\(\int \frac {(e+f x)^2 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [210]

Optimal result

Integrand size = 28, antiderivative size = 392 \[ \int \frac {(e+f x)^2 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 i (e+f x)^2}{a d}-\frac {3 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {f^2 \text {arctanh}(\cos (c+d x))}{a d^3}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {f (e+f x) \csc (c+d x)}{a d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}+\frac {4 i f^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}+\frac {i f^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac {3 f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {3 f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {4631, 4271, 3855, 4268, 2611, 2320, 6724, 4269, 3798, 2221, 2317, 2438, 3399} \[ \int \frac {(e+f x)^2 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {f^2 \text {arctanh}(\cos (c+d x))}{a d^3}-\frac {3 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {4 i f^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac {i f^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac {3 f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {3 f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {f (e+f x) \csc (c+d x)}{a d^2}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{a d}+\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {2 i (e+f x)^2}{a d} \]

[In]

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

[Out]

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

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 3855

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

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^2 \csc ^3(c+d x) \, dx}{a}-\int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx \\ & = -\frac {f (e+f x) \csc (c+d x)}{a d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\int (e+f x)^2 \csc (c+d x) \, dx}{2 a}-\frac {\int (e+f x)^2 \csc ^2(c+d x) \, dx}{a}+\frac {f^2 \int \csc (c+d x) \, dx}{a d^2}+\int \frac {(e+f x)^2 \csc (c+d x)}{a+a \sin (c+d x)} \, dx \\ & = -\frac {(e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {f^2 \text {arctanh}(\cos (c+d x))}{a d^3}+\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {f (e+f x) \csc (c+d x)}{a d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\int (e+f x)^2 \csc (c+d x) \, dx}{a}-\frac {f \int (e+f x) \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d}+\frac {f \int (e+f x) \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d}-\frac {(2 f) \int (e+f x) \cot (c+d x) \, dx}{a d}-\int \frac {(e+f x)^2}{a+a \sin (c+d x)} \, dx \\ & = \frac {i (e+f x)^2}{a d}-\frac {3 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {f^2 \text {arctanh}(\cos (c+d x))}{a d^3}+\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {f (e+f x) \csc (c+d x)}{a d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {i f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {i f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {\int (e+f x)^2 \csc ^2\left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {d x}{2}\right ) \, dx}{2 a}+\frac {(4 i f) \int \frac {e^{2 i (c+d x)} (e+f x)}{1-e^{2 i (c+d x)}} \, dx}{a d}-\frac {(2 f) \int (e+f x) \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d}+\frac {(2 f) \int (e+f x) \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d}-\frac {\left (i f^2\right ) \int \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (i f^2\right ) \int \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) \, dx}{a d^2} \\ & = \frac {i (e+f x)^2}{a d}-\frac {3 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {f^2 \text {arctanh}(\cos (c+d x))}{a d^3}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {f (e+f x) \csc (c+d x)}{a d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {(2 f) \int (e+f x) \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx}{a d}-\frac {f^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^3}+\frac {f^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^3}-\frac {\left (2 i f^2\right ) \int \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (2 i f^2\right ) \int \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (2 f^2\right ) \int \log \left (1-e^{2 i (c+d x)}\right ) \, dx}{a d^2} \\ & = \frac {2 i (e+f x)^2}{a d}-\frac {3 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {f^2 \text {arctanh}(\cos (c+d x))}{a d^3}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {f (e+f x) \csc (c+d x)}{a d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}-\frac {(4 f) \int \frac {e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )} (e+f x)}{1-i e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}} \, dx}{a d}-\frac {\left (i f^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{a d^3}-\frac {\left (2 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^3}+\frac {\left (2 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^3} \\ & = \frac {2 i (e+f x)^2}{a d}-\frac {3 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {f^2 \text {arctanh}(\cos (c+d x))}{a d^3}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {f (e+f x) \csc (c+d x)}{a d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}+\frac {i f^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac {3 f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {3 f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac {\left (4 f^2\right ) \int \log \left (1-i e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^2} \\ & = \frac {2 i (e+f x)^2}{a d}-\frac {3 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {f^2 \text {arctanh}(\cos (c+d x))}{a d^3}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {f (e+f x) \csc (c+d x)}{a d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}+\frac {i f^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac {3 f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {3 f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}-\frac {\left (4 i f^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{a d^3} \\ & = \frac {2 i (e+f x)^2}{a d}-\frac {3 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {f^2 \text {arctanh}(\cos (c+d x))}{a d^3}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {f (e+f x) \csc (c+d x)}{a d^2}-\frac {(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 a d}-\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}+\frac {4 i f^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {3 i f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}+\frac {i f^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac {3 f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {3 f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3} \\ \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. \(951\) vs. \(2(392)=784\).

