3.178 \(\int (c+d x)^4 \cot ^3(a+b x) \, dx\)
Optimal. Leaf size=302 \[ \frac {3 d^4 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{b^5}+\frac {3 d^4 \text {Li}_5\left (e^{2 i (a+b x)}\right )}{2 b^5}-\frac {6 i d^3 (c+d x) \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^4}-\frac {3 i d^3 (c+d x) \text {Li}_4\left (e^{2 i (a+b x)}\right )}{b^4}-\frac {3 d^2 (c+d x)^2 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^3}+\frac {2 i d (c+d x)^3 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}-\frac {2 d (c+d x)^3 \cot (a+b x)}{b^2}-\frac {(c+d x)^4 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {(c+d x)^4 \cot ^2(a+b x)}{2 b}-\frac {2 i d (c+d x)^3}{b^2}-\frac {(c+d x)^4}{2 b}+\frac {i (c+d x)^5}{5 d} \]
[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] time = 0.46, antiderivative size = 302, normalized size of antiderivative = 1.00,
number of steps used = 15, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used
= {3720, 3717, 2190, 2531, 2282, 6589, 32, 6609} \[ -\frac {3 d^2 (c+d x)^2 \text {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}-\frac {6 i d^3 (c+d x) \text {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^4}-\frac {3 i d^3 (c+d x) \text {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \text {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}+\frac {3 d^4 \text {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^5}+\frac {3 d^4 \text {PolyLog}\left (5,e^{2 i (a+b x)}\right )}{2 b^5}+\frac {6 d^2 (c+d x)^2 \log \left (1-e^{2 i (a+b x)}\right )}{b^3}-\frac {2 d (c+d x)^3 \cot (a+b x)}{b^2}-\frac {(c+d x)^4 \log \left (1-e^{2 i (a+b x)}\right )}{b}-\frac {(c+d x)^4 \cot ^2(a+b x)}{2 b}-\frac {2 i d (c+d x)^3}{b^2}-\frac {(c+d x)^4}{2 b}+\frac {i (c+d x)^5}{5 d} \]
Antiderivative was successfully verified.
[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 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 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 3720
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 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 \cot ^3(a+b x) \, dx &=-\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 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}-\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 (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) \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^4}+\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 {\left (6 d^3\right ) \int (c+d x) \text {Li}_3\left (e^{2 i (a+b x)}\right ) \, dx}{b^3}+\frac {\left (6 i d^4\right ) \int \text {Li}_2\left (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) \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^4}+\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 {\left (3 d^4\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{b^5}+\frac {\left (3 i d^4\right ) \int \text {Li}_4\left (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) \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}+\frac {3 d^4 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{b^5}-\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 {\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 {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) \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^2}+\frac {3 d^4 \text {Li}_3\left (e^{2 i (a+b x)}\right )}{b^5}-\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}\\ \end {align*}
