3.42 \(\int (c+d x)^3 (a+b \cot (e+f x))^2 \, dx\)

Optimal. Leaf size=295 \[ \frac{3 a b d^2 (c+d x) \text{PolyLog}\left (3,e^{2 i (e+f x)}\right )}{f^3}-\frac{3 i a b d (c+d x)^2 \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac{3 i a b d^3 \text{PolyLog}\left (4,e^{2 i (e+f x)}\right )}{2 f^4}-\frac{3 i b^2 d^2 (c+d x) \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^3}+\frac{3 b^2 d^3 \text{PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^4}+\frac{a^2 (c+d x)^4}{4 d}+\frac{2 a b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{i a b (c+d x)^4}{2 d}+\frac{3 b^2 d (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f^2}-\frac{b^2 (c+d x)^3 \cot (e+f x)}{f}-\frac{i b^2 (c+d x)^3}{f}-\frac{b^2 (c+d x)^4}{4 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.531047, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {3722, 3717, 2190, 2531, 6609, 2282, 6589, 3720, 32} \[ \frac{3 a b d^2 (c+d x) \text{PolyLog}\left (3,e^{2 i (e+f x)}\right )}{f^3}-\frac{3 i a b d (c+d x)^2 \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac{3 i a b d^3 \text{PolyLog}\left (4,e^{2 i (e+f x)}\right )}{2 f^4}-\frac{3 i b^2 d^2 (c+d x) \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^3}+\frac{3 b^2 d^3 \text{PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^4}+\frac{a^2 (c+d x)^4}{4 d}+\frac{2 a b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{i a b (c+d x)^4}{2 d}+\frac{3 b^2 d (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f^2}-\frac{b^2 (c+d x)^3 \cot (e+f x)}{f}-\frac{i b^2 (c+d x)^3}{f}-\frac{b^2 (c+d x)^4}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*(a + b*Cot[e + f*x])^2,x]

[Out]

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

Rule 3722

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 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 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 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 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]

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 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 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 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]

Rubi steps

\begin{align*} \int (c+d x)^3 (a+b \cot (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^3+2 a b (c+d x)^3 \cot (e+f x)+b^2 (c+d x)^3 \cot ^2(e+f x)\right ) \, dx\\ &=\frac{a^2 (c+d x)^4}{4 d}+(2 a b) \int (c+d x)^3 \cot (e+f x) \, dx+b^2 \int (c+d x)^3 \cot ^2(e+f x) \, dx\\ &=\frac{a^2 (c+d x)^4}{4 d}-\frac{i a b (c+d x)^4}{2 d}-\frac{b^2 (c+d x)^3 \cot (e+f x)}{f}-(4 i a b) \int \frac{e^{2 i (e+f x)} (c+d x)^3}{1-e^{2 i (e+f x)}} \, dx-b^2 \int (c+d x)^3 \, dx+\frac{\left (3 b^2 d\right ) \int (c+d x)^2 \cot (e+f x) \, dx}{f}\\ &=-\frac{i b^2 (c+d x)^3}{f}+\frac{a^2 (c+d x)^4}{4 d}-\frac{i a b (c+d x)^4}{2 d}-\frac{b^2 (c+d