Time = 12.37 (sec) , antiderivative size = 951, normalized size of antiderivative = 2.43 \[ \int \frac {(e+f x)^2 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {8 \left (2 i d^2 e f x+i d^2 f^2 x^2-3 d^2 e^2 \text {arctanh}(\cos (c+d x)+i \sin (c+d x))-2 f^2 \text {arctanh}(\cos (c+d x)+i \sin (c+d x))-6 d^2 e f x \text {arctanh}(\cos (c+d x)+i \sin (c+d x))-3 d^2 f^2 x^2 \text {arctanh}(\cos (c+d x)+i \sin (c+d x))+2 d^2 e f x \cot (c)+d^2 f^2 x^2 \cot (c)-2 d e f \log (1-\cos (2 (c+d x))-i \sin (2 (c+d x)))-2 d f^2 x \log (1-\cos (2 (c+d x))-i \sin (2 (c+d x)))+3 i d f (e+f x) \operatorname {PolyLog}(2,-\cos (c+d x)-i \sin (c+d x))-3 i d f (e+f x) \operatorname {PolyLog}(2,\cos (c+d x)+i \sin (c+d x))+i f^2 \operatorname {PolyLog}(2,\cos (2 (c+d x))+i \sin (2 (c+d x)))-3 f^2 \operatorname {PolyLog}(3,-\cos (c+d x)-i \sin (c+d x))+3 f^2 \operatorname {PolyLog}(3,\cos (c+d x)+i \sin (c+d x))\right )+\frac {32 d^2 f (\cos (c)+i \sin (c)) \left (\frac {(e+f x)^2 (\cos (c)-i \sin (c))}{2 f}-\frac {(e+f x) \log (1+i \cos (c+d x)+\sin (c+d x)) (1+i \cos (c)+\sin (c))}{d}+\frac {f \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i (1+\sin (c)))}{d^2}\right )}{\cos (c)+i (1+\sin (c))}-\frac {d (e+f x) \csc (c) \csc ^2(c+d x) \left (2 f \cos \left (\frac {d x}{2}\right )+2 f \cos \left (\frac {3 d x}{2}\right )+5 d e \cos \left (c-\frac {d x}{2}\right )+5 d f x \cos \left (c-\frac {d x}{2}\right )-d e \cos \left (c+\frac {d x}{2}\right )-d f x \cos \left (c+\frac {d x}{2}\right )-2 f \cos \left (2 c+\frac {d x}{2}\right )+d e \cos \left (c+\frac {3 d x}{2}\right )+d f x \cos \left (c+\frac {3 d x}{2}\right )-2 f \cos \left (2 c+\frac {3 d x}{2}\right )-3 d e \cos \left (3 c+\frac {3 d x}{2}\right )-3 d f x \cos \left (3 c+\frac {3 d x}{2}\right )-4 d e \cos \left (c+\frac {5 d x}{2}\right )-4 d f x \cos \left (c+\frac {5 d x}{2}\right )+2 d e \cos \left (3 c+\frac {5 d x}{2}\right )+2 d f x \cos \left (3 c+\frac {5 d x}{2}\right )+d e \sin \left (\frac {d x}{2}\right )+d f x \sin \left (\frac {d x}{2}\right )+d e \sin \left (\frac {3 d x}{2}\right )+d f x \sin \left (\frac {3 d x}{2}\right )+2 f \sin \left (c-\frac {d x}{2}\right )+2 f \sin \left (c+\frac {d x}{2}\right )+3 d e \sin \left (2 c+\frac {d x}{2}\right )+3 d f x \sin \left (2 c+\frac {d x}{2}\right )+2 f \sin \left (c+\frac {3 d x}{2}\right )+d e \sin \left (2 c+\frac {3 d x}{2}\right )+d f x \sin \left (2 c+\frac {3 d x}{2}\right )-2 f \sin \left (3 c+\frac {3 d x}{2}\right )-2 d e \sin \left (2 c+\frac {5 d x}{2}\right )-2 d f x \sin \left (2 c+\frac {5 d x}{2}\right )\right )}{\left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}}{8 a d^3} \]

