\(\int \frac {(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [209]

Optimal result

Integrand size = 28, antiderivative size = 600 \[ \int \frac {(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f^3 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{2 a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {3 i f^3 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{2 a d^2}+\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}-\frac {12 f^3 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {9 i f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4} \]

[Out]

2*I*(f*x+e)^3/a/d-6*f^2*(f*x+e)*arctanh(exp(I*(d*x+c)))/a/d^3-3*(f*x+e)^3*arctanh(exp(I*(d*x+c)))/a/d+(f*x+e)^
3*cot(1/2*c+1/4*Pi+1/2*d*x)/a/d+(f*x+e)^3*cot(d*x+c)/a/d-3/2*f*(f*x+e)^2*csc(d*x+c)/a/d^2-1/2*(f*x+e)^3*cot(d*
x+c)*csc(d*x+c)/a/d-6*f*(f*x+e)^2*ln(1-I*exp(I*(d*x+c)))/a/d^2-3*f*(f*x+e)^2*ln(1-exp(2*I*(d*x+c)))/a/d^2-9/2*
I*f*(f*x+e)^2*polylog(2,exp(I*(d*x+c)))/a/d^2-3*I*f^3*polylog(2,exp(I*(d*x+c)))/a/d^4+3*I*f^2*(f*x+e)*polylog(
2,exp(2*I*(d*x+c)))/a/d^3+12*I*f^2*(f*x+e)*polylog(2,I*exp(I*(d*x+c)))/a/d^3+9*I*f^3*polylog(4,exp(I*(d*x+c)))
/a/d^4+3*I*f^3*polylog(2,-exp(I*(d*x+c)))/a/d^4-9*f^2*(f*x+e)*polylog(3,-exp(I*(d*x+c)))/a/d^3-12*f^3*polylog(
3,I*exp(I*(d*x+c)))/a/d^4+9*f^2*(f*x+e)*polylog(3,exp(I*(d*x+c)))/a/d^3-3/2*f^3*polylog(3,exp(2*I*(d*x+c)))/a/
d^4-9*I*f^3*polylog(4,-exp(I*(d*x+c)))/a/d^4+9/2*I*f*(f*x+e)^2*polylog(2,-exp(I*(d*x+c)))/a/d^2

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.00, number of steps used = 40, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {4631, 4271, 4268, 2317, 2438, 2611, 6744, 2320, 6724, 4269, 3798, 2221, 3399} \[ \int \frac {(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {3 i f^3 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^4}-\frac {3 i f^3 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^4}-\frac {12 f^3 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {9 i f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{2 a d^2}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{2 a d^2}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {2 i (e+f x)^3}{a d} \]

[In]

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

[Out]

((2*I)*(e + f*x)^3)/(a*d) - (6*f^2*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/(a*d^3) - (3*(e + f*x)^3*ArcTanh[E^(I*(
c + d*x))])/(a*d) + ((e + f*x)^3*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) + ((e + f*x)^3*Cot[c + d*x])/(a*d) - (3*f*(e
+ f*x)^2*Csc[c + d*x])/(2*a*d^2) - ((e + f*x)^3*Cot[c + d*x]*Csc[c + d*x])/(2*a*d) - (6*f*(e + f*x)^2*Log[1 -
I*E^(I*(c + d*x))])/(a*d^2) - (3*f*(e + f*x)^2*Log[1 - E^((2*I)*(c + d*x))])/(a*d^2) + ((3*I)*f^3*PolyLog[2,
-E^(I*(c + d*x))])/(a*d^4) + (((9*I)/2)*f*(e + f*x)^2*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) + ((12*I)*f^2*(e +
f*x)*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3) - ((3*I)*f^3*PolyLog[2, E^(I*(c + d*x))])/(a*d^4) - (((9*I)/2)*f*
(e + f*x)^2*PolyLog[2, E^(I*(c + d*x))])/(a*d^2) + ((3*I)*f^2*(e + f*x)*PolyLog[2, E^((2*I)*(c + d*x))])/(a*d^
3) - (9*f^2*(e + f*x)*PolyLog[3, -E^(I*(c + d*x))])/(a*d^3) - (12*f^3*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^4) +
(9*f^2*(e + f*x)*PolyLog[3, E^(I*(c + d*x))])/(a*d^3) - (3*f^3*PolyLog[3, E^((2*I)*(c + d*x))])/(2*a*d^4) - (
(9*I)*f^3*PolyLog[4, -E^(I*(c + d*x))])/(a*d^4) + ((9*I)*f^3*PolyLog[4, E^(I*(c + d*x))])/(a*d^4)

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 2317

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

Rule 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 2438

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

Rule 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 3399

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

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 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4271

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-b^2)*(c + d*x)^m*Cot[e
+ f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))), Int[(c +
d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)^m*(b*Csc[e + f*x])^
(n - 2), x], x] - Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 4631

Int[(Csc[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbo
l] :> Dist[1/a, Int[(e + f*x)^m*Csc[c + d*x]^n, x], x] - Dist[b/a, Int[(e + f*x)^m*(Csc[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]

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 \csc ^3(c+d x) \, dx}{a}-\int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx \\ & = -\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\int (e+f x)^3 \csc (c+d x) \, dx}{2 a}-\frac {\int (e+f x)^3 \csc ^2(c+d x) \, dx}{a}+\frac {\left (3 f^2\right ) \int (e+f x) \csc (c+d x) \, dx}{a d^2}+\int \frac {(e+f x)^3 \csc (c+d x)}{a+a \sin (c+d x)} \, dx \\ & = -\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {(e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\int (e+f x)^3 \csc (c+d x) \, dx}{a}-\frac {(3 f) \int (e+f x)^2 \log \left (1-e^{i (c+d x)}\right ) \, dx}{2 a d}+\frac {(3 f) \int (e+f x)^2 \log \left (1+e^{i (c+d x)}\right ) \, dx}{2 a d}-\frac {(3 f) \int (e+f x)^2 \cot (c+d x) \, dx}{a d}-\frac {\left (3 f^3\right ) \int \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d^3}+\frac {\left (3 f^3\right ) \int \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d^3}-\int \frac {(e+f x)^3}{a+a \sin (c+d x)} \, dx \\ & = \frac {i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{2 a d^2}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{2 a d^2}-\frac {\int (e+f x)^3 \csc ^2\left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {d x}{2}\right ) \, dx}{2 a}+\frac {(6 i f) \int \frac {e^{2 i (c+d x)} (e+f x)^2}{1-e^{2 i (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 i f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (3 i f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (3 i f^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}-\frac {\left (3 i f^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4} \\ & = \frac {i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f^3 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{2 a d^2}-\frac {3 i f^3 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{2 a d^2}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}-\frac {(3 f) \int (e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx}{a d}-\frac {\left (6 i f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (6 i f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (6 f^2\right ) \int (e+f x) \log \left (1-e^{2 i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (3 f^3\right ) \int \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right ) \, dx}{a d^3}-\frac {\left (3 f^3\right ) \int \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right ) \, dx}{a d^3} \\ & = \frac {2 i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f^3 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{2 a d^2}-\frac {3 i f^3 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{2 a d^2}+\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}-\frac {(6 f) \int \frac {e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )} (e+f x)^2}{1-i e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}} \, dx}{a d}-\frac {\left (3 i f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac {\left (3 i f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}-\frac {\left (3 i f^3\right ) \int \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right ) \, dx}{a d^3}+\frac {\left (6 f^3\right ) \int \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right ) \, dx}{a d^3}-\frac {\left (6 f^3\right ) \int \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right ) \, dx}{a d^3} \\ & = \frac {2 i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f^3 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{2 a d^2}-\frac {3 i f^3 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{2 a d^2}+\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}-\frac {3 i f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}+\frac {3 i f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4}+\frac {\left (12 f^2\right ) \int (e+f x) \log \left (1-i e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^2}-\frac {\left (6 i f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac {\left (6 i f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}-\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{2 a d^4} \\ & = \frac {2 i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f^3 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{2 a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {3 i f^3 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{2 a d^2}+\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {9 i f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4}-\frac {\left (12 i f^3\right ) \int \operatorname {PolyLog}\left (2,i e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^3} \\ & = \frac {2 i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f^3 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{2 a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {3 i f^3 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{2 a d^2}+\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {9 i f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4}-\frac {\left (12 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{a d^4} \\ & = \frac {2 i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f^3 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{2 a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {3 i f^3 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{2 a d^2}+\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}-\frac {12 f^3 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {9 i f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}+\frac {9 i f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4} \\ \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. \(1493\) vs. \(2(600)=1200\).

