) - 6*(-I*b*d^4*x - I*b*c*d^3 + (I*b*d^4*x
+ I*b*c*d^3)*cos(2*b*x + 2*a))*polylog(4, cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a)) - 6*(I*b*d^4*x + I*b*c*d^3 +
(-I*b*d^4*x - I*b*c*d^3)*cos(2*b*x + 2*a))*polylog(4, cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a)) + 6*(b^2*d^4*x^2
+ 2*b^2*c*d^3*x + b^2*c^2*d^2 - d^4 - (b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2 - d^4)*cos(2*b*x + 2*a))*pol
ylog(3, cos(2*b*x + 2*a) + I*sin(2*b*x + 2*a)) + 6*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2 - d^4 - (b^2*d^4
*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2 - d^4)*cos(2*b*x + 2*a))*polylog(3, cos(2*b*x + 2*a) - I*sin(2*b*x + 2*a))
+ 8*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*b^3*c^2*d^2*x + b^3*c^3*d)*sin(2*b*x + 2*a))/(b^5*cos(2*b*x + 2*a) - b^
5)

Sympy [F]

\[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=\int \left (c + d x\right )^{4} \cot ^{3}{\left (a + b x \right )}\, dx \]

[In]

integrate((d*x+c)**4*cot(b*x+a)**3,x)

[Out]

Integral((c + d*x)**4*cot(a + b*x)**3, 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. 7158 vs. \(2 (266) = 532\).