x)^4}{4 d}-\frac{b^2 (c+d x)^3 \cot (e+f x)}{f}+\frac{2 a b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{(6 a b d) \int (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f}-\frac{\left (6 i b^2 d\right ) \int \frac{e^{2 i (e+f x)} (c+d x)^2}{1-e^{2 i (e+f x)}} \, dx}{f}\\ &=-\frac{i b^2 (c+d x)^3}{f}+\frac{a^2 (c+d x)^4}{4 d}-\frac{i a b (c+d x)^4}{2 d}-\frac{b^2 (c+d x)^4}{4 d}-\frac{b^2 (c+d x)^3 \cot (e+f x)}{f}+\frac{3 b^2 d (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f^2}+\frac{2 a b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{3 i a b d (c+d x)^2 \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac{\left (6 i a b d^2\right ) \int (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right ) \, dx}{f^2}-\frac{\left (6 b^2 d^2\right ) \int (c+d x) \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f^2}\\ &=-\frac{i b^2 (c+d x)^3}{f}+\frac{a^2 (c+d x)^4}{4 d}-\frac{i a b (c+d x)^4}{2 d}-\frac{b^2 (c+d x)^4}{4 d}-\frac{b^2 (c+d x)^3 \cot (e+f x)}{f}+\frac{3 b^2 d (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f^2}+\frac{2 a b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{3 i b^2 d^2 (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^3}-\frac{3 i a b d (c+d x)^2 \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac{3 a b d^2 (c+d x) \text{Li}_3\left (e^{2 i (e+f x)}\right )}{f^3}-\frac{\left (3 a b d^3\right ) \int \text{Li}_3\left (e^{2 i (e+f x)}\right ) \, dx}{f^3}+\frac{\left (3 i b^2 d^3\right ) \int \text{Li}_2\left (e^{2 i (e+f x)}\right ) \, dx}{f^3}\\ &=-\frac{i b^2 (c+d x)^3}{f}+\frac{a^2 (c+d x)^4}{4 d}-\frac{i a b (c+d x)^4}{2 d}-\frac{b^2 (c+d x)^4}{4 d}-\frac{b^2 (c+d x)^3 \cot (e+f x)}{f}+\frac{3 b^2 d (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f^2}+\frac{2 a b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{3 i b^2 d^2 (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^3}-\frac{3 i a b d (c+d x)^2 \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac{3 a b d^2 (c+d x) \text{Li}_3\left (e^{2 i (e+f x)}\right )}{f^3}+\frac{\left (3 i a b d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^4}+\frac{\left (3 b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^4}\\ &=-\frac{i b^2 (c+d x)^3}{f}+\frac{a^2 (c+d x)^4}{4 d}-\frac{i a b (c+d x)^4}{2 d}-\frac{b^2 (c+d x)^4}{4 d}-\frac{b^2 (c+d x)^3 \cot (e+f x)}{f}+\frac{3 b^2 d (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f^2}+\frac{2 a b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{3 i b^2 d^2 (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^3}-\frac{3 i a b d (c+d x)^2 \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac{3 b^2 d^3 \text{Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^4}+\frac{3 a b d^2 (c+d x) \text{Li}_3\left (e^{2 i (e+f x)}\right )}{f^3}+\frac{3 i a b d^3 \text{Li}_4\left (e^{2 i (e+f x)}\right )}{2 f^4}\\ \end{align*}

