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

Optimal. Leaf size=1138 \[ \text{result too large to display} \]

[Out]

(3*e*f^2*x)/(4*b*d^2) + (3*f^3*x^2)/(8*b*d^2) - (e + f*x)^4/(8*b*f) + ((a^2 - b^2)*(e + f*x)^4)/(4*b^3*f) - (2
*(e + f*x)^3*ArcTanh[E^(I*(c + d*x))])/(a*d) - (6*f^2*(e + f*x)*Cos[c + d*x])/(a*d^3) - (6*(a^2 - b^2)*f^2*(e
+ f*x)*Cos[c + d*x])/(a*b^2*d^3) + ((e + f*x)^3*Cos[c + d*x])/(a*d) + ((a^2 - b^2)*(e + f*x)^3*Cos[c + d*x])/(
a*b^2*d) + (3*f^3*Cos[c + d*x]^2)/(8*b*d^4) - (3*f*(e + f*x)^2*Cos[c + d*x]^2)/(4*b*d^2) + (I*(a^2 - b^2)^(3/2
)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^3*d) - (I*(a^2 - b^2)^(3/2)*(e + f*x)
^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^3*d) + ((3*I)*f*(e + f*x)^2*PolyLog[2, -E^(I*(c
+ d*x))])/(a*d^2) - ((3*I)*f*(e + f*x)^2*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) + (3*(a^2 - b^2)^(3/2)*f*(e + f*
x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^3*d^2) - (3*(a^2 - b^2)^(3/2)*f*(e + f*x)^2
*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^3*d^2) - (6*f^2*(e + f*x)*PolyLog[3, -E^(I*(c +
d*x))])/(a*d^3) + (6*f^2*(e + f*x)*PolyLog[3, E^(I*(c + d*x))])/(a*d^3) + ((6*I)*(a^2 - b^2)^(3/2)*f^2*(e + f
*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^3*d^3) - ((6*I)*(a^2 - b^2)^(3/2)*f^2*(e + f
*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^3*d^3) - ((6*I)*f^3*PolyLog[4, -E^(I*(c + d*
x))])/(a*d^4) + ((6*I)*f^3*PolyLog[4, E^(I*(c + d*x))])/(a*d^4) - (6*(a^2 - b^2)^(3/2)*f^3*PolyLog[4, (I*b*E^(
I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^3*d^4) + (6*(a^2 - b^2)^(3/2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/
(a + Sqrt[a^2 - b^2])])/(a*b^3*d^4) + (6*f^3*Sin[c + d*x])/(a*d^4) + (6*(a^2 - b^2)*f^3*Sin[c + d*x])/(a*b^2*d
^4) - (3*f*(e + f*x)^2*Sin[c + d*x])/(a*d^2) - (3*(a^2 - b^2)*f*(e + f*x)^2*Sin[c + d*x])/(a*b^2*d^2) + (3*f^2
*(e + f*x)*Cos[c + d*x]*Sin[c + d*x])/(4*b*d^3) - ((e + f*x)^3*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)

