\(\int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [337]

Optimal result

Integrand size = 34, antiderivative size = 852 \[ \int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {i b (e+f x)^4}{4 a^2 f}+\frac {i \left (a^2-b^2\right ) (e+f x)^4}{4 a^2 b f}-\frac {6 f (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^2}-\frac {(e+f x)^3 \csc (c+d x)}{a 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^2 b 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^2 b d}-\frac {b (e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^3}+\frac {3 i \left (a^2-b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^2}+\frac {3 i \left (a^2-b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac {6 f^3 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^4}-\frac {6 \left (a^2-b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^3}-\frac {6 \left (a^2-b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d^3}-\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a^2 d^3}-\frac {6 i \left (a^2-b^2\right ) f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^4}-\frac {6 i \left (a^2-b^2\right ) f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d^4}-\frac {3 i b f^3 \operatorname {PolyLog}\left (4,e^{2 i (c+d x)}\right )}{4 a^2 d^4} \]

[Out]

3*I*(a^2-b^2)*f*(f*x+e)^2*polylog(2,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/b/d^2-3/4*I*b*f^3*polylog(4,ex
p(2*I*(d*x+c)))/a^2/d^4-6*f*(f*x+e)^2*arctanh(exp(I*(d*x+c)))/a/d^2-(f*x+e)^3*csc(d*x+c)/a/d-b*(f*x+e)^3*ln(1-
exp(2*I*(d*x+c)))/a^2/d-(a^2-b^2)*(f*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/b/d-(a^2-b^2)*(f*
x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/b/d-6*I*(a^2-b^2)*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a+
(a^2-b^2)^(1/2)))/a^2/b/d^4+1/4*I*b*(f*x+e)^4/a^2/f+3/2*I*b*f*(f*x+e)^2*polylog(2,exp(2*I*(d*x+c)))/a^2/d^2+3*
I*(a^2-b^2)*f*(f*x+e)^2*polylog(2,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/b/d^2+1/4*I*(a^2-b^2)*(f*x+e)^4/
a^2/b/f-6*f^3*polylog(3,-exp(I*(d*x+c)))/a/d^4+6*f^3*polylog(3,exp(I*(d*x+c)))/a/d^4-3/2*b*f^2*(f*x+e)*polylog
(3,exp(2*I*(d*x+c)))/a^2/d^3-6*(a^2-b^2)*f^2*(f*x+e)*polylog(3,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/b/d
^3-6*(a^2-b^2)*f^2*(f*x+e)*polylog(3,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/b/d^3-6*I*f^2*(f*x+e)*polylog
(2,exp(I*(d*x+c)))/a/d^3-6*I*(a^2-b^2)*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/b/d^4+6*I*f^2
*(f*x+e)*polylog(2,-exp(I*(d*x+c)))/a/d^3

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 852, normalized size of antiderivative = 1.00, number of steps used = 48, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.559, Rules used = {4639, 4493, 3377, 2718, 4495, 4268, 2611, 2320, 6724, 4489, 3392, 32, 2715, 8, 3798, 2221, 6744, 4621, 4615} \[ \int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {i b (e+f x)^4}{4 a^2 f}+\frac {i \left (a^2-b^2\right ) (e+f x)^4}{4 a^2 b f}-\frac {\csc (c+d x) (e+f x)^3}{a d}-\frac {\left (a^2-b^2\right ) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)^3}{a^2 b d}-\frac {\left (a^2-b^2\right ) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)^3}{a^2 b d}-\frac {b \log \left (1-e^{2 i (c+d x)}\right ) (e+f x)^3}{a^2 d}-\frac {6 f \text {arctanh}\left (e^{i (c+d x)}\right ) (e+f x)^2}{a d^2}+\frac {3 i \left (a^2-b^2\right ) f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)^2}{a^2 b d^2}+\frac {3 i \left (a^2-b^2\right ) f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)^2}{a^2 b d^2}+\frac {3 i b f \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right ) (e+f x)^2}{2 a^2 d^2}+\frac {6 i f^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) (e+f x)}{a d^3}-\frac {6 i f^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) (e+f x)}{a d^3}-\frac {6 \left (a^2-b^2\right ) f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) (e+f x)}{a^2 b d^3}-\frac {6 \left (a^2-b^2\right ) f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) (e+f x)}{a^2 b d^3}-\frac {3 b f^2 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right ) (e+f x)}{2 a^2 d^3}-\frac {6 f^3 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^4}-\frac {6 i \left (a^2-b^2\right ) f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^4}-\frac {6 i \left (a^2-b^2\right ) f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d^4}-\frac {3 i b f^3 \operatorname {PolyLog}\left (4,e^{2 i (c+d x)}\right )}{4 a^2 d^4} \]

[In]

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

[Out]

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

Rule 8

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

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2320

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 2611

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 2715

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] && Integ
erQ[2*n]

Rule 2718

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

Rule 3377

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 3392

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 3798

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 4268

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 4489

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 4493

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 4495

Int[Cot[(a_.) + (b_.)*(x_)]^(p_.)*Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[
(-(c + d*x)^m)*(Csc[a + b*x]^n/(b*n)), x] + Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /;
FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 4615

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]

Rule 4621

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 4639

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 6724

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 6744

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

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3039\) vs. \(2(852)=1704\).

Time = 11.37 (sec) , antiderivative size = 3039, normalized size of antiderivative = 3.57 \[ \int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (f x +e \right )^{3} \cos \left (d x +c \right ) \left (\cot ^{2}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   Too many variables

Sympy [F]

\[ \int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{3} \cos {\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

Giac [F]

\[ \int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cos \left (d x + c\right ) \cot \left (d x + c\right )^{2}}{b \sin \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \]

[In]

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

[Out]

\text{Hanged}