Mathematica [B]  time = 7.37825, size = 1611, normalized size = 5.46 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + d*x)^3*(a + b*Cot[e + f*x])^2,x]

[Out]

-(b^2*d^3*E^(I*e)*Csc[e]*((2*f^3*x^3)/E^((2*I)*e) + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 - E^((-I)*(e + f*x)
)] + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 + E^((-I)*(e + f*x))] - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, -E^(
(-I)*(e + f*x))] - I*PolyLog[3, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, E^(
(-I)*(e + f*x))] - I*PolyLog[3, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/(2*f^4) - (a*b*c*d^2*E^(I*e)*Csc[e]*((2*f^
3*x^3)/E^((2*I)*e) + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 - E^((-I)*(e + f*x))] + (3*I)*(1 - E^((-2*I)*e))*f
^2*x^2*Log[1 + E^((-I)*(e + f*x))] - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, -E^((-I)*(e + f*x))] - I*PolyLog[3,
-E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, E^((-I)*(e + f*x))] - I*PolyLog[3,
E^((-I)*(e + f*x))]))/E^((2*I)*e)))/f^3 - (a*b*d^3*E^(I*e)*Csc[e]*((f^4*x^4)/E^((2*I)*e) + (2*I)*(1 - E^((-2*
I)*e))*f^3*x^3*Log[1 - E^((-I)*(e + f*x))] + (2*I)*(1 - E^((-2*I)*e))*f^3*x^3*Log[1 + E^((-I)*(e + f*x))] - (6
*(-1 + E^((2*I)*e))*(f^2*x^2*PolyLog[2, -E^((-I)*(e + f*x))] - (2*I)*f*x*PolyLog[3, -E^((-I)*(e + f*x))] - 2*P
olyLog[4, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f^2*x^2*PolyLog[2, E^((-I)*(e + f*x))] -
(2*I)*f*x*PolyLog[3, E^((-I)*(e + f*x))] - 2*PolyLog[4, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/(2*f^4) + (3*b^2*
c^2*d*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f^2*(Cos[e]^2 + Sin[e]^2)) + (2
*a*b*c^3*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) + (
Csc[e]*Csc[e + f*x]*(4*a^2*c^3*f*x*Cos[f*x] - 4*b^2*c^3*f*x*Cos[f*x] + 6*a^2*c^2*d*f*x^2*Cos[f*x] - 6*b^2*c^2*
d*f*x^2*Cos[f*x] + 4*a^2*c*d^2*f*x^3*Cos[f*x] - 4*b^2*c*d^2*f*x^3*Cos[f*x] + a^2*d^3*f*x^4*Cos[f*x] - b^2*d^3*
f*x^4*Cos[f*x] - 4*a^2*c^3*f*x*Cos[2*e + f*x] + 4*b^2*c^3*f*x*Cos[2*e + f*x] - 6*a^2*c^2*d*f*x^2*Cos[2*e + f*x
] + 6*b^2*c^2*d*f*x^2*Cos[2*e + f*x] - 4*a^2*c*d^2*f*x^3*Cos[2*e + f*x] + 4*b^2*c*d^2*f*x^3*Cos[2*e + f*x] - a
^2*d^3*f*x^4*Cos[2*e + f*x] + b^2*d^3*f*x^4*Cos[2*e + f*x] + 8*b^2*c^3*Sin[f*x] + 24*b^2*c^2*d*x*Sin[f*x] + 8*
a*b*c^3*f*x*Sin[f*x] + 24*b^2*c*d^2*x^2*Sin[f*x] + 12*a*b*c^2*d*f*x^2*Sin[f*x] + 8*b^2*d^3*x^3*Sin[f*x] + 8*a*
b*c*d^2*f*x^3*Sin[f*x] + 2*a*b*d^3*f*x^4*Sin[f*x] + 8*a*b*c^3*f*x*Sin[2*e + f*x] + 12*a*b*c^2*d*f*x^2*Sin[2*e
+ f*x] + 8*a*b*c*d^2*f*x^3*Sin[2*e + f*x] + 2*a*b*d^3*f*x^4*Sin[2*e + f*x]))/(8*f) - (3*b^2*c*d^2*Csc[e]*Sec[e
]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + Arc
Tan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + Arc
Tan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(f^3*Sqrt[Sec[e]^
2*(Cos[e]^2 + Sin[e]^2)]) - (3*a*b*c^2*d*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan
[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] +
Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e
]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(f^2*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)])

