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

Optimal. Leaf size=433 \[ -\frac{3 i a^2 b d (c+d x) \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac{3 a^2 b d^2 \text{PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}-\frac{3 i a b^2 d^2 \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^3}+\frac{i b^3 d (c+d x) \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}-\frac{b^3 d^2 \text{PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}+\frac{3 a^2 b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{i a^2 b (c+d x)^3}{d}+\frac{a^3 (c+d x)^3}{3 d}+\frac{6 a b^2 d (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^2}-\frac{3 a b^2 (c+d x)^2 \cot (e+f x)}{f}-\frac{3 i a b^2 (c+d x)^2}{f}-\frac{a b^2 (c+d x)^3}{d}-\frac{b^3 d (c+d x) \cot (e+f x)}{f^2}-\frac{b^3 (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{b^3 (c+d x)^2 \cot ^2(e+f x)}{2 f}-\frac{b^3 c d x}{f}+\frac{i b^3 (c+d x)^3}{3 d}+\frac{b^3 d^2 \log (\sin (e+f x))}{f^3}-\frac{b^3 d^2 x^2}{2 f} \]

[Out]

-((b^3*c*d*x)/f) - (b^3*d^2*x^2)/(2*f) - ((3*I)*a*b^2*(c + d*x)^2)/f + (a^3*(c + d*x)^3)/(3*d) - (I*a^2*b*(c +
d*x)^3)/d - (a*b^2*(c + d*x)^3)/d + ((I/3)*b^3*(c + d*x)^3)/d - (b^3*d*(c + d*x)*Cot[e + f*x])/f^2 - (3*a*b^2
*(c + d*x)^2*Cot[e + f*x])/f - (b^3*(c + d*x)^2*Cot[e + f*x]^2)/(2*f) + (6*a*b^2*d*(c + d*x)*Log[1 - E^((2*I)*
(e + f*x))])/f^2 + (3*a^2*b*(c + d*x)^2*Log[1 - E^((2*I)*(e + f*x))])/f - (b^3*(c + d*x)^2*Log[1 - E^((2*I)*(e
+ f*x))])/f + (b^3*d^2*Log[Sin[e + f*x]])/f^3 - ((3*I)*a*b^2*d^2*PolyLog[2, E^((2*I)*(e + f*x))])/f^3 - ((3*I
)*a^2*b*d*(c + d*x)*PolyLog[2, E^((2*I)*(e + f*x))])/f^2 + (I*b^3*d*(c + d*x)*PolyLog[2, E^((2*I)*(e + f*x))])
/f^2 + (3*a^2*b*d^2*PolyLog[3, E^((2*I)*(e + f*x))])/(2*f^3) - (b^3*d^2*PolyLog[3, E^((2*I)*(e + f*x))])/(2*f^
3)

________________________________________________________________________________________

Rubi [A]  time = 0.648633, antiderivative size = 433, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {3722, 3717, 2190, 2531, 2282, 6589, 3720, 2279, 2391, 32, 3475} \[ -\frac{3 i a^2 b d (c+d x) \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac{3 a^2 b d^2 \text{PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}-\frac{3 i a b^2 d^2 \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^3}+\frac{i b^3 d (c+d x) \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}-\frac{b^3 d^2 \text{PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}+\frac{3 a^2 b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{i a^2 b (c+d x)^3}{d}+\frac{a^3 (c+d x)^3}{3 d}+\frac{6 a b^2 d (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^2}-\frac{3 a b^2 (c+d x)^2 \cot (e+f x)}{f}-\frac{3 i a b^2 (c+d x)^2}{f}-\frac{a b^2 (c+d x)^3}{d}-\frac{b^3 d (c+d x) \cot (e+f x)}{f^2}-\frac{b^3 (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{b^3 (c+d x)^2 \cot ^2(e+f x)}{2 f}-\frac{b^3 c d x}{f}+\frac{i b^3 (c+d x)^3}{3 d}+\frac{b^3 d^2 \log (\sin (e+f x))}{f^3}-\frac{b^3 d^2 x^2}{2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^3*c*d*x)/f) - (b^3*d^2*x^2)/(2*f) - ((3*I)*a*b^2*(c + d*x)^2)/f + (a^3*(c + d*x)^3)/(3*d) - (I*a^2*b*(c +
d*x)^3)/d - (a*b^2*(c + d*x)^3)/d + ((I/3)*b^3*(c + d*x)^3)/d - (b^3*d*(c + d*x)*Cot[e + f*x])/f^2 - (3*a*b^2
*(c + d*x)^2*Cot[e + f*x])/f - (b^3*(c + d*x)^2*Cot[e + f*x]^2)/(2*f) + (6*a*b^2*d*(c + d*x)*Log[1 - E^((2*I)*
(e + f*x))])/f^2 + (3*a^2*b*(c + d*x)^2*Log[1 - E^((2*I)*(e + f*x))])/f - (b^3*(c + d*x)^2*Log[1 - E^((2*I)*(e
+ f*x))])/f + (b^3*d^2*Log[Sin[e + f*x]])/f^3 - ((3*I)*a*b^2*d^2*PolyLog[2, E^((2*I)*(e + f*x))])/f^3 - ((3*I
)*a^2*b*d*(c + d*x)*PolyLog[2, E^((2*I)*(e + f*x))])/f^2 + (I*b^3*d*(c + d*x)*PolyLog[2, E^((2*I)*(e + f*x))])
/f^2 + (3*a^2*b*d^2*PolyLog[3, E^((2*I)*(e + f*x))])/(2*f^3) - (b^3*d^2*PolyLog[3, E^((2*I)*(e + f*x))])/(2*f^
3)

