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

Optimal. Leaf size=392 \[ \frac{3 i f (e+f x) \text{PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac{3 i f (e+f x) \text{PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}+\frac{4 i f^2 \text{PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac{i f^2 \text{PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac{3 f^2 \text{PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac{3 f^2 \text{PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}-\frac{4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac{2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac{f (e+f x) \csc (c+d x)}{a d^2}-\frac{f^2 \tanh ^{-1}(\cos (c+d x))}{a d^3}+\frac{(e+f x)^2 \cot \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right )}{a d}+\frac{(e+f x)^2 \cot (c+d x)}{a d}-\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac{(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac{2 i (e+f x)^2}{a d} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.722168, antiderivative size = 392, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464, Rules used = {4535, 4186, 3770, 4183, 2531, 2282, 6589, 4184, 3717, 2190, 2279, 2391, 3318} \[ \frac{3 i f (e+f x) \text{PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac{3 i f (e+f x) \text{PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}+\frac{4 i f^2 \text{PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac{i f^2 \text{PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac{3 f^2 \text{PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac{3 f^2 \text{PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}-\frac{4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac{2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac{f (e+f x) \csc (c+d x)}{a d^2}-\frac{f^2 \tanh ^{-1}(\cos (c+d x))}{a d^3}+\frac{(e+f x)^2 \cot \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right )}{a d}+\frac{(e+f x)^2 \cot (c+d x)}{a d}-\frac{3 (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac{(e+f x)^2 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac{2 i (e+f x)^2}{a d} \]

Antiderivative was successfully verified.

[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 4535

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 4186

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]) /; FreeQ[
{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 3770

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

Rule 4183

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

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 4184

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 3717

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 2190

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

Rule 2279

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 2391

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 3318

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*(e + (Pi*a)/(2*b)))/2 + (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])

Rubi steps

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

Mathematica [B]  time = 17.4976, size = 951, normalized size = 2.43 \[ \frac{\frac{32 f (\cos (c)+i \sin (c)) \left (\frac{(\cos (c)-i \sin (c)) (e+f x)^2}{2 f}-\frac{\log (i \cos (c+d x)+\sin (c+d x)+1) (i \cos (c)+\sin (c)+1) (e+f x)}{d}+\frac{f \text{PolyLog}(2,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i (\sin (c)+1))}{d^2}\right ) d^2}{\cos (c)+i (\sin (c)+1)}-\frac{(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 ) d}{\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 \left (i f^2 x^2 d^2+2 i e f x d^2-3 e^2 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x)) d^2-3 f^2 x^2 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x)) d^2-6 e f x \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x)) d^2+f^2 x^2 \cot (c) d^2+2 e f x \cot (c) d^2-2 e f \log (-\cos (2 (c+d x))-i \sin (2 (c+d x))+1) d-2 f^2 x \log (-\cos (2 (c+d x))-i \sin (2 (c+d x))+1) d+3 i f (e+f x) \text{PolyLog}(2,-\cos (c+d x)-i \sin (c+d x)) d-3 i f (e+f x) \text{PolyLog}(2,\cos (c+d x)+i \sin (c+d x)) d-2 f^2 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x))+i f^2 \text{PolyLog}(2,\cos (2 (c+d x))+i \sin (2 (c+d x)))-3 f^2 \text{PolyLog}(3,-\cos (c+d x)-i \sin (c+d x))+3 f^2 \text{PolyLog}(3,\cos (c+d x)+i \sin (c+d x))\right )}{8 a d^3} \]

Warning: Unable to verify antiderivative.