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Maple [B]  time = 0.584, size = 1571, normalized size = 5.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3*(a+b*cot(f*x+e))^2,x)

[Out]

12*b/f^3*a*c*d^2*polylog(3,-exp(I*(f*x+e)))+4*b/f^4*a*d^3*e^3*ln(exp(I*(f*x+e)))+12*b/f^3*a*c*d^2*polylog(3,ex
p(I*(f*x+e)))+12*b^2/f^3*c*d^2*e*ln(exp(I*(f*x+e)))+6*b^2/f^2*c*d^2*ln(1-exp(I*(f*x+e)))*x+6*b^2/f^3*c*d^2*ln(
1-exp(I*(f*x+e)))*e+12*b/f^3*a*d^3*polylog(3,-exp(I*(f*x+e)))*x+12*b/f^3*a*d^3*polylog(3,exp(I*(f*x+e)))*x+6*b
^2/f^2*c*d^2*ln(exp(I*(f*x+e))+1)*x+6*b^2/f^4*d^3*polylog(3,-exp(I*(f*x+e)))+6*b^2/f^4*d^3*polylog(3,exp(I*(f*
x+e)))+3*b^2/f^2*d^3*ln(exp(I*(f*x+e))+1)*x^2-6*b^2/f^4*d^3*e^2*ln(exp(I*(f*x+e)))+3*b^2/f^4*d^3*e^2*ln(exp(I*
(f*x+e))-1)+3*b^2/f^2*c^2*d*ln(exp(I*(f*x+e))+1)+3*b^2/f^2*c^2*d*ln(exp(I*(f*x+e))-1)+2*b/f*a*c^3*ln(exp(I*(f*
x+e))+1)+2*b/f*a*c^3*ln(exp(I*(f*x+e))-1)-6*b^2/f^2*c^2*d*ln(exp(I*(f*x+e)))-b^2*c*d^2*x^3+3/2*a^2*c^2*d*x^2-3
/2*b^2*c^2*d*x^2+1/4*a^2*d^3*x^4-1/4*b^2*d^3*x^4+a^2*c^3*x-b^2*c^3*x+a^2*c*d^2*x^3-1/2*I*a*b*d^3*x^4-12*I*b/f^
2*a*c*d^2*polylog(2,-exp(I*(f*x+e)))*x-12*I*b/f^2*a*c*d^2*polylog(2,exp(I*(f*x+e)))*x+12*I*b/f^2*a*c*d^2*e^2*x
-12*I*b/f*a*c^2*d*e*x+2*I*a*b*c^3*x+6*b/f^3*a*c*d^2*e^2*ln(exp(I*(f*x+e))-1)+6*b/f*a*c*d^2*ln(exp(I*(f*x+e))+1
)*x^2+6*b/f*ln(1-exp(I*(f*x+e)))*a*c^2*d*x+6*b/f*ln(exp(I*(f*x+e))+1)*a*c^2*d*x-6*b/f^3*a*c*d^2*e^2*ln(1-exp(I
*(f*x+e)))+6*b/f^2*ln(1-exp(I*(f*x+e)))*a*c^2*d*e-12*b/f^3*a*c*d^2*e^2*ln(exp(I*(f*x+e)))-2*I*a*b*c*d^2*x^3-3*
I*a*b*c^2*d*x^2-6*b/f^2*a*c^2*d*e*ln(exp(I*(f*x+e))-1)+6*b/f*a*c*d^2*ln(1-exp(I*(f*x+e)))*x^2+2*b/f^4*a*d^3*ln
(1-exp(I*(f*x+e)))*e^3+2*b/f*a*d^3*ln(1-exp(I*(f*x+e)))*x^3+12*b/f^2*a*c^2*d*e*ln(exp(I*(f*x+e)))+2*b/f*a*d^3*
ln(exp(I*(f*x+e))+1)*x^3+8*I*b/f^3*a*c*d^2*e^3-4*I*b/f^3*a*d^3*e^3*x-12*I*b^2/f^2*c*d^2*e*x-6*I*b/f^2*a*c^2*d*
polylog(2,-exp(I*(f*x+e)))-6*I*b/f^2*a*c^2*d*polylog(2,exp(I*(f*x+e)))-6*I*b/f^2*a*c^2*d*e^2-6*I*b/f^2*a*d^3*p
olylog(2,-exp(I*(f*x+e)))*x^2-6*I*b/f^2*a*d^3*polylog(2,exp(I*(f*x+e)))*x^2-2*I*b^2*(d^3*x^3+3*c*d^2*x^2+3*c^2
*d*x+c^3)/f/(exp(2*I*(f*x+e))-1)-6*I*b^2/f^3*d^3*polylog(2,exp(I*(f*x+e)))*x+6*I*b^2/f^3*d^3*e^2*x-6*I*b^2/f*c
*d^2*x^2-6*I*b^2/f^3*c*d^2*e^2+12*I*b/f^4*a*d^3*polylog(4,-exp(I*(f*x+e)))+12*I*b/f^4*a*d^3*polylog(4,exp(I*(f
*x+e)))-6*I*b^2/f^3*c*d^2*polylog(2,-exp(I*(f*x+e)))-6*I*b^2/f^3*c*d^2*polylog(2,exp(I*(f*x+e)))-4*b/f*a*c^3*l
n(exp(I*(f*x+e)))-3*b^2/f^4*d^3*e^2*ln(1-exp(I*(f*x+e)))+3*b^2/f^2*d^3*ln(1-exp(I*(f*x+e)))*x^2+4*I*b^2/f^4*d^
3*e^3-2*I*b^2/f*d^3*x^3-6*b^2/f^3*c*d^2*e*ln(exp(I*(f*x+e))-1)-2*b/f^4*a*d^3*e^3*ln(exp(I*(f*x+e))-1)-3*I*b/f^
4*a*d^3*e^4-6*I*b^2/f^3*d^3*polylog(2,-exp(I*(f*x+e)))*x

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Maxima [B]  time = 9.78436, size = 5420, normalized size = 18.37 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*(a+b*cot(f*x+e))^2,x, algorithm="maxima")

[Out]