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 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 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int (c+d x)^2 (a+b \cot (e+f x))^3 \, dx &=\int \left (a^3 (c+d x)^2+3 a^2 b (c+d x)^2 \cot (e+f x)+3 a b^2 (c+d x)^2 \cot ^2(e+f x)+b^3 (c+d x)^2 \cot ^3(e+f x)\right ) \, dx\\ &=\frac{a^3 (c+d x)^3}{3 d}+\left (3 a^2 b\right ) \int (c+d x)^2 \cot (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x)^2 \cot ^2(e+f x) \, dx+b^3 \int (c+d x)^2 \cot ^3(e+f x) \, dx\\ &=\frac{a^3 (c+d x)^3}{3 d}-\frac{i a^2 b (c+d x)^3}{d}-\frac{3 a b^2 (c+d x)^2 \cot (e+f x)}{f}-\frac{b^3 (c+d x)^2 \cot ^2(e+f x)}{2 f}-\left (6 i a^2 b\right ) \int \frac{e^{2 i (e+f x)} (c+d x)^2}{1-e^{2 i (e+f x)}} \, dx-\left (3 a b^2\right ) \int (c+d x)^2 \, dx-b^3 \int (c+d x)^2 \cot (e+f x) \, dx+\frac{\left (6 a b^2 d\right ) \int (c+d x) \cot (e+f x) \, dx}{f}+\frac{\left (b^3 d\right ) \int (c+d x) \cot ^2(e+f x) \, dx}{f}\\ &=-\frac{3 i a b^2 (c+d x)^2}{f}+\frac{a^3 (c+d x)^3}{3 d}-\frac{i a^2 b (c+d x)^3}{d}-\frac{a b^2 (c+d x)^3}{d}+\frac{i b^3 (c+d x)^3}{3 d}-\frac{b^3 d (c+d x) \cot (e+f x)}{f^2}-\frac{3 a b^2 (c+d x)^2 \cot (e+f x)}{f}-\frac{b^3 (c+d x)^2 \cot ^2(e+f x)}{2 f}+\frac{3 a^2 b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}+\left (2 i b^3\right ) \int \frac{e^{2 i (e+f x)} (c+d x)^2}{1-e^{2 i (e+f x)}} \, dx+\frac{\left (b^3 d^2\right ) \int \cot (e+f x) \, dx}{f^2}-\frac{\left (6 a^2 b d\right ) \int (c+d x) \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f}-\frac{\left (12 i a b^2 d\right ) \int \frac{e^{2 i (e+f x)} (c+d x)}{1-e^{2 i (e+f x)}} \, dx}{f}-\frac{\left (b^3 d\right ) \int (c+d x) \, dx}{f}\\ &=-\frac{b^3 c d x}{f}-\frac{b^3 d^2 x^2}{2 f}-\frac{3 i a b^2 (c+d x)^2}{f}+\frac{a^3 (c+d x)^3}{3 d}-\frac{i a^2 b (c+d x)^3}{d}-\frac{a b^2 (c+d x)^3}{d}+\frac{i b^3 (c+d x)^3}{3 d}-\frac{b^3 d (c+d x) \cot (e+f x)}{f^2}-\frac{3 a b^2 (c+d x)^2 \cot (e+f x)}{f}-\frac{b^3 (c+d x)^2 \cot ^2(e+f x)}{2 f}+\frac{6 a b^2 d (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^2}+\frac{3 a^2 b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{b^3 (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}+\frac{b^3 d^2 \log (\sin (e+f x))}{f^3}-\frac{3 i a^2 b d (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac{\left (3 i a^2 b d^2\right ) \int \text{Li}_2\left (e^{2 i (e+f x)}\right ) \, dx}{f^2}-\frac{\left (6 a b^2 d^2\right ) \int \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f^2}+\frac{\left (2 b^3 d\right ) \int (c+d x) \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f}\\ &=-\frac{b^3 c d x}{f}-\frac{b^3 d^2 x^2}{2 f}-\frac{3 i a b^2 (c+d x)^2}{f}+\frac{a^3 (c+d x)^3}{3 d}-\frac{i a^2 b (c+d x)^3}{d}-\frac{a b^2 (c+d x)^3}{d}+\frac{i b^3 (c+d x)^3}{3 d}-\frac{b^3 d (c+d x) \cot (e+f x)}{f^2}-\frac{3 a b^2 (c+d x)^2 \cot (e+f x)}{f}-\frac{b^3 (c+d x)^2 \cot ^2(e+f x)}{2 f}+\frac{6 a b^2 d (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^2}+\frac{3 a^2 b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{b^3 (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}+\frac{b^3 d^2 \log (\sin (e+f x))}{f^3}-\frac{3 i a^2 b d (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac{i b^3 d (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac{\left (3 a^2 b d^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^3}+\frac{\left (3 i a b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{f^3}-\frac{\left (i b^3 d^2\right ) \int \text{Li}_2\left (e^{2 i (e+f x)}\right ) \, dx}{f^2}\\ &=-\frac{b^3 c d x}{f}-\frac{b^3 d^2 x^2}{2 f}-\frac{3 i a b^2 (c+d x)^2}{f}+\frac{a^3 (c+d x)^3}{3 d}-\frac{i a^2 b (c+d x)^3}{d}-\frac{a b^2 (c+d x)^3}{d}+\frac{i b^3 (c+d x)^3}{3 d}-\frac{b^3 d (c+d x) \cot (e+f x)}{f^2}-\frac{3 a b^2 (c+d x)^2 \cot (e+f x)}{f}-\frac{b^3 (c+d x)^2 \cot ^2(e+f x)}{2 f}+\frac{6 a b^2 d (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^2}+\frac{3 a^2 b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{b^3 (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}+\frac{b^3 d^2 \log (\sin (e+f x))}{f^3}-\frac{3 i a b^2 d^2 \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^3}-\frac{3 i a^2 b d (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac{i b^3 d (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac{3 a^2 b d^2 \text{Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^3}-\frac{\left (b^3 d^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^3}\\ &=-\frac{b^3 c d x}{f}-\frac{b^3 d^2 x^2}{2 f}-\frac{3 i a b^2 (c+d x)^2}{f}+\frac{a^3 (c+d x)^3}{3 d}-\frac{i a^2 b (c+d x)^3}{d}-\frac{a b^2 (c+d x)^3}{d}+\frac{i b^3 (c+d x)^3}{3 d}-\frac{b^3 d (c+d x) \cot (e+f x)}{f^2}-\frac{3 a b^2 (c+d x)^2 \cot (e+f x)}{f}-\frac{b^3 (c+d x)^2 \cot ^2(e+f x)}{2 f}+\frac{6 a b^2 d (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^2}+\frac{3 a^2 b (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{b^3 (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f}+\frac{b^3 d^2 \log (\sin (e+f x))}{f^3}-\frac{3 i a b^2 d^2 \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^3}-\frac{3 i a^2 b d (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac{i b^3 d (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^2}+\frac{3 a^2 b d^2 \text{Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^3}-\frac{b^3 d^2 \text{Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^3}\\ \end{align*}