________________________________________________________________________________________

Rubi [A]  time = 2.10748, antiderivative size = 1138, normalized size of antiderivative = 1., number of steps used = 53, number of rules used = 18, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529, Rules used = {4543, 4408, 4405, 3311, 3296, 2637, 2633, 4183, 2531, 6609, 2282, 6589, 4525, 32, 3310, 3323, 2264, 2190} \[ \frac{\left (a^2-b^2\right ) (e+f x)^4}{4 b^3 f}-\frac{(e+f x)^4}{8 b f}-\frac{2 \tanh ^{-1}\left (e^{i (c+d x)}\right ) (e+f x)^3}{a d}+\frac{\left (a^2-b^2\right ) \cos (c+d x) (e+f x)^3}{a b^2 d}+\frac{\cos (c+d x) (e+f x)^3}{a d}+\frac{i \left (a^2-b^2\right )^{3/2} \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) (e+f x)^3}{a b^3 d}-\frac{i \left (a^2-b^2\right )^{3/2} \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) (e+f x)^3}{a b^3 d}-\frac{\cos (c+d x) \sin (c+d x) (e+f x)^3}{2 b d}-\frac{3 f \cos ^2(c+d x) (e+f x)^2}{4 b d^2}+\frac{3 i f \text{PolyLog}\left (2,-e^{i (c+d x)}\right ) (e+f x)^2}{a d^2}-\frac{3 i f \text{PolyLog}\left (2,e^{i (c+d x)}\right ) (e+f x)^2}{a d^2}+\frac{3 \left (a^2-b^2\right )^{3/2} f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) (e+f x)^2}{a b^3 d^2}-\frac{3 \left (a^2-b^2\right )^{3/2} f \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) (e+f x)^2}{a b^3 d^2}-\frac{3 \left (a^2-b^2\right ) f \sin (c+d x) (e+f x)^2}{a b^2 d^2}-\frac{3 f \sin (c+d x) (e+f x)^2}{a d^2}-\frac{6 \left (a^2-b^2\right ) f^2 \cos (c+d x) (e+f x)}{a b^2 d^3}-\frac{6 f^2 \cos (c+d x) (e+f x)}{a d^3}-\frac{6 f^2 \text{PolyLog}\left (3,-e^{i (c+d x)}\right ) (e+f x)}{a d^3}+\frac{6 f^2 \text{PolyLog}\left (3,e^{i (c+d x)}\right ) (e+f x)}{a d^3}+\frac{6 i \left (a^2-b^2\right )^{3/2} f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right ) (e+f x)}{a b^3 d^3}-\frac{6 i \left (a^2-b^2\right )^{3/2} f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) (e+f x)}{a b^3 d^3}+\frac{3 f^2 \cos (c+d x) \sin (c+d x) (e+f x)}{4 b d^3}+\frac{3 f^3 x^2}{8 b d^2}+\frac{3 f^3 \cos ^2(c+d x)}{8 b d^4}+\frac{3 e f^2 x}{4 b d^2}-\frac{6 i f^3 \text{PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}+\frac{6 i f^3 \text{PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4}-\frac{6 \left (a^2-b^2\right )^{3/2} f^3 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^3 d^4}+\frac{6 \left (a^2-b^2\right )^{3/2} f^3 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{a b^3 d^4}+\frac{6 \left (a^2-b^2\right ) f^3 \sin (c+d x)}{a b^2 d^4}+\frac{6 f^3 \sin (c+d x)}{a d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*e*f^2*x)/(4*b*d^2) + (3*f^3*x^2)/(8*b*d^2) - (e + f*x)^4/(8*b*f) + ((a^2 - b^2)*(e + f*x)^4)/(4*b^3*f) - (2
*(e + f*x)^3*ArcTanh[E^(I*(c + d*x))])/(a*d) - (6*f^2*(e + f*x)*Cos[c + d*x])/(a*d^3) - (6*(a^2 - b^2)*f^2*(e
+ f*x)*Cos[c + d*x])/(a*b^2*d^3) + ((e + f*x)^3*Cos[c + d*x])/(a*d) + ((a^2 - b^2)*(e + f*x)^3*Cos[c + d*x])/(
a*b^2*d) + (3*f^3*Cos[c + d*x]^2)/(8*b*d^4) - (3*f*(e + f*x)^2*Cos[c + d*x]^2)/(4*b*d^2) + (I*(a^2 - b^2)^(3/2
)*(e + f*x)^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^3*d) - (I*(a^2 - b^2)^(3/2)*(e + f*x)
^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^3*d) + ((3*I)*f*(e + f*x)^2*PolyLog[2, -E^(I*(c
+ d*x))])/(a*d^2) - ((3*I)*f*(e + f*x)^2*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) + (3*(a^2 - b^2)^(3/2)*f*(e + f*
x)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^3*d^2) - (3*(a^2 - b^2)^(3/2)*f*(e + f*x)^2
*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^3*d^2) - (6*f^2*(e + f*x)*PolyLog[3, -E^(I*(c +
d*x))])/(a*d^3) + (6*f^2*(e + f*x)*PolyLog[3, E^(I*(c + d*x))])/(a*d^3) + ((6*I)*(a^2 - b^2)^(3/2)*f^2*(e + f
*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^3*d^3) - ((6*I)*(a^2 - b^2)^(3/2)*f^2*(e + f
*x)*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a*b^3*d^3) - ((6*I)*f^3*PolyLog[4, -E^(I*(c + d*
x))])/(a*d^4) + ((6*I)*f^3*PolyLog[4, E^(I*(c + d*x))])/(a*d^4) - (6*(a^2 - b^2)^(3/2)*f^3*PolyLog[4, (I*b*E^(
I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(a*b^3*d^4) + (6*(a^2 - b^2)^(3/2)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/
(a + Sqrt[a^2 - b^2])])/(a*b^3*d^4) + (6*f^3*Sin[c + d*x])/(a*d^4) + (6*(a^2 - b^2)*f^3*Sin[c + d*x])/(a*b^2*d
^4) - (3*f*(e + f*x)^2*Sin[c + d*x])/(a*d^2) - (3*(a^2 - b^2)*f*(e + f*x)^2*Sin[c + d*x])/(a*b^2*d^2) + (3*f^2
*(e + f*x)*Cos[c + d*x]*Sin[c + d*x])/(4*b*d^3) - ((e + f*x)^3*Cos[c + d*x]*Sin[c + d*x])/(2*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 4405

Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)], x_Symbol] :> -Simp[((c +
d*x)^m*Cos[a + b*x]^(n + 1))/(b*(n + 1)), x] + Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Cos[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 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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 4183

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E^(I*(e + f*
x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, 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 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 3310

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

Rule 3323

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

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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]