Time = 20.46 (sec) , antiderivative size = 1493, normalized size of antiderivative = 2.49 \[ \int \frac {(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 e^3 \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )}{2 a d}+\frac {3 e f^2 \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d^3}+\frac {9 e^2 f \left ((c+d x) \left (\log \left (1-e^{i (c+d x)}\right )-\log \left (1+e^{i (c+d x)}\right )\right )-c \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )-\operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )\right )\right )}{2 a d^2}+\frac {3 f^3 \left ((c+d x) \left (\log \left (1-e^{i (c+d x)}\right )-\log \left (1+e^{i (c+d x)}\right )\right )-c \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )-\operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )\right )\right )}{a d^4}+\frac {e^{i c} f^3 \csc (c) \left (2 d^3 e^{-2 i c} x^3+3 i d^2 \left (1-e^{-2 i c}\right ) x^2 \log \left (1-e^{-i (c+d x)}\right )+3 i d^2 \left (1-e^{-2 i c}\right ) x^2 \log \left (1+e^{-i (c+d x)}\right )-6 d \left (1-e^{-2 i c}\right ) x \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )-6 d \left (1-e^{-2 i c}\right ) x \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )+6 i \left (1-e^{-2 i c}\right ) \operatorname {PolyLog}\left (3,-e^{-i (c+d x)}\right )+6 i \left (1-e^{-2 i c}\right ) \operatorname {PolyLog}\left (3,e^{-i (c+d x)}\right )\right )}{2 a d^4}-\frac {9 e f^2 \left (d^2 x^2 \text {arctanh}(\cos (c+d x)+i \sin (c+d x))-i d x \operatorname {PolyLog}(2,-\cos (c+d x)-i \sin (c+d x))+i d x \operatorname {PolyLog}(2,\cos (c+d x)+i \sin (c+d x))+\operatorname {PolyLog}(3,-\cos (c+d x)-i \sin (c+d x))-\operatorname {PolyLog}(3,\cos (c+d x)+i \sin (c+d x))\right )}{a d^3}+\frac {3 f^3 \left (-2 d^3 x^3 \text {arctanh}(\cos (c+d x)+i \sin (c+d x))+3 i d^2 x^2 \operatorname {PolyLog}(2,-\cos (c+d x)-i \sin (c+d x))-3 i d^2 x^2 \operatorname {PolyLog}(2,\cos (c+d x)+i \sin (c+d x))-6 d x \operatorname {PolyLog}(3,-\cos (c+d x)-i \sin (c+d x))+6 d x \operatorname {PolyLog}(3,\cos (c+d x)+i \sin (c+d x))-6 i \operatorname {PolyLog}(4,-\cos (c+d x)-i \sin (c+d x))+6 i \operatorname {PolyLog}(4,\cos (c+d x)+i \sin (c+d x))\right )}{2 a d^4}-\frac {3 e^2 f \csc (c) (-d x \cos (c)+\log (\cos (d x) \sin (c)+\cos (c) \sin (d x)) \sin (c))}{a d^2 \left (\cos ^2(c)+\sin ^2(c)\right )}+\frac {6 f (\cos (c)+i \sin (c)) \left (\frac {(e+f x)^3 (\cos (c)-i \sin (c))}{3 f}-\frac {(e+f x)^2 \log (1+i \cos (c+d x)+\sin (c+d x)) (1+i \cos (c)+\sin (c))}{d}+\frac {2 f (d (e+f x) \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x))-i f \operatorname {PolyLog}(3,-i \cos (c+d x)-\sin (c+d x))) (\cos (c)-i (1+\sin (c)))}{d^3}\right )}{a d (\cos (c)+i (1+\sin (c)))}+\frac {\csc (c) \csc ^2(c+d x) \left (e^3 \sin (d x)+3 e^2 f x \sin (d x)+3 e f^2 x^2 \sin (d x)+f^3 x^3 \sin (d x)\right )}{2 a d}+\frac {\csc (c) \csc (c+d x) \left (-d e^3 \cos (c)-3 d e^2 f x \cos (c)-3 d e f^2 x^2 \cos (c)-d f^3 x^3 \cos (c)-3 e^2 f \sin (c)-6 e f^2 x \sin (c)-3 f^3 x^2 \sin (c)-2 d e^3 \sin (d x)-6 d e^2 f x \sin (d x)-6 d e f^2 x^2 \sin (d x)-2 d f^3 x^3 \sin (d x)\right )}{2 a d^2}-\frac {2 \left (e^3 \sin \left (\frac {d x}{2}\right )+3 e^2 f x \sin \left (\frac {d x}{2}\right )+3 e f^2 x^2 \sin \left (\frac {d x}{2}\right )+f^3 x^3 \sin \left (\frac {d x}{2}\right )\right )}{a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {3 e f^2 \csc (c) \sec (c) \left (d^2 e^{i \arctan (\tan (c))} x^2+\frac {\left (i d x (-\pi +2 \arctan (\tan (c)))-\pi \log \left (1+e^{-2 i d x}\right )-2 (d x+\arctan (\tan (c))) \log \left (1-e^{2 i (d x+\arctan (\tan (c)))}\right )+\pi \log (\cos (d x))+2 \arctan (\tan (c)) \log (\sin (d x+\arctan (\tan (c))))+i \operatorname {PolyLog}\left (2,e^{2 i (d x+\arctan (\tan (c)))}\right )\right ) \tan (c)}{\sqrt {1+\tan ^2(c)}}\right )}{a d^3 \sqrt {\sec ^2(c) \left (\cos ^2(c)+\sin ^2(c)\right )}} \]

[In]

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

[Out]

(3*e^3*Log[Tan[(c + d*x)/2]])/(2*a*d) + (3*e*f^2*Log[Tan[(c + d*x)/2]])/(a*d^3) + (9*e^2*f*((c + d*x)*(Log[1 -
E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) - c*Log[Tan[(c + d*x)/2]] + I*(PolyLog[2, -E^(I*(c + d*x))] - Po
lyLog[2, E^(I*(c + d*x))])))/(2*a*d^2) + (3*f^3*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))
]) - c*Log[Tan[(c + d*x)/2]] + I*(PolyLog[2, -E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))])))/(a*d^4) + (E^(
I*c)*f^3*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))]))/(2*a*d^4) - (9*e*f^2*(d^2*x^2*ArcTanh
[Cos[c + d*x] + I*Sin[c + d*x]] - I*d*x*PolyLog[2, -Cos[c + d*x] - I*Sin[c + d*x]] + I*d*x*PolyLog[2, Cos[c +
d*x] + I*Sin[c + d*x]] + PolyLog[3, -Cos[c + d*x] - I*Sin[c + d*x]] - PolyLog[3, Cos[c + d*x] + I*Sin[c + d*x]
]))/(a*d^3) + (3*f^3*(-2*d^3*x^3*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] + (3*I)*d^2*x^2*PolyLog[2, -Cos[c + d*
x] - I*Sin[c + d*x]] - (3*I)*d^2*x^2*PolyLog[2, Cos[c + d*x] + I*Sin[c + d*x]] - 6*d*x*PolyLog[3, -Cos[c + d*x
] - I*Sin[c + d*x]] + 6*d*x*PolyLog[3, Cos[c + d*x] + I*Sin[c + d*x]] - (6*I)*PolyLog[4, -Cos[c + d*x] - I*Sin
[c + d*x]] + (6*I)*PolyLog[4, Cos[c + d*x] + I*Sin[c + d*x]]))/(2*a*d^4) - (3*e^2*f*Csc[c]*(-(d*x*Cos[c]) + Lo
g[Cos[d*x]*Sin[c] + Cos[c]*Sin[d*x]]*Sin[c]))/(a*d^2*(Cos[c]^2 + Sin[c]^2)) + (6*f*(Cos[c] + I*Sin[c])*(((e +
f*x)^3*(Cos[c] - I*Sin[c]))/(3*f) - ((e + f*x)^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]
))/d + (2*f*(d*(e + f*x)*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*f*PolyLog[3, (-I)*Cos[c + d*x] - Sin
[c + d*x]])*(Cos[c] - I*(1 + Sin[c])))/d^3))/(a*d*(Cos[c] + I*(1 + Sin[c]))) + (Csc[c]*Csc[c + d*x]^2*(e^3*Sin
[d*x] + 3*e^2*f*x*Sin[d*x] + 3*e*f^2*x^2*Sin[d*x] + f^3*x^3*Sin[d*x]))/(2*a*d) + (Csc[c]*Csc[c + d*x]*(-(d*e^3
*Cos[c]) - 3*d*e^2*f*x*Cos[c] - 3*d*e*f^2*x^2*Cos[c] - d*f^3*x^3*Cos[c] - 3*e^2*f*Sin[c] - 6*e*f^2*x*Sin[c] -
3*f^3*x^2*Sin[c] - 2*d*e^3*Sin[d*x] - 6*d*e^2*f*x*Sin[d*x] - 6*d*e*f^2*x^2*Sin[d*x] - 2*d*f^3*x^3*Sin[d*x]))/(
2*a*d^2) - (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]))/(
a*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])) + (3*e*f^2*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[Tan[c]]*Log[Sin[d*x + ArcTan[Tan[c]]]] + I*
PolyLog[2, E^((2*I)*(d*x + ArcTan[Tan[c]]))])*Tan[c])/Sqrt[1 + Tan[c]^2]))/(a*d^3*Sqrt[Sec[c]^2*(Cos[c]^2 + Si
n[c]^2)])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2325 vs. \(2 (540 ) = 1080\).