1/4*(4*(f*x + e)*a^2*c^3 + (f*x + e)^4*a^2*d^3/f^3 - 4*(f*x + e)^3*a^2*d^3*e/f^3 + 6*(f*x + e)^2*a^2*d^3*e^2/f
^3 - 4*(f*x + e)*a^2*d^3*e^3/f^3 + 4*(f*x + e)^3*a^2*c*d^2/f^2 - 12*(f*x + e)^2*a^2*c*d^2*e/f^2 + 12*(f*x + e)
*a^2*c*d^2*e^2/f^2 + 6*(f*x + e)^2*a^2*c^2*d/f - 12*(f*x + e)*a^2*c^2*d*e/f + 8*a*b*c^3*log(sin(f*x + e)) - 8*
a*b*d^3*e^3*log(sin(f*x + e))/f^3 + 24*a*b*c*d^2*e^2*log(sin(f*x + e))/f^2 - 24*a*b*c^2*d*e*log(sin(f*x + e))/
f + 4*((2*a*b - I*b^2)*(f*x + e)^4*d^3 + 8*b^2*d^3*e^3 - 24*b^2*c*d^2*e^2*f + 24*b^2*c^2*d*e*f^2 - 8*b^2*c^3*f
^3 - 4*((2*a*b - I*b^2)*d^3*e - (2*a*b - I*b^2)*c*d^2*f)*(f*x + e)^3 + 6*((2*a*b - I*b^2)*d^3*e^2 - 2*(2*a*b -
I*b^2)*c*d^2*e*f + (2*a*b - I*b^2)*c^2*d*f^2)*(f*x + e)^2 + (4*I*b^2*d^3*e^3 - 12*I*b^2*c*d^2*e^2*f + 12*I*b^
2*c^2*d*e*f^2 - 4*I*b^2*c^3*f^3)*(f*x + e) - (8*(f*x + e)^3*a*b*d^3 + 12*b^2*d^3*e^2 - 24*b^2*c*d^2*e*f + 12*b
^2*c^2*d*f^2 - 12*(2*a*b*d^3*e - 2*a*b*c*d^2*f - b^2*d^3)*(f*x + e)^2 + 24*(a*b*d^3*e^2 + a*b*c^2*d*f^2 - b^2*
d^3*e - (2*a*b*c*d^2*e - b^2*c*d^2)*f)*(f*x + e) - 4*(2*(f*x + e)^3*a*b*d^3 + 3*b^2*d^3*e^2 - 6*b^2*c*d^2*e*f
+ 3*b^2*c^2*d*f^2 - 3*(2*a*b*d^3*e - 2*a*b*c*d^2*f - b^2*d^3)*(f*x + e)^2 + 6*(a*b*d^3*e^2 + a*b*c^2*d*f^2 - b
^2*d^3*e - (2*a*b*c*d^2*e - b^2*c*d^2)*f)*(f*x + e))*cos(2*f*x + 2*e) - (8*I*(f*x + e)^3*a*b*d^3 + 12*I*b^2*d^
3*e^2 - 24*I*b^2*c*d^2*e*f + 12*I*b^2*c^2*d*f^2 + (-24*I*a*b*d^3*e + 24*I*a*b*c*d^2*f + 12*I*b^2*d^3)*(f*x + e
)^2 + (24*I*a*b*d^3*e^2 + 24*I*a*b*c^2*d*f^2 - 24*I*b^2*d^3*e + (-48*I*a*b*c*d^2*e + 24*I*b^2*c*d^2)*f)*(f*x +
e))*sin(2*f*x + 2*e))*arctan2(sin(f*x + e), cos(f*x + e) + 1) - (12*b^2*d^3*e^2 - 24*b^2*c*d^2*e*f + 12*b^2*c
^2*d*f^2 - 12*(b^2*d^3*e^2 - 2*b^2*c*d^2*e*f + b^2*c^2*d*f^2)*cos(2*f*x + 2*e) - (12*I*b^2*d^3*e^2 - 24*I*b^2*
c*d^2*e*f + 12*I*b^2*c^2*d*f^2)*sin(2*f*x + 2*e))*arctan2(sin(f*x + e), cos(f*x + e) - 1) + (8*(f*x + e)^3*a*b
*d^3 - 12*(2*a*b*d^3*e - 2*a*b*c*d^2*f - b^2*d^3)*(f*x + e)^2 + 24*(a*b*d^3*e^2 + a*b*c^2*d*f^2 - b^2*d^3*e -
(2*a*b*c*d^2*e - b^2*c*d^2)*f)*(f*x + e) - 4*(2*(f*x + e)^3*a*b*d^3 - 3*(2*a*b*d^3*e - 2*a*b*c*d^2*f - b^2*d^3
)*(f*x + e)^2 + 6*(a*b*d^3*e^2 + a*b*c^2*d*f^2 - b^2*d^3*e - (2*a*b*c*d^2*e - b^2*c*d^2)*f)*(f*x + e))*cos(2*f
*x + 2*e) + (-8*I*(f*x + e)^3*a*b*d^3 + (24*I*a*b*d^3*e - 24*I*a*b*c*d^2*f - 12*I*b^2*d^3)*(f*x + e)^2 + (-24*