Rubi steps

\begin{align*} \int \frac{(e+f x)^3 \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\int (e+f x)^3 \cos ^3(c+d x) \cot (c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x)^3 \cos ^4(c+d x)}{a+b \sin (c+d x)} \, dx}{a}\\ &=\frac{\int (e+f x)^3 \cos (c+d x) \cot (c+d x) \, dx}{a}-\frac{\int (e+f x)^3 \cos ^2(c+d x) \, dx}{b}+\left (\frac{a}{b}-\frac{b}{a}\right ) \int \frac{(e+f x)^3 \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx\\ &=-\frac{3 f (e+f x)^2 \cos ^2(c+d x)}{4 b d^2}-\frac{(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac{\int (e+f x)^3 \csc (c+d x) \, dx}{a}-\frac{\int (e+f x)^3 \sin (c+d x) \, dx}{a}+\left (\frac{1}{a}-\frac{a}{b^2}\right ) \int (e+f x)^3 \sin (c+d x) \, dx-\frac{\int (e+f x)^3 \, dx}{2 b}+\frac{\left (a^2-b^2\right ) \int (e+f x)^3 \, dx}{b^3}-\frac{\left (a^2-b^2\right )^2 \int \frac{(e+f x)^3}{a+b \sin (c+d x)} \, dx}{a b^3}+\frac{\left (3 f^2\right ) \int (e+f x) \cos ^2(c+d x) \, dx}{2 b d^2}\\ &=-\frac{(e+f x)^4}{8 b f}+\frac{\left (a^2-b^2\right ) (e+f x)^4}{4 b^3 f}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac{(e+f x)^3 \cos (c+d x)}{a d}-\frac{\left (\frac{1}{a}-\frac{a}{b^2}\right ) (e+f x)^3 \cos (c+d x)}{d}+\frac{3 f^3 \cos ^2(c+d x)}{8 b d^4}-\frac{3 f (e+f x)^2 \cos ^2(c+d x)}{4 b d^2}+\frac{3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac{\left (2 \left (a^2-b^2\right )^2\right ) \int \frac{e^{i (c+d x)} (e+f x)^3}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{a b^3}-\frac{(3 f) \int (e+f x)^2 \cos (c+d x) \, dx}{a d}-\frac{(3 f) \int (e+f x)^2 \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d}+\frac{(3 f) \int (e+f x)^2 \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d}+\frac{\left (3 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f\right ) \int (e+f x)^2 \cos (c+d x) \, dx}{d}+\frac{\left (3 f^2\right ) \int (e+f x) \, dx}{4 b d^2}\\ &=\frac{3 e f^2 x}{4 b d^2}+\frac{3 f^3 x^2}{8 b d^2}-\frac{(e+f x)^4}{8 b f}+\frac{\left (a^2-b^2\right ) (e+f x)^4}{4 b^3 f}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac{(e+f x)^3 \cos (c+d x)}{a d}-\frac{\left (\frac{1}{a}-\frac{a}{b^2}\right ) (e+f x)^3 \cos (c+d x)}{d}+\frac{3 f^3 \cos ^2(c+d x)}{8 b d^4}-\frac{3 f (e+f x)^2 \cos ^2(c+d x)}{4 b d^2}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{a d^2}-\frac{3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac{3 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f (e+f x)^2 \sin (c+d x)}{d^2}+\frac{3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac{\left (2 i \left (a^2-b^2\right )^{3/2}\right ) \int \frac{e^{i (c+d x)} (e+f x)^3}{2 a-2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a b^2}-\frac{\left (2 i \left (a^2-b^2\right )^{3/2}\right ) \int \frac{e^{i (c+d x)} (e+f x)^3}{2 a+2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a b^2}-\frac{\left (6 i f^2\right ) \int (e+f x) \text{Li}_2\left (-e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac{\left (6 i f^2\right ) \int (e+f x) \text{Li}_2\left (e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac{\left (6 f^2\right ) \int (e+f x) \sin (c+d x) \, dx}{a d^2}-\frac{\left (6 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f^2\right ) \int (e+f x) \sin (c+d x) \, dx}{d^2}\\ &=\frac{3 e f^2 x}{4 b d^2}+\frac{3 f^3 x^2}{8 b d^2}-\frac{(e+f x)^4}{8 b f}+\frac{\left (a^2-b^2\right ) (e+f x)^4}{4 b^3 f}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac{6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac{6 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f^2 (e+f x) \cos (c+d x)}{d^3}+\frac{(e+f x)^3 \cos (c+d x)}{a d}-\frac{\left (\frac{1}{a}-\frac{a}{b^2}\right ) (e+f x)^3 \cos (c+d x)}{d}+\frac{3 f^3 \cos ^2(c+d x)}{8 b d^4}-\frac{3 f (e+f x)^2 \cos ^2(c+d x)}{4 b d^2}+\frac{i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^3 d}-\frac{i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{a b^3 d}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{a d^2}-\frac{6 f^2 (e+f x) \text{Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\frac{3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac{3 