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Mathematica [B] time = 7.12, size = 1534, normalized size = 5.08 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
[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*(-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 - (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*(-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^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*(-1 +
E^((2*I)*a))*(b^2*x^2*PolyLog[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) - (6*(-1 + E^((2*I)*a))*(b^2*x^2*PolyLog[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]*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[Ta
n[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)]) - (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)])
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fricas [C] time = 0.58, size = 1747, normalized size = 5.78 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[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 -
12*I*b^3*c*d^3*x^2 - 4*I*b^3*c^3*d + 12*I*b*c*d^3 - 12*I*(b^3*c^2*d^2 - b*d^4)*x + (4*I*b^3*d^4*x^3 + 12*I*b^3
*c*d^3*x^2 + 4*I*b^3*c^3*d - 12*I*b*c*d^3 + 12*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 + 12*I*b^3*c*d^3*x^2 + 4*I*b^3*c^3*d - 12*I*b*c*d^3 + 12*I*(b^3*
c^2*d^2 - b*d^4)*x + (-4*I*b^3*d^4*x^3 - 12*I*b^3*c*d^3*x^2 - 4*I*b^3*c^3*d + 12*I*b*c*d^3 - 12*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*cos(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 +
6*I*b*c*d^3 + (-6*I*b*d^4*x - 6*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 - 6*I*b*c*d^3 + (6*I*b*d^4*x + 6*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))*polylog(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)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{4} \cot \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[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)
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maple [B] time = 0.20, size = 1868, normalized size = 6.19 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^4*cot(b*x+a)^3,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+1/5*I*d^4*x^5+I*c*d^3*x^4-24*I/b^3*d^3*c*a*x+12*d^4*polylog(
3,-exp(I*(b*x+a)))/b^5+12*d^4*polylog(3,exp(I*(b*x+a)))/b^5+24*d^4*polylog(5,-exp(I*(b*x+a)))/b^5+24*d^4*polyl
og(5,exp(I*(b*x+a)))/b^5+8*I/b*a*c^3*d*x-12*I/b^2*a^2*c^2*d^2*x+8*I/b^3*c*d^3*a^3*x+12*I/b^2*polylog(2,exp(I*(
b*x+a)))*c^2*d^2*x+12*I/b^2*c*d^3*polylog(2,exp(I*(b*x+a)))*x^2+6*I/b^4*c*d^3*a^4-24*I/b^4*c*d^3*polylog(4,exp