I*a*b*d^3*e^2 - 24*I*a*b*c^2*d*f^2 + 24*I*b^2*d^3*e + (48*I*a*b*c*d^2*e - 24*I*b^2*c*d^2)*f)*(f*x + e))*sin(2*
f*x + 2*e))*arctan2(sin(f*x + e), -cos(f*x + e) + 1) - ((2*a*b - I*b^2)*(f*x + e)^4*d^3 + 4*(2*b^2*d^3 - (2*a*
b - I*b^2)*d^3*e + (2*a*b - I*b^2)*c*d^2*f)*(f*x + e)^3 - 6*(4*b^2*d^3*e - (2*a*b - I*b^2)*d^3*e^2 - (2*a*b -
I*b^2)*c^2*d*f^2 - 2*(2*b^2*c*d^2 - (2*a*b - I*b^2)*c*d^2*e)*f)*(f*x + e)^2 - (-4*I*b^2*d^3*e^3 + 4*I*b^2*c^3*
f^3 - 24*b^2*d^3*e^2 - 12*(I*b^2*c^2*d*e + 2*b^2*c^2*d)*f^2 - 12*(-I*b^2*c*d^2*e^2 - 4*b^2*c*d^2*e)*f)*(f*x +
e))*cos(2*f*x + 2*e) + (24*(f*x + e)^2*a*b*d^3 + 24*a*b*d^3*e^2 + 24*a*b*c^2*d*f^2 - 24*b^2*d^3*e - 24*(2*a*b*
d^3*e - 2*a*b*c*d^2*f - b^2*d^3)*(f*x + e) - 24*(2*a*b*c*d^2*e - b^2*c*d^2)*f - 24*((f*x + e)^2*a*b*d^3 + a*b*
d^3*e^2 + a*b*c^2*d*f^2 - b^2*d^3*e - (2*a*b*d^3*e - 2*a*b*c*d^2*f - b^2*d^3)*(f*x + e) - (2*a*b*c*d^2*e - b^2
*c*d^2)*f)*cos(2*f*x + 2*e) + (-24*I*(f*x + e)^2*a*b*d^3 - 24*I*a*b*d^3*e^2 - 24*I*a*b*c^2*d*f^2 + 24*I*b^2*d^
3*e + (48*I*a*b*d^3*e - 48*I*a*b*c*d^2*f - 24*I*b^2*d^3)*(f*x + e) + (48*I*a*b*c*d^2*e - 24*I*b^2*c*d^2)*f)*si
n(2*f*x + 2*e))*dilog(-e^(I*f*x + I*e)) + (24*(f*x + e)^2*a*b*d^3 + 24*a*b*d^3*e^2 + 24*a*b*c^2*d*f^2 - 24*b^2
*d^3*e - 24*(2*a*b*d^3*e - 2*a*b*c*d^2*f - b^2*d^3)*(f*x + e) - 24*(2*a*b*c*d^2*e - b^2*c*d^2)*f - 24*((f*x +
e)^2*a*b*d^3 + a*b*d^3*e^2 + a*b*c^2*d*f^2 - b^2*d^3*e - (2*a*b*d^3*e - 2*a*b*c*d^2*f - b^2*d^3)*(f*x + e) - (
2*a*b*c*d^2*e - b^2*c*d^2)*f)*cos(2*f*x + 2*e) + (-24*I*(f*x + e)^2*a*b*d^3 - 24*I*a*b*d^3*e^2 - 24*I*a*b*c^2*
d*f^2 + 24*I*b^2*d^3*e + (48*I*a*b*d^3*e - 48*I*a*b*c*d^2*f - 24*I*b^2*d^3)*(f*x + e) + (48*I*a*b*c*d^2*e - 24
*I*b^2*c*d^2)*f)*sin(2*f*x + 2*e))*dilog(e^(I*f*x + I*e)) + (4*I*(f*x + e)^3*a*b*d^3 + 6*I*b^2*d^3*e^2 - 12*I*
b^2*c*d^2*e*f + 6*I*b^2*c^2*d*f^2 + (-12*I*a*b*d^3*e + 12*I*a*b*c*d^2*f + 6*I*b^2*d^3)*(f*x + e)^2 + (12*I*a*b
*d^3*e^2 + 12*I*a*b*c^2*d*f^2 - 12*I*b^2*d^3*e + (-24*I*a*b*c*d^2*e + 12*I*b^2*c*d^2)*f)*(f*x + e) + (-4*I*(f*
x + e)^3*a*b*d^3 - 6*I*b^2*d^3*e^2 + 12*I*b^2*c*d^2*e*f - 6*I*b^2*c^2*d*f^2 + (12*I*a*b*d^3*e - 12*I*a*b*c*d^2
*f - 6*I*b^2*d^3)*(f*x + e)^2 + (-12*I*a*b*d^3*e^2 - 12*I*a*b*c^2*d*f^2 + 12*I*b^2*d^3*e + (24*I*a*b*c*d^2*e -
12*I*b^2*c*d^2)*f)*(f*x + e))*cos(2*f*x + 2*e) + 2*(2*(f*x + e)^3*a*b*d^3 + 3*b^2*d^3*e^2 - 6*b^2*c*d^2*e*f +