[In]

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

[Out]

(8*((2*I)*d^2*e*f*x + I*d^2*f^2*x^2 - 3*d^2*e^2*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] - 2*f^2*ArcTanh[Cos[c +
d*x] + I*Sin[c + d*x]] - 6*d^2*e*f*x*ArcTanh[Cos[c + d*x] + I*Sin[c + d*x]] - 3*d^2*f^2*x^2*ArcTanh[Cos[c + d
*x] + I*Sin[c + d*x]] + 2*d^2*e*f*x*Cot[c] + d^2*f^2*x^2*Cot[c] - 2*d*e*f*Log[1 - Cos[2*(c + d*x)] - I*Sin[2*(
c + d*x)]] - 2*d*f^2*x*Log[1 - Cos[2*(c + d*x)] - I*Sin[2*(c + d*x)]] + (3*I)*d*f*(e + f*x)*PolyLog[2, -Cos[c
+ d*x] - I*Sin[c + d*x]] - (3*I)*d*f*(e + f*x)*PolyLog[2, Cos[c + d*x] + I*Sin[c + d*x]] + I*f^2*PolyLog[2, Co
s[2*(c + d*x)] + I*Sin[2*(c + d*x)]] - 3*f^2*PolyLog[3, -Cos[c + d*x] - I*Sin[c + d*x]] + 3*f^2*PolyLog[3, Cos
[c + d*x] + I*Sin[c + d*x]]) + (32*d^2*f*(Cos[c] + I*Sin[c])*(((e + f*x)^2*(Cos[c] - I*Sin[c]))/(2*f) - ((e +
f*x)*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (f*PolyLog[2, (-I)*Cos[c + d*x] - Sin
[c + d*x]]*(Cos[c] - I*(1 + Sin[c])))/d^2))/(Cos[c] + I*(1 + Sin[c])) - (d*(e + f*x)*Csc[c]*Csc[c + d*x]^2*(2*
f*Cos[(d*x)/2] + 2*f*Cos[(3*d*x)/2] + 5*d*e*Cos[c - (d*x)/2] + 5*d*f*x*Cos[c - (d*x)/2] - d*e*Cos[c + (d*x)/2]
- d*f*x*Cos[c + (d*x)/2] - 2*f*Cos[2*c + (d*x)/2] + d*e*Cos[c + (3*d*x)/2] + d*f*x*Cos[c + (3*d*x)/2] - 2*f*C
os[2*c + (3*d*x)/2] - 3*d*e*Cos[3*c + (3*d*x)/2] - 3*d*f*x*Cos[3*c + (3*d*x)/2] - 4*d*e*Cos[c + (5*d*x)/2] - 4
*d*f*x*Cos[c + (5*d*x)/2] + 2*d*e*Cos[3*c + (5*d*x)/2] + 2*d*f*x*Cos[3*c + (5*d*x)/2] + d*e*Sin[(d*x)/2] + d*f
*x*Sin[(d*x)/2] + d*e*Sin[(3*d*x)/2] + d*f*x*Sin[(3*d*x)/2] + 2*f*Sin[c - (d*x)/2] + 2*f*Sin[c + (d*x)/2] + 3*
d*e*Sin[2*c + (d*x)/2] + 3*d*f*x*Sin[2*c + (d*x)/2] + 2*f*Sin[c + (3*d*x)/2] + d*e*Sin[2*c + (3*d*x)/2] + d*f*
x*Sin[2*c + (3*d*x)/2] - 2*f*Sin[3*c + (3*d*x)/2] - 2*d*e*Sin[2*c + (5*d*x)/2] - 2*d*f*x*Sin[2*c + (5*d*x)/2])
)/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(8*a*d^3)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1256 vs. \(2 (359 ) = 718\).