[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)

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Maple [B]  time = 0.231, size = 1215, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a/d^3*f^2*ln(exp(I*(d*x+c))-1)-1/a/d^3*f^2*ln(exp(I*(d*x+c))+1)+8*f/d^2/a*ln(exp(I*(d*x+c)))*e-4*f/d^2/a*ln(
exp(I*(d*x+c))+I)*e-4*f^2/d^2/a*ln(1-I*exp(I*(d*x+c)))*x-4*f^2/d^3/a*ln(1-I*exp(I*(d*x+c)))*c-8*f^2/d^3/a*c*ln
(exp(I*(d*x+c)))+4*f^2/d^3/a*c*ln(exp(I*(d*x+c))+I)+3/2/a/d^3*f^2*c^2*ln(exp(I*(d*x+c))-1)+3/2/a/d*f^2*ln(1-ex
p(I*(d*x+c)))*x^2-3/2/a/d^3*f^2*ln(1-exp(I*(d*x+c)))*c^2-3/2/a/d*f^2*ln(exp(I*(d*x+c))+1)*x^2+3/2/a/d*e^2*ln(e
xp(I*(d*x+c))-1)-3/2/a/d*e^2*ln(exp(I*(d*x+c))+1)+(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))-I*d*f^2*x^2*exp(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))+2*I*f^2*x*exp(2*I*(d*x+c))+2*I*e*f*exp(2*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))-I*d*e^2*exp(I*(d*x+c))+3*I*d*f^2*x^2*exp(3*I*(d*x+c))+6*I*d*e*f*x*exp(3*I*(d*x+c))+8*d*
e*f*x-2*f^2*x*exp(I*(d*x+c))-2*I*d*e*f*x*exp(I*(d*x+c))+3*I*d*e^2*exp(3*I*(d*x+c))+4*d*e^2-2*exp(I*(d*x+c))*e*
f-2*I*f^2*x*exp(4*I*(d*x+c))-2*I*e*f*exp(4*I*(d*x+c)))/(exp(2*I*(d*x+c))-1)^2/d^2/(exp(I*(d*x+c))+I)/a+8*I/a/d
^2*c*f^2*x-3*I/a/d^2*polylog(2,exp(I*(d*x+c)))*f^2*x-3*I/a/d^2*e*f*polylog(2,exp(I*(d*x+c)))+3*I/a/d^2*e*f*pol
ylog(2,-exp(I*(d*x+c)))+3*I/a/d^2*polylog(2,-exp(I*(d*x+c)))*f^2*x+4*I*f^2*polylog(2,I*exp(I*(d*x+c)))/a/d^3+3
/a/d^2*ln(1-exp(I*(d*x+c)))*c*e*f+3/a/d*ln(1-exp(I*(d*x+c)))*e*f*x-3/a/d*ln(exp(I*(d*x+c))+1)*e*f*x-3/a/d^2*e*
f*c*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+2*I*f^2*
polylog(2,-exp(I*(d*x+c)))/a/d^3+2*I/a/d^3*f^2*polylog(2,exp(I*(d*x+c)))+4*I/a/d^3*c^2*f^2+4*I/a/d*f^2*x^2-2/a
/d^2*f^2*ln(exp(I*(d*x+c))+1)*x+2/a/d^3*f^2*c*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^3*f^2*ln(1-exp(I*(d*x+c)))*c