Mathematica [B]  time = 7.79398, size = 2013, normalized size = 4.65 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(a^2*b*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)))/(2*f^3) + (b^3*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)))/(6*f^3) + (b^3*d^2*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*S
in[f*x]]*Sin[e]))/(f^3*(Cos[e]^2 + Sin[e]^2)) + (6*a*b^2*c*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)) + (3*a^2*b*c^2*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)) - (b^3*c^2*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]^2*(6*b^3*c*d*Cos[e] + 18*a*b^2*c^2
*f*Cos[e] + 6*b^3*d^2*x*Cos[e] + 36*a*b^2*c*d*f*x*Cos[e] + 18*a^2*b*c^2*f^2*x*Cos[e] - 6*b^3*c^2*f^2*x*Cos[e]
+ 18*a*b^2*d^2*f*x^2*Cos[e] + 18*a^2*b*c*d*f^2*x^2*Cos[e] - 6*b^3*c*d*f^2*x^2*Cos[e] + 6*a^2*b*d^2*f^2*x^3*Cos
[e] - 2*b^3*d^2*f^2*x^3*Cos[e] - 6*b^3*c*d*Cos[e + 2*f*x] - 18*a*b^2*c^2*f*Cos[e + 2*f*x] - 6*b^3*d^2*x*Cos[e
+ 2*f*x] - 36*a*b^2*c*d*f*x*Cos[e + 2*f*x] - 9*a^2*b*c^2*f^2*x*Cos[e + 2*f*x] + 3*b^3*c^2*f^2*x*Cos[e + 2*f*x]
- 18*a*b^2*d^2*f*x^2*Cos[e + 2*f*x] - 9*a^2*b*c*d*f^2*x^2*Cos[e + 2*f*x] + 3*b^3*c*d*f^2*x^2*Cos[e + 2*f*x] -
3*a^2*b*d^2*f^2*x^3*Cos[e + 2*f*x] + b^3*d^2*f^2*x^3*Cos[e + 2*f*x] - 9*a^2*b*c^2*f^2*x*Cos[3*e + 2*f*x] + 3*
b^3*c^2*f^2*x*Cos[3*e + 2*f*x] - 9*a^2*b*c*d*f^2*x^2*Cos[3*e + 2*f*x] + 3*b^3*c*d*f^2*x^2*Cos[3*e + 2*f*x] - 3
*a^2*b*d^2*f^2*x^3*Cos[3*e + 2*f*x] + b^3*d^2*f^2*x^3*Cos[3*e + 2*f*x] - 6*b^3*c^2*f*Sin[e] - 12*b^3*c*d*f*x*S
in[e] + 6*a^3*c^2*f^2*x*Sin[e] - 18*a*b^2*c^2*f^2*x*Sin[e] - 6*b^3*d^2*f*x^2*Sin[e] + 6*a^3*c*d*f^2*x^2*Sin[e]
- 18*a*b^2*c*d*f^2*x^2*Sin[e] + 2*a^3*d^2*f^2*x^3*Sin[e] - 6*a*b^2*d^2*f^2*x^3*Sin[e] + 3*a^3*c^2*f^2*x*Sin[e
+ 2*f*x] - 9*a*b^2*c^2*f^2*x*Sin[e + 2*f*x] + 3*a^3*c*d*f^2*x^2*Sin[e + 2*f*x] - 9*a*b^2*c*d*f^2*x^2*Sin[e +
2*f*x] + a^3*d^2*f^2*x^3*Sin[e + 2*f*x] - 3*a*b^2*d^2*f^2*x^3*Sin[e + 2*f*x] - 3*a^3*c^2*f^2*x*Sin[3*e + 2*f*x
] + 9*a*b^2*c^2*f^2*x*Sin[3*e + 2*f*x] - 3*a^3*c*d*f^2*x^2*Sin[3*e + 2*f*x] + 9*a*b^2*c*d*f^2*x^2*Sin[3*e + 2*
f*x] - a^3*d^2*f^2*x^3*Sin[3*e + 2*f*x] + 3*a*b^2*d^2*f^2*x^3*Sin[3*e + 2*f*x]))/(12*f^2) - (3*a*b^2*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 + 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^3*Sqrt[
Sec[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (3*a^2*b*c*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 + ArcTa
n[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(f^2*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)]) + (b^3*c*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 + ArcT
an[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 + ArcT
an[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)])

________________________________________________________________________________________

Maple [B]  time = 0.597, size = 1740, normalized size = 4. \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)^2*(a+b*cot(f*x+e))^3,x)

[Out]

