3.329 \(\int \frac{(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=763 \[ \frac{6 f^2 \left (a^2-b^2\right ) (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^2 d^3}+\frac{6 f^2 \left (a^2-b^2\right ) (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b^2 d^3}-\frac{3 i f \left (a^2-b^2\right ) (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^2 d^2}-\frac{3 i f \left (a^2-b^2\right ) (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b^2 d^2}+\frac{6 i f^3 \left (a^2-b^2\right ) \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^2 d^4}+\frac{6 i f^3 \left (a^2-b^2\right ) \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b^2 d^4}+\frac{3 f^2 (e+f x) \text{PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^3}-\frac{3 i f (e+f x)^2 \text{PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a d^2}+\frac{3 i f^3 \text{PolyLog}\left (4,e^{2 i (c+d x)}\right )}{4 a d^4}+\frac{\left (a^2-b^2\right ) (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^2 d}+\frac{\left (a^2-b^2\right ) (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b^2 d}-\frac{i \left (a^2-b^2\right ) (e+f x)^4}{4 a b^2 f}+\frac{(e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac{i (e+f x)^4}{4 a f}+\frac{6 f^2 (e+f x) \sin (c+d x)}{b d^3}-\frac{3 f (e+f x)^2 \cos (c+d x)}{b d^2}+\frac{6 f^3 \cos (c+d x)}{b d^4}-\frac{(e+f x)^3 \sin (c+d x)}{b d} \]

[Out]

((-I/4)*(e + f*x)^4)/(a*f) - ((I/4)*(a^2 - b^2)*(e + f*x)^4)/(a*b^2*f) + (6*f^3*Cos[c + d*x])/(b*d^4) - (3*f*(
e + f*x)^2*Cos[c + d*x])/(b*d^2) + ((a^2 - b^2)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2]
)])/(a*b^2*d) + ((a^2 - b^2)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^2*d) + ((e
+ f*x)^3*Log[1 - E^((2*I)*(c + d*x))])/(a*d) - ((3*I)*(a^2 - b^2)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x
)))/(a - Sqrt[a^2 - b^2])])/(a*b^2*d^2) - ((3*I)*(a^2 - b^2)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a
+ Sqrt[a^2 - b^2])])/(a*b^2*d^2) - (((3*I)/2)*f*(e + f*x)^2*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^2) + (6*(a^
2 - b^2)*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^2*d^3) + (6*(a^2 - b^2)*f
^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^2*d^3) + (3*f^2*(e + f*x)*PolyLog[3
, E^((2*I)*(c + d*x))])/(2*a*d^3) + ((6*I)*(a^2 - b^2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^
2])])/(a*b^2*d^4) + ((6*I)*(a^2 - b^2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^2*d^4
) + (((3*I)/4)*f^3*PolyLog[4, E^((2*I)*(c + d*x))])/(a*d^4) + (6*f^2*(e + f*x)*Sin[c + d*x])/(b*d^3) - ((e + f
*x)^3*Sin[c + d*x])/(b*d)