Time = 3.68 (sec) , antiderivative size = 7158, normalized size of antiderivative = 23.70 \[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^4*cot(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2*(c^4*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2)) - 4*a*c^3*d*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2))/b + 6*
a^2*c^2*d^2*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2))/b^2 - 4*a^3*c*d^3*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2)
)/b^3 + a^4*d^4*(1/sin(b*x + a)^2 + log(sin(b*x + a)^2))/b^4 - 2*(2*(b*x + a)^5*d^4 + 40*b^3*c^3*d - 120*a*b^2
*c^2*d^2 + 120*a^2*b*c*d^3 - 40*a^3*d^4 + 10*(b*c*d^3 - a*d^4)*(b*x + a)^4 + 20*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a
^2*d^4)*(b*x + a)^3 + 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)*(b*x + a)^2 - 10*((b*x + a)^4
*d^4 - 6*b^2*c^2*d^2 + 12*a*b*c*d^3 - 6*a^2*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 - 1)*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 1)*b*c*d^3 - (a^3 - 3*a)*d^4)*(b*x
+ a) + ((b*x + a)^4*d^4 - 6*b^2*c^2*d^2 + 12*a*b*c*d^3 - 6*a^2*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 - 1)*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 1)*b*c*d^3 - (
a^3 - 3*a)*d^4)*(b*x + a))*cos(4*b*x + 4*a) - 2*((b*x + a)^4*d^4 - 6*b^2*c^2*d^2 + 12*a*b*c*d^3 - 6*a^2*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 - 1)*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3
*a*b^2*c^2*d^2 + 3*(a^2 - 1)*b*c*d^3 - (a^3 - 3*a)*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (-I*(b*x + a)^4*d^4 + 6*
I*b^2*c^2*d^2 - 12*I*a*b*c*d^3 + 6*I*a^2*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 + I)*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 + I)*b*c*d^3 + (I*
a^3 - 3*I*a)*d^4)*(b*x + a))*sin(4*b*x + 4*a) - 2*(I*(b*x + a)^4*d^4 - 6*I*b^2*c^2*d^2 + 12*I*a*b*c*d^3 - 6*I*
a^2*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 - I)*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 - I)*b*c*d^3 + (-I*a^3 + 3*I*a)*d^4)*(b*x + a))*sin(2*b*x +
2*a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) + 60*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4 + (b^2*c^2*d^2 - 2*a*b
*c*d^3 + a^2*d^4)*cos(4*b*x + 4*a) - 2*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*cos(2*b*x + 2*a) + (I*b^2*c^2*d^2
- 2*I*a*b*c*d^3 + I*a^2*d^4)*sin(4*b*x + 4*a) + 2*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*sin(2*b*x + 2*
a))*arctan2(sin(b*x + a), cos(b*x + a) - 1) + 10*((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 - 1)*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 1)*b*c*d^3 - (a^
3 - 3*a)*d^4)*(b*x + a) + ((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 - 1)*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 1)*b*c*d^3 - (a^3 - 3*a)*d^4)*(b*x + a)
)*cos(4*b*x + 4*a) - 2*((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 - 1)*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 1)*b*c*d^3 - (a^3 - 3*a)*d^4)*(b*x + a))*c
os(2*b*x + 2*a) + (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 - I)*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 - I)*b*c*d^3 + (-I*a^3 + 3*I*a)
*d^4)*(b*x + a))*sin(4*b*x + 4*a) + 2*(-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 + I)*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 + I)*
b*c*d^3 + (I*a^3 - 3*I*a)*d^4)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), -cos(b*x + a) + 1) + 2*((b*x
+ a)^5*d^4 + 5*(b*c*d^3 - a*d^4)*(b*x + a)^4 + 10*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (a^2 - 2)*d^4)*(b*x + a)^3 + 1
0*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 2)*b*c*d^3 - (a^3 - 6*a)*d^4)*(b*x + a)^2 - 60*(b^2*c^2*d^2 - 2*a*b*
c*d^3 + a^2*d^4)*(b*x + a))*cos(4*b*x + 4*a) - 4*((b*x + a)^5*d^4 + 10*b^3*c^3*d - 30*a*b^2*c^2*d^2 + 30*a^2*b
*c*d^3 - 10*a^3*d^4 + 5*(b*c*d^3 - (a - I)*d^4)*(b*x + a)^4 + 10*(b^2*c^2*d^2 - 2*(a - I)*b*c*d^3 + (a^2 - 2*I
*a - 1)*d^4)*(b*x + a)^3 + 10*(b^3*c^3*d - 3*(a - I)*b^2*c^2*d^2 + 3*(a^2 - 2*I*a - 1)*b*c*d^3 - (a^3 - 3*I*a^
2 - 3*a)*d^4)*(b*x + a)^2 - 10*(-2*I*b^3*c^3*d + 3*(2*I*a + 1)*b^2*c^2*d^2 + 6*(-I*a^2 - a)*b*c*d^3 + (2*I*a^3