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Maxima [B]  time = 13.3787, size = 8266, normalized size = 21.09 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^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 - 16*I*c*f^2 + 16*(d*e*f -
c*f^2)*cos(5*d*x + 5*c) + (16*I*d*e*f - 16*I*c*f^2)*cos(4*d*x + 4*c) - 32*(d*e*f - c*f^2)*cos(3*d*x + 3*c) +
(-32*I*d*e*f + 32*I*c*f^2)*cos(2*d*x + 2*c) + 16*(d*e*f - c*f^2)*cos(d*x + c) + (16*I*d*e*f - 16*I*c*f^2)*sin(
5*d*x + 5*c) - 16*(d*e*f - c*f^2)*sin(4*d*x + 4*c) + (-32*I*d*e*f + 32*I*c*f^2)*sin(3*d*x + 3*c) + 32*(d*e*f -
c*f^2)*sin(2*d*x + 2*c) + (16*I*d*e*f - 16*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) + 16*I*(d*x + c)*f^2*cos(4*d*x + 4*c) - 32*(d*x + c)*f^2*cos(3*d*x + 3*c) -
32*I*(d*x + c)*f^2*cos(2*d*x + 2*c) + 16*(d*x + c)*f^2*cos(d*x + c) + 16*I*(d*x + c)*f^2*sin(5*d*x + 5*c) - 16
*(d*x + c)*f^2*sin(4*d*x + 4*c) - 32*I*(d*x + c)*f^2*sin(3*d*x + 3*c) + 32*(d*x + c)*f^2*sin(2*d*x + 2*c) + 16
*I*(d*x + c)*f^2*sin(d*x + c) + 16*I*(d*x + c)*f^2)*arctan2(cos(d*x + c), sin(d*x + c) + 1) + (6*I*(d*x + c)^2
*f^2 + 8*I*d*e*f + (6*I*c^2 - 8*I*c + 4*I)*f^2 + (12*I*d*e*f + (-12*I*c + 8*I)*f^2)*(d*x + 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(5*d*x + 5*c) + (6*I*(d*x
+ c)^2*f^2 + 8*I*d*e*f + (6*I*c^2 - 8*I*c + 4*I)*f^2 + (12*I*d*e*f + (-12*I*c + 8*I)*f^2)*(d*x + c))*cos(4*d*
x + 4*c) - 4*(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) + (-12*I*(d*x + c)^2*f^2 - 16*I*d*e*f + (-12*I*c^2 + 16*I*c - 8*I)*f^2 + (-24*I*d*e*f + (24*I*c
- 16*I)*f^2)*(d*x + c))*cos(2*d*x + 2*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(d*x + c) + (6*I*(d*x + c)^2*f^2 + 8*I*d*e*f + (6*I*c^2 - 8*I*c + 4*I)*f^2 + (
12*I*d*e*f + (-12*I*c + 8*I)*f^2)*(d*x + c))*sin(5*d*x + 5*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(4*d*x + 4*c) + (-12*I*(d*x + c)^2*f^2 - 16*I*d*e*f + (-1
2*I*c^2 + 16*I*c - 8*I)*f^2 + (-24*I*d*e*f + (24*I*c - 16*I)*f^2)*(d*x + c))*sin(3*d*x + 3*c) + 4*(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) + (6*I*(d*x
+ c)^2*f^2 + 8*I*d*e*f + (6*I*c^2 - 8*I*c + 4*I)*f^2 + (12*I*d*e*f + (-12*I*c + 8*I)*f^2)*(d*x + c))*sin(d*x
+ c))*arctan2(sin(d*x + c), cos(d*x + c) + 1) + (8*I*d*e*f + (-6*I*c^2 - 8*I*c - 4*I)*f^2 + 2*(4*d*e*f - (3*c^
2 + 4*c + 2)*f^2)*cos(5*d*x + 5*c) + (8*I*d*e*f + (-6*I*c^2 - 8*I*c - 4*I)*f^2)*cos(4*d*x + 4*c) - 4*(4*d*e*f
- (3*c^2 + 4*c + 2)*f^2)*cos(3*d*x + 3*c) + (-16*I*d*e*f + (12*I*c^2 + 16*I*c + 8*I)*f^2)*cos(2*d*x + 2*c) + 2