(I*(b*x+a)))+4*I/b^2*c^3*d*polylog(2,exp(I*(b*x+a)))-8*I/b^3*a^3*c^2*d^2+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-2*I/b^4*d^4*a^4*x+4*I/b^2*a^2*c^3*d-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)+6/b^5*d^4*a^2*ln(exp(I*(b*x+a))-1)-12/b^5*d^4
*a^2*ln(exp(I*(b*x+a)))+6/b^3*d^2*c^2*ln(exp(I*(b*x+a))-1)+6/b^3*d^2*c^2*ln(exp(I*(b*x+a))+1)-12/b^3*d^2*c^2*l
n(exp(I*(b*x+a)))-6/b^5*d^4*a^2*ln(1-exp(I*(b*x+a)))+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^2-4*I/b^2*d^4*x^3+8*I/b^5*d^4*a^3-24*I/b^4*d^4*polylog(4,-exp(I*(b*x+a)))*x-24*I/b^4*c*d^3*polylog(
4,-exp(I*(b*x+a)))+4*I/b^2*c^3*d*polylog(2,-exp(I*(b*x+a)))+4*I/b^2*d^4*polylog(2,-exp(I*(b*x+a)))*x^3+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)^2-4/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-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^4-4/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+2*I*c^2*d^2*x^3+2*I*c^3*d*x^2+12/b^3*d^3*c*ln(exp(I*(b*x+a))+1)*x+12/b^3*d^3*c
*ln(1-exp(I*(b*x+a)))*x+12/b^4*d^3*c*ln(1-exp(I*(b*x+a)))*a+24/b^4*d^3*c*a*ln(exp(I*(b*x+a)))+12*I/b^2*c*d^3*p
olylog(2,-exp(I*(b*x+a)))*x^2+12*I/b^2*c^2*d^2*polylog(2,-exp(I*(b*x+a)))*x-8/5*I/b^5*d^4*a^5-12/b^4*d^3*c*a*l
n(exp(I*(b*x+a))-1)-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)))+12*I/b^
4*d^4*a^2*x-12*I/b^2*d^3*c*x^2-12*I/b^4*d^3*c*a^2-12*I/b^4*d^4*polylog(2,-exp(I*(b*x+a)))*x-12*I/b^4*d^4*polyl
og(2,exp(I*(b*x+a)))*x
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maxima [B] time = 4.85, size = 7111, normalized size = 23.55 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[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 - 60*b^2*c^2*d^2 + 120*a*b*c*d^3 - 60*a^2*d^4 + 40*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 60*(b^2*c^2*d^2 - 2*a*
b*c*d^3 + (a^2 - 1)*d^4)*(b*x + a)^2 + 40*(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) + 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))*cos(4*b*x + 4*a) - 20*((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) + (10*I*(b*x + a
)^4*d^4 - 60*I*b^2*c^2*d^2 + 120*I*a*b*c*d^3 - 60*I*a^2*d^4 + (40*I*b*c*d^3 - 40*I*a*d^4)*(b*x + a)^3 + (60*I*
b^2*c^2*d^2 - 120*I*a*b*c*d^3 + (60*I*a^2 - 60*I)*d^4)*(b*x + a)^2 + (40*I*b^3*c^3*d - 120*I*a*b^2*c^2*d^2 + (
120*I*a^2 - 120*I)*b*c*d^3 + (-40*I*a^3 + 120*I*a)*d^4)*(b*x + a))*sin(4*b*x + 4*a) + (-20*I*(b*x + a)^4*d^4 +
120*I*b^2*c^2*d^2 - 240*I*a*b*c*d^3 + 120*I*a^2*d^4 + (-80*I*b*c*d^3 + 80*I*a*d^4)*(b*x + a)^3 + (-120*I*b^2*
c^2*d^2 + 240*I*a*b*c*d^3 + (-120*I*a^2 + 120*I)*d^4)*(b*x + a)^2 + (-80*I*b^3*c^3*d + 240*I*a*b^2*c^2*d^2 + (
-240*I*a^2 + 240*I)*b*c*d^3 + (80*I*a^3 - 240*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 - 120*a*b*c*d^3 + 60*a^2*d^4 + 60*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*cos(4
*b*x + 4*a) - 120*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*cos(2*b*x + 2*a) - (-60*I*b^2*c^2*d^2 + 120*I*a*b*c*d^
3 - 60*I*a^2*d^4)*sin(4*b*x + 4*a) - (120*I*b^2*c^2*d^2 - 240*I*a*b*c*d^3 + 120*I*a^2*d^4)*sin(2*b*x + 2*a))*a