Time = 0.72 (sec) , antiderivative size = 2326, normalized size of antiderivative = 3.88

method	result	size
risch	\(\text {Expression too large to display}\)	\(2326\)

[In]

int((f*x+e)^3*csc(d*x+c)^3/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

9/2/a/d*f^2*e*ln(1-exp(I*(d*x+c)))*x^2-6/a/d^3*f^2*e*ln(1-exp(I*(d*x+c)))*c-12/a/d^3*f^2*e*ln(1-I*exp(I*(d*x+c
)))*c-6/a/d^2*f^2*e*ln(1-exp(I*(d*x+c)))*x-12/a/d^2*f^2*e*ln(1-I*exp(I*(d*x+c)))*x-6/a/d^2*f^2*e*ln(exp(I*(d*x
+c))+1)*x-9/2/a/d^2*c*e^2*f*ln(exp(I*(d*x+c))-1)-24/a/d^3*e*f^2*c*ln(exp(I*(d*x+c)))+9/2/a/d^2*e^2*f*ln(1-exp(
I*(d*x+c)))*c-9/2/a/d*e^2*f*ln(exp(I*(d*x+c))+1)*x+9/2/a/d*e^2*f*ln(1-exp(I*(d*x+c)))*x-9/2/a/d^3*c^2*f^2*e*ln
(1-exp(I*(d*x+c)))+9/2/a/d^3*c^2*f^2*e*ln(exp(I*(d*x+c))-1)+6/a/d^3*c*f^2*e*ln(exp(I*(d*x+c))-1)+6/a/d^3*c*f^2
*e*ln(1+exp(2*I*(d*x+c)))-9/2/a/d*f^2*e*ln(exp(I*(d*x+c))+1)*x^2+9*I/a/d^2*e*f^2*polylog(2,-exp(I*(d*x+c)))*x-
9*I/a/d^2*e*f^2*polylog(2,exp(I*(d*x+c)))*x-12*I/a/d^3*e*f^2*c*arctan(exp(I*(d*x+c)))+24*I/a/d^2*e*f^2*c*x+3/2
/a/d*e^3*ln(exp(I*(d*x+c))-1)-3/2/a/d*e^3*ln(exp(I*(d*x+c))+1)+(3*I*exp(2*I*(d*x+c))*f*e^2-3*I*f^3*x^2*exp(4*I
*(d*x+c))+3*I*d*e^3*exp(3*I*(d*x+c))-5*d*f^3*x^3*exp(2*I*(d*x+c))+3*d*f^3*x^3*exp(4*I*(d*x+c))+6*e*f^2*x*exp(3
*I*(d*x+c))-I*d*f^3*x^3*exp(I*(d*x+c))+9*d*e*f^2*x^2*exp(4*I*(d*x+c))+9*d*e^2*f*x*exp(4*I*(d*x+c))-15*d*e*f^2*
x^2*exp(2*I*(d*x+c))-6*e*f^2*x*exp(I*(d*x+c))-I*d*e^3*exp(I*(d*x+c))+9*I*d*e*f^2*x^2*exp(3*I*(d*x+c))+9*I*d*e^
2*f*x*exp(3*I*(d*x+c))+3*I*f^3*x^2*exp(2*I*(d*x+c))-3*I*exp(4*I*(d*x+c))*f*e^2+12*d*e*f^2*x^2+12*d*e^2*f*x-3*f
^3*x^2*exp(I*(d*x+c))-3*e^2*f*exp(I*(d*x+c))+3*f^3*x^2*exp(3*I*(d*x+c))+3*d*e^3*exp(4*I*(d*x+c))+3*exp(3*I*(d*
x+c))*f*e^2+3*I*d*f^3*x^3*exp(3*I*(d*x+c))-5*d*e^3*exp(2*I*(d*x+c))-6*I*e*f^2*x*exp(4*I*(d*x+c))+6*I*e*f^2*x*e
xp(2*I*(d*x+c))-3*I*d*e*f^2*x^2*exp(I*(d*x+c))-3*I*d*e^2*f*x*exp(I*(d*x+c))-15*d*e^2*f*x*exp(2*I*(d*x+c))+4*d*
f^3*x^3+4*d*e^3)/(exp(2*I*(d*x+c))-1)^2/d^2/(exp(I*(d*x+c))+I)/a-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/a/d^3*f^3*ln(exp(I*(d*x+c))+1)*x+3/a/d^4*f^3*ln(1-exp(I*(d*x+c)))*c+3/a/d^3*
f^3*ln(1-exp(I*(d*x+c)))*x-3/a/d^4*c*f^3*ln(exp(I*(d*x+c))-1)-9*I*f^3*polylog(4,-exp(I*(d*x+c)))/a/d^4-3*I*f^3
*polylog(2,exp(I*(d*x+c)))/a/d^4+12/a/d^4*f^3*c^2*ln(exp(I*(d*x+c)))-3/a/d^2*e^2*f*ln(exp(I*(d*x+c))-1)-3/a/d^
2*e^2*f*ln(1+exp(2*I*(d*x+c)))-3/a/d^2*e^2*f*ln(exp(I*(d*x+c))+1)+3/a/d^3*e*f^2*ln(exp(I*(d*x+c))-1)-3/a/d^3*e
*f^2*ln(exp(I*(d*x+c))+1)-8*I/a/d^4*c^3*f^3+4*I/a/d*f^3*x^3+3*I*f^3*polylog(2,-exp(I*(d*x+c)))/a/d^4+9*I*f^3*p
olylog(4,exp(I*(d*x+c)))/a/d^4-12*f^3*polylog(3,I*exp(I*(d*x+c)))/a/d^4+6*I/a/d^4*c^2*f^3*arctan(exp(I*(d*x+c)
))+12*I/a/d^3*e*f^2*polylog(2,I*exp(I*(d*x+c)))+6*I/a/d^3*e*f^2*polylog(2,-exp(I*(d*x+c)))+6*I/a/d^3*e*f^2*pol
ylog(2,exp(I*(d*x+c)))+9/2*I/a/d^2*f^3*polylog(2,-exp(I*(d*x+c)))*x^2+12*I/a/d*e*f^2*x^2+6*I/a/d^3*f^3*polylog
(2,-exp(I*(d*x+c)))*x+12*I/a/d^3*f^3*polylog(2,I*exp(I*(d*x+c)))*x+6*I/a/d^3*f^3*polylog(2,exp(I*(d*x+c)))*x+6
*I/a/d^2*e^2*f*arctan(exp(I*(d*x+c)))-12*I/a/d^3*c^2*f^3*x-9/2*I/a/d^2*e^2*f*polylog(2,exp(I*(d*x+c)))+9/2*I/a
/d^2*e^2*f*polylog(2,-exp(I*(d*x+c)))+12*I/a/d^3*e*f^2*c^2-9/2*I/a/d^2*f^3*polylog(2,exp(I*(d*x+c)))*x^2-3/a/d
^2*f^3*ln(exp(I*(d*x+c))+1)*x^2-3/a/d^2*f^3*ln(1-exp(I*(d*x+c)))*x^2+3/a/d^4*c^2*f^3*ln(1-exp(I*(d*x+c)))+6/a/
d^4*c^2*f^3*ln(1-I*exp(I*(d*x+c)))-3/a/d^4*c^2*f^3*ln(exp(I*(d*x+c))-1)-3/a/d^4*c^2*f^3*ln(1+exp(2*I*(d*x+c)))
-6/a/d^2*f^3*ln(1-I*exp(I*(d*x+c)))*x^2+12/a/d^2*e^2*f*ln(exp(I*(d*x+c)))+9/a/d^3*f^2*e*polylog(3,exp(I*(d*x+c
)))-9/a/d^3*f^2*e*polylog(3,-exp(I*(d*x+c)))-3/2/a/d*f^3*ln(exp(I*(d*x+c))+1)*x^3-9/a/d^3*f^3*polylog(3,-exp(I
*(d*x+c)))*x+3/2/a/d*f^3*ln(1-exp(I*(d*x+c)))*x^3+9/a/d^3*f^3*polylog(3,exp(I*(d*x+c)))*x-3/2/a/d^4*c^3*f^3*ln
(exp(I*(d*x+c))-1)+3/2/a/d^4*c^3*f^3*ln(1-exp(I*(d*x+c)))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 7842 vs. \(2 (522) = 1044\).

Time = 0.55 (sec) , antiderivative size = 7842, normalized size of antiderivative = 13.07 \[ \int \frac {(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

\[ \int \frac {(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {e^{3} \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{3} x^{3} \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e f^{2} x^{2} \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e^{2} f x \csc ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]

[In]

integrate((f*x+e)**3*csc(d*x+c)**3/(a+a*sin(d*x+c)),x)

[Out]

(Integral(e**3*csc(c + d*x)**3/(sin(c + d*x) + 1), x) + Integral(f**3*x**3*csc(c + d*x)**3/(sin(c + d*x) + 1),
x) + Integral(3*e*f**2*x**2*csc(c + d*x)**3/(sin(c + d*x) + 1), x) + Integral(3*e**2*f*x*csc(c + d*x)**3/(sin
(c + d*x) + 1), x))/a

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 12815 vs. \(2 (522) = 1044\).