________________________________________________________________________________________

Rubi [A]  time = 1.35906, antiderivative size = 763, normalized size of antiderivative = 1., number of steps used = 34, number of rules used = 17, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4543, 4408, 4404, 3311, 32, 2635, 8, 3717, 2190, 2531, 6609, 2282, 6589, 4525, 3296, 2638, 4519} \[ \frac{6 f^2 \left (a^2-b^2\right ) (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^2 d^3}+\frac{6 f^2 \left (a^2-b^2\right ) (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b^2 d^3}-\frac{3 i f \left (a^2-b^2\right ) (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^2 d^2}-\frac{3 i f \left (a^2-b^2\right ) (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b^2 d^2}+\frac{6 i f^3 \left (a^2-b^2\right ) \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^2 d^4}+\frac{6 i f^3 \left (a^2-b^2\right ) \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b^2 d^4}+\frac{3 f^2 (e+f x) \text{PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^3}-\frac{3 i f (e+f x)^2 \text{PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a d^2}+\frac{3 i f^3 \text{PolyLog}\left (4,e^{2 i (c+d x)}\right )}{4 a d^4}+\frac{\left (a^2-b^2\right ) (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^2 d}+\frac{\left (a^2-b^2\right ) (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b^2 d}-\frac{i \left (a^2-b^2\right ) (e+f x)^4}{4 a b^2 f}+\frac{(e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac{i (e+f x)^4}{4 a f}+\frac{6 f^2 (e+f x) \sin (c+d x)}{b d^3}-\frac{3 f (e+f x)^2 \cos (c+d x)}{b d^2}+\frac{6 f^3 \cos (c+d x)}{b d^4}-\frac{(e+f x)^3 \sin (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/4)*(e + f*x)^4)/(a*f) - ((I/4)*(a^2 - b^2)*(e + f*x)^4)/(a*b^2*f) + (6*f^3*Cos[c + d*x])/(b*d^4) - (3*f*(
e + f*x)^2*Cos[c + d*x])/(b*d^2) + ((a^2 - b^2)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2]
)])/(a*b^2*d) + ((a^2 - b^2)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^2*d) + ((e
+ f*x)^3*Log[1 - E^((2*I)*(c + d*x))])/(a*d) - ((3*I)*(a^2 - b^2)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x
)))/(a - Sqrt[a^2 - b^2])])/(a*b^2*d^2) - ((3*I)*(a^2 - b^2)*f*(e + f*x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a
+ Sqrt[a^2 - b^2])])/(a*b^2*d^2) - (((3*I)/2)*f*(e + f*x)^2*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^2) + (6*(a^
2 - b^2)*f^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^2*d^3) + (6*(a^2 - b^2)*f
^2*(e + f*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^2*d^3) + (3*f^2*(e + f*x)*PolyLog[3
, E^((2*I)*(c + d*x))])/(2*a*d^3) + ((6*I)*(a^2 - b^2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^
2])])/(a*b^2*d^4) + ((6*I)*(a^2 - b^2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^2*d^4
) + (((3*I)/4)*f^3*PolyLog[4, E^((2*I)*(c + d*x))])/(a*d^4) + (6*f^2*(e + f*x)*Sin[c + d*x])/(b*d^3) - ((e + f
*x)^3*Sin[c + d*x])/(b*d)

Rule 4543

Int[(Cos[(c_.) + (d_.)*(x_)]^(p_.)*Cot[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin
[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Cos[c + d*x]^p*Cot[c + d*x]^n, x], x] - Dist[b/a
, Int[((e + f*x)^m*Cos[c + d*x]^(p + 1)*Cot[c + d*x]^(n - 1))/(a + b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 4408

Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*Cot[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Int[
(c + d*x)^m*Cos[a + b*x]^n*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cos[a + b*x]^(n - 2)*Cot[a + b*x]^p, x]
/; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 4404

Int[Cos[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[((c +
d*x)^m*Sin[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n +
1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 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 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 4525

Int[(Cos[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol
] :> Dist[a/b^2, Int[(e + f*x)^m*Cos[c + d*x]^(n - 2), x], x] + (-Dist[1/b, Int[(e + f*x)^m*Cos[c + d*x]^(n -
2)*Sin[c + d*x], x], x] - Dist[(a^2 - b^2)/b^2, Int[((e + f*x)^m*Cos[c + d*x]^(n - 2))/(a + b*Sin[c + d*x]), x
], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

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

Rule 4519

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
-Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(I*(c + d*x)))/(a - Rt[a^2 - b^2, 2] - I*b
*E^(I*(c + d*x))), x] + Int[((e + f*x)^m*E^(I*(c + d*x)))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x))), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]