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f (e+f x)^2 \sin (c+d x)}{d^2}+\frac{3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac{\left (3 i \left (a^2-b^2\right )^{3/2} f\right ) \int (e+f x)^2 \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{a b^3 d}+\frac{\left (3 i \left (a^2-b^2\right )^{3/2} f\right ) \int (e+f x)^2 \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{a b^3 d}+\frac{\left (6 f^3\right ) \int \cos (c+d x) \, dx}{a d^3}+\frac{\left (6 f^3\right ) \int \text{Li}_3\left (-e^{i (c+d x)}\right ) \, dx}{a d^3}-\frac{\left (6 f^3\right ) \int \text{Li}_3\left (e^{i (c+d x)}\right ) \, dx}{a d^3}-\frac{\left (6 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f^3\right ) \int \cos (c+d x) \, dx}{d^3}\\ &=\frac{3 e f^2 x}{4 b d^2}+\frac{3 f^3 x^2}{8 b d^2}-\frac{(e+f x)^4}{8 b f}+\frac{\left (a^2-b^2\right ) (e+f x)^4}{4 b^3 f}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac{6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac{6 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f^2 (e+f x) \cos (c+d x)}{d^3}+\frac{(e+f x)^3 \cos (c+d x)}{a d}-\frac{\left (\frac{1}{a}-\frac{a}{b^2}\right ) (e+f x)^3 \cos (c+d x)}{d}+\frac{3 f^3 \cos ^2(c+d x)}{8 b d^4}-\frac{3 f (e+f x)^2 \cos ^2(c+d x)}{4 b d^2}+\frac{i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^3 d}-\frac{i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{a b^3 d}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac{3 \left (a^2-b^2\right )^{3/2} 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^3 d^2}-\frac{3 \left (a^2-b^2\right )^{3/2} 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^3 d^2}-\frac{6 f^2 (e+f x) \text{Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\frac{6 f^3 \sin (c+d x)}{a d^4}-\frac{6 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f^3 \sin (c+d x)}{d^4}-\frac{3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac{3 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f (e+f x)^2 \sin (c+d x)}{d^2}+\frac{3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac{\left (6 \left (a^2-b^2\right )^{3/2} f^2\right ) \int (e+f x) \text{Li}_2\left (\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{a b^3 d^2}+\frac{\left (6 \left (a^2-b^2\right )^{3/2} f^2\right ) \int (e+f x) \text{Li}_2\left (\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{a b^3 d^2}-\frac{\left (6 i f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac{\left (6 i f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}\\ &=\frac{3 e f^2 x}{4 b d^2}+\frac{3 f^3 x^2}{8 b d^2}-\frac{(e+f x)^4}{8 b f}+\frac{\left (a^2-b^2\right ) (e+f x)^4}{4 b^3 f}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac{6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac{6 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f^2 (e+f x) \cos (c+d x)}{d^3}+\frac{(e+f x)^3 \cos (c+d x)}{a d}-\frac{\left (\frac{1}{a}-\frac{a}{b^2}\right ) (e+f x)^3 \cos (c+d x)}{d}+\frac{3 f^3 \cos ^2(c+d x)}{8 b d^4}-\frac{3 f (e+f x)^2 \cos ^2(c+d x)}{4 b d^2}+\frac{i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^3 d}-\frac{i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{a b^3 d}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac{3 \left (a^2-b^2\right )^{3/2} 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^3 d^2}-\frac{3 \left (a^2-b^2\right )^{3/2} 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^3 d^2}-\frac{6 f^2 (e+f x) \text{Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\frac{6 i \left (a^2-b^2\right )^{3/2} 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^3 d^3}-\frac{6 i \left (a^2-b^2\right )^{3/2} 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^3 d^3}-\frac{6 i f^3 \text{Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac{6 i f^3 \text{Li}_4\left (e^{i (c+d x)}\right )}{a d^4}+\frac{6 f^3 \sin (c+d x)}{a d^4}-\frac{6 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f^3 \sin (c+d x)}{d^4}-\frac{3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac{3 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f (e+f x)^2 \sin (c+d