rctan2(sin(b*x + a), cos(b*x + a) - 1) + (10*(b*x + a)^4*d^4 + 40*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 60*(b^2*c^2*
d^2 - 2*a*b*c*d^3 + (a^2 - 1)*d^4)*(b*x + a)^2 + 40*(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) + 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
))*cos(4*b*x + 4*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 - 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) - (-10*I*(b*x + a)^4*d^4 + (-40*I*b*c*d^3 + 40*I*a*d^4)*(b*x + a)^3 + (-60*I*b^2*c^2*d^2 + 1
20*I*a*b*c*d^3 + (-60*I*a^2 + 60*I)*d^4)*(b*x + a)^2 + (-40*I*b^3*c^3*d + 120*I*a*b^2*c^2*d^2 + (-120*I*a^2 +
120*I)*b*c*d^3 + (40*I*a^3 - 120*I*a)*d^4)*(b*x + a))*sin(4*b*x + 4*a) - (20*I*(b*x + a)^4*d^4 + (80*I*b*c*d^3
- 80*I*a*d^4)*(b*x + a)^3 + (120*I*b^2*c^2*d^2 - 240*I*a*b*c*d^3 + (120*I*a^2 - 120*I)*d^4)*(b*x + a)^2 + (80
*I*b^3*c^3*d - 240*I*a*b^2*c^2*d^2 + (240*I*a^2 - 240*I)*b*c*d^3 + (-80*I*a^3 + 240*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 + 10*(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 + 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 + (20*b*c*d^3 - (20*a - 2
0*I)*d^4)*(b*x + a)^4 + (40*b^2*c^2*d^2 - (80*a - 80*I)*b*c*d^3 + 40*(a^2 - 2*I*a - 1)*d^4)*(b*x + a)^3 + (40*
b^3*c^3*d - (120*a - 120*I)*b^2*c^2*d^2 + 120*(a^2 - 2*I*a - 1)*b*c*d^3 - (40*a^3 - 120*I*a^2 - 120*a)*d^4)*(b
*x + a)^2 + (80*I*b^3*c^3*d - 120*(2*I*a + 1)*b^2*c^2*d^2 + (240*I*a^2 + 240*a)*b*c*d^3 + (-80*I*a^3 - 120*a^2
)*d^4)*(b*x + a))*cos(2*b*x + 2*a) + (40*b^3*c^3*d - 120*a*b^2*c^2*d^2 + 40*(b*x + a)^3*d^4 + 120*(a^2 - 1)*b*
c*d^3 - 40*(a^3 - 3*a)*d^4 + 120*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 120*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (a^2 - 1)*d^
4)*(b*x + 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))*cos(4*b*x + 4*a) - 80*
(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) - (-40*I*b^3*c^3*d + 12
0*I*a*b^2*c^2*d^2 - 40*I*(b*x + a)^3*d^4 + (-120*I*a^2 + 120*I)*b*c*d^3 + (40*I*a^3 - 120*I*a)*d^4 + (-120*I*b
*c*d^3 + 120*I*a*d^4)*(b*x + a)^2 + (-120*I*b^2*c^2*d^2 + 240*I*a*b*c*d^3 + (-120*I*a^2 + 120*I)*d^4)*(b*x + a
))*sin(4*b*x + 4*a) - (80*I*b^3*c^3*d - 240*I*a*b^2*c^2*d^2 + 80*I*(b*x + a)^3*d^4 + (240*I*a^2 - 240*I)*b*c*d
^3 + (-80*I*a^3 + 240*I*a)*d^4 + (240*I*b*c*d^3 - 240*I*a*d^4)*(b*x + a)^2 + (240*I*b^2*c^2*d^2 - 480*I*a*b*c*
d^3 + (240*I*a^2 - 240*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 - 120*a*b^
2*c^2*d^2 + 40*(b*x + a)^3*d^4 + 120*(a^2 - 1)*b*c*d^3 - 40*(a^3 - 3*a)*d^4 + 120*(b*c*d^3 - a*d^4)*(b*x + a)^
2 + 120*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (a^2 - 1)*d^4)*(b*x + 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))*cos(4*b*x + 4*a) - 80*(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) - (-40*I*b^3*c^3*d + 120*I*a*b^2*c^2*d^2 - 40*I*(b*x + a)^3*d^4 + (-120*I*a^2 + 120*
I)*b*c*d^3 + (40*I*a^3 - 120*I*a)*d^4 + (-120*I*b*c*d^3 + 120*I*a*d^4)*(b*x + a)^2 + (-120*I*b^2*c^2*d^2 + 240