Time = 0.52 (sec) , antiderivative size = 1257, normalized size of antiderivative = 3.21

method	result	size
risch	\(\text {Expression too large to display}\)	\(1257\)

[In]

int((f*x+e)^2*csc(d*x+c)^3/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

3/2/d^3/a*c^2*f^2*ln(exp(I*(d*x+c))-1)-3/2/d/a*f^2*ln(exp(I*(d*x+c))+1)*x^2+3/2/d/a*f^2*ln(1-exp(I*(d*x+c)))*x
^2+1/a/d^3*f^2*ln(exp(I*(d*x+c))-1)+3*I/a/d^2*e*f*polylog(2,-exp(I*(d*x+c)))-3*I/a/d^2*f^2*polylog(2,exp(I*(d*
x+c)))*x+3*I/a/d^2*f^2*polylog(2,-exp(I*(d*x+c)))*x-1/a/d^3*f^2*ln(exp(I*(d*x+c))+1)-2/a/d^2*e*f*ln(exp(I*(d*x
+c))-1)-2/a/d^2*e*f*ln(exp(I*(d*x+c))+1)-2/a/d^2*f^2*ln(1-exp(I*(d*x+c)))*x-2/a/d^2*f^2*ln(exp(I*(d*x+c))+1)*x
-2/a/d^3*f^2*ln(1-exp(I*(d*x+c)))*c+2/a/d^3*c*f^2*ln(exp(I*(d*x+c))-1)+4*I/a/d^3*f^2*c^2+4*I/a/d*f^2*x^2+2*I/a
/d^3*f^2*polylog(2,exp(I*(d*x+c)))-4*I/a/d^3*f^2*c*arctan(exp(I*(d*x+c)))+8*I/a/d^2*c*f^2*x+4*I/a/d^2*e*f*arct
an(exp(I*(d*x+c)))-3*I/a/d^2*e*f*polylog(2,exp(I*(d*x+c)))+(3*d*f^2*x^2*exp(4*I*(d*x+c))+6*d*e*f*x*exp(4*I*(d*
x+c))+3*d*e^2*exp(4*I*(d*x+c))-5*d*f^2*x^2*exp(2*I*(d*x+c))+6*I*d*e*f*x*exp(3*I*(d*x+c))-10*d*e*f*x*exp(2*I*(d
*x+c))+2*f^2*x*exp(3*I*(d*x+c))+3*I*d*f^2*x^2*exp(3*I*(d*x+c))-2*I*d*e*f*x*exp(I*(d*x+c))-5*d*e^2*exp(2*I*(d*x
+c))+4*d*f^2*x^2+2*e*f*exp(3*I*(d*x+c))+2*I*exp(2*I*(d*x+c))*f*e+2*I*f^2*x*exp(2*I*(d*x+c))-2*I*f^2*x*exp(4*I*
(d*x+c))+8*d*e*f*x-2*f^2*x*exp(I*(d*x+c))-I*d*e^2*exp(I*(d*x+c))-I*d*f^2*x^2*exp(I*(d*x+c))+4*d*e^2-2*e*f*exp(
I*(d*x+c))-2*I*exp(4*I*(d*x+c))*f*e+3*I*d*e^2*exp(3*I*(d*x+c)))/(exp(2*I*(d*x+c))-1)^2/d^2/(exp(I*(d*x+c))+I)/
a+2*I*f^2*polylog(2,-exp(I*(d*x+c)))/a/d^3+4*I*f^2*polylog(2,I*exp(I*(d*x+c)))/a/d^3-2/a/d^2*f*e*ln(1+exp(2*I*
(d*x+c)))+8/a/d^2*f*e*ln(exp(I*(d*x+c)))-4/a/d^2*f^2*ln(1-I*exp(I*(d*x+c)))*x-4/a/d^3*f^2*ln(1-I*exp(I*(d*x+c)
))*c+2/a/d^3*f^2*c*ln(1+exp(2*I*(d*x+c)))-8/a/d^3*f^2*c*ln(exp(I*(d*x+c)))+3/2/d/a*e^2*ln(exp(I*(d*x+c))-1)-3/
2/d/a*e^2*ln(exp(I*(d*x+c))+1)-3/2/d^3/a*f^2*ln(1-exp(I*(d*x+c)))*c^2-3/d/a*e*f*ln(exp(I*(d*x+c))+1)*x+3/d/a*e
*f*ln(1-exp(I*(d*x+c)))*x+3/d^2/a*e*f*ln(1-exp(I*(d*x+c)))*c-3/d^2/a*c*e*f*ln(exp(I*(d*x+c))-1)-3*f^2*polylog(
3,-exp(I*(d*x+c)))/a/d^3+3*f^2*polylog(3,exp(I*(d*x+c)))/a/d^3