x)}{d^2}+\frac{3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac{\left (6 i \left (a^2-b^2\right )^{3/2} f^3\right ) \int \text{Li}_3\left (\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{a b^3 d^3}+\frac{\left (6 i \left (a^2-b^2\right )^{3/2} f^3\right ) \int \text{Li}_3\left (\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{a b^3 d^3}\\ &=\frac{3 e f^2 x}{4 b d^2}+\frac{3 f^3 x^2}{8 b d^2}-\frac{(e+f x)^4}{8 b f}+\frac{\left (a^2-b^2\right ) (e+f x)^4}{4 b^3 f}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac{6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac{6 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f^2 (e+f x) \cos (c+d x)}{d^3}+\frac{(e+f x)^3 \cos (c+d x)}{a d}-\frac{\left (\frac{1}{a}-\frac{a}{b^2}\right ) (e+f x)^3 \cos (c+d x)}{d}+\frac{3 f^3 \cos ^2(c+d x)}{8 b d^4}-\frac{3 f (e+f x)^2 \cos ^2(c+d x)}{4 b d^2}+\frac{i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^3 d}-\frac{i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{a b^3 d}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac{3 \left (a^2-b^2\right )^{3/2} 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^3 d^2}-\frac{3 \left (a^2-b^2\right )^{3/2} 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^3 d^2}-\frac{6 f^2 (e+f x) \text{Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\frac{6 i \left (a^2-b^2\right )^{3/2} 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^3 d^3}-\frac{6 i \left (a^2-b^2\right )^{3/2} 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^3 d^3}-\frac{6 i f^3 \text{Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac{6 i f^3 \text{Li}_4\left (e^{i (c+d x)}\right )}{a d^4}+\frac{6 f^3 \sin (c+d x)}{a d^4}-\frac{6 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f^3 \sin (c+d x)}{d^4}-\frac{3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac{3 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f (e+f x)^2 \sin (c+d x)}{d^2}+\frac{3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac{\left (6 \left (a^2-b^2\right )^{3/2} 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^3 d^4}+\frac{\left (6 \left (a^2-b^2\right )^{3/2} 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^3 d^4}\\ &=\frac{3 e f^2 x}{4 b d^2}+\frac{3 f^3 x^2}{8 b d^2}-\frac{(e+f x)^4}{8 b f}+\frac{\left (a^2-b^2\right ) (e+f x)^4}{4 b^3 f}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac{6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac{6 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f^2 (e+f x) \cos (c+d x)}{d^3}+\frac{(e+f x)^3 \cos (c+d x)}{a d}-\frac{\left (\frac{1}{a}-\frac{a}{b^2}\right ) (e+f x)^3 \cos (c+d x)}{d}+\frac{3 f^3 \cos ^2(c+d x)}{8 b d^4}-\frac{3 f (e+f x)^2 \cos ^2(c+d x)}{4 b d^2}+\frac{i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^3 d}-\frac{i \left (a^2-b^2\right )^{3/2} (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{a b^3 d}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac{3 \left (a^2-b^2\right )^{3/2} 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^3 d^2}-\frac{3 \left (a^2-b^2\right )^{3/2} 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^3 d^2}-\frac{6 f^2 (e+f x) \text{Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\frac{6 i \left (a^2-b^2\right )^{3/2} 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^3 d^3}-\frac{6 i \left (a^2-b^2\right )^{3/2} 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^3 d^3}-\frac{6 i f^3 \text{Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac{6 i f^3 \text{Li}_4\left (e^{i (c+d x)}\right )}{a d^4}-\frac{6 \left (a^2-b^2\right )^{3/2} f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b^3 d^4}+\frac{6 \left (a^2-b^2\right )^{3/2} f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{a b^3 d^4}+\frac{6 f^3 \sin (c+d x)}{a d^4}-\frac{6 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f^3 \sin (c+d x)}{d^4}-\frac{3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac{3 \left (\frac{1}{a}-\frac{a}{b^2}\right ) f (e+f x)^2 \sin (c+d x)}{d^2}+\frac{3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac{(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}\\ \end{align*}