-6*I*b/f^2*polylog(2,exp(I*(f*x+e)))*a^2*d^2*x-12*I*b^2/f^2*a*d^2*e*x+6*I*b/f^2*a^2*d^2*e^2*x-6*I*b/f^2*a^2*c*
d*e^2-6*I*b/f^2*a^2*c*d*polylog(2,-exp(I*(f*x+e)))-6*I*b/f^2*a^2*c*d*polylog(2,exp(I*(f*x+e)))-6*I*b/f^2*polyl
og(2,-exp(I*(f*x+e)))*a^2*d^2*x+4*I*b^3/f*c*d*e*x+2*b^2*(-3*I*a*d^2*f*x^2*exp(2*I*(f*x+e))-6*I*a*c*d*f*x*exp(2
*I*(f*x+e))+b*d^2*f*x^2*exp(2*I*(f*x+e))-3*I*a*c^2*f*exp(2*I*(f*x+e))+3*I*a*d^2*f*x^2-I*b*d^2*x*exp(2*I*(f*x+e
))+2*b*c*d*f*x*exp(2*I*(f*x+e))+6*I*a*c*d*f*x-I*b*c*d*exp(2*I*(f*x+e))+b*c^2*f*exp(2*I*(f*x+e))+3*I*a*c^2*f+I*
b*d^2*x+I*b*c*d)/f^2/(exp(2*I*(f*x+e))-1)^2-3*a*b^2*c*d*x^2+1/3*I*b^3*d^2*x^3+I*b^3*c*d*x^2+2*b^3/f*c^2*ln(exp
(I*(f*x+e)))-2*b^3/f^3*d^2*ln(exp(I*(f*x+e)))+b^3/f^3*d^2*ln(exp(I*(f*x+e))+1)+b^3/f^3*d^2*ln(exp(I*(f*x+e))-1
)-2*b^3/f^3*d^2*polylog(3,-exp(I*(f*x+e)))-2*b^3/f^3*d^2*polylog(3,exp(I*(f*x+e)))-b^3/f*c^2*ln(exp(I*(f*x+e))
-1)-b^3/f*c^2*ln(exp(I*(f*x+e))+1)-a*b^2*d^2*x^3-3*b^2*a*c^2*x-12*I*b/f*a^2*c*d*e*x-I*b^3*c^2*x+a^3*c*d*x^2+6*
b/f*ln(1-exp(I*(f*x+e)))*a^2*c*d*x+6*b/f^2*ln(1-exp(I*(f*x+e)))*a^2*c*d*e+6*b/f*ln(exp(I*(f*x+e))+1)*a^2*c*d*x
-6*b/f^2*a^2*c*d*e*ln(exp(I*(f*x+e))-1)+12*b/f^2*a^2*c*d*e*ln(exp(I*(f*x+e)))-3*I*a^2*b*c*d*x^2+3*I*a^2*b*c^2*
x+1/3*a^3*d^2*x^3+a^3*c^2*x+3*b/f*ln(exp(I*(f*x+e))+1)*a^2*d^2*x^2-3*b/f^3*ln(1-exp(I*(f*x+e)))*a^2*d^2*e^2+6*
b^2/f^2*a*c*d*ln(exp(I*(f*x+e))+1)+6*b^2/f^2*a*c*d*ln(exp(I*(f*x+e))-1)+2*b^3/f^2*c*d*e*ln(exp(I*(f*x+e))-1)-4
*b^3/f^2*c*d*e*ln(exp(I*(f*x+e)))-12*b^2/f^2*a*c*d*ln(exp(I*(f*x+e)))+12*b^2/f^3*a*d^2*e*ln(exp(I*(f*x+e)))-6*
b^2/f^3*a*d^2*e*ln(exp(I*(f*x+e))-1)-6*b/f^3*a^2*d^2*e^2*ln(exp(I*(f*x+e)))-I*a^2*b*d^2*x^3+3*b/f^3*a^2*d^2*e^
2*ln(exp(I*(f*x+e))-1)-6*I*b^2/f^3*a*d^2*polylog(2,-exp(I*(f*x+e)))-6*I*b^2/f^3*a*d^2*polylog(2,exp(I*(f*x+e))
)+2*I*b^3/f^2*polylog(2,-exp(I*(f*x+e)))*d^2*x+2*I*b^3/f^2*polylog(2,exp(I*(f*x+e)))*d^2*x-6*I*b^2/f^3*a*d^2*e
^2-6*I*b^2/f*a*d^2*x^2+2*I*b^3/f^2*c*d*polylog(2,-exp(I*(f*x+e)))+2*I*b^3/f^2*c*d*polylog(2,exp(I*(f*x+e)))-2*
I*b^3/f^2*d^2*e^2*x+4*I*b/f^3*a^2*d^2*e^3+2*I*b^3/f^2*c*d*e^2-2*b^3/f*ln(exp(I*(f*x+e))+1)*c*d*x-b^3/f*ln(exp(
I*(f*x+e))+1)*d^2*x^2-b^3/f*ln(1-exp(I*(f*x+e)))*d^2*x^2+b^3/f^3*ln(1-exp(I*(f*x+e)))*d^2*e^2+6*b/f^3*a^2*d^2*
polylog(3,-exp(I*(f*x+e)))+6*b/f^3*a^2*d^2*polylog(3,exp(I*(f*x+e)))+2*b^3/f^3*d^2*e^2*ln(exp(I*(f*x+e)))-6*b/
f*a^2*c^2*ln(exp(I*(f*x+e)))-b^3/f^3*d^2*e^2*ln(exp(I*(f*x+e))-1)+3*b/f*a^2*c^2*ln(exp(I*(f*x+e))+1)+3*b/f*a^2
*c^2*ln(exp(I*(f*x+e))-1)-4/3*I*b^3/f^3*d^2*e^3+3*b/f*ln(1-exp(I*(f*x+e)))*a^2*d^2*x^2+6*b^2/f^3*ln(1-exp(I*(f
*x+e)))*a*d^2*e-2*b^3/f*ln(1-exp(I*(f*x+e)))*c*d*x-2*b^3/f^2*ln(1-exp(I*(f*x+e)))*c*d*e+6*b^2/f^2*ln(exp(I*(f*
x+e))+1)*a*d^2*x+6*b^2/f^2*ln(1-exp(I*(f*x+e)))*a*d^2*x

________________________________________________________________________________________

Maxima [B]  time = 16.6731, size = 7318, normalized size = 16.9 \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)^2*(a+b*cot(f*x+e))^3,x, algorithm="maxima")

[Out]