*I*a*b*c*d^3 + (-120*I*a^2 + 120*I)*d^4)*(b*x + a))*sin(4*b*x + 4*a) - (80*I*b^3*c^3*d - 240*I*a*b^2*c^2*d^2 +
80*I*(b*x + a)^3*d^4 + (240*I*a^2 - 240*I)*b*c*d^3 + (-80*I*a^3 + 240*I*a)*d^4 + (240*I*b*c*d^3 - 240*I*a*d^4
)*(b*x + a)^2 + (240*I*b^2*c^2*d^2 - 480*I*a*b*c*d^3 + (240*I*a^2 - 240*I)*d^4)*(b*x + a))*sin(2*b*x + 2*a))*d
ilog(e^(I*b*x + I*a)) - (-5*I*(b*x + a)^4*d^4 + 30*I*b^2*c^2*d^2 - 60*I*a*b*c*d^3 + 30*I*a^2*d^4 + (-20*I*b*c*
d^3 + 20*I*a*d^4)*(b*x + a)^3 + (-30*I*b^2*c^2*d^2 + 60*I*a*b*c*d^3 + (-30*I*a^2 + 30*I)*d^4)*(b*x + a)^2 + (-
20*I*b^3*c^3*d + 60*I*a*b^2*c^2*d^2 + (-60*I*a^2 + 60*I)*b*c*d^3 + (20*I*a^3 - 60*I*a)*d^4)*(b*x + a) + (-5*I*
(b*x + a)^4*d^4 + 30*I*b^2*c^2*d^2 - 60*I*a*b*c*d^3 + 30*I*a^2*d^4 + (-20*I*b*c*d^3 + 20*I*a*d^4)*(b*x + a)^3
+ (-30*I*b^2*c^2*d^2 + 60*I*a*b*c*d^3 + (-30*I*a^2 + 30*I)*d^4)*(b*x + a)^2 + (-20*I*b^3*c^3*d + 60*I*a*b^2*c^
2*d^2 + (-60*I*a^2 + 60*I)*b*c*d^3 + (20*I*a^3 - 60*I*a)*d^4)*(b*x + a))*cos(4*b*x + 4*a) + (10*I*(b*x + a)^4*
d^4 - 60*I*b^2*c^2*d^2 + 120*I*a*b*c*d^3 - 60*I*a^2*d^4 + (40*I*b*c*d^3 - 40*I*a*d^4)*(b*x + a)^3 + (60*I*b^2*
c^2*d^2 - 120*I*a*b*c*d^3 + (60*I*a^2 - 60*I)*d^4)*(b*x + a)^2 + (40*I*b^3*c^3*d - 120*I*a*b^2*c^2*d^2 + (120*
I*a^2 - 120*I)*b*c*d^3 + (-40*I*a^3 + 120*I*a)*d^4)*(b*x + a))*cos(2*b*x + 2*a) + 5*((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) - 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))*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 + 30*I*b^2*c^2*d^2 - 60*I*a*b*c*d^3 + 30*I*a^2*d^4 + (-20*I*b*c*d^3 + 20*I*a*d^4)*(b*x
+ a)^3 + (-30*I*b^2*c^2*d^2 + 60*I*a*b*c*d^3 + (-30*I*a^2 + 30*I)*d^4)*(b*x + a)^2 + (-20*I*b^3*c^3*d + 60*I*
a*b^2*c^2*d^2 + (-60*I*a^2 + 60*I)*b*c*d^3 + (20*I*a^3 - 60*I*a)*d^4)*(b*x + a) + (-5*I*(b*x + a)^4*d^4 + 30*I
*b^2*c^2*d^2 - 60*I*a*b*c*d^3 + 30*I*a^2*d^4 + (-20*I*b*c*d^3 + 20*I*a*d^4)*(b*x + a)^3 + (-30*I*b^2*c^2*d^2 +
60*I*a*b*c*d^3 + (-30*I*a^2 + 30*I)*d^4)*(b*x + a)^2 + (-20*I*b^3*c^3*d + 60*I*a*b^2*c^2*d^2 + (-60*I*a^2 + 6
0*I)*b*c*d^3 + (20*I*a^3 - 60*I*a)*d^4)*(b*x + a))*cos(4*b*x + 4*a) + (10*I*(b*x + a)^4*d^4 - 60*I*b^2*c^2*d^2
+ 120*I*a*b*c*d^3 - 60*I*a^2*d^4 + (40*I*b*c*d^3 - 40*I*a*d^4)*(b*x + a)^3 + (60*I*b^2*c^2*d^2 - 120*I*a*b*c*
d^3 + (60*I*a^2 - 60*I)*d^4)*(b*x + a)^2 + (40*I*b^3*c^3*d - 120*I*a*b^2*c^2*d^2 + (120*I*a^2 - 120*I)*b*c*d^3
+ (-40*I*a^3 + 120*I*a)*d^4)*(b*x + a))*cos(2*b*x + 2*a) + 5*((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) - 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))*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) - 480*I*d^4*cos(2*b*x + 2*a) - 240*d^4*sin(4*b*x + 4*a) + 480*d^4*sin(2*b*x + 2*a) + 240*I*d^4)*polylo
g(5, -e^(I*b*x + I*a)) - (240*I*d^4*cos(4*b*x + 4*a) - 480*I*d^4*cos(2*b*x + 2*a) - 240*d^4*sin(4*b*x + 4*a) +
480*d^4*sin(2*b*x + 2*a) + 240*I*d^4)*polylog(5, e^(I*b*x + I*a)) - (240*b*c*d^3 + 240*(b*x + a)*d^4 - 240*a*
d^4 + 240*(b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(4*b*x + 4*a) - 480*(b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(2*b*x