+ 3*a^2)*d^4)*(b*x + a))*cos(2*b*x + 2*a) + 40*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + (b*x + a)^3*d^4 + 3*(a^2 - 1)*b
*c*d^3 - (a^3 - 3*a)*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 - 1)*d^4)*(b*
x + a) + (b^3*c^3*d - 3*a*b^2*c^2*d^2 + (b*x + a)^3*d^4 + 3*(a^2 - 1)*b*c*d^3 - (a^3 - 3*a)*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 - 1)*d^4)*(b*x + a))*cos(4*b*x + 4*a) - 2*(b^3*c^3*d
- 3*a*b^2*c^2*d^2 + (b*x + a)^3*d^4 + 3*(a^2 - 1)*b*c*d^3 - (a^3 - 3*a)*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 - 1)*d^4)*(b*x + a))*cos(2*b*x + 2*a) + (I*b^3*c^3*d - 3*I*a*b^2*c^2*d^
2 + I*(b*x + a)^3*d^4 + 3*(I*a^2 - I)*b*c*d^3 + (-I*a^3 + 3*I*a)*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 - I)*d^4)*(b*x + a))*sin(4*b*x + 4*a) + 2*(-I*b^3*c^3*d + 3*I*a*b^2*c
^2*d^2 - I*(b*x + a)^3*d^4 + 3*(-I*a^2 + I)*b*c*d^3 + (I*a^3 - 3*I*a)*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 + I)*d^4)*(b*x + a))*sin(2*b*x + 2*a))*dilog(-e^(I*b*x + I*a)
) + 40*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + (b*x + a)^3*d^4 + 3*(a^2 - 1)*b*c*d^3 - (a^3 - 3*a)*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 - 1)*d^4)*(b*x + a) + (b^3*c^3*d - 3*a*b^2*c^2*d^2 + (
b*x + a)^3*d^4 + 3*(a^2 - 1)*b*c*d^3 - (a^3 - 3*a)*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 - 1)*d^4)*(b*x + a))*cos(4*b*x + 4*a) - 2*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + (b*x + a)^3*d^4 + 3*
(a^2 - 1)*b*c*d^3 - (a^3 - 3*a)*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 -
1)*d^4)*(b*x + a))*cos(2*b*x + 2*a) + (I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + I*(b*x + a)^3*d^4 + 3*(I*a^2 - I)*b*c
*d^3 + (-I*a^3 + 3*I*a)*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
- I)*d^4)*(b*x + a))*sin(4*b*x + 4*a) + 2*(-I*b^3*c^3*d + 3*I*a*b^2*c^2*d^2 - I*(b*x + a)^3*d^4 + 3*(-I*a^2 +
I)*b*c*d^3 + (I*a^3 - 3*I*a)*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 + I)*d^4)*(b*x + a))*sin(2*b*x + 2*a))*dilog(e^(I*b*x + I*a)) + 5*(I*(b*x + a)^4*d^4 - 6*I*b^2*c^2*d^2
+ 12*I*a*b*c*d^3 - 6*I*a^2*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 - I)*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 - I)*b*c*d^3 + (-I*a^3 + 3*I*a)*d^4)
*(b*x + a) + (I*(b*x + a)^4*d^4 - 6*I*b^2*c^2*d^2 + 12*I*a*b*c*d^3 - 6*I*a^2*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 - I)*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 - I)*b*c*d^3 + (-I*a^3 + 3*I*a)*d^4)*(b*x + a))*cos(4*b*x + 4*a) + 2*(-I*(b*x + a)^4*d^4 + 6*I*b^
2*c^2*d^2 - 12*I*a*b*c*d^3 + 6*I*a^2*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 + I)*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 + I)*b*c*d^3 + (I*a^3
- 3*I*a)*d^4)*(b*x + a))*cos(2*b*x + 2*a) - ((b*x + a)^4*d^4 - 6*b^2*c^2*d^2 + 12*a*b*c*d^3 - 6*a^2*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 - 1)*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b
^2*c^2*d^2 + 3*(a^2 - 1)*b*c*d^3 - (a^3 - 3*a)*d^4)*(b*x + a))*sin(4*b*x + 4*a) + 2*((b*x + a)^4*d^4 - 6*b^2*c
^2*d^2 + 12*a*b*c*d^3 - 6*a^2*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 - 1)
*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 1)*b*c*d^3 - (a^3 - 3*a)*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) + 5*(I*(b*x + a)^4*d^4 - 6*I*b^2*c^2*d^2
+ 12*I*a*b*c*d^3 - 6*I*a^2*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 - I)*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 - I)*b*c*d^3 + (-I*a^3 + 3*I*a)*d^4)*
(b*x + a) + (I*(b*x + a)^4*d^4 - 6*I*b^2*c^2*d^2 + 12*I*a*b*c*d^3 - 6*I*a^2*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 - I)*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 - I)*b*c*d^3 + (-I*a^3 + 3*I*a)*d^4)*(b*x + a))*cos(4*b*x + 4*a) + 2*(-I*(b*x + a)^4*d^4 + 6*I*b^2
*c^2*d^2 - 12*I*a*b*c*d^3 + 6*I*a^2*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 + I)*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 + I)*b*c*d^3 + (I*a^3 -
3*I*a)*d^4)*(b*x + a))*cos(2*b*x + 2*a) - ((b*x + a)^4*d^4 - 6*b^2*c^2*d^2 + 12*a*b*c*d^3 - 6*a^2*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 - 1)*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^