3*b^2*c^2*d*f^2 - 3*(2*a*b*d^3*e - 2*a*b*c*d^2*f - b^2*d^3)*(f*x + e)^2 + 6*(a*b*d^3*e^2 + a*b*c^2*d*f^2 - b^
2*d^3*e - (2*a*b*c*d^2*e - b^2*c*d^2)*f)*(f*x + e))*sin(2*f*x + 2*e))*log(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*
cos(f*x + e) + 1) + (4*I*(f*x + e)^3*a*b*d^3 + 6*I*b^2*d^3*e^2 - 12*I*b^2*c*d^2*e*f + 6*I*b^2*c^2*d*f^2 + (-12
*I*a*b*d^3*e + 12*I*a*b*c*d^2*f + 6*I*b^2*d^3)*(f*x + e)^2 + (12*I*a*b*d^3*e^2 + 12*I*a*b*c^2*d*f^2 - 12*I*b^2
*d^3*e + (-24*I*a*b*c*d^2*e + 12*I*b^2*c*d^2)*f)*(f*x + e) + (-4*I*(f*x + e)^3*a*b*d^3 - 6*I*b^2*d^3*e^2 + 12*
I*b^2*c*d^2*e*f - 6*I*b^2*c^2*d*f^2 + (12*I*a*b*d^3*e - 12*I*a*b*c*d^2*f - 6*I*b^2*d^3)*(f*x + e)^2 + (-12*I*a
*b*d^3*e^2 - 12*I*a*b*c^2*d*f^2 + 12*I*b^2*d^3*e + (24*I*a*b*c*d^2*e - 12*I*b^2*c*d^2)*f)*(f*x + e))*cos(2*f*x
+ 2*e) + 2*(2*(f*x + e)^3*a*b*d^3 + 3*b^2*d^3*e^2 - 6*b^2*c*d^2*e*f + 3*b^2*c^2*d*f^2 - 3*(2*a*b*d^3*e - 2*a*
b*c*d^2*f - b^2*d^3)*(f*x + e)^2 + 6*(a*b*d^3*e^2 + a*b*c^2*d*f^2 - b^2*d^3*e - (2*a*b*c*d^2*e - b^2*c*d^2)*f)
*(f*x + e))*sin(2*f*x + 2*e))*log(cos(f*x + e)^2 + sin(f*x + e)^2 - 2*cos(f*x + e) + 1) + 48*(a*b*d^3*cos(2*f*
x + 2*e) + I*a*b*d^3*sin(2*f*x + 2*e) - a*b*d^3)*polylog(4, -e^(I*f*x + I*e)) + 48*(a*b*d^3*cos(2*f*x + 2*e) +
I*a*b*d^3*sin(2*f*x + 2*e) - a*b*d^3)*polylog(4, e^(I*f*x + I*e)) + (48*I*(f*x + e)*a*b*d^3 - 48*I*a*b*d^3*e
+ 48*I*a*b*c*d^2*f + 24*I*b^2*d^3 + (-48*I*(f*x + e)*a*b*d^3 + 48*I*a*b*d^3*e - 48*I*a*b*c*d^2*f - 24*I*b^2*d^
3)*cos(2*f*x + 2*e) + 24*(2*(f*x + e)*a*b*d^3 - 2*a*b*d^3*e + 2*a*b*c*d^2*f + b^2*d^3)*sin(2*f*x + 2*e))*polyl
og(3, -e^(I*f*x + I*e)) + (48*I*(f*x + e)*a*b*d^3 - 48*I*a*b*d^3*e + 48*I*a*b*c*d^2*f + 24*I*b^2*d^3 + (-48*I*
(f*x + e)*a*b*d^3 + 48*I*a*b*d^3*e - 48*I*a*b*c*d^2*f - 24*I*b^2*d^3)*cos(2*f*x + 2*e) + 24*(2*(f*x + e)*a*b*d
^3 - 2*a*b*d^3*e + 2*a*b*c*d^2*f + b^2*d^3)*sin(2*f*x + 2*e))*polylog(3, e^(I*f*x + I*e)) + ((-2*I*a*b - b^2)*
(f*x + e)^4*d^3 + (-8*I*b^2*d^3 + (8*I*a*b + 4*b^2)*d^3*e + (-8*I*a*b - 4*b^2)*c*d^2*f)*(f*x + e)^3 + (24*I*b^
2*d^3*e + (-12*I*a*b - 6*b^2)*d^3*e^2 + (-12*I*a*b - 6*b^2)*c^2*d*f^2 + (-24*I*b^2*c*d^2 + (24*I*a*b + 12*b^2)
*c*d^2*e)*f)*(f*x + e)^2 + (4*b^2*d^3*e^3 - 4*b^2*c^3*f^3 - 24*I*b^2*d^3*e^2 + (12*b^2*c^2*d*e - 24*I*b^2*c^2*
d)*f^2 - (12*b^2*c*d^2*e^2 - 48*I*b^2*c*d^2*e)*f)*(f*x + e))*sin(2*f*x + 2*e))/(-4*I*f^3*cos(2*f*x + 2*e) + 4*
f^3*sin(2*f*x + 2*e) + 4*I*f^3))/f