*(4*d*e*f - (3*c^2 + 4*c + 2)*f^2)*cos(d*x + c) + (8*I*d*e*f + (-6*I*c^2 - 8*I*c - 4*I)*f^2)*sin(5*d*x + 5*c)
- 2*(4*d*e*f - (3*c^2 + 4*c + 2)*f^2)*sin(4*d*x + 4*c) + (-16*I*d*e*f + (12*I*c^2 + 16*I*c + 8*I)*f^2)*sin(3*d
*x + 3*c) + 4*(4*d*e*f - (3*c^2 + 4*c + 2)*f^2)*sin(2*d*x + 2*c) + (8*I*d*e*f + (-6*I*c^2 - 8*I*c - 4*I)*f^2)*
sin(d*x + c))*arctan2(sin(d*x + c), cos(d*x + c) - 1) + (6*I*(d*x + c)^2*f^2 + (12*I*d*e*f + (-12*I*c - 8*I)*f
^2)*(d*x + c) + 2*(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) + (6*I*(d*x + c
)^2*f^2 + (12*I*d*e*f + (-12*I*c - 8*I)*f^2)*(d*x + c))*cos(4*d*x + 4*c) - 4*(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) + (-12*I*(d*x + c)^2*f^2 + (-24*I*d*e*f + (24*I*c + 16*I)*f^2)*(d*
x + c))*cos(2*d*x + 2*c) + 2*(3*(d*x + c)^2*f^2 + 2*(3*d*e*f - (3*c + 2)*f^2)*(d*x + c))*cos(d*x + c) + (6*I*(
d*x + c)^2*f^2 + (12*I*d*e*f + (-12*I*c - 8*I)*f^2)*(d*x + c))*sin(5*d*x + 5*c) - 2*(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) + (-12*I*(d*x + c)^2*f^2 + (-24*I*d*e*f + (24*I*c + 16*I)*f
^2)*(d*x + c))*sin(3*d*x + 3*c) + 4*(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) + (6*I*(d*x + c)^2*f^2 + (12*I*d*e*f + (-12*I*c - 8*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))*cos(5*d*x + 5*c) + (-4*I*(d*x + c)^2*f
^2 + 8*d*e*f + (12*I*c^2 - 8*c)*f^2 - 8*(I*d*e*f + (-I*c - 1)*f^2)*(d*x + c))*cos(4*d*x + 4*c) + (20*(d*x + c)
^2*f^2 + 8*I*d*e*f - 4*(3*c^2 + 2*I*c)*f^2 + (40*d*e*f - (40*c - 8*I)*f^2)*(d*x + c))*cos(3*d*x + 3*c) + (12*I
*(d*x + c)^2*f^2 - 8*d*e*f + (-20*I*c^2 + 8*c)*f^2 - 8*(-3*I*d*e*f + (3*I*c + 1)*f^2)*(d*x + c))*cos(2*d*x + 2
*c) - (12*(d*x + c)^2*f^2 + 8*I*d*e*f - 4*(c^2 + 2*I*c)*f^2 + (24*d*e*f - (24*c - 8*I)*f^2)*(d*x + c))*cos(d*x
+ c) - (16*f^2*cos(5*d*x + 5*c) + 16*I*f^2*cos(4*d*x + 4*c) - 32*f^2*cos(3*d*x + 3*c) - 32*I*f^2*cos(2*d*x +
2*c) + 16*f^2*cos(d*x + c) + 16*I*f^2*sin(5*d*x + 5*c) - 16*f^2*sin(4*d*x + 4*c) - 32*I*f^2*sin(3*d*x + 3*c) +
32*f^2*sin(2*d*x + 2*c) + 16*I*f^2*sin(d*x + c) + 16*I*f^2)*dilog(I*e^(I*d*x + I*c)) + (-12*I*d*e*f - 12*I*(d
*x + c)*f^2 + (12*I*c - 8*I)*f^2 - 4*(3*d*e*f + 3*(d*x + c)*f^2 - (3*c - 2)*f^2)*cos(5*d*x + 5*c) + (-12*I*d*e
*f - 12*I*(d*x + c)*f^2 + (12*I*c - 8*I)*f^2)*cos(4*d*x + 4*c) + 8*(3*d*e*f + 3*(d*x + c)*f^2 - (3*c - 2)*f^2)
*cos(3*d*x + 3*c) + (24*I*d*e*f + 24*I*(d*x + c)*f^2 + (-24*I*c + 16*I)*f^2)*cos(2*d*x + 2*c) - 4*(3*d*e*f + 3