2*c^2*d^2 + 3*(a^2 - 1)*b*c*d^3 - (a^3 - 3*a)*d^4)*(b*x + a))*sin(4*b*x + 4*a) + 2*((b*x + a)^4*d^4 - 6*b^2*c^
2*d^2 + 12*a*b*c*d^3 - 6*a^2*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 - 1)*
d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 1)*b*c*d^3 - (a^3 - 3*a)*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) + 240*(-I*d^4*cos(4*b*x + 4*a) + 2*I*d^4*c
os(2*b*x + 2*a) + d^4*sin(4*b*x + 4*a) - 2*d^4*sin(2*b*x + 2*a) - I*d^4)*polylog(5, -e^(I*b*x + I*a)) + 240*(-
I*d^4*cos(4*b*x + 4*a) + 2*I*d^4*cos(2*b*x + 2*a) + d^4*sin(4*b*x + 4*a) - 2*d^4*sin(2*b*x + 2*a) - I*d^4)*pol
ylog(5, e^(I*b*x + I*a)) - 240*(b*c*d^3 + (b*x + a)*d^4 - a*d^4 + (b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(4*b*x
+ 4*a) - 2*(b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(2*b*x + 2*a) - (-I*b*c*d^3 - I*(b*x + a)*d^4 + I*a*d^4)*sin(4
*b*x + 4*a) - 2*(I*b*c*d^3 + I*(b*x + a)*d^4 - I*a*d^4)*sin(2*b*x + 2*a))*polylog(4, -e^(I*b*x + I*a)) - 240*(
b*c*d^3 + (b*x + a)*d^4 - a*d^4 + (b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(4*b*x + 4*a) - 2*(b*c*d^3 + (b*x + a)*
d^4 - a*d^4)*cos(2*b*x + 2*a) - (-I*b*c*d^3 - I*(b*x + a)*d^4 + I*a*d^4)*sin(4*b*x + 4*a) - 2*(I*b*c*d^3 + I*(
b*x + a)*d^4 - I*a*d^4)*sin(2*b*x + 2*a))*polylog(4, e^(I*b*x + I*a)) + 120*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I
*(b*x + a)^2*d^4 + (I*a^2 - I)*d^4 + 2*(I*b*c*d^3 - I*a*d^4)*(b*x + 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 - I)*d^4 + 2*(I*b*c*d^3 - I*a*d^4)*(b*x + a))*cos(4*b*x + 4*a) + 2*(-I*b^2*c^2*d^2 + 2*
I*a*b*c*d^3 - I*(b*x + a)^2*d^4 + (-I*a^2 + I)*d^4 + 2*(-I*b*c*d^3 + I*a*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (b
^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + (a^2 - 1)*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*sin(4*b*x + 4*a) +
2*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + (a^2 - 1)*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*sin(2*b*x + 2
*a))*polylog(3, -e^(I*b*x + I*a)) + 120*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*(b*x + a)^2*d^4 + (I*a^2 - I)*d^4 +
2*(I*b*c*d^3 - I*a*d^4)*(b*x + 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 - I)*d^4 + 2*
(I*b*c*d^3 - I*a*d^4)*(b*x + a))*cos(4*b*x + 4*a) + 2*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*(b*x + a)^2*d^4 + (-
I*a^2 + I)*d^4 + 2*(-I*b*c*d^3 + I*a*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)
^2*d^4 + (a^2 - 1)*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*sin(4*b*x + 4*a) + 2*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x
+ a)^2*d^4 + (a^2 - 1)*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*polylog(3, e^(I*b*x + I*a)) + 2
*(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 - 2*I)*
d^4)*(b*x + a)^3 + 10*(I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + 3*(I*a^2 - 2*I)*b*c*d^3 + (-I*a^3 + 6*I*a)*d^4)*(b*x
+ a)^2 + 60*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a))*sin(4*b*x + 4*a) + 4*(-I*(b*x + a)^5*d^4 -
10*I*b^3*c^3*d + 30*I*a*b^2*c^2*d^2 - 30*I*a^2*b*c*d^3 + 10*I*a^3*d^4 + 5*(-I*b*c*d^3 + (I*a + 1)*d^4)*(b*x +
a)^4 + 10*(-I*b^2*c^2*d^2 + 2*(I*a + 1)*b*c*d^3 + (-I*a^2 - 2*a + I)*d^4)*(b*x + a)^3 + 10*(-I*b^3*c^3*d + 3*
(I*a + 1)*b^2*c^2*d^2 + 3*(-I*a^2 - 2*a + I)*b*c*d^3 + (I*a^3 + 3*a^2 - 3*I*a)*d^4)*(b*x + a)^2 + 10*(2*b^3*c^
3*d - 3*(2*a - I)*b^2*c^2*d^2 + 6*(a^2 - I*a)*b*c*d^3 - (2*a^3 - 3*I*a^2)*d^4)*(b*x + a))*sin(2*b*x + 2*a))/(-
10*I*b^4*cos(4*b*x + 4*a) + 20*I*b^4*cos(2*b*x + 2*a) + 10*b^4*sin(4*b*x + 4*a) - 20*b^4*sin(2*b*x + 2*a) - 10
*I*b^4))/b

Giac [F]

\[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{4} \cot \left (b x + a\right )^{3} \,d x } \]

[In]

integrate((d*x+c)^4*cot(b*x+a)^3,x, algorithm="giac")

[Out]

integrate((d*x + c)^4*cot(b*x + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^4 \cot ^3(a+b x) \, dx=\int {\mathrm {cot}\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^4 \,d x \]

[In]

int(cot(a + b*x)^3*(c + d*x)^4,x)

[Out]

int(cot(a + b*x)^3*(c + d*x)^4, x)