Mathematica [A]  time = 6.63455, size = 1181, normalized size = 1.04 \[ \frac{2 a \left (2 a^2-3 b^2\right ) f^3 x^4 d^4+8 a \left (2 a^2-3 b^2\right ) e f^2 x^3 d^4+12 a \left (2 a^2-3 b^2\right ) e^2 f x^2 d^4+8 a \left (2 a^2-3 b^2\right ) e^3 x d^4-32 b^3 (e+f x)^3 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x)) d^3+16 a^2 b (e+f x)^3 \cos (c+d x) d^3-4 a b^2 (e+f x)^3 \sin (2 (c+d x)) d^3-6 a b^2 f (e+f x)^2 \cos (2 (c+d x)) d^2+48 \left (a^2-b^2\right )^{3/2} f (e+f x)^2 \text{PolyLog}\left (2,-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right ) d^2-48 a^2 b f (e+f x)^2 \sin (c+d x) d^2-96 a^2 b f^2 (e+f x) \cos (c+d x) d+6 a b^2 f^2 (e+f x) \sin (2 (c+d x)) d+3 a b^2 f^3 \cos (2 (c+d x))+16 i \left (a^2-b^2\right )^{3/2} \left (2 i e^3 \tan ^{-1}\left (\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right ) d^3+f^3 x^3 \log \left (\frac{i e^{i (c+d x)} b}{\sqrt{a^2-b^2}-a}+1\right ) d^3+3 e f^2 x^2 \log \left (\frac{i e^{i (c+d x)} b}{\sqrt{a^2-b^2}-a}+1\right ) d^3+3 e^2 f x \log \left (\frac{i e^{i (c+d x)} b}{\sqrt{a^2-b^2}-a}+1\right ) d^3-f^3 x^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) d^3-3 e f^2 x^2 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) d^3-3 e^2 f x \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) d^3+3 i f (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) d^2+6 f^2 (e+f x) \text{PolyLog}\left (3,-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right ) d-6 e f^2 \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) d-6 f^3 x \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right ) d+6 i f^3 \text{PolyLog}\left (4,-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}-a}\right )-6 i f^3 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )\right )+48 i b^3 f \left (-2 \text{PolyLog}(4,-\cos (c+d x)-i \sin (c+d x)) f^2+2 i d (e+f x) \text{PolyLog}(3,-\cos (c+d x)-i \sin (c+d x)) f+d^2 (e+f x)^2 \text{PolyLog}(2,-\cos (c+d x)-i \sin (c+d x))\right )-48 i b^3 f \left (-2 \text{PolyLog}(4,\cos (c+d x)+i \sin (c+d x)) f^2+2 i d (e+f x) \text{PolyLog}(3,\cos (c+d x)+i \sin (c+d x)) f+d^2 (e+f x)^2 \text{PolyLog}(2,\cos (c+d x)+i \sin (c+d x))\right )+96 a^2 b f^3 \sin (c+d x)}{16 a b^3 d^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(8*a*(2*a^2 - 3*b^2)*d^4*e^3*x + 12*a*(2*a^2 - 3*b^2)*d^4*e^2*f*x^2 + 8*a*(2*a^2 - 3*b^2)*d^4*e*f^2*x^3 + 2*a*
(2*a^2 - 3*b^2)*d^4*f^3*x^4 - 32*b^3*d^3*(e + f*x)^3*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] - 96*a^2*b*d*f^2*(
e + f*x)*Cos[c + d*x] + 16*a^2*b*d^3*(e + f*x)^3*Cos[c + d*x] + 3*a*b^2*f^3*Cos[2*(c + d*x)] - 6*a*b^2*d^2*f*(
e + f*x)^2*Cos[2*(c + d*x)] + 48*(a^2 - b^2)^(3/2)*d^2*f*(e + f*x)^2*PolyLog[2, ((-I)*b*E^(I*(c + d*x)))/(-a +
Sqrt[a^2 - b^2])] + (16*I)*(a^2 - b^2)^(3/2)*((2*I)*d^3*e^3*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]]
+ 3*d^3*e^2*f*x*Log[1 + (I*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] + 3*d^3*e*f^2*x^2*Log[1 + (I*b*E^(I*(c
+ d*x)))/(-a + Sqrt[a^2 - b^2])] + d^3*f^3*x^3*Log[1 + (I*b*E^(I*(c + d*x)))/(-a + Sqrt[a^2 - b^2])] - 3*d^3*e
^2*f*x*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] - 3*d^3*e*f^2*x^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a
+ Sqrt[a^2 - b^2])] - d^3*f^3*x^3*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + (3*I)*d^2*f*(e + f*x
)^2*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + 6*d*f^2*(e + f*x)*PolyLog[3, ((-I)*b*E^(I*(c + d
*x)))/(-a + Sqrt[a^2 - b^2])] - 6*d*e*f^2*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] - 6*d*f^3*x*
PolyLog[3, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])] + (6*I)*f^3*PolyLog[4, ((-I)*b*E^(I*(c + d*x)))/(-a +
Sqrt[a^2 - b^2])] - (6*I)*f^3*PolyLog[4, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])]) + (48*I)*b^3*f*(d^2*(e
+ f*x)^2*PolyLog[2, -Cos[c + d*x] - I*Sin[c + d*x]] + (2*I)*d*f*(e + f*x)*PolyLog[3, -Cos[c + d*x] - I*Sin[c +
d*x]] - 2*f^2*PolyLog[4, -Cos[c + d*x] - I*Sin[c + d*x]]) - (48*I)*b^3*f*(d^2*(e + f*x)^2*PolyLog[2, Cos[c +
d*x] + I*Sin[c + d*x]] + (2*I)*d*f*(e + f*x)*PolyLog[3, Cos[c + d*x] + I*Sin[c + d*x]] - 2*f^2*PolyLog[4, Cos[
c + d*x] + I*Sin[c + d*x]]) + 96*a^2*b*f^3*Sin[c + d*x] - 48*a^2*b*d^2*f*(e + f*x)^2*Sin[c + d*x] + 6*a*b^2*d*
f^2*(e + f*x)*Sin[2*(c + d*x)] - 4*a*b^2*d^3*(e + f*x)^3*Sin[2*(c + d*x)])/(16*a*b^3*d^4)