Time = 10.83 (sec) , antiderivative size = 12815, normalized size of antiderivative = 21.36 \[ \int \frac {(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/8*(3*c*e^2*f*((3*sin(d*x + c)/(cos(d*x + c) + 1) + 20*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1)/(a*d*sin(d*x
+ c)^2/(cos(d*x + c) + 1)^2 + a*d*sin(d*x + c)^3/(cos(d*x + c) + 1)^3) - (4*sin(d*x + c)/(cos(d*x + c) + 1) -
sin(d*x + c)^2/(cos(d*x + c) + 1)^2)/(a*d) + 12*log(sin(d*x + c)/(cos(d*x + c) + 1))/(a*d)) + e^3*((4*sin(d*x
+ c)/(cos(d*x + c) + 1) - sin(d*x + c)^2/(cos(d*x + c) + 1)^2)/a - (3*sin(d*x + c)/(cos(d*x + c) + 1) + 20*si
n(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1)/(a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a*sin(d*x + c)^3/(cos(d*x + c)
+ 1)^3) - 12*log(sin(d*x + c)/(cos(d*x + c) + 1))/a) + 8*(48*I*c^2*d*e*f^2 - 16*I*c^3*f^3 - 24*(-I*d^2*e^2*f
+ 2*I*c*d*e*f^2 - I*c^2*f^3 - (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(5*d*x + 5*c) + (-I*d^2*e^2*f + 2*I*c*d*e
*f^2 - I*c^2*f^3)*cos(4*d*x + 4*c) + 2*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(3*d*x + 3*c) + 2*(I*d^2*e^2*f -
2*I*c*d*e*f^2 + I*c^2*f^3)*cos(2*d*x + 2*c) - (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x + c) + (-I*d^2*e^2*
f + 2*I*c*d*e*f^2 - I*c^2*f^3)*sin(5*d*x + 5*c) + (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*sin(4*d*x + 4*c) + 2*(I*
d^2*e^2*f - 2*I*c*d*e*f^2 + I*c^2*f^3)*sin(3*d*x + 3*c) - 2*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*sin(2*d*x + 2*
c) + (-I*d^2*e^2*f + 2*I*c*d*e*f^2 - I*c^2*f^3)*sin(d*x + c))*arctan2(sin(d*x + c) + 1, cos(d*x + c)) - 24*(I*
(d*x + c)^2*f^3 + 2*(I*d*e*f^2 - I*c*f^3)*(d*x + c) + ((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*cos(5*
d*x + 5*c) + (I*(d*x + c)^2*f^3 + 2*(I*d*e*f^2 - I*c*f^3)*(d*x + c))*cos(4*d*x + 4*c) - 2*((d*x + c)^2*f^3 + 2
*(d*e*f^2 - c*f^3)*(d*x + c))*cos(3*d*x + 3*c) + 2*(-I*(d*x + c)^2*f^3 + 2*(-I*d*e*f^2 + I*c*f^3)*(d*x + c))*c
os(2*d*x + 2*c) + ((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*cos(d*x + c) + (I*(d*x + c)^2*f^3 + 2*(I*d
*e*f^2 - I*c*f^3)*(d*x + c))*sin(5*d*x + 5*c) - ((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*sin(4*d*x +
4*c) + 2*(-I*(d*x + c)^2*f^3 + 2*(-I*d*e*f^2 + I*c*f^3)*(d*x + c))*sin(3*d*x + 3*c) + 2*((d*x + c)^2*f^3 + 2*(
d*e*f^2 - c*f^3)*(d*x + c))*sin(2*d*x + 2*c) + (I*(d*x + c)^2*f^3 + 2*(I*d*e*f^2 - I*c*f^3)*(d*x + c))*sin(d*x
+ c))*arctan2(cos(d*x + c), sin(d*x + c) + 1) - 6*(-I*(d*x + c)^3*f^3 - 2*I*d^2*e^2*f + (-3*I*c^2 + 4*I*c - 2
*I)*d*e*f^2 + (I*c^3 - 2*I*c^2 + 2*I*c)*f^3 + (-3*I*d*e*f^2 + (3*I*c - 2*I)*f^3)*(d*x + c)^2 + (-3*I*d^2*e^2*f
+ 2*(3*I*c - 2*I)*d*e*f^2 + (-3*I*c^2 + 4*I*c - 2*I)*f^3)*(d*x + c) - ((d*x + c)^3*f^3 + 2*d^2*e^2*f + (3*c^2
- 4*c + 2)*d*e*f^2 - (c^3 - 2*c^2 + 2*c)*f^3 + (3*d*e*f^2 - (3*c - 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*
c - 2)*d*e*f^2 + (3*c^2 - 4*c + 2)*f^3)*(d*x + c))*cos(5*d*x + 5*c) + (-I*(d*x + c)^3*f^3 - 2*I*d^2*e^2*f + (-
3*I*c^2 + 4*I*c - 2*I)*d*e*f^2 + (I*c^3 - 2*I*c^2 + 2*I*c)*f^3 + (-3*I*d*e*f^2 + (3*I*c - 2*I)*f^3)*(d*x + c)^
2 + (-3*I*d^2*e^2*f + 2*(3*I*c - 2*I)*d*e*f^2 + (-3*I*c^2 + 4*I*c - 2*I)*f^3)*(d*x + c))*cos(4*d*x + 4*c) + 2*
((d*x + c)^3*f^3 + 2*d^2*e^2*f + (3*c^2 - 4*c + 2)*d*e*f^2 - (c^3 - 2*c^2 + 2*c)*f^3 + (3*d*e*f^2 - (3*c - 2)*
f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c - 2)*d*e*f^2 + (3*c^2 - 4*c + 2)*f^3)*(d*x + c))*cos(3*d*x + 3*c) + 2
*(I*(d*x + c)^3*f^3 + 2*I*d^2*e^2*f + (3*I*c^2 - 4*I*c + 2*I)*d*e*f^2 + (-I*c^3 + 2*I*c^2 - 2*I*c)*f^3 + (3*I*
d*e*f^2 + (-3*I*c + 2*I)*f^3)*(d*x + c)^2 + (3*I*d^2*e^2*f + 2*(-3*I*c + 2*I)*d*e*f^2 + (3*I*c^2 - 4*I*c + 2*I
)*f^3)*(d*x + c))*cos(2*d*x + 2*c) - ((d*x + c)^3*f^3 + 2*d^2*e^2*f + (3*c^2 - 4*c + 2)*d*e*f^2 - (c^3 - 2*c^2
+ 2*c)*f^3 + (3*d*e*f^2 - (3*c - 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c - 2)*d*e*f^2 + (3*c^2 - 4*c + 2)
*f^3)*(d*x + c))*cos(d*x + c) + (-I*(d*x + c)^3*f^3 - 2*I*d^2*e^2*f + (-3*I*c^2 + 4*I*c - 2*I)*d*e*f^2 + (I*c^
3 - 2*I*c^2 + 2*I*c)*f^3 + (-3*I*d*e*f^2 + (3*I*c - 2*I)*f^3)*(d*x + c)^2 + (-3*I*d^2*e^2*f + 2*(3*I*c - 2*I)*
d*e*f^2 + (-3*I*c^2 + 4*I*c - 2*I)*f^3)*(d*x + c))*sin(5*d*x + 5*c) + ((d*x + c)^3*f^3 + 2*d^2*e^2*f + (3*c^2
- 4*c + 2)*d*e*f^2 - (c^3 - 2*c^2 + 2*c)*f^3 + (3*d*e*f^2 - (3*c - 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c
- 2)*d*e*f^2 + (3*c^2 - 4*c + 2)*f^3)*(d*x + c))*sin(4*d*x + 4*c) + 2*(I*(d*x + c)^3*f^3 + 2*I*d^2*e^2*f + (3
*I*c^2 - 4*I*c + 2*I)*d*e*f^2 + (-I*c^3 + 2*I*c^2 - 2*I*c)*f^3 + (3*I*d*e*f^2 + (-3*I*c + 2*I)*f^3)*(d*x + c)^
2 + (3*I*d^2*e^2*f + 2*(-3*I*c + 2*I)*d*e*f^2 + (3*I*c^2 - 4*I*c + 2*I)*f^3)*(d*x + c))*sin(3*d*x + 3*c) - 2*(
(d*x + c)^3*f^3 + 2*d^2*e^2*f + (3*c^2 - 4*c + 2)*d*e*f^2 - (c^3 - 2*c^2 + 2*c)*f^3 + (3*d*e*f^2 - (3*c - 2)*f