+ 2*a) + (240*I*b*c*d^3 + 240*I*(b*x + a)*d^4 - 240*I*a*d^4)*sin(4*b*x + 4*a) + (-480*I*b*c*d^3 - 480*I*(b*x
+ a)*d^4 + 480*I*a*d^4)*sin(2*b*x + 2*a))*polylog(4, -e^(I*b*x + I*a)) - (240*b*c*d^3 + 240*(b*x + a)*d^4 - 24
0*a*d^4 + 240*(b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(4*b*x + 4*a) - 480*(b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(2
*b*x + 2*a) + (240*I*b*c*d^3 + 240*I*(b*x + a)*d^4 - 240*I*a*d^4)*sin(4*b*x + 4*a) + (-480*I*b*c*d^3 - 480*I*(
b*x + a)*d^4 + 480*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 + 240*I*a*b*c*
d^3 - 120*I*(b*x + a)^2*d^4 + (-120*I*a^2 + 120*I)*d^4 + (-240*I*b*c*d^3 + 240*I*a*d^4)*(b*x + a) + (-120*I*b^
2*c^2*d^2 + 240*I*a*b*c*d^3 - 120*I*(b*x + a)^2*d^4 + (-120*I*a^2 + 120*I)*d^4 + (-240*I*b*c*d^3 + 240*I*a*d^4
)*(b*x + a))*cos(4*b*x + 4*a) + (240*I*b^2*c^2*d^2 - 480*I*a*b*c*d^3 + 240*I*(b*x + a)^2*d^4 + (240*I*a^2 - 24
0*I)*d^4 + (480*I*b*c*d^3 - 480*I*a*d^4)*(b*x + a))*cos(2*b*x + 2*a) + 120*(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) - 240*(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 + 240*I*a*b*c*d^3 - 120*I*(b*x + a)^2*d^4 + (-120*I*a^2 + 120*I)*d^4 + (-240*I*b*c*d^
3 + 240*I*a*d^4)*(b*x + a) + (-120*I*b^2*c^2*d^2 + 240*I*a*b*c*d^3 - 120*I*(b*x + a)^2*d^4 + (-120*I*a^2 + 120
*I)*d^4 + (-240*I*b*c*d^3 + 240*I*a*d^4)*(b*x + a))*cos(4*b*x + 4*a) + (240*I*b^2*c^2*d^2 - 480*I*a*b*c*d^3 +
240*I*(b*x + a)^2*d^4 + (240*I*a^2 - 240*I)*d^4 + (480*I*b*c*d^3 - 480*I*a*d^4)*(b*x + a))*cos(2*b*x + 2*a) +
120*(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) - 240*(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 + (-10*I*b*c*d^3 + 10*I*a*d^4)*(b*x + a)^4 +
(-20*I*b^2*c^2*d^2 + 40*I*a*b*c*d^3 + (-20*I*a^2 + 40*I)*d^4)*(b*x + a)^3 + (-20*I*b^3*c^3*d + 60*I*a*b^2*c^2*
d^2 + (-60*I*a^2 + 120*I)*b*c*d^3 + (20*I*a^3 - 120*I*a)*d^4)*(b*x + a)^2 + (120*I*b^2*c^2*d^2 - 240*I*a*b*c*d
^3 + 120*I*a^2*d^4)*(b*x + a))*sin(4*b*x + 4*a) - (4*I*(b*x + a)^5*d^4 + 40*I*b^3*c^3*d - 120*I*a*b^2*c^2*d^2
+ 120*I*a^2*b*c*d^3 - 40*I*a^3*d^4 + (20*I*b*c*d^3 - 20*(I*a + 1)*d^4)*(b*x + a)^4 + (40*I*b^2*c^2*d^2 - 80*(I
*a + 1)*b*c*d^3 + (40*I*a^2 + 80*a - 40*I)*d^4)*(b*x + a)^3 + (40*I*b^3*c^3*d - 120*(I*a + 1)*b^2*c^2*d^2 + (1
20*I*a^2 + 240*a - 120*I)*b*c*d^3 + (-40*I*a^3 - 120*a^2 + 120*I*a)*d^4)*(b*x + a)^2 - (80*b^3*c^3*d - (240*a
- 120*I)*b^2*c^2*d^2 + 240*(a^2 - I*a)*b*c*d^3 - 40*(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
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {cot}\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^4 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(a + b*x)^3*(c + d*x)^4,x)
[Out]
int(cot(a + b*x)^3*(c + d*x)^4, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right )^{4} \cot ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)**4*cot(b*x+a)**3,x)
[Out]
Integral((c + d*x)**4*cot(a + b*x)**3, x)
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