1/3*(3*(f*x + e)*a^3*c^2 + (f*x + e)^3*a^3*d^2/f^2 - 3*(f*x + e)^2*a^3*d^2*e/f^2 + 3*(f*x + e)*a^3*d^2*e^2/f^2
+ 3*(f*x + e)^2*a^3*c*d/f - 6*(f*x + e)*a^3*c*d*e/f + 9*a^2*b*c^2*log(sin(f*x + e)) + 9*a^2*b*d^2*e^2*log(sin
(f*x + e))/f^2 - 18*a^2*b*c*d*e*log(sin(f*x + e))/f + 3*(36*a*b^2*d^2*e^2 + 36*a*b^2*c^2*f^2 - (6*a^2*b - 6*I*
a*b^2 - 2*b^3)*(f*x + e)^3*d^2 - 12*b^3*d^2*e + ((18*a^2*b - 18*I*a*b^2 - 6*b^3)*d^2*e - (18*a^2*b - 18*I*a*b^
2 - 6*b^3)*c*d*f)*(f*x + e)^2 - ((-18*I*a*b^2 - 6*b^3)*d^2*e^2 + (36*I*a*b^2 + 12*b^3)*c*d*e*f + (-18*I*a*b^2
- 6*b^3)*c^2*f^2)*(f*x + e) - 12*(6*a*b^2*c*d*e - b^3*c*d)*f - (6*b^3*d^2*e^2 + 6*b^3*c^2*f^2 + 36*a*b^2*d^2*e
- 6*(3*a^2*b - b^3)*(f*x + e)^2*d^2 - 6*b^3*d^2 - 12*(3*a*b^2*d^2 - (3*a^2*b - b^3)*d^2*e + (3*a^2*b - b^3)*c
*d*f)*(f*x + e) - 12*(b^3*c*d*e + 3*a*b^2*c*d)*f + 6*(b^3*d^2*e^2 + b^3*c^2*f^2 + 6*a*b^2*d^2*e - (3*a^2*b - b
^3)*(f*x + e)^2*d^2 - b^3*d^2 - 2*(3*a*b^2*d^2 - (3*a^2*b - b^3)*d^2*e + (3*a^2*b - b^3)*c*d*f)*(f*x + e) - 2*
(b^3*c*d*e + 3*a*b^2*c*d)*f)*cos(4*f*x + 4*e) - 12*(b^3*d^2*e^2 + b^3*c^2*f^2 + 6*a*b^2*d^2*e - (3*a^2*b - b^3
)*(f*x + e)^2*d^2 - b^3*d^2 - 2*(3*a*b^2*d^2 - (3*a^2*b - b^3)*d^2*e + (3*a^2*b - b^3)*c*d*f)*(f*x + e) - 2*(b
^3*c*d*e + 3*a*b^2*c*d)*f)*cos(2*f*x + 2*e) + (6*I*b^3*d^2*e^2 + 6*I*b^3*c^2*f^2 + 36*I*a*b^2*d^2*e + (-18*I*a
^2*b + 6*I*b^3)*(f*x + e)^2*d^2 - 6*I*b^3*d^2 + (-36*I*a*b^2*d^2 + (36*I*a^2*b - 12*I*b^3)*d^2*e + (-36*I*a^2*
b + 12*I*b^3)*c*d*f)*(f*x + e) + (-12*I*b^3*c*d*e - 36*I*a*b^2*c*d)*f)*sin(4*f*x + 4*e) + (-12*I*b^3*d^2*e^2 -
12*I*b^3*c^2*f^2 - 72*I*a*b^2*d^2*e + (36*I*a^2*b - 12*I*b^3)*(f*x + e)^2*d^2 + 12*I*b^3*d^2 + (72*I*a*b^2*d^
2 + (-72*I*a^2*b + 24*I*b^3)*d^2*e + (72*I*a^2*b - 24*I*b^3)*c*d*f)*(f*x + e) + (24*I*b^3*c*d*e + 72*I*a*b^2*c
*d)*f)*sin(2*f*x + 2*e))*arctan2(sin(f*x + e), cos(f*x + e) + 1) - (6*b^3*d^2*e^2 + 6*b^3*c^2*f^2 + 36*a*b^2*d
^2*e - 6*b^3*d^2 - 12*(b^3*c*d*e + 3*a*b^2*c*d)*f + 6*(b^3*d^2*e^2 + b^3*c^2*f^2 + 6*a*b^2*d^2*e - b^3*d^2 - 2
*(b^3*c*d*e + 3*a*b^2*c*d)*f)*cos(4*f*x + 4*e) - 12*(b^3*d^2*e^2 + b^3*c^2*f^2 + 6*a*b^2*d^2*e - b^3*d^2 - 2*(
b^3*c*d*e + 3*a*b^2*c*d)*f)*cos(2*f*x + 2*e) + (6*I*b^3*d^2*e^2 + 6*I*b^3*c^2*f^2 + 36*I*a*b^2*d^2*e - 6*I*b^3
*d^2 + (-12*I*b^3*c*d*e - 36*I*a*b^2*c*d)*f)*sin(4*f*x + 4*e) + (-12*I*b^3*d^2*e^2 - 12*I*b^3*c^2*f^2 - 72*I*a
*b^2*d^2*e + 12*I*b^3*d^2 + (24*I*b^3*c*d*e + 72*I*a*b^2*c*d)*f)*sin(2*f*x + 2*e))*arctan2(sin(f*x + e), cos(f
*x + e) - 1) - (6*(3*a^2*b - b^3)*(f*x + e)^2*d^2 + 12*(3*a*b^2*d^2 - (3*a^2*b - b^3)*d^2*e + (3*a^2*b - b^3)*
c*d*f)*(f*x + e) + 6*((3*a^2*b - b^3)*(f*x + e)^2*d^2 + 2*(3*a*b^2*d^2 - (3*a^2*b - b^3)*d^2*e + (3*a^2*b - b^
3)*c*d*f)*(f*x + e))*cos(4*f*x + 4*e) - 12*((3*a^2*b - b^3)*(f*x + e)^2*d^2 + 2*(3*a*b^2*d^2 - (3*a^2*b - b^3)
*d^2*e + (3*a^2*b - b^3)*c*d*f)*(f*x + e))*cos(2*f*x + 2*e) + ((18*I*a^2*b - 6*I*b^3)*(f*x + e)^2*d^2 + (36*I*
a*b^2*d^2 + (-36*I*a^2*b + 12*I*b^3)*d^2*e + (36*I*a^2*b - 12*I*b^3)*c*d*f)*(f*x + e))*sin(4*f*x + 4*e) + ((-3
6*I*a^2*b + 12*I*b^3)*(f*x + e)^2*d^2 + (-72*I*a*b^2*d^2 + (72*I*a^2*b - 24*I*b^3)*d^2*e + (-72*I*a^2*b + 24*I