^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c - 2)*d*e*f^2 + (3*c^2 - 4*c + 2)*f^3)*(d*x + c))*sin(2*d*x + 2*c) + (-
I*(d*x + c)^3*f^3 - 2*I*d^2*e^2*f + (-3*I*c^2 + 4*I*c - 2*I)*d*e*f^2 + (I*c^3 - 2*I*c^2 + 2*I*c)*f^3 + (-3*I*d
*e*f^2 + (3*I*c - 2*I)*f^3)*(d*x + c)^2 + (-3*I*d^2*e^2*f + 2*(3*I*c - 2*I)*d*e*f^2 + (-3*I*c^2 + 4*I*c - 2*I)
*f^3)*(d*x + c))*sin(d*x + c))*arctan2(sin(d*x + c), cos(d*x + c) + 1) - 6*(-2*I*d^2*e^2*f + (3*I*c^2 + 4*I*c
+ 2*I)*d*e*f^2 + (-I*c^3 - 2*I*c^2 - 2*I*c)*f^3 - (2*d^2*e^2*f - (3*c^2 + 4*c + 2)*d*e*f^2 + (c^3 + 2*c^2 + 2*
c)*f^3)*cos(5*d*x + 5*c) + (-2*I*d^2*e^2*f + (3*I*c^2 + 4*I*c + 2*I)*d*e*f^2 + (-I*c^3 - 2*I*c^2 - 2*I*c)*f^3)
*cos(4*d*x + 4*c) + 2*(2*d^2*e^2*f - (3*c^2 + 4*c + 2)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3)*cos(3*d*x + 3*c) + 2
*(2*I*d^2*e^2*f + (-3*I*c^2 - 4*I*c - 2*I)*d*e*f^2 + (I*c^3 + 2*I*c^2 + 2*I*c)*f^3)*cos(2*d*x + 2*c) - (2*d^2*
e^2*f - (3*c^2 + 4*c + 2)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3)*cos(d*x + c) + (-2*I*d^2*e^2*f + (3*I*c^2 + 4*I*c
+ 2*I)*d*e*f^2 + (-I*c^3 - 2*I*c^2 - 2*I*c)*f^3)*sin(5*d*x + 5*c) + (2*d^2*e^2*f - (3*c^2 + 4*c + 2)*d*e*f^2
+ (c^3 + 2*c^2 + 2*c)*f^3)*sin(4*d*x + 4*c) + 2*(2*I*d^2*e^2*f + (-3*I*c^2 - 4*I*c - 2*I)*d*e*f^2 + (I*c^3 + 2
*I*c^2 + 2*I*c)*f^3)*sin(3*d*x + 3*c) - 2*(2*d^2*e^2*f - (3*c^2 + 4*c + 2)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3)*
sin(2*d*x + 2*c) + (-2*I*d^2*e^2*f + (3*I*c^2 + 4*I*c + 2*I)*d*e*f^2 + (-I*c^3 - 2*I*c^2 - 2*I*c)*f^3)*sin(d*x
+ c))*arctan2(sin(d*x + c), cos(d*x + c) - 1) - 6*(-I*(d*x + c)^3*f^3 + (-3*I*d*e*f^2 + (3*I*c + 2*I)*f^3)*(d
*x + c)^2 + (-3*I*d^2*e^2*f + 2*(3*I*c + 2*I)*d*e*f^2 + (-3*I*c^2 - 4*I*c - 2*I)*f^3)*(d*x + c) - ((d*x + c)^3
*f^3 + (3*d*e*f^2 - (3*c + 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c + 2)*d*e*f^2 + (3*c^2 + 4*c + 2)*f^3)*(
d*x + c))*cos(5*d*x + 5*c) + (-I*(d*x + c)^3*f^3 + (-3*I*d*e*f^2 + (3*I*c + 2*I)*f^3)*(d*x + c)^2 + (-3*I*d^2*
e^2*f + 2*(3*I*c + 2*I)*d*e*f^2 + (-3*I*c^2 - 4*I*c - 2*I)*f^3)*(d*x + c))*cos(4*d*x + 4*c) + 2*((d*x + c)^3*f
^3 + (3*d*e*f^2 - (3*c + 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c + 2)*d*e*f^2 + (3*c^2 + 4*c + 2)*f^3)*(d*
x + c))*cos(3*d*x + 3*c) + 2*(I*(d*x + c)^3*f^3 + (3*I*d*e*f^2 + (-3*I*c - 2*I)*f^3)*(d*x + c)^2 + (3*I*d^2*e^
2*f + 2*(-3*I*c - 2*I)*d*e*f^2 + (3*I*c^2 + 4*I*c + 2*I)*f^3)*(d*x + c))*cos(2*d*x + 2*c) - ((d*x + c)^3*f^3 +
(3*d*e*f^2 - (3*c + 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c + 2)*d*e*f^2 + (3*c^2 + 4*c + 2)*f^3)*(d*x +
c))*cos(d*x + c) + (-I*(d*x + c)^3*f^3 + (-3*I*d*e*f^2 + (3*I*c + 2*I)*f^3)*(d*x + c)^2 + (-3*I*d^2*e^2*f + 2*
(3*I*c + 2*I)*d*e*f^2 + (-3*I*c^2 - 4*I*c - 2*I)*f^3)*(d*x + c))*sin(5*d*x + 5*c) + ((d*x + c)^3*f^3 + (3*d*e*
f^2 - (3*c + 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c + 2)*d*e*f^2 + (3*c^2 + 4*c + 2)*f^3)*(d*x + c))*sin(
4*d*x + 4*c) + 2*(I*(d*x + c)^3*f^3 + (3*I*d*e*f^2 + (-3*I*c - 2*I)*f^3)*(d*x + c)^2 + (3*I*d^2*e^2*f + 2*(-3*
I*c - 2*I)*d*e*f^2 + (3*I*c^2 + 4*I*c + 2*I)*f^3)*(d*x + c))*sin(3*d*x + 3*c) - 2*((d*x + c)^3*f^3 + (3*d*e*f^
2 - (3*c + 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c + 2)*d*e*f^2 + (3*c^2 + 4*c + 2)*f^3)*(d*x + c))*sin(2*
d*x + 2*c) + (-I*(d*x + c)^3*f^3 + (-3*I*d*e*f^2 + (3*I*c + 2*I)*f^3)*(d*x + c)^2 + (-3*I*d^2*e^2*f + 2*(3*I*c
+ 2*I)*d*e*f^2 + (-3*I*c^2 - 4*I*c - 2*I)*f^3)*(d*x + c))*sin(d*x + c))*arctan2(sin(d*x + c), -cos(d*x + c) +
1) - 16*((d*x + c)^3*f^3 + 3*(d*e*f^2 - c*f^3)*(d*x + c)^2 + 3*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*(d*x + c))
*cos(5*d*x + 5*c) - 4*(I*(d*x + c)^3*f^3 - 3*d^2*e^2*f + 3*(-3*I*c^2 + 2*c)*d*e*f^2 + 3*(I*c^3 - c^2)*f^3 + 3*
(I*d*e*f^2 + (-I*c - 1)*f^3)*(d*x + c)^2 + 3*(I*d^2*e^2*f + 2*(-I*c - 1)*d*e*f^2 + (I*c^2 + 2*c)*f^3)*(d*x + c
))*cos(4*d*x + 4*c) + 4*(5*(d*x + c)^3*f^3 + 3*I*d^2*e^2*f - 3*(3*c^2 + 2*I*c)*d*e*f^2 + 3*(c^3 + I*c^2)*f^3 +
3*(5*d*e*f^2 - (5*c - I)*f^3)*(d*x + c)^2 + 3*(5*d^2*e^2*f - 2*(5*c - I)*d*e*f^2 + (5*c^2 - 2*I*c)*f^3)*(d*x
+ c))*cos(3*d*x + 3*c) - 4*(-3*I*(d*x + c)^3*f^3 + 3*d^2*e^2*f + 3*(5*I*c^2 - 2*c)*d*e*f^2 + (-5*I*c^3 + 3*c^2
)*f^3 + 3*(-3*I*d*e*f^2 + (3*I*c + 1)*f^3)*(d*x + c)^2 + 3*(-3*I*d^2*e^2*f + 2*(3*I*c + 1)*d*e*f^2 + (-3*I*c^2
- 2*c)*f^3)*(d*x + c))*cos(2*d*x + 2*c) - 4*(3*(d*x + c)^3*f^3 + 3*I*d^2*e^2*f - 3*(c^2 + 2*I*c)*d*e*f^2 + (c
^3 + 3*I*c^2)*f^3 + 3*(3*d*e*f^2 - (3*c - I)*f^3)*(d*x + c)^2 + 3*(3*d^2*e^2*f - 2*(3*c - I)*d*e*f^2 + (3*c^2
- 2*I*c)*f^3)*(d*x + c))*cos(d*x + c) - 48*(I*d*e*f^2 + I*(d*x + c)*f^3 - I*c*f^3 + (d*e*f^2 + (d*x + c)*f^3 -
c*f^3)*cos(5*d*x + 5*c) + (I*d*e*f^2 + I*(d*x + c)*f^3 - I*c*f^3)*cos(4*d*x + 4*c) - 2*(d*e*f^2 + (d*x + c)*f
^3 - c*f^3)*cos(3*d*x + 3*c) + 2*(-I*d*e*f^2 - I*(d*x + c)*f^3 + I*c*f^3)*cos(2*d*x + 2*c) + (d*e*f^2 + (d*x +
c)*f^3 - c*f^3)*cos(d*x + c) + (I*d*e*f^2 + I*(d*x + c)*f^3 - I*c*f^3)*sin(5*d*x + 5*c) - (d*e*f^2 + (d*x + c