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Maple [F]  time = 3.191, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\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)^3*cot(d*x+c)/(a+b*sin(d*x+c)),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 7.69813, size = 9441, normalized size = 8.3 \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)^3*cot(d*x+c)/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/8*((2*a^3 - 3*a*b^2)*d^4*f^3*x^4 + 4*(2*a^3 - 3*a*b^2)*d^4*e*f^2*x^3 + 24*I*b^3*f^3*polylog(4, cos(d*x + c)
+ I*sin(d*x + c)) - 24*I*b^3*f^3*polylog(4, cos(d*x + c) - I*sin(d*x + c)) + 24*I*b^3*f^3*polylog(4, -cos(d*x
+ c) + I*sin(d*x + c)) - 24*I*b^3*f^3*polylog(4, -cos(d*x + c) - I*sin(d*x + c)) + 24*I*(a^2*b - b^3)*f^3*sqrt
(-(a^2 - b^2)/b^2)*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) - 24*I*(a^2*b - b^3)*f^3*sqrt(-(a^2 - b^2)/b^2)*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) + 24*I*(a^2*b - b^3)*
f^3*sqrt(-(a^2 - b^2)/b^2)*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) - 24*I*(a^2*b - b^3)*f^3*sqrt(-(a^2 - b^2)/b^2)*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) + 3*(2*(2*a^3 - 3*a*b^2)*d^4*e^
2*f + a*b^2*d^2*f^3)*x^2 - 3*(2*a*b^2*d^2*f^3*x^2 + 4*a*b^2*d^2*e*f^2*x + 2*a*b^2*d^2*e^2*f - a*b^2*f^3)*cos(d
*x + c)^2 + 2*(-6*I*(a^2*b - b^3)*d^2*f^3*x^2 - 12*I*(a^2*b - b^3)*d^2*e*f^2*x - 6*I*(a^2*b - b^3)*d^2*e^2*f)*
sqrt(-(a^2 - b^2)/b^2)*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) + 2*(6*I*(a^2*b - b^3)*d^2*f^3*x^2 + 12*I*(a^2*b - b^3)*d^2*e*f^2*x +
6*I*(a^2*b - b^3)*d^2*e^2*f)*sqrt(-(a^2 - b^2)/b^2)*dilog(-1/2*(2*I*a*cos(d*x + c) + 2*a*sin(d*x + c) - 2*(b*c
os(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1) + 2*(6*I*(a^2*b - b^3)*d^2*f^3*x^2 + 12*I
*(a^2*b - b^3)*d^2*e*f^2*x + 6*I*(a^2*b - b^3)*d^2*e^2*f)*sqrt(-(a^2 - b^2)/b^2)*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) + 2*(-6*I*(
a^2*b - b^3)*d^2*f^3*x^2 - 12*I*(a^2*b - b^3)*d^2*e*f^2*x - 6*I*(a^2*b - b^3)*d^2*e^2*f)*sqrt(-(a^2 - b^2)/b^2
)*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) - 4*((a^2*b - b^3)*d^3*e^3 - 3*(a^2*b - b^3)*c*d^2*e^2*f + 3*(a^2*b - b^3)*c^2*d*e*f^2 - (
a^2*b - b^3)*c^3*f^3)*sqrt(-(a^2 - b^2)/b^2)*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) - 4*((a^2*b - b^3)*d^3*e^3 - 3*(a^2*b - b^3)*c*d^2*e^2*f + 3*(a^2*b - b^3)*c^2*d*e*f^2 - (a^2*b
- b^3)*c^3*f^3)*sqrt(-(a^2 - b^2)/b^2)*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) + 4*((a^2*b - b^3)*d^3*e^3 - 3*(a^2*b - b^3)*c*d^2*e^2*f + 3*(a^2*b - b^3)*c^2*d*e*f^2 - (a^2*b - b^
3)*c^3*f^3)*sqrt(-(a^2 - b^2)/b^2)*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) + 4*((a^2*b - b^3)*d^3*e^3 - 3*(a^2*b - b^3)*c*d^2*e^2*f + 3*(a^2*b - b^3)*c^2*d*e*f^2 - (a^2*b - b^3)*c
^3*f^3)*sqrt(-(a^2 - b^2)/b^2)*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
) - 4*((a^2*b - b^3)*d^3*f^3*x^3 + 3*(a^2*b - b^3)*d^3*e*f^2*x^2 + 3*(a^2*b - b^3)*d^3*e^2*f*x + 3*(a^2*b - b^
3)*c*d^2*e^2*f - 3*(a^2*b - b^3)*c^2*d*e*f^2 + (a^2*b - b^3)*c^3*f^3)*sqrt(-(a^2 - b^2)/b^2)*log(1/2*(2*I*a*co
s(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) + 4*((a
^2*b - b^3)*d^3*f^3*x^3 + 3*(a^2*b - b^3)*d^3*e*f^2*x^2 + 3*(a^2*b - b^3)*d^3*e^2*f*x + 3*(a^2*b - b^3)*c*d^2*
e^2*f - 3*(a^2*b - b^3)*c^2*d*e*f^2 + (a^2*b - b^3)*c^3*f^3)*sqrt(-(a^2 - b^2)/b^2)*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) - 4*((a^2*b - b^