*b^3)*c*d*f)*(f*x + e))*sin(2*f*x + 2*e))*arctan2(sin(f*x + e), -cos(f*x + e) + 1) - ((6*a^2*b - 6*I*a*b^2 - 2
*b^3)*(f*x + e)^3*d^2 + (36*a*b^2*d^2 - (18*a^2*b - 18*I*a*b^2 - 6*b^3)*d^2*e + (18*a^2*b - 18*I*a*b^2 - 6*b^3
)*c*d*f)*(f*x + e)^2 - (72*a*b^2*d^2*e - 12*b^3*d^2 - (-18*I*a*b^2 - 6*b^3)*d^2*e^2 - (-18*I*a*b^2 - 6*b^3)*c^
2*f^2 - (72*a*b^2*c*d + (36*I*a*b^2 + 12*b^3)*c*d*e)*f)*(f*x + e))*cos(4*f*x + 4*e) + ((12*a^2*b - 12*I*a*b^2
- 4*b^3)*(f*x + e)^3*d^2 + 12*b^3*d^2*e - 12*(3*a*b^2 + I*b^3)*d^2*e^2 - 12*(3*a*b^2 + I*b^3)*c^2*f^2 - ((36*a
^2*b - 36*I*a*b^2 - 12*b^3)*d^2*e - (36*a^2*b - 36*I*a*b^2 - 12*b^3)*c*d*f - 12*(3*a*b^2 - I*b^3)*d^2)*(f*x +
e)^2 + (12*b^3*d^2 - (36*I*a*b^2 + 12*b^3)*d^2*e^2 - (36*I*a*b^2 + 12*b^3)*c^2*f^2 - 24*(3*a*b^2 - I*b^3)*d^2*
e - ((-72*I*a*b^2 - 24*b^3)*c*d*e - 24*(3*a*b^2 - I*b^3)*c*d)*f)*(f*x + e) - 12*(b^3*c*d - 2*(3*a*b^2 + I*b^3)
*c*d*e)*f)*cos(2*f*x + 2*e) - (36*a*b^2*d^2 + 12*(3*a^2*b - b^3)*(f*x + e)*d^2 - 12*(3*a^2*b - b^3)*d^2*e + 12
*(3*a^2*b - b^3)*c*d*f + 12*(3*a*b^2*d^2 + (3*a^2*b - b^3)*(f*x + e)*d^2 - (3*a^2*b - b^3)*d^2*e + (3*a^2*b -
b^3)*c*d*f)*cos(4*f*x + 4*e) - 24*(3*a*b^2*d^2 + (3*a^2*b - b^3)*(f*x + e)*d^2 - (3*a^2*b - b^3)*d^2*e + (3*a^
2*b - b^3)*c*d*f)*cos(2*f*x + 2*e) + (36*I*a*b^2*d^2 + (36*I*a^2*b - 12*I*b^3)*(f*x + e)*d^2 + (-36*I*a^2*b +
12*I*b^3)*d^2*e + (36*I*a^2*b - 12*I*b^3)*c*d*f)*sin(4*f*x + 4*e) + (-72*I*a*b^2*d^2 + (-72*I*a^2*b + 24*I*b^3
)*(f*x + e)*d^2 + (72*I*a^2*b - 24*I*b^3)*d^2*e + (-72*I*a^2*b + 24*I*b^3)*c*d*f)*sin(2*f*x + 2*e))*dilog(-e^(
I*f*x + I*e)) - (36*a*b^2*d^2 + 12*(3*a^2*b - b^3)*(f*x + e)*d^2 - 12*(3*a^2*b - b^3)*d^2*e + 12*(3*a^2*b - b^
3)*c*d*f + 12*(3*a*b^2*d^2 + (3*a^2*b - b^3)*(f*x + e)*d^2 - (3*a^2*b - b^3)*d^2*e + (3*a^2*b - b^3)*c*d*f)*co
s(4*f*x + 4*e) - 24*(3*a*b^2*d^2 + (3*a^2*b - b^3)*(f*x + e)*d^2 - (3*a^2*b - b^3)*d^2*e + (3*a^2*b - b^3)*c*d
*f)*cos(2*f*x + 2*e) + (36*I*a*b^2*d^2 + (36*I*a^2*b - 12*I*b^3)*(f*x + e)*d^2 + (-36*I*a^2*b + 12*I*b^3)*d^2*
e + (36*I*a^2*b - 12*I*b^3)*c*d*f)*sin(4*f*x + 4*e) + (-72*I*a*b^2*d^2 + (-72*I*a^2*b + 24*I*b^3)*(f*x + e)*d^
2 + (72*I*a^2*b - 24*I*b^3)*d^2*e + (-72*I*a^2*b + 24*I*b^3)*c*d*f)*sin(2*f*x + 2*e))*dilog(e^(I*f*x + I*e)) -
(-3*I*b^3*d^2*e^2 - 3*I*b^3*c^2*f^2 - 18*I*a*b^2*d^2*e + (9*I*a^2*b - 3*I*b^3)*(f*x + e)^2*d^2 + 3*I*b^3*d^2
+ (18*I*a*b^2*d^2 + (-18*I*a^2*b + 6*I*b^3)*d^2*e + (18*I*a^2*b - 6*I*b^3)*c*d*f)*(f*x + e) + (6*I*b^3*c*d*e +
18*I*a*b^2*c*d)*f + (-3*I*b^3*d^2*e^2 - 3*I*b^3*c^2*f^2 - 18*I*a*b^2*d^2*e + (9*I*a^2*b - 3*I*b^3)*(f*x + e)^
2*d^2 + 3*I*b^3*d^2 + (18*I*a*b^2*d^2 + (-18*I*a^2*b + 6*I*b^3)*d^2*e + (18*I*a^2*b - 6*I*b^3)*c*d*f)*(f*x + e
) + (6*I*b^3*c*d*e + 18*I*a*b^2*c*d)*f)*cos(4*f*x + 4*e) + (6*I*b^3*d^2*e^2 + 6*I*b^3*c^2*f^2 + 36*I*a*b^2*d^2
*e + (-18*I*a^2*b + 6*I*b^3)*(f*x + e)^2*d^2 - 6*I*b^3*d^2 + (-36*I*a*b^2*d^2 + (36*I*a^2*b - 12*I*b^3)*d^2*e
+ (-36*I*a^2*b + 12*I*b^3)*c*d*f)*(f*x + e) + (-12*I*b^3*c*d*e - 36*I*a*b^2*c*d)*f)*cos(2*f*x + 2*e) + 3*(b^3*
d^2*e^2 + b^3*c^2*f^2 + 6*a*b^2*d^2*e - (3*a^2*b - b^3)*(f*x + e)^2*d^2 - b^3*d^2 - 2*(3*a*b^2*d^2 - (3*a^2*b