)*f^3 - c*f^3)*sin(4*d*x + 4*c) + 2*(-I*d*e*f^2 - I*(d*x + c)*f^3 + I*c*f^3)*sin(3*d*x + 3*c) + 2*(d*e*f^2 + (
d*x + c)*f^3 - c*f^3)*sin(2*d*x + 2*c) + (I*d*e*f^2 + I*(d*x + c)*f^3 - I*c*f^3)*sin(d*x + c))*dilog(I*e^(I*d*
x + I*c)) - 6*(3*I*d^2*e^2*f + 2*(-3*I*c + 2*I)*d*e*f^2 + 3*I*(d*x + c)^2*f^3 + (3*I*c^2 - 4*I*c + 2*I)*f^3 +
2*(3*I*d*e*f^2 + (-3*I*c + 2*I)*f^3)*(d*x + c) + (3*d^2*e^2*f - 2*(3*c - 2)*d*e*f^2 + 3*(d*x + c)^2*f^3 + (3*c
^2 - 4*c + 2)*f^3 + 2*(3*d*e*f^2 - (3*c - 2)*f^3)*(d*x + c))*cos(5*d*x + 5*c) + (3*I*d^2*e^2*f + 2*(-3*I*c + 2
*I)*d*e*f^2 + 3*I*(d*x + c)^2*f^3 + (3*I*c^2 - 4*I*c + 2*I)*f^3 + 2*(3*I*d*e*f^2 + (-3*I*c + 2*I)*f^3)*(d*x +
c))*cos(4*d*x + 4*c) - 2*(3*d^2*e^2*f - 2*(3*c - 2)*d*e*f^2 + 3*(d*x + c)^2*f^3 + (3*c^2 - 4*c + 2)*f^3 + 2*(3
*d*e*f^2 - (3*c - 2)*f^3)*(d*x + c))*cos(3*d*x + 3*c) + 2*(-3*I*d^2*e^2*f + 2*(3*I*c - 2*I)*d*e*f^2 - 3*I*(d*x
+ c)^2*f^3 + (-3*I*c^2 + 4*I*c - 2*I)*f^3 + 2*(-3*I*d*e*f^2 + (3*I*c - 2*I)*f^3)*(d*x + c))*cos(2*d*x + 2*c)
+ (3*d^2*e^2*f - 2*(3*c - 2)*d*e*f^2 + 3*(d*x + c)^2*f^3 + (3*c^2 - 4*c + 2)*f^3 + 2*(3*d*e*f^2 - (3*c - 2)*f^
3)*(d*x + c))*cos(d*x + c) + (3*I*d^2*e^2*f + 2*(-3*I*c + 2*I)*d*e*f^2 + 3*I*(d*x + c)^2*f^3 + (3*I*c^2 - 4*I*
c + 2*I)*f^3 + 2*(3*I*d*e*f^2 + (-3*I*c + 2*I)*f^3)*(d*x + c))*sin(5*d*x + 5*c) - (3*d^2*e^2*f - 2*(3*c - 2)*d
*e*f^2 + 3*(d*x + c)^2*f^3 + (3*c^2 - 4*c + 2)*f^3 + 2*(3*d*e*f^2 - (3*c - 2)*f^3)*(d*x + c))*sin(4*d*x + 4*c)
+ 2*(-3*I*d^2*e^2*f + 2*(3*I*c - 2*I)*d*e*f^2 - 3*I*(d*x + c)^2*f^3 + (-3*I*c^2 + 4*I*c - 2*I)*f^3 + 2*(-3*I*
d*e*f^2 + (3*I*c - 2*I)*f^3)*(d*x + c))*sin(3*d*x + 3*c) + 2*(3*d^2*e^2*f - 2*(3*c - 2)*d*e*f^2 + 3*(d*x + c)^
2*f^3 + (3*c^2 - 4*c + 2)*f^3 + 2*(3*d*e*f^2 - (3*c - 2)*f^3)*(d*x + c))*sin(2*d*x + 2*c) + (3*I*d^2*e^2*f + 2
*(-3*I*c + 2*I)*d*e*f^2 + 3*I*(d*x + c)^2*f^3 + (3*I*c^2 - 4*I*c + 2*I)*f^3 + 2*(3*I*d*e*f^2 + (-3*I*c + 2*I)*
f^3)*(d*x + c))*sin(d*x + c))*dilog(-e^(I*d*x + I*c)) - 6*(-3*I*d^2*e^2*f + 2*(3*I*c + 2*I)*d*e*f^2 - 3*I*(d*x
+ c)^2*f^3 + (-3*I*c^2 - 4*I*c - 2*I)*f^3 + 2*(-3*I*d*e*f^2 + (3*I*c + 2*I)*f^3)*(d*x + c) - (3*d^2*e^2*f - 2
*(3*c + 2)*d*e*f^2 + 3*(d*x + c)^2*f^3 + (3*c^2 + 4*c + 2)*f^3 + 2*(3*d*e*f^2 - (3*c + 2)*f^3)*(d*x + c))*cos(
5*d*x + 5*c) + (-3*I*d^2*e^2*f + 2*(3*I*c + 2*I)*d*e*f^2 - 3*I*(d*x + c)^2*f^3 + (-3*I*c^2 - 4*I*c - 2*I)*f^3
+ 2*(-3*I*d*e*f^2 + (3*I*c + 2*I)*f^3)*(d*x + c))*cos(4*d*x + 4*c) + 2*(3*d^2*e^2*f - 2*(3*c + 2)*d*e*f^2 + 3*
(d*x + c)^2*f^3 + (3*c^2 + 4*c + 2)*f^3 + 2*(3*d*e*f^2 - (3*c + 2)*f^3)*(d*x + c))*cos(3*d*x + 3*c) + 2*(3*I*d
^2*e^2*f + 2*(-3*I*c - 2*I)*d*e*f^2 + 3*I*(d*x + c)^2*f^3 + (3*I*c^2 + 4*I*c + 2*I)*f^3 + 2*(3*I*d*e*f^2 + (-3
*I*c - 2*I)*f^3)*(d*x + c))*cos(2*d*x + 2*c) - (3*d^2*e^2*f - 2*(3*c + 2)*d*e*f^2 + 3*(d*x + c)^2*f^3 + (3*c^2
+ 4*c + 2)*f^3 + 2*(3*d*e*f^2 - (3*c + 2)*f^3)*(d*x + c))*cos(d*x + c) + (-3*I*d^2*e^2*f + 2*(3*I*c + 2*I)*d*
e*f^2 - 3*I*(d*x + c)^2*f^3 + (-3*I*c^2 - 4*I*c - 2*I)*f^3 + 2*(-3*I*d*e*f^2 + (3*I*c + 2*I)*f^3)*(d*x + c))*s
in(5*d*x + 5*c) + (3*d^2*e^2*f - 2*(3*c + 2)*d*e*f^2 + 3*(d*x + c)^2*f^3 + (3*c^2 + 4*c + 2)*f^3 + 2*(3*d*e*f^
2 - (3*c + 2)*f^3)*(d*x + c))*sin(4*d*x + 4*c) + 2*(3*I*d^2*e^2*f + 2*(-3*I*c - 2*I)*d*e*f^2 + 3*I*(d*x + c)^2
*f^3 + (3*I*c^2 + 4*I*c + 2*I)*f^3 + 2*(3*I*d*e*f^2 + (-3*I*c - 2*I)*f^3)*(d*x + c))*sin(3*d*x + 3*c) - 2*(3*d
^2*e^2*f - 2*(3*c + 2)*d*e*f^2 + 3*(d*x + c)^2*f^3 + (3*c^2 + 4*c + 2)*f^3 + 2*(3*d*e*f^2 - (3*c + 2)*f^3)*(d*
x + c))*sin(2*d*x + 2*c) + (-3*I*d^2*e^2*f + 2*(3*I*c + 2*I)*d*e*f^2 - 3*I*(d*x + c)^2*f^3 + (-3*I*c^2 - 4*I*c
- 2*I)*f^3 + 2*(-3*I*d*e*f^2 + (3*I*c + 2*I)*f^3)*(d*x + c))*sin(d*x + c))*dilog(e^(I*d*x + I*c)) + 3*((d*x +
c)^3*f^3 + 2*d^2*e^2*f + (3*c^2 - 4*c + 2)*d*e*f^2 - (c^3 - 2*c^2 + 2*c)*f^3 + (3*d*e*f^2 - (3*c - 2)*f^3)*(d
*x + c)^2 + (3*d^2*e^2*f - 2*(3*c - 2)*d*e*f^2 + (3*c^2 - 4*c + 2)*f^3)*(d*x + c) - (I*(d*x + c)^3*f^3 + 2*I*d
^2*e^2*f + (3*I*c^2 - 4*I*c + 2*I)*d*e*f^2 + (-I*c^3 + 2*I*c^2 - 2*I*c)*f^3 + (3*I*d*e*f^2 + (-3*I*c + 2*I)*f^
3)*(d*x + c)^2 + (3*I*d^2*e^2*f + 2*(-3*I*c + 2*I)*d*e*f^2 + (3*I*c^2 - 4*I*c + 2*I)*f^3)*(d*x + c))*cos(5*d*x
+ 5*c) + ((d*x + c)^3*f^3 + 2*d^2*e^2*f + (3*c^2 - 4*c + 2)*d*e*f^2 - (c^3 - 2*c^2 + 2*c)*f^3 + (3*d*e*f^2 -
(3*c - 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c - 2)*d*e*f^2 + (3*c^2 - 4*c + 2)*f^3)*(d*x + c))*cos(4*d*x
+ 4*c) - 2*(-I*(d*x + c)^3*f^3 - 2*I*d^2*e^2*f + (-3*I*c^2 + 4*I*c - 2*I)*d*e*f^2 + (I*c^3 - 2*I*c^2 + 2*I*c)*
f^3 + (-3*I*d*e*f^2 + (3*I*c - 2*I)*f^3)*(d*x + c)^2 + (-3*I*d^2*e^2*f + 2*(3*I*c - 2*I)*d*e*f^2 + (-3*I*c^2 +
4*I*c - 2*I)*f^3)*(d*x + c))*cos(3*d*x + 3*c) - 2*((d*x + c)^3*f^3 + 2*d^2*e^2*f + (3*c^2 - 4*c + 2)*d*e*f^2
- (c^3 - 2*c^2 + 2*c)*f^3 + (3*d*e*f^2 - (3*c - 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c - 2)*d*e*f^2 + (3*