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. 4026 vs. \(2 (348) = 696\).

Time = 0.45 (sec) , antiderivative size = 4026, normalized size of antiderivative = 10.27 \[ \int \frac {(e+f x)^2 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/4*(4*d^2*f^2*x^2 + 4*d^2*e^2 - 8*(d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^2)*cos(d*x + c)^3 - 4*d*e*f - 2*(3*d^2*
f^2*x^2 + 3*d^2*e^2 - 2*d*e*f + 2*(3*d^2*e*f - d*f^2)*x)*cos(d*x + c)^2 + 4*(2*d^2*e*f - d*f^2)*x + 6*(d^2*f^2
*x^2 + 2*d^2*e*f*x + d^2*e^2)*cos(d*x + c) + 2*(-3*I*d*f^2*x + (3*I*d*f^2*x + 3*I*d*e*f - 2*I*f^2)*cos(d*x + c
)^3 - 3*I*d*e*f + (3*I*d*f^2*x + 3*I*d*e*f - 2*I*f^2)*cos(d*x + c)^2 + 2*I*f^2 + (-3*I*d*f^2*x - 3*I*d*e*f + 2
*I*f^2)*cos(d*x + c) + (-3*I*d*f^2*x - 3*I*d*e*f + (3*I*d*f^2*x + 3*I*d*e*f - 2*I*f^2)*cos(d*x + c)^2 + 2*I*f^
2)*sin(d*x + c))*dilog(cos(d*x + c) + I*sin(d*x + c)) + 2*(3*I*d*f^2*x + (-3*I*d*f^2*x - 3*I*d*e*f + 2*I*f^2)*
cos(d*x + c)^3 + 3*I*d*e*f + (-3*I*d*f^2*x - 3*I*d*e*f + 2*I*f^2)*cos(d*x + c)^2 - 2*I*f^2 + (3*I*d*f^2*x + 3*
I*d*e*f - 2*I*f^2)*cos(d*x + c) + (3*I*d*f^2*x + 3*I*d*e*f + (-3*I*d*f^2*x - 3*I*d*e*f + 2*I*f^2)*cos(d*x + c)
^2 - 2*I*f^2)*sin(d*x + c))*dilog(cos(d*x + c) - I*sin(d*x + c)) + 8*(-I*f^2*cos(d*x + c)^3 - I*f^2*cos(d*x +
c)^2 + I*f^2*cos(d*x + c) + I*f^2 + (-I*f^2*cos(d*x + c)^2 + I*f^2)*sin(d*x + c))*dilog(I*cos(d*x + c) - sin(d
*x + c)) + 8*(I*f^2*cos(d*x + c)^3 + I*f^2*cos(d*x + c)^2 - I*f^2*cos(d*x + c) - I*f^2 + (I*f^2*cos(d*x + c)^2
- I*f^2)*sin(d*x + c))*dilog(-I*cos(d*x + c) - sin(d*x + c)) + 2*(-3*I*d*f^2*x + (3*I*d*f^2*x + 3*I*d*e*f + 2
*I*f^2)*cos(d*x + c)^3 - 3*I*d*e*f + (3*I*d*f^2*x + 3*I*d*e*f + 2*I*f^2)*cos(d*x + c)^2 - 2*I*f^2 + (-3*I*d*f^
2*x - 3*I*d*e*f - 2*I*f^2)*cos(d*x + c) + (-3*I*d*f^2*x - 3*I*d*e*f + (3*I*d*f^2*x + 3*I*d*e*f + 2*I*f^2)*cos(
d*x + c)^2 - 2*I*f^2)*sin(d*x + c))*dilog(-cos(d*x + c) + I*sin(d*x + c)) + 2*(3*I*d*f^2*x + (-3*I*d*f^2*x - 3