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Fricas [C]  time = 2.2122, size = 2641, normalized size = 8.95 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*(a+b*cot(f*x+e))^2,x, algorithm="fricas")

[Out]

-1/4*(4*b^2*d^3*f^3*x^3 + 12*b^2*c*d^2*f^3*x^2 + 12*b^2*c^2*d*f^3*x + 4*b^2*c^3*f^3 - 3*I*a*b*d^3*polylog(4, c
os(2*f*x + 2*e) + I*sin(2*f*x + 2*e))*sin(2*f*x + 2*e) + 3*I*a*b*d^3*polylog(4, cos(2*f*x + 2*e) - I*sin(2*f*x
+ 2*e))*sin(2*f*x + 2*e) - (-6*I*a*b*d^3*f^2*x^2 - 6*I*a*b*c^2*d*f^2 - 6*I*b^2*c*d^2*f - 6*I*(2*a*b*c*d^2*f^2
+ b^2*d^3*f)*x)*dilog(cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e))*sin(2*f*x + 2*e) - (6*I*a*b*d^3*f^2*x^2 + 6*I*a*
b*c^2*d*f^2 + 6*I*b^2*c*d^2*f + 6*I*(2*a*b*c*d^2*f^2 + b^2*d^3*f)*x)*dilog(cos(2*f*x + 2*e) - I*sin(2*f*x + 2*
e))*sin(2*f*x + 2*e) + 2*(2*a*b*d^3*e^3 - 2*a*b*c^3*f^3 - 3*b^2*d^3*e^2 + 3*(2*a*b*c^2*d*e - b^2*c^2*d)*f^2 -
6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f)*log(-1/2*cos(2*f*x + 2*e) + 1/2*I*sin(2*f*x + 2*e) + 1/2)*sin(2*f*x + 2*e)
+ 2*(2*a*b*d^3*e^3 - 2*a*b*c^3*f^3 - 3*b^2*d^3*e^2 + 3*(2*a*b*c^2*d*e - b^2*c^2*d)*f^2 - 6*(a*b*c*d^2*e^2 - b^
2*c*d^2*e)*f)*log(-1/2*cos(2*f*x + 2*e) - 1/2*I*sin(2*f*x + 2*e) + 1/2)*sin(2*f*x + 2*e) - 2*(2*a*b*d^3*f^3*x^
3 + 2*a*b*d^3*e^3 + 6*a*b*c^2*d*e*f^2 - 3*b^2*d^3*e^2 + 3*(2*a*b*c*d^2*f^3 + b^2*d^3*f^2)*x^2 - 6*(a*b*c*d^2*e
^2 - b^2*c*d^2*e)*f + 6*(a*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*log(-cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e) + 1)*sin
(2*f*x + 2*e) - 2*(2*a*b*d^3*f^3*x^3 + 2*a*b*d^3*e^3 + 6*a*b*c^2*d*e*f^2 - 3*b^2*d^3*e^2 + 3*(2*a*b*c*d^2*f^3
+ b^2*d^3*f^2)*x^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f + 6*(a*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*log(-cos(2*f*x +
2*e) - I*sin(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e) - 3*(2*a*b*d^3*f*x + 2*a*b*c*d^2*f + b^2*d^3)*polylog(3, cos(
2*f*x + 2*e) + I*sin(2*f*x + 2*e))*sin(2*f*x + 2*e) - 3*(2*a*b*d^3*f*x + 2*a*b*c*d^2*f + b^2*d^3)*polylog(3, c
os(2*f*x + 2*e) - I*sin(2*f*x + 2*e))*sin(2*f*x + 2*e) + 4*(b^2*d^3*f^3*x^3 + 3*b^2*c*d^2*f^3*x^2 + 3*b^2*c^2*
d*f^3*x + b^2*c^3*f^3)*cos(2*f*x + 2*e) - ((a^2 - b^2)*d^3*f^4*x^4 + 4*(a^2 - b^2)*c*d^2*f^4*x^3 + 6*(a^2 - b^
2)*c^2*d*f^4*x^2 + 4*(a^2 - b^2)*c^3*f^4*x)*sin(2*f*x + 2*e))/(f^4*sin(2*f*x + 2*e))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \cot{\left (e + f x \right )}\right )^{2} \left (c + d x\right )^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3*(a+b*cot(f*x+e))**2,x)

[Out]

Integral((a + b*cot(e + f*x))**2*(c + d*x)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{3}{\left (b \cot \left (f x + e\right ) + a\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*(a+b*cot(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^3*(b*cot(f*x + e) + a)^2, x)