c^2 - 4*c + 2)*f^3)*(d*x + c))*cos(2*d*x + 2*c) - (I*(d*x + c)^3*f^3 + 2*I*d^2*e^2*f + (3*I*c^2 - 4*I*c + 2*I)
*d*e*f^2 + (-I*c^3 + 2*I*c^2 - 2*I*c)*f^3 + (3*I*d*e*f^2 + (-3*I*c + 2*I)*f^3)*(d*x + c)^2 + (3*I*d^2*e^2*f +
2*(-3*I*c + 2*I)*d*e*f^2 + (3*I*c^2 - 4*I*c + 2*I)*f^3)*(d*x + c))*cos(d*x + c) + ((d*x + c)^3*f^3 + 2*d^2*e^2
*f + (3*c^2 - 4*c + 2)*d*e*f^2 - (c^3 - 2*c^2 + 2*c)*f^3 + (3*d*e*f^2 - (3*c - 2)*f^3)*(d*x + c)^2 + (3*d^2*e^
2*f - 2*(3*c - 2)*d*e*f^2 + (3*c^2 - 4*c + 2)*f^3)*(d*x + c))*sin(5*d*x + 5*c) - (-I*(d*x + c)^3*f^3 - 2*I*d^2
*e^2*f + (-3*I*c^2 + 4*I*c - 2*I)*d*e*f^2 + (I*c^3 - 2*I*c^2 + 2*I*c)*f^3 + (-3*I*d*e*f^2 + (3*I*c - 2*I)*f^3)
*(d*x + c)^2 + (-3*I*d^2*e^2*f + 2*(3*I*c - 2*I)*d*e*f^2 + (-3*I*c^2 + 4*I*c - 2*I)*f^3)*(d*x + c))*sin(4*d*x
+ 4*c) - 2*((d*x + c)^3*f^3 + 2*d^2*e^2*f + (3*c^2 - 4*c + 2)*d*e*f^2 - (c^3 - 2*c^2 + 2*c)*f^3 + (3*d*e*f^2 -
(3*c - 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c - 2)*d*e*f^2 + (3*c^2 - 4*c + 2)*f^3)*(d*x + c))*sin(3*d*x
+ 3*c) - 2*(I*(d*x + c)^3*f^3 + 2*I*d^2*e^2*f + (3*I*c^2 - 4*I*c + 2*I)*d*e*f^2 + (-I*c^3 + 2*I*c^2 - 2*I*c)*
f^3 + (3*I*d*e*f^2 + (-3*I*c + 2*I)*f^3)*(d*x + c)^2 + (3*I*d^2*e^2*f + 2*(-3*I*c + 2*I)*d*e*f^2 + (3*I*c^2 -
4*I*c + 2*I)*f^3)*(d*x + c))*sin(2*d*x + 2*c) + ((d*x + c)^3*f^3 + 2*d^2*e^2*f + (3*c^2 - 4*c + 2)*d*e*f^2 - (
c^3 - 2*c^2 + 2*c)*f^3 + (3*d*e*f^2 - (3*c - 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c - 2)*d*e*f^2 + (3*c^2
- 4*c + 2)*f^3)*(d*x + c))*sin(d*x + c))*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1) - 3*((d*x
+ c)^3*f^3 - 2*d^2*e^2*f + (3*c^2 + 4*c + 2)*d*e*f^2 - (c^3 + 2*c^2 + 2*c)*f^3 + (3*d*e*f^2 - (3*c + 2)*f^3)*(
d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c + 2)*d*e*f^2 + (3*c^2 + 4*c + 2)*f^3)*(d*x + c) + (-I*(d*x + c)^3*f^3 + 2*I
*d^2*e^2*f + (-3*I*c^2 - 4*I*c - 2*I)*d*e*f^2 + (I*c^3 + 2*I*c^2 + 2*I*c)*f^3 + (-3*I*d*e*f^2 + (3*I*c + 2*I)*
f^3)*(d*x + c)^2 + (-3*I*d^2*e^2*f + 2*(3*I*c + 2*I)*d*e*f^2 + (-3*I*c^2 - 4*I*c - 2*I)*f^3)*(d*x + c))*cos(5*
d*x + 5*c) + ((d*x + c)^3*f^3 - 2*d^2*e^2*f + (3*c^2 + 4*c + 2)*d*e*f^2 - (c^3 + 2*c^2 + 2*c)*f^3 + (3*d*e*f^2
- (3*c + 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c + 2)*d*e*f^2 + (3*c^2 + 4*c + 2)*f^3)*(d*x + c))*cos(4*d
*x + 4*c) + 2*(I*(d*x + c)^3*f^3 - 2*I*d^2*e^2*f + (3*I*c^2 + 4*I*c + 2*I)*d*e*f^2 + (-I*c^3 - 2*I*c^2 - 2*I*c
)*f^3 + (3*I*d*e*f^2 + (-3*I*c - 2*I)*f^3)*(d*x + c)^2 + (3*I*d^2*e^2*f + 2*(-3*I*c - 2*I)*d*e*f^2 + (3*I*c^2
+ 4*I*c + 2*I)*f^3)*(d*x + c))*cos(3*d*x + 3*c) - 2*((d*x + c)^3*f^3 - 2*d^2*e^2*f + (3*c^2 + 4*c + 2)*d*e*f^2
- (c^3 + 2*c^2 + 2*c)*f^3 + (3*d*e*f^2 - (3*c + 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c + 2)*d*e*f^2 + (3
*c^2 + 4*c + 2)*f^3)*(d*x + c))*cos(2*d*x + 2*c) + (-I*(d*x + c)^3*f^3 + 2*I*d^2*e^2*f + (-3*I*c^2 - 4*I*c - 2
*I)*d*e*f^2 + (I*c^3 + 2*I*c^2 + 2*I*c)*f^3 + (-3*I*d*e*f^2 + (3*I*c + 2*I)*f^3)*(d*x + c)^2 + (-3*I*d^2*e^2*f
+ 2*(3*I*c + 2*I)*d*e*f^2 + (-3*I*c^2 - 4*I*c - 2*I)*f^3)*(d*x + c))*cos(d*x + c) + ((d*x + c)^3*f^3 - 2*d^2*
e^2*f + (3*c^2 + 4*c + 2)*d*e*f^2 - (c^3 + 2*c^2 + 2*c)*f^3 + (3*d*e*f^2 - (3*c + 2)*f^3)*(d*x + c)^2 + (3*d^2
*e^2*f - 2*(3*c + 2)*d*e*f^2 + (3*c^2 + 4*c + 2)*f^3)*(d*x + c))*sin(5*d*x + 5*c) + (I*(d*x + c)^3*f^3 - 2*I*d
^2*e^2*f + (3*I*c^2 + 4*I*c + 2*I)*d*e*f^2 + (-I*c^3 - 2*I*c^2 - 2*I*c)*f^3 + (3*I*d*e*f^2 + (-3*I*c - 2*I)*f^
3)*(d*x + c)^2 + (3*I*d^2*e^2*f + 2*(-3*I*c - 2*I)*d*e*f^2 + (3*I*c^2 + 4*I*c + 2*I)*f^3)*(d*x + c))*sin(4*d*x
+ 4*c) - 2*((d*x + c)^3*f^3 - 2*d^2*e^2*f + (3*c^2 + 4*c + 2)*d*e*f^2 - (c^3 + 2*c^2 + 2*c)*f^3 + (3*d*e*f^2
- (3*c + 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c + 2)*d*e*f^2 + (3*c^2 + 4*c + 2)*f^3)*(d*x + c))*sin(3*d*
x + 3*c) + 2*(-I*(d*x + c)^3*f^3 + 2*I*d^2*e^2*f + (-3*I*c^2 - 4*I*c - 2*I)*d*e*f^2 + (I*c^3 + 2*I*c^2 + 2*I*c
)*f^3 + (-3*I*d*e*f^2 + (3*I*c + 2*I)*f^3)*(d*x + c)^2 + (-3*I*d^2*e^2*f + 2*(3*I*c + 2*I)*d*e*f^2 + (-3*I*c^2
- 4*I*c - 2*I)*f^3)*(d*x + c))*sin(2*d*x + 2*c) + ((d*x + c)^3*f^3 - 2*d^2*e^2*f + (3*c^2 + 4*c + 2)*d*e*f^2
- (c^3 + 2*c^2 + 2*c)*f^3 + (3*d*e*f^2 - (3*c + 2)*f^3)*(d*x + c)^2 + (3*d^2*e^2*f - 2*(3*c + 2)*d*e*f^2 + (3*
c^2 + 4*c + 2)*f^3)*(d*x + c))*sin(d*x + c))*log(cos(d*x + c)^2 + sin(d*x + c)^2 - 2*cos(d*x + c) + 1) + 12*(d
^2*e^2*f - 2*c*d*e*f^2 + (d*x + c)^2*f^3 + c^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c) - (I*d^2*e^2*f - 2*I*c*d*e*
f^2 + I*(d*x + c)^2*f^3 + I*c^2*f^3 + 2*(I*d*e*f^2 - I*c*f^3)*(d*x + c))*cos(5*d*x + 5*c) + (d^2*e^2*f - 2*c*d
*e*f^2 + (d*x + c)^2*f^3 + c^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*cos(4*d*x + 4*c) - 2*(-I*d^2*e^2*f + 2*I*c
*d*e*f^2 - I*(d*x + c)^2*f^3 - I*c^2*f^3 + 2*(-I*d*e*f^2 + I*c*f^3)*(d*x + c))*cos(3*d*x + 3*c) - 2*(d^2*e^2*f