3)*d^3*f^3*x^3 + 3*(a^2*b - b^3)*d^3*e*f^2*x^2 + 3*(a^2*b - b^3)*d^3*e^2*f*x + 3*(a^2*b - b^3)*c*d^2*e^2*f - 3
*(a^2*b - b^3)*c^2*d*e*f^2 + (a^2*b - b^3)*c^3*f^3)*sqrt(-(a^2 - b^2)/b^2)*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) + 4*((a^2*b - b^3)*d^3*f
^3*x^3 + 3*(a^2*b - b^3)*d^3*e*f^2*x^2 + 3*(a^2*b - b^3)*d^3*e^2*f*x + 3*(a^2*b - b^3)*c*d^2*e^2*f - 3*(a^2*b
- b^3)*c^2*d*e*f^2 + (a^2*b - b^3)*c^3*f^3)*sqrt(-(a^2 - b^2)/b^2)*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) + 24*((a^2*b - b^3)*d*f^3*x + (a
^2*b - b^3)*d*e*f^2)*sqrt(-(a^2 - b^2)/b^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) - 24*((a^2*b - b^3)*d*f^3*x + (a^2*b - b^3)*d*e*f^2)*sq
rt(-(a^2 - b^2)/b^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) - 24*((a^2*b - b^3)*d*f^3*x + (a^2*b - b^3)*d*e*f^2)*sqrt(-(a^2 - b^2)/b^2)*po
lylog(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)
+ 24*((a^2*b - b^3)*d*f^3*x + (a^2*b - b^3)*d*e*f^2)*sqrt(-(a^2 - b^2)/b^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) + 2*(2*(2*a^3 - 3*a*b^2)*d^4*e^3
+ 3*a*b^2*d^2*e*f^2)*x + 8*(a^2*b*d^3*f^3*x^3 + 3*a^2*b*d^3*e*f^2*x^2 + a^2*b*d^3*e^3 - 6*a^2*b*d*e*f^2 + 3*(
a^2*b*d^3*e^2*f - 2*a^2*b*d*f^3)*x)*cos(d*x + c) + (-12*I*b^3*d^2*f^3*x^2 - 24*I*b^3*d^2*e*f^2*x - 12*I*b^3*d^
2*e^2*f)*dilog(cos(d*x + c) + I*sin(d*x + c)) + (12*I*b^3*d^2*f^3*x^2 + 24*I*b^3*d^2*e*f^2*x + 12*I*b^3*d^2*e^
2*f)*dilog(cos(d*x + c) - I*sin(d*x + c)) + (-12*I*b^3*d^2*f^3*x^2 - 24*I*b^3*d^2*e*f^2*x - 12*I*b^3*d^2*e^2*f
)*dilog(-cos(d*x + c) + I*sin(d*x + c)) + (12*I*b^3*d^2*f^3*x^2 + 24*I*b^3*d^2*e*f^2*x + 12*I*b^3*d^2*e^2*f)*d
ilog(-cos(d*x + c) - I*sin(d*x + c)) - 4*(b^3*d^3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2 + 3*b^3*d^3*e^2*f*x + b^3*d^3*
e^3)*log(cos(d*x + c) + I*sin(d*x + c) + 1) - 4*(b^3*d^3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2 + 3*b^3*d^3*e^2*f*x + b
^3*d^3*e^3)*log(cos(d*x + c) - I*sin(d*x + c) + 1) + 4*(b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^3*c^2*d*e*f^2 -
b^3*c^3*f^3)*log(-1/2*cos(d*x + c) + 1/2*I*sin(d*x + c) + 1/2) + 4*(b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^3*c^
2*d*e*f^2 - b^3*c^3*f^3)*log(-1/2*cos(d*x + c) - 1/2*I*sin(d*x + c) + 1/2) + 4*(b^3*d^3*f^3*x^3 + 3*b^3*d^3*e*
f^2*x^2 + 3*b^3*d^3*e^2*f*x + 3*b^3*c*d^2*e^2*f - 3*b^3*c^2*d*e*f^2 + b^3*c^3*f^3)*log(-cos(d*x + c) + I*sin(d
*x + c) + 1) + 4*(b^3*d^3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2 + 3*b^3*d^3*e^2*f*x + 3*b^3*c*d^2*e^2*f - 3*b^3*c^2*d*
e*f^2 + b^3*c^3*f^3)*log(-cos(d*x + c) - I*sin(d*x + c) + 1) + 24*(b^3*d*f^3*x + b^3*d*e*f^2)*polylog(3, cos(d
*x + c) + I*sin(d*x + c)) + 24*(b^3*d*f^3*x + b^3*d*e*f^2)*polylog(3, cos(d*x + c) - I*sin(d*x + c)) - 24*(b^3
*d*f^3*x + b^3*d*e*f^2)*polylog(3, -cos(d*x + c) + I*sin(d*x + c)) - 24*(b^3*d*f^3*x + b^3*d*e*f^2)*polylog(3,
-cos(d*x + c) - I*sin(d*x + c)) - 2*(12*a^2*b*d^2*f^3*x^2 + 24*a^2*b*d^2*e*f^2*x + 12*a^2*b*d^2*e^2*f - 24*a^
2*b*f^3 + (2*a*b^2*d^3*f^3*x^3 + 6*a*b^2*d^3*e*f^2*x^2 + 2*a*b^2*d^3*e^3 - 3*a*b^2*d*e*f^2 + 3*(2*a*b^2*d^3*e^
2*f - a*b^2*d*f^3)*x)*cos(d*x + c))*sin(d*x + c))/(a*b^3*d^4)

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Sympy [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)**3*cot(d*x+c)/(a+b*sin(d*x+c)),x)

[Out]

Timed out

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

[Out]

Timed out