Rubi steps

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

Mathematica [B]  time = 10.6296, size = 4014, normalized size = 5.26 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 4.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}\cot \left ( dx+c \right ) }{a+b\sin \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [C]  time = 5.83696, size = 7961, normalized size = 10.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(6*I*b^2*f^3*polylog(4, cos(d*x + c) + I*sin(d*x + c)) - 6*I*b^2*f^3*polylog(4, cos(d*x + c) - I*sin(d*x +
c)) - 6*I*b^2*f^3*polylog(4, -cos(d*x + c) + I*sin(d*x + c)) + 6*I*b^2*f^3*polylog(4, -cos(d*x + c) - I*sin(d
*x + c)) + 6*I*(a^2 - b^2)*f^3*polylog(4, 1/2*(2*I*a*cos(d*x + c) - 2*a*sin(d*x + c) + 2*(b*cos(d*x + c) + I*b
*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) + 6*I*(a^2 - b^2)*f^3*polylog(4, 1/2*(2*I*a*cos(d*x + c) - 2*a*sin(d
*x + c) - 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) - 6*I*(a^2 - b^2)*f^3*polylog(4, -(
I*a*cos(d*x + c) + a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) - 6*I*(a^2
- b^2)*f^3*polylog(4, -(I*a*cos(d*x + c) + a*sin(d*x + c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b
^2)/b^2))/b) - 6*(a*b*d^2*f^3*x^2 + 2*a*b*d^2*e*f^2*x + a*b*d^2*e^2*f - 2*a*b*f^3)*cos(d*x + c) + (3*I*(a^2 -
b^2)*d^2*f^3*x^2 + 6*I*(a^2 - b^2)*d^2*e*f^2*x + 3*I*(a^2 - b^2)*d^2*e^2*f)*dilog(-1/2*(2*I*a*cos(d*x + c) + 2
*a*sin(d*x + c) + 2*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1) + (3*I*(a^2 - b^2
)*d^2*f^3*x^2 + 6*I*(a^2 - b^2)*d^2*e*f^2*x + 3*I*(a^2 - b^2)*d^2*e^2*f)*dilog(-1/2*(2*I*a*cos(d*x + c) + 2*a*
sin(d*x + c) - 2*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1) + (-3*I*(a^2 - b^2)*
d^2*f^3*x^2 - 6*I*(a^2 - b^2)*d^2*e*f^2*x - 3*I*(a^2 - b^2)*d^2*e^2*f)*dilog(-1/2*(-2*I*a*cos(d*x + c) + 2*a*s
in(d*x + c) + 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1) + (-3*I*(a^2 - b^2)*d
^2*f^3*x^2 - 6*I*(a^2 - b^2)*d^2*e*f^2*x - 3*I*(a^2 - b^2)*d^2*e^2*f)*dilog(-1/2*(-2*I*a*cos(d*x + c) + 2*a*si
n(d*x + c) - 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1) + (-3*I*b^2*d^2*f^3*x^
2 - 6*I*b^2*d^2*e*f^2*x - 3*I*b^2*d^2*e^2*f)*dilog(cos(d*x + c) + I*sin(d*x + c)) + (3*I*b^2*d^2*f^3*x^2 + 6*I
*b^2*d^2*e*f^2*x + 3*I*b^2*d^2*e^2*f)*dilog(cos(d*x + c) - I*sin(d*x + c)) + (3*I*b^2*d^2*f^3*x^2 + 6*I*b^2*d^
2*e*f^2*x + 3*I*b^2*d^2*e^2*f)*dilog(-cos(d*x + c) + I*sin(d*x + c)) + (-3*I*b^2*d^2*f^3*x^2 - 6*I*b^2*d^2*e*f
^2*x - 3*I*b^2*d^2*e^2*f)*dilog(-cos(d*x + c) - I*sin(d*x + c)) + ((a^2 - b^2)*d^3*e^3 - 3*(a^2 - b^2)*c*d^2*e
^2*f + 3*(a^2 - b^2)*c^2*d*e*f^2 - (a^2 - b^2)*c^3*f^3)*log(2*b*cos(d*x + c) + 2*I*b*sin(d*x + c) + 2*b*sqrt(-
(a^2 - b^2)/b^2) + 2*I*a) + ((a^2 - b^2)*d^3*e^3 - 3*(a^2 - b^2)*c*d^2*e^2*f + 3*(a^2 - b^2)*c^2*d*e*f^2 - (a^
2 - b^2)*c^3*f^3)*log(2*b*cos(d*x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a) + ((a^2 - b^
2)*d^3*e^3 - 3*(a^2 - b^2)*c*d^2*e^2*f + 3*(a^2 - b^2)*c^2*d*e*f^2 - (a^2 - b^2)*c^3*f^3)*log(-2*b*cos(d*x + c
) + 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a) + ((a^2 - b^2)*d^3*e^3 - 3*(a^2 - b^2)*c*d^2*e^2*
f + 3*(a^2 - b^2)*c^2*d*e*f^2 - (a^2 - b^2)*c^3*f^3)*log(-2*b*cos(d*x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a