*I*d*e*f - 2*I*f^2)*cos(d*x + c)^3 + 3*I*d*e*f + (-3*I*d*f^2*x - 3*I*d*e*f - 2*I*f^2)*cos(d*x + c)^2 + 2*I*f^2
+ (3*I*d*f^2*x + 3*I*d*e*f + 2*I*f^2)*cos(d*x + c) + (3*I*d*f^2*x + 3*I*d*e*f + (-3*I*d*f^2*x - 3*I*d*e*f - 2
*I*f^2)*cos(d*x + c)^2 + 2*I*f^2)*sin(d*x + c))*dilog(-cos(d*x + c) - I*sin(d*x + c)) - (3*d^2*f^2*x^2 + 3*d^2
*e^2 - (3*d^2*f^2*x^2 + 3*d^2*e^2 + 4*d*e*f + 2*f^2 + 2*(3*d^2*e*f + 2*d*f^2)*x)*cos(d*x + c)^3 + 4*d*e*f - (3
*d^2*f^2*x^2 + 3*d^2*e^2 + 4*d*e*f + 2*f^2 + 2*(3*d^2*e*f + 2*d*f^2)*x)*cos(d*x + c)^2 + 2*f^2 + 2*(3*d^2*e*f
+ 2*d*f^2)*x + (3*d^2*f^2*x^2 + 3*d^2*e^2 + 4*d*e*f + 2*f^2 + 2*(3*d^2*e*f + 2*d*f^2)*x)*cos(d*x + c) + (3*d^2
*f^2*x^2 + 3*d^2*e^2 + 4*d*e*f - (3*d^2*f^2*x^2 + 3*d^2*e^2 + 4*d*e*f + 2*f^2 + 2*(3*d^2*e*f + 2*d*f^2)*x)*cos
(d*x + c)^2 + 2*f^2 + 2*(3*d^2*e*f + 2*d*f^2)*x)*sin(d*x + c))*log(cos(d*x + c) + I*sin(d*x + c) + 1) + 8*((d*
e*f - c*f^2)*cos(d*x + c)^3 - d*e*f + c*f^2 + (d*e*f - c*f^2)*cos(d*x + c)^2 - (d*e*f - c*f^2)*cos(d*x + c) -
(d*e*f - c*f^2 - (d*e*f - c*f^2)*cos(d*x + c)^2)*sin(d*x + c))*log(cos(d*x + c) + I*sin(d*x + c) + I) - (3*d^2
*f^2*x^2 + 3*d^2*e^2 - (3*d^2*f^2*x^2 + 3*d^2*e^2 + 4*d*e*f + 2*f^2 + 2*(3*d^2*e*f + 2*d*f^2)*x)*cos(d*x + c)^
3 + 4*d*e*f - (3*d^2*f^2*x^2 + 3*d^2*e^2 + 4*d*e*f + 2*f^2 + 2*(3*d^2*e*f + 2*d*f^2)*x)*cos(d*x + c)^2 + 2*f^2
+ 2*(3*d^2*e*f + 2*d*f^2)*x + (3*d^2*f^2*x^2 + 3*d^2*e^2 + 4*d*e*f + 2*f^2 + 2*(3*d^2*e*f + 2*d*f^2)*x)*cos(d
*x + c) + (3*d^2*f^2*x^2 + 3*d^2*e^2 + 4*d*e*f - (3*d^2*f^2*x^2 + 3*d^2*e^2 + 4*d*e*f + 2*f^2 + 2*(3*d^2*e*f +
2*d*f^2)*x)*cos(d*x + c)^2 + 2*f^2 + 2*(3*d^2*e*f + 2*d*f^2)*x)*sin(d*x + c))*log(cos(d*x + c) - I*sin(d*x +
c) + 1) - 8*(d*f^2*x - (d*f^2*x + c*f^2)*cos(d*x + c)^3 + c*f^2 - (d*f^2*x + c*f^2)*cos(d*x + c)^2 + (d*f^2*x
+ c*f^2)*cos(d*x + c) + (d*f^2*x + c*f^2 - (d*f^2*x + c*f^2)*cos(d*x + c)^2)*sin(d*x + c))*log(I*cos(d*x + c)
+ sin(d*x + c) + 1) - 8*(d*f^2*x - (d*f^2*x + c*f^2)*cos(d*x + c)^3 + c*f^2 - (d*f^2*x + c*f^2)*cos(d*x + c)^2