- b^3)*d^2*e + (3*a^2*b - b^3)*c*d*f)*(f*x + e) - 2*(b^3*c*d*e + 3*a*b^2*c*d)*f)*sin(4*f*x + 4*e) - 6*(b^3*d^2
*e^2 + b^3*c^2*f^2 + 6*a*b^2*d^2*e - (3*a^2*b - b^3)*(f*x + e)^2*d^2 - b^3*d^2 - 2*(3*a*b^2*d^2 - (3*a^2*b - b
^3)*d^2*e + (3*a^2*b - b^3)*c*d*f)*(f*x + e) - 2*(b^3*c*d*e + 3*a*b^2*c*d)*f)*sin(2*f*x + 2*e))*log(cos(f*x +
e)^2 + sin(f*x + e)^2 + 2*cos(f*x + e) + 1) - (-3*I*b^3*d^2*e^2 - 3*I*b^3*c^2*f^2 - 18*I*a*b^2*d^2*e + (9*I*a^
2*b - 3*I*b^3)*(f*x + e)^2*d^2 + 3*I*b^3*d^2 + (18*I*a*b^2*d^2 + (-18*I*a^2*b + 6*I*b^3)*d^2*e + (18*I*a^2*b -
6*I*b^3)*c*d*f)*(f*x + e) + (6*I*b^3*c*d*e + 18*I*a*b^2*c*d)*f + (-3*I*b^3*d^2*e^2 - 3*I*b^3*c^2*f^2 - 18*I*a
*b^2*d^2*e + (9*I*a^2*b - 3*I*b^3)*(f*x + e)^2*d^2 + 3*I*b^3*d^2 + (18*I*a*b^2*d^2 + (-18*I*a^2*b + 6*I*b^3)*d
^2*e + (18*I*a^2*b - 6*I*b^3)*c*d*f)*(f*x + e) + (6*I*b^3*c*d*e + 18*I*a*b^2*c*d)*f)*cos(4*f*x + 4*e) + (6*I*b
^3*d^2*e^2 + 6*I*b^3*c^2*f^2 + 36*I*a*b^2*d^2*e + (-18*I*a^2*b + 6*I*b^3)*(f*x + e)^2*d^2 - 6*I*b^3*d^2 + (-36
*I*a*b^2*d^2 + (36*I*a^2*b - 12*I*b^3)*d^2*e + (-36*I*a^2*b + 12*I*b^3)*c*d*f)*(f*x + e) + (-12*I*b^3*c*d*e -
36*I*a*b^2*c*d)*f)*cos(2*f*x + 2*e) + 3*(b^3*d^2*e^2 + b^3*c^2*f^2 + 6*a*b^2*d^2*e - (3*a^2*b - b^3)*(f*x + e)
^2*d^2 - b^3*d^2 - 2*(3*a*b^2*d^2 - (3*a^2*b - b^3)*d^2*e + (3*a^2*b - b^3)*c*d*f)*(f*x + e) - 2*(b^3*c*d*e +
3*a*b^2*c*d)*f)*sin(4*f*x + 4*e) - 6*(b^3*d^2*e^2 + b^3*c^2*f^2 + 6*a*b^2*d^2*e - (3*a^2*b - b^3)*(f*x + e)^2*
d^2 - b^3*d^2 - 2*(3*a*b^2*d^2 - (3*a^2*b - b^3)*d^2*e + (3*a^2*b - b^3)*c*d*f)*(f*x + e) - 2*(b^3*c*d*e + 3*a
*b^2*c*d)*f)*sin(2*f*x + 2*e))*log(cos(f*x + e)^2 + sin(f*x + e)^2 - 2*cos(f*x + e) + 1) - ((36*I*a^2*b - 12*I
*b^3)*d^2*cos(4*f*x + 4*e) + (-72*I*a^2*b + 24*I*b^3)*d^2*cos(2*f*x + 2*e) - 12*(3*a^2*b - b^3)*d^2*sin(4*f*x
+ 4*e) + 24*(3*a^2*b - b^3)*d^2*sin(2*f*x + 2*e) + (36*I*a^2*b - 12*I*b^3)*d^2)*polylog(3, -e^(I*f*x + I*e)) -
((36*I*a^2*b - 12*I*b^3)*d^2*cos(4*f*x + 4*e) + (-72*I*a^2*b + 24*I*b^3)*d^2*cos(2*f*x + 2*e) - 12*(3*a^2*b -
b^3)*d^2*sin(4*f*x + 4*e) + 24*(3*a^2*b - b^3)*d^2*sin(2*f*x + 2*e) + (36*I*a^2*b - 12*I*b^3)*d^2)*polylog(3,
e^(I*f*x + I*e)) - ((6*I*a^2*b + 6*a*b^2 - 2*I*b^3)*(f*x + e)^3*d^2 + (36*I*a*b^2*d^2 + (-18*I*a^2*b - 18*a*b
^2 + 6*I*b^3)*d^2*e + (18*I*a^2*b + 18*a*b^2 - 6*I*b^3)*c*d*f)*(f*x + e)^2 + (-72*I*a*b^2*d^2*e + 12*I*b^3*d^2
+ 6*(3*a*b^2 - I*b^3)*d^2*e^2 + 6*(3*a*b^2 - I*b^3)*c^2*f^2 - 12*(-6*I*a*b^2*c*d + (3*a*b^2 - I*b^3)*c*d*e)*f
)*(f*x + e))*sin(4*f*x + 4*e) - ((-12*I*a^2*b - 12*a*b^2 + 4*I*b^3)*(f*x + e)^3*d^2 - 12*I*b^3*d^2*e + (36*I*a
*b^2 - 12*b^3)*d^2*e^2 + (36*I*a*b^2 - 12*b^3)*c^2*f^2 + ((36*I*a^2*b + 36*a*b^2 - 12*I*b^3)*d^2*e + (-36*I*a^
2*b - 36*a*b^2 + 12*I*b^3)*c*d*f + (-36*I*a*b^2 - 12*b^3)*d^2)*(f*x + e)^2 + (-12*I*b^3*d^2 - 12*(3*a*b^2 - I*
b^3)*d^2*e^2 - 12*(3*a*b^2 - I*b^3)*c^2*f^2 + (72*I*a*b^2 + 24*b^3)*d^2*e + (24*(3*a*b^2 - I*b^3)*c*d*e + (-72
*I*a*b^2 - 24*b^3)*c*d)*f)*(f*x + e) + (12*I*b^3*c*d + (-72*I*a*b^2 + 24*b^3)*c*d*e)*f)*sin(2*f*x + 2*e))/(-6*
I*f^2*cos(4*f*x + 4*e) + 12*I*f^2*cos(2*f*x + 2*e) + 6*f^2*sin(4*f*x + 4*e) - 12*f^2*sin(2*f*x + 2*e) - 6*I*f^
2))/f

