3.4.0 ∫ (e+fx)3 cos2(c+dx) cot(c+dx) a+b sin(c+dx) dx [329]

Integrand size = 34, antiderivative size = 763 \[ \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {i (e+f x)^4}{4 a f}-\frac {i \left (a^2-b^2\right ) (e+f x)^4}{4 a b^2 f}+\frac {6 f^3 \cos (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cos (c+d x)}{b d^2}+\frac {\left (a^2-b^2\right ) (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {\left (a^2-b^2\right ) (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {(e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac {3 i \left (a^2-b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^2}-\frac {3 i \left (a^2-b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^2}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a d^2}+\frac {6 \left (a^2-b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^3}+\frac {6 \left (a^2-b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^3}+\frac {6 i \left (a^2-b^2\right ) f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^4}+\frac {6 i \left (a^2-b^2\right ) f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^4}+\frac {3 i f^3 \operatorname {PolyLog}\left (4,e^{2 i (c+d x)}\right )}{4 a d^4}+\frac {6 f^2 (e+f x) \sin (c+d x)}{b d^3}-\frac {(e+f x)^3 \sin (c+d x)}{b d} \]

-1/4*I*(f*x+e)^4/a/f+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/b^2/d^4+6*f^3*cos(d

*x+c)/b/d^4-3*f*(f*x+e)^2*cos(d*x+c)/b/d^2+(f*x+e)^3*ln(1-exp(2*I*(d*x+c)))/a/d+(a^2-b^2)*(f*x+e)^3*ln(1-I*b*e

xp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a/b^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/

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

)/a/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/b^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/b^2/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/b^2/d^4+3/4*I*f^3*polylog(4,exp(2*I*(d*x+c)))/a/d^4-3*I*(a^2-b^2)*f*(f*x+e)^2*polylo

g(2,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a/b^2/d^2+6*f^2*(f*x+e)*sin(d*x+c)/b/d^3-(f*x+e)^3*sin(d*x+c)/b/d

Rubi [A] (veriﬁed)

Time = 1.34 (sec) , antiderivative size = 763, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4639, 4493, 4489, 3392, 32, 2715, 8, 3798, 2221, 2611, 6744, 2320, 6724, 4621, 3377, 2718, 4615} \[ \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {6 i f^3 \left (a^2-b^2\right ) \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^4}+\frac {6 i f^3 \left (a^2-b^2\right ) \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^4}+\frac {6 f^2 \left (a^2-b^2\right ) (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^3}+\frac {6 f^2 \left (a^2-b^2\right ) (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^3}-\frac {3 i f \left (a^2-b^2\right ) (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^2}-\frac {3 i f \left (a^2-b^2\right ) (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^2}+\frac {\left (a^2-b^2\right ) (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {\left (a^2-b^2\right ) (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a b^2 d}-\frac {i \left (a^2-b^2\right ) (e+f x)^4}{4 a b^2 f}+\frac {3 i f^3 \operatorname {PolyLog}\left (4,e^{2 i (c+d x)}\right )}{4 a d^4}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^3}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a d^2}+\frac {(e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac {i (e+f x)^4}{4 a f}+\frac {6 f^3 \cos (c+d x)}{b d^4}+\frac {6 f^2 (e+f x) \sin (c+d x)}{b d^3}-\frac {3 f (e+f x)^2 \cos (c+d x)}{b d^2}-\frac {(e+f x)^3 \sin (c+d x)}{b d} \]

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

((-1/4*I)*(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 +

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

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N

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]

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

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]

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

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

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]

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]

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]

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]

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]

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]

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]

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]

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]

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,

Rubi steps \begin{align*} \text {integral}& = \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 \operatorname {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 \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 b^2 d^2}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,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) \operatorname {PolyLog}\left (2,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) \operatorname {PolyLog}\left (2,\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) \operatorname {PolyLog}\left (2,\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 \operatorname {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 \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 b^2 d^2}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a d^2}+\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 b^2 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 b^2 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,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 \operatorname {PolyLog}\left (3,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 \operatorname {PolyLog}\left (3,\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 \operatorname {PolyLog}\left (3,\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 \operatorname {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 \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 b^2 d^2}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a d^2}+\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 b^2 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 b^2 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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 ) \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 )}{a b^2 d^4}+\frac {\left (6 i \left (a^2-b^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 )}{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 \operatorname {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 \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 b^2 d^2}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a d^2}+\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 b^2 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 b^2 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a 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 b^2 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 b^2 d^4}+\frac {3 i f^3 \operatorname {PolyLog}\left (4,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] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(4052\) vs. \(2(763)=1526\).

Time = 9.84 (sec) , antiderivative size = 4052, normalized size of antiderivative = 5.31 \[ \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Result too large to show} \]

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

-1/2*(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*d*(1 - E^((-2*I)*c))*x*PolyLog[2, -E^(

(-I)*(c + d*x))] - 6*d*(1 - E^((-2*I)*c))*x*PolyLog[2, E^((-I)*(c + d*x))] + (6*I)*(1 - E^((-2*I)*c))*PolyLog[

3, -E^((-I)*(c + d*x))] + (6*I)*(1 - E^((-2*I)*c))*PolyLog[3, E^((-I)*(c + d*x))]))/(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*Log[1 + E^((-I)*(c + d*x))] - 6*d^2*(1 - E^((-2*I)*c))*x^2*PolyLog[2, -E^((-I)*(c + d*x))] - 6*

d^2*(1 - E^((-2*I)*c))*x^2*PolyLog[2, E^((-I)*(c + d*x))] + (12*I)*d*(1 - E^((-2*I)*c))*x*PolyLog[3, -E^((-I)*

(c + d*x))] + (12*I)*d*(1 - E^((-2*I)*c))*x*PolyLog[3, E^((-I)*(c + d*x))] + 12*(1 - E^((-2*I)*c))*PolyLog[4,

-E^((-I)*(c + d*x))] + 12*(1 - E^((-2*I)*c))*PolyLog[4, E^((-I)*(c + d*x))]))/(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*Sqr

t[(-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*Pol

yLog[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*PolyL

og[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] - 12*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] - 2

4*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 + ArcTan[Tan[c]]))] + Pi*Log[Cos[d*x]] + 2*ArcTan[T

an[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]

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

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

Fricas [F(-2)]

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

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

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

Sympy [F]

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

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

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

Maxima [F(-2)]

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

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

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?`

Giac [F]

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

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

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

Mupad [F(-1)]

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

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

\(\int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx\) [329]

Optimal result