^2 - b^2)/b^2) - 2*I*a) + ((a^2 - b^2)*d^3*f^3*x^3 + 3*(a^2 - b^2)*d^3*e*f^2*x^2 + 3*(a^2 - b^2)*d^3*e^2*f*x +
3*(a^2 - b^2)*c*d^2*e^2*f - 3*(a^2 - b^2)*c^2*d*e*f^2 + (a^2 - b^2)*c^3*f^3)*log(1/2*(2*I*a*cos(d*x + c) + 2*
a*sin(d*x + c) + 2*(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b) + ((a^2 - b^2)*d^3*f^3
*x^3 + 3*(a^2 - b^2)*d^3*e*f^2*x^2 + 3*(a^2 - b^2)*d^3*e^2*f*x + 3*(a^2 - b^2)*c*d^2*e^2*f - 3*(a^2 - b^2)*c^2
*d*e*f^2 + (a^2 - b^2)*c^3*f^3)*log(1/2*(2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) - 2*(b*cos(d*x + c) - I*b*sin(d
*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b) + ((a^2 - b^2)*d^3*f^3*x^3 + 3*(a^2 - b^2)*d^3*e*f^2*x^2 + 3*(a^2 -
b^2)*d^3*e^2*f*x + 3*(a^2 - b^2)*c*d^2*e^2*f - 3*(a^2 - b^2)*c^2*d*e*f^2 + (a^2 - b^2)*c^3*f^3)*log(1/2*(-2*I*
a*cos(d*x + c) + 2*a*sin(d*x + c) + 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b) + (
(a^2 - b^2)*d^3*f^3*x^3 + 3*(a^2 - b^2)*d^3*e*f^2*x^2 + 3*(a^2 - b^2)*d^3*e^2*f*x + 3*(a^2 - b^2)*c*d^2*e^2*f
- 3*(a^2 - b^2)*c^2*d*e*f^2 + (a^2 - b^2)*c^3*f^3)*log(1/2*(-2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) - 2*(b*cos(
d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b) + (b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 + 3*b^2
*d^3*e^2*f*x + b^2*d^3*e^3)*log(cos(d*x + c) + I*sin(d*x + c) + 1) + (b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 +
3*b^2*d^3*e^2*f*x + b^2*d^3*e^3)*log(cos(d*x + c) - I*sin(d*x + c) + 1) + (b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3
*b^2*c^2*d*e*f^2 - b^2*c^3*f^3)*log(-1/2*cos(d*x + c) + 1/2*I*sin(d*x + c) + 1/2) + (b^2*d^3*e^3 - 3*b^2*c*d^2
*e^2*f + 3*b^2*c^2*d*e*f^2 - b^2*c^3*f^3)*log(-1/2*cos(d*x + c) - 1/2*I*sin(d*x + c) + 1/2) + (b^2*d^3*f^3*x^3
+ 3*b^2*d^3*e*f^2*x^2 + 3*b^2*d^3*e^2*f*x + 3*b^2*c*d^2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3)*log(-cos(d*x
+ c) + I*sin(d*x + c) + 1) + (b^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 + 3*b^2*d^3*e^2*f*x + 3*b^2*c*d^2*e^2*f -
3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3)*log(-cos(d*x + c) - I*sin(d*x + c) + 1) + 6*((a^2 - b^2)*d*f^3*x + (a^2 - b^
2)*d*e*f^2)*polylog(3, 1/2*(2*I*a*cos(d*x + c) - 2*a*sin(d*x + c) + 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt
(-(a^2 - b^2)/b^2))/b) + 6*((a^2 - b^2)*d*f^3*x + (a^2 - b^2)*d*e*f^2)*polylog(3, 1/2*(2*I*a*cos(d*x + c) - 2*
a*sin(d*x + c) - 2*(b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) + 6*((a^2 - b^2)*d*f^3*x + (
a^2 - b^2)*d*e*f^2)*polylog(3, -(I*a*cos(d*x + c) + a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(
-(a^2 - b^2)/b^2))/b) + 6*((a^2 - b^2)*d*f^3*x + (a^2 - b^2)*d*e*f^2)*polylog(3, -(I*a*cos(d*x + c) + a*sin(d*
x + c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2))/b) + 6*(b^2*d*f^3*x + b^2*d*e*f^2)*polylo
g(3, cos(d*x + c) + I*sin(d*x + c)) + 6*(b^2*d*f^3*x + b^2*d*e*f^2)*polylog(3, cos(d*x + c) - I*sin(d*x + c))
+ 6*(b^2*d*f^3*x + b^2*d*e*f^2)*polylog(3, -cos(d*x + c) + I*sin(d*x + c)) + 6*(b^2*d*f^3*x + b^2*d*e*f^2)*pol
ylog(3, -cos(d*x + c) - I*sin(d*x + c)) - 2*(a*b*d^3*f^3*x^3 + 3*a*b*d^3*e*f^2*x^2 + a*b*d^3*e^3 - 6*a*b*d*e*f
^2 + 3*(a*b*d^3*e^2*f - 2*a*b*d*f^3)*x)*sin(d*x + c))/(a*b^2*d^4)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out