________________________________________________________________________________________

Fricas [C]  time = 2.33852, size = 3310, normalized size = 7.64 \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)^2*(a+b*cot(f*x+e))^3,x, algorithm="fricas")

[Out]

-1/12*(4*(a^3 - 3*a*b^2)*d^2*f^3*x^3 - 12*b^3*c^2*f^2 - 12*(b^3*d^2*f^2 - (a^3 - 3*a*b^2)*c*d*f^3)*x^2 - 12*(2
*b^3*c*d*f^2 - (a^3 - 3*a*b^2)*c^2*f^3)*x - 4*((a^3 - 3*a*b^2)*d^2*f^3*x^3 + 3*(a^3 - 3*a*b^2)*c*d*f^3*x^2 + 3
*(a^3 - 3*a*b^2)*c^2*f^3*x)*cos(2*f*x + 2*e) - (18*I*a*b^2*d^2 + 6*I*(3*a^2*b - b^3)*d^2*f*x + 6*I*(3*a^2*b -
b^3)*c*d*f + (-18*I*a*b^2*d^2 - 6*I*(3*a^2*b - b^3)*d^2*f*x - 6*I*(3*a^2*b - b^3)*c*d*f)*cos(2*f*x + 2*e))*dil
og(cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e)) - (-18*I*a*b^2*d^2 - 6*I*(3*a^2*b - b^3)*d^2*f*x - 6*I*(3*a^2*b - b^
3)*c*d*f + (18*I*a*b^2*d^2 + 6*I*(3*a^2*b - b^3)*d^2*f*x + 6*I*(3*a^2*b - b^3)*c*d*f)*cos(2*f*x + 2*e))*dilog(
cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e)) - 6*(6*a*b^2*d^2*e - b^3*d^2 - (3*a^2*b - b^3)*d^2*e^2 - (3*a^2*b - b^3
)*c^2*f^2 - 2*(3*a*b^2*c*d - (3*a^2*b - b^3)*c*d*e)*f - (6*a*b^2*d^2*e - b^3*d^2 - (3*a^2*b - b^3)*d^2*e^2 - (
3*a^2*b - b^3)*c^2*f^2 - 2*(3*a*b^2*c*d - (3*a^2*b - b^3)*c*d*e)*f)*cos(2*f*x + 2*e))*log(-1/2*cos(2*f*x + 2*e
) + 1/2*I*sin(2*f*x + 2*e) + 1/2) - 6*(6*a*b^2*d^2*e - b^3*d^2 - (3*a^2*b - b^3)*d^2*e^2 - (3*a^2*b - b^3)*c^2
*f^2 - 2*(3*a*b^2*c*d - (3*a^2*b - b^3)*c*d*e)*f - (6*a*b^2*d^2*e - b^3*d^2 - (3*a^2*b - b^3)*d^2*e^2 - (3*a^2
*b - b^3)*c^2*f^2 - 2*(3*a*b^2*c*d - (3*a^2*b - b^3)*c*d*e)*f)*cos(2*f*x + 2*e))*log(-1/2*cos(2*f*x + 2*e) - 1
/2*I*sin(2*f*x + 2*e) + 1/2) + 6*((3*a^2*b - b^3)*d^2*f^2*x^2 + 6*a*b^2*d^2*e - (3*a^2*b - b^3)*d^2*e^2 + 2*(3
*a^2*b - b^3)*c*d*e*f + 2*(3*a*b^2*d^2*f + (3*a^2*b - b^3)*c*d*f^2)*x - ((3*a^2*b - b^3)*d^2*f^2*x^2 + 6*a*b^2
*d^2*e - (3*a^2*b - b^3)*d^2*e^2 + 2*(3*a^2*b - b^3)*c*d*e*f + 2*(3*a*b^2*d^2*f + (3*a^2*b - b^3)*c*d*f^2)*x)*
cos(2*f*x + 2*e))*log(-cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e) + 1) + 6*((3*a^2*b - b^3)*d^2*f^2*x^2 + 6*a*b^2*d
^2*e - (3*a^2*b - b^3)*d^2*e^2 + 2*(3*a^2*b - b^3)*c*d*e*f + 2*(3*a*b^2*d^2*f + (3*a^2*b - b^3)*c*d*f^2)*x - (
(3*a^2*b - b^3)*d^2*f^2*x^2 + 6*a*b^2*d^2*e - (3*a^2*b - b^3)*d^2*e^2 + 2*(3*a^2*b - b^3)*c*d*e*f + 2*(3*a*b^2
*d^2*f + (3*a^2*b - b^3)*c*d*f^2)*x)*cos(2*f*x + 2*e))*log(-cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e) + 1) - 3*((3
*a^2*b - b^3)*d^2*cos(2*f*x + 2*e) - (3*a^2*b - b^3)*d^2)*polylog(3, cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e)) -
3*((3*a^2*b - b^3)*d^2*cos(2*f*x + 2*e) - (3*a^2*b - b^3)*d^2)*polylog(3, cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e
)) - 12*(3*a*b^2*d^2*f^2*x^2 + 3*a*b^2*c^2*f^2 + b^3*c*d*f + (6*a*b^2*c*d*f^2 + b^3*d^2*f)*x)*sin(2*f*x + 2*e)
)/(f^3*cos(2*f*x + 2*e) - f^3)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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