+ (d*f^2*x + c*f^2)*cos(d*x + c) + (d*f^2*x + c*f^2 - (d*f^2*x + c*f^2)*cos(d*x + c)^2)*sin(d*x + c))*log(-I*
cos(d*x + c) + sin(d*x + c) + 1) + (3*d^2*e^2 - 2*(3*c + 2)*d*e*f - (3*d^2*e^2 - 2*(3*c + 2)*d*e*f + (3*c^2 +
4*c + 2)*f^2)*cos(d*x + c)^3 + (3*c^2 + 4*c + 2)*f^2 - (3*d^2*e^2 - 2*(3*c + 2)*d*e*f + (3*c^2 + 4*c + 2)*f^2)
*cos(d*x + c)^2 + (3*d^2*e^2 - 2*(3*c + 2)*d*e*f + (3*c^2 + 4*c + 2)*f^2)*cos(d*x + c) + (3*d^2*e^2 - 2*(3*c +
2)*d*e*f + (3*c^2 + 4*c + 2)*f^2 - (3*d^2*e^2 - 2*(3*c + 2)*d*e*f + (3*c^2 + 4*c + 2)*f^2)*cos(d*x + c)^2)*si
n(d*x + c))*log(-1/2*cos(d*x + c) + 1/2*I*sin(d*x + c) + 1/2) + (3*d^2*e^2 - 2*(3*c + 2)*d*e*f - (3*d^2*e^2 -
2*(3*c + 2)*d*e*f + (3*c^2 + 4*c + 2)*f^2)*cos(d*x + c)^3 + (3*c^2 + 4*c + 2)*f^2 - (3*d^2*e^2 - 2*(3*c + 2)*d
*e*f + (3*c^2 + 4*c + 2)*f^2)*cos(d*x + c)^2 + (3*d^2*e^2 - 2*(3*c + 2)*d*e*f + (3*c^2 + 4*c + 2)*f^2)*cos(d*x
+ c) + (3*d^2*e^2 - 2*(3*c + 2)*d*e*f + (3*c^2 + 4*c + 2)*f^2 - (3*d^2*e^2 - 2*(3*c + 2)*d*e*f + (3*c^2 + 4*c
+ 2)*f^2)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) - 1/2*I*sin(d*x + c) + 1/2) + (3*d^2*f^2*x^2 +
6*c*d*e*f - (3*d^2*f^2*x^2 + 6*c*d*e*f - (3*c^2 + 4*c)*f^2 + 2*(3*d^2*e*f - 2*d*f^2)*x)*cos(d*x + c)^3 - (3*c^
2 + 4*c)*f^2 - (3*d^2*f^2*x^2 + 6*c*d*e*f - (3*c^2 + 4*c)*f^2 + 2*(3*d^2*e*f - 2*d*f^2)*x)*cos(d*x + c)^2 + 2*
(3*d^2*e*f - 2*d*f^2)*x + (3*d^2*f^2*x^2 + 6*c*d*e*f - (3*c^2 + 4*c)*f^2 + 2*(3*d^2*e*f - 2*d*f^2)*x)*cos(d*x
+ c) + (3*d^2*f^2*x^2 + 6*c*d*e*f - (3*c^2 + 4*c)*f^2 - (3*d^2*f^2*x^2 + 6*c*d*e*f - (3*c^2 + 4*c)*f^2 + 2*(3*
d^2*e*f - 2*d*f^2)*x)*cos(d*x + c)^2 + 2*(3*d^2*e*f - 2*d*f^2)*x)*sin(d*x + c))*log(-cos(d*x + c) + I*sin(d*x
+ c) + 1) + 8*((d*e*f - c*f^2)*cos(d*x + c)^3 - d*e*f + c*f^2 + (d*e*f - c*f^2)*cos(d*x + c)^2 - (d*e*f - c*f^
2)*cos(d*x + c) - (d*e*f - c*f^2 - (d*e*f - c*f^2)*cos(d*x + c)^2)*sin(d*x + c))*log(-cos(d*x + c) + I*sin(d*x
+ c) + I) + (3*d^2*f^2*x^2 + 6*c*d*e*f - (3*d^2*f^2*x^2 + 6*c*d*e*f - (3*c^2 + 4*c)*f^2 + 2*(3*d^2*e*f - 2*d*