- 2*c*d*e*f^2 + (d*x + c)^2*f^3 + c^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*cos(2*d*x + 2*c) - (I*d^2*e^2*f -
2*I*c*d*e*f^2 + I*(d*x + c)^2*f^3 + I*c^2*f^3 + 2*(I*d*e*f^2 - I*c*f^3)*(d*x + c))*cos(d*x + c) + (d^2*e^2*f -
2*c*d*e*f^2 + (d*x + c)^2*f^3 + c^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*sin(5*d*x + 5*c) - (-I*d^2*e^2*f + 2
*I*c*d*e*f^2 - I*(d*x + c)^2*f^3 - I*c^2*f^3 + 2*(-I*d*e*f^2 + I*c*f^3)*(d*x + c))*sin(4*d*x + 4*c) - 2*(d^2*e
^2*f - 2*c*d*e*f^2 + (d*x + c)^2*f^3 + c^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*sin(3*d*x + 3*c) - 2*(I*d^2*e^
2*f - 2*I*c*d*e*f^2 + I*(d*x + c)^2*f^3 + I*c^2*f^3 + 2*(I*d*e*f^2 - I*c*f^3)*(d*x + c))*sin(2*d*x + 2*c) + (d
^2*e^2*f - 2*c*d*e*f^2 + (d*x + c)^2*f^3 + c^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*sin(d*x + c))*log(cos(d*x
+ c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1) + 36*(f^3*cos(5*d*x + 5*c) + I*f^3*cos(4*d*x + 4*c) - 2*f^3*cos(
3*d*x + 3*c) - 2*I*f^3*cos(2*d*x + 2*c) + f^3*cos(d*x + c) + I*f^3*sin(5*d*x + 5*c) - f^3*sin(4*d*x + 4*c) - 2
*I*f^3*sin(3*d*x + 3*c) + 2*f^3*sin(2*d*x + 2*c) + I*f^3*sin(d*x + c) + I*f^3)*polylog(4, -e^(I*d*x + I*c)) -
36*(f^3*cos(5*d*x + 5*c) + I*f^3*cos(4*d*x + 4*c) - 2*f^3*cos(3*d*x + 3*c) - 2*I*f^3*cos(2*d*x + 2*c) + f^3*co
s(d*x + c) + I*f^3*sin(5*d*x + 5*c) - f^3*sin(4*d*x + 4*c) - 2*I*f^3*sin(3*d*x + 3*c) + 2*f^3*sin(2*d*x + 2*c)
+ I*f^3*sin(d*x + c) + I*f^3)*polylog(4, e^(I*d*x + I*c)) - 48*(I*f^3*cos(5*d*x + 5*c) - f^3*cos(4*d*x + 4*c)
- 2*I*f^3*cos(3*d*x + 3*c) + 2*f^3*cos(2*d*x + 2*c) + I*f^3*cos(d*x + c) - f^3*sin(5*d*x + 5*c) - I*f^3*sin(4
*d*x + 4*c) + 2*f^3*sin(3*d*x + 3*c) + 2*I*f^3*sin(2*d*x + 2*c) - f^3*sin(d*x + c) - f^3)*polylog(3, I*e^(I*d*
x + I*c)) + 12*(3*d*e*f^2 + 3*(d*x + c)*f^3 - (3*c - 2)*f^3 - (3*I*d*e*f^2 + 3*I*(d*x + c)*f^3 + (-3*I*c + 2*I
)*f^3)*cos(5*d*x + 5*c) + (3*d*e*f^2 + 3*(d*x + c)*f^3 - (3*c - 2)*f^3)*cos(4*d*x + 4*c) - 2*(-3*I*d*e*f^2 - 3
*I*(d*x + c)*f^3 + (3*I*c - 2*I)*f^3)*cos(3*d*x + 3*c) - 2*(3*d*e*f^2 + 3*(d*x + c)*f^3 - (3*c - 2)*f^3)*cos(2
*d*x + 2*c) - (3*I*d*e*f^2 + 3*I*(d*x + c)*f^3 + (-3*I*c + 2*I)*f^3)*cos(d*x + c) + (3*d*e*f^2 + 3*(d*x + c)*f
^3 - (3*c - 2)*f^3)*sin(5*d*x + 5*c) - (-3*I*d*e*f^2 - 3*I*(d*x + c)*f^3 + (3*I*c - 2*I)*f^3)*sin(4*d*x + 4*c)
- 2*(3*d*e*f^2 + 3*(d*x + c)*f^3 - (3*c - 2)*f^3)*sin(3*d*x + 3*c) - 2*(3*I*d*e*f^2 + 3*I*(d*x + c)*f^3 + (-3
*I*c + 2*I)*f^3)*sin(2*d*x + 2*c) + (3*d*e*f^2 + 3*(d*x + c)*f^3 - (3*c - 2)*f^3)*sin(d*x + c))*polylog(3, -e^
(I*d*x + I*c)) - 12*(3*d*e*f^2 + 3*(d*x + c)*f^3 - (3*c + 2)*f^3 + (-3*I*d*e*f^2 - 3*I*(d*x + c)*f^3 + (3*I*c
+ 2*I)*f^3)*cos(5*d*x + 5*c) + (3*d*e*f^2 + 3*(d*x + c)*f^3 - (3*c + 2)*f^3)*cos(4*d*x + 4*c) + 2*(3*I*d*e*f^2
+ 3*I*(d*x + c)*f^3 + (-3*I*c - 2*I)*f^3)*cos(3*d*x + 3*c) - 2*(3*d*e*f^2 + 3*(d*x + c)*f^3 - (3*c + 2)*f^3)*
cos(2*d*x + 2*c) + (-3*I*d*e*f^2 - 3*I*(d*x + c)*f^3 + (3*I*c + 2*I)*f^3)*cos(d*x + c) + (3*d*e*f^2 + 3*(d*x +
c)*f^3 - (3*c + 2)*f^3)*sin(5*d*x + 5*c) + (3*I*d*e*f^2 + 3*I*(d*x + c)*f^3 + (-3*I*c - 2*I)*f^3)*sin(4*d*x +
4*c) - 2*(3*d*e*f^2 + 3*(d*x + c)*f^3 - (3*c + 2)*f^3)*sin(3*d*x + 3*c) + 2*(-3*I*d*e*f^2 - 3*I*(d*x + c)*f^3
+ (3*I*c + 2*I)*f^3)*sin(2*d*x + 2*c) + (3*d*e*f^2 + 3*(d*x + c)*f^3 - (3*c + 2)*f^3)*sin(d*x + c))*polylog(3
, e^(I*d*x + I*c)) - 16*(I*(d*x + c)^3*f^3 + 3*(I*d*e*f^2 - I*c*f^3)*(d*x + c)^2 + 3*(I*d^2*e^2*f - 2*I*c*d*e*
f^2 + I*c^2*f^3)*(d*x + c))*sin(5*d*x + 5*c) + 4*((d*x + c)^3*f^3 + 3*I*d^2*e^2*f - 3*(3*c^2 + 2*I*c)*d*e*f^2
+ 3*(c^3 + I*c^2)*f^3 + 3*(d*e*f^2 - (c - I)*f^3)*(d*x + c)^2 + 3*(d^2*e^2*f - 2*(c - I)*d*e*f^2 + (c^2 - 2*I*
c)*f^3)*(d*x + c))*sin(4*d*x + 4*c) - 4*(-5*I*(d*x + c)^3*f^3 + 3*d^2*e^2*f + 3*(3*I*c^2 - 2*c)*d*e*f^2 + 3*(-
I*c^3 + c^2)*f^3 + 3*(-5*I*d*e*f^2 + (5*I*c + 1)*f^3)*(d*x + c)^2 + 3*(-5*I*d^2*e^2*f + 2*(5*I*c + 1)*d*e*f^2
+ (-5*I*c^2 - 2*c)*f^3)*(d*x + c))*sin(3*d*x + 3*c) - 4*(3*(d*x + c)^3*f^3 + 3*I*d^2*e^2*f - 3*(5*c^2 + 2*I*c)
*d*e*f^2 + (5*c^3 + 3*I*c^2)*f^3 + 3*(3*d*e*f^2 - (3*c - I)*f^3)*(d*x + c)^2 + 3*(3*d^2*e^2*f - 2*(3*c - I)*d*
e*f^2 + (3*c^2 - 2*I*c)*f^3)*(d*x + c))*sin(2*d*x + 2*c) - 4*(3*I*(d*x + c)^3*f^3 - 3*d^2*e^2*f + 3*(-I*c^2 +
2*c)*d*e*f^2 + (I*c^3 - 3*c^2)*f^3 + 3*(3*I*d*e*f^2 + (-3*I*c - 1)*f^3)*(d*x + c)^2 + 3*(3*I*d^2*e^2*f + 2*(-3
*I*c - 1)*d*e*f^2 + (3*I*c^2 + 2*c)*f^3)*(d*x + c))*sin(d*x + c))/(-4*I*a*d^3*cos(5*d*x + 5*c) + 4*a*d^3*cos(4
*d*x + 4*c) + 8*I*a*d^3*cos(3*d*x + 3*c) - 8*a*d^3*cos(2*d*x + 2*c) - 4*I*a*d^3*cos(d*x + c) + 4*a*d^3*sin(5*d
*x + 5*c) + 4*I*a*d^3*sin(4*d*x + 4*c) - 8*a*d^3*sin(3*d*x + 3*c) - 8*I*a*d^3*sin(2*d*x + 2*c) + 4*a*d^3*sin(d
*x + c) + 4*a*d^3))/d

Giac [F]

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

[In]

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

[Out]

integrate((f*x + e)^3*csc(d*x + c)^3/(a*sin(d*x + c) + a), x)

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}