f^2)*x)*cos(d*x + c)^3 - (3*c^2 + 4*c)*f^2 - (3*d^2*f^2*x^2 + 6*c*d*e*f - (3*c^2 + 4*c)*f^2 + 2*(3*d^2*e*f - 2
*d*f^2)*x)*cos(d*x + c)^2 + 2*(3*d^2*e*f - 2*d*f^2)*x + (3*d^2*f^2*x^2 + 6*c*d*e*f - (3*c^2 + 4*c)*f^2 + 2*(3*
d^2*e*f - 2*d*f^2)*x)*cos(d*x + c) + (3*d^2*f^2*x^2 + 6*c*d*e*f - (3*c^2 + 4*c)*f^2 - (3*d^2*f^2*x^2 + 6*c*d*e
*f - (3*c^2 + 4*c)*f^2 + 2*(3*d^2*e*f - 2*d*f^2)*x)*cos(d*x + c)^2 + 2*(3*d^2*e*f - 2*d*f^2)*x)*sin(d*x + c))*
log(-cos(d*x + c) - I*sin(d*x + c) + 1) - 6*(f^2*cos(d*x + c)^3 + f^2*cos(d*x + c)^2 - f^2*cos(d*x + c) - f^2
+ (f^2*cos(d*x + c)^2 - f^2)*sin(d*x + c))*polylog(3, cos(d*x + c) + I*sin(d*x + c)) - 6*(f^2*cos(d*x + c)^3 +
f^2*cos(d*x + c)^2 - f^2*cos(d*x + c) - f^2 + (f^2*cos(d*x + c)^2 - f^2)*sin(d*x + c))*polylog(3, cos(d*x + c
) - I*sin(d*x + c)) + 6*(f^2*cos(d*x + c)^3 + f^2*cos(d*x + c)^2 - f^2*cos(d*x + c) - f^2 + (f^2*cos(d*x + c)^
2 - f^2)*sin(d*x + c))*polylog(3, -cos(d*x + c) + I*sin(d*x + c)) + 6*(f^2*cos(d*x + c)^3 + f^2*cos(d*x + c)^2
- f^2*cos(d*x + c) - f^2 + (f^2*cos(d*x + c)^2 - f^2)*sin(d*x + c))*polylog(3, -cos(d*x + c) - I*sin(d*x + c)
) - 2*(2*d^2*f^2*x^2 + 2*d^2*e^2 + 2*d*e*f - 4*(d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^2)*cos(d*x + c)^2 + 2*(2*d^2
*e*f + d*f^2)*x - (d^2*f^2*x^2 + d^2*e^2 - 2*d*e*f + 2*(d^2*e*f - d*f^2)*x)*cos(d*x + c))*sin(d*x + c))/(a*d^3
*cos(d*x + c)^3 + a*d^3*cos(d*x + c)^2 - a*d^3*cos(d*x + c) - a*d^3 + (a*d^3*cos(d*x + c)^2 - a*d^3)*sin(d*x +
c))

Sympy [F]

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

[In]

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

[Out]

(Integral(e**2*csc(c + d*x)**3/(sin(c + d*x) + 1), x) + Integral(f**2*x**2*csc(c + d*x)**3/(sin(c + d*x) + 1),
x) + Integral(2*e*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. 6160 vs. \(2 (348) = 696\).

Time = 2.29 (sec) , antiderivative size = 6160, normalized size of antiderivative = 15.71 \[ \int \frac {(e+f x)^2 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}