*(d*x + c)*f^2 - (3*c - 2)*f^2)*cos(d*x + c) + (-12*I*d*e*f - 12*I*(d*x + c)*f^2 + (12*I*c - 8*I)*f^2)*sin(5*d
*x + 5*c) + 4*(3*d*e*f + 3*(d*x + c)*f^2 - (3*c - 2)*f^2)*sin(4*d*x + 4*c) + (24*I*d*e*f + 24*I*(d*x + c)*f^2
+ (-24*I*c + 16*I)*f^2)*sin(3*d*x + 3*c) - 8*(3*d*e*f + 3*(d*x + c)*f^2 - (3*c - 2)*f^2)*sin(2*d*x + 2*c) + (-
12*I*d*e*f - 12*I*(d*x + c)*f^2 + (12*I*c - 8*I)*f^2)*sin(d*x + c))*dilog(-e^(I*d*x + I*c)) + (12*I*d*e*f + 12
*I*(d*x + c)*f^2 + (-12*I*c - 8*I)*f^2 + 4*(3*d*e*f + 3*(d*x + c)*f^2 - (3*c + 2)*f^2)*cos(5*d*x + 5*c) + (12*
I*d*e*f + 12*I*(d*x + c)*f^2 + (-12*I*c - 8*I)*f^2)*cos(4*d*x + 4*c) - 8*(3*d*e*f + 3*(d*x + c)*f^2 - (3*c + 2
)*f^2)*cos(3*d*x + 3*c) + (-24*I*d*e*f - 24*I*(d*x + c)*f^2 + (24*I*c + 16*I)*f^2)*cos(2*d*x + 2*c) + 4*(3*d*e
*f + 3*(d*x + c)*f^2 - (3*c + 2)*f^2)*cos(d*x + c) + (12*I*d*e*f + 12*I*(d*x + c)*f^2 + (-12*I*c - 8*I)*f^2)*s
in(5*d*x + 5*c) - 4*(3*d*e*f + 3*(d*x + c)*f^2 - (3*c + 2)*f^2)*sin(4*d*x + 4*c) + (-24*I*d*e*f - 24*I*(d*x +
c)*f^2 + (24*I*c + 16*I)*f^2)*sin(3*d*x + 3*c) + 8*(3*d*e*f + 3*(d*x + c)*f^2 - (3*c + 2)*f^2)*sin(2*d*x + 2*c
) + (12*I*d*e*f + 12*I*(d*x + c)*f^2 + (-12*I*c - 8*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 + (-6*I*d*e*f + (6*I*c - 4*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) + (6*
I*(d*x + c)^2*f^2 + 8*I*d*e*f + (6*I*c^2 - 8*I*c + 4*I)*f^2 + (12*I*d*e*f + (-12*I*c + 8*I)*f^2)*(d*x + c))*co
s(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 + (-6*I*d*e*f + (6*I*c
- 4*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 + (6*I
*d*e*f + (-6*I*c + 4*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) + (-6*I*(d*x + c)^2*f^2 - 8*I*d*e*f + (-6*I*c^2
+ 8*I*c - 4*I)*f^2 + (-12*I*d*e*f + (12*I*c - 8*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 + (6*I*d*e*f + (-6*I*c - 4*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) - (-6*I*(d*x + c)^2*f^2 + 8*I*d*e*f + (-6*I*c^2 - 8*I*c - 4*I)*f^2 + (-1
2*I*d*e*f + (12*I*c + 8*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 + (6*I*d*e*f + (-6*I*c - 4*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 + (-6*I*d*e*f + (6*I*c + 4*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) - (
6*I*(d*x + c)^2*f^2 - 8*I*d*e*f + (6*I*c^2 + 8*I*c + 4*I)*f^2 + (12*I*d*e*f + (-12*I*c - 8*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 + 8*(d*x + c)*f^2 - 8*c*
f^2 + (-8*I*d*e*f - 8*I*(d*x + c)*f^2 + 8*I*c*f^2)*cos(5*d*x + 5*c) + 8*(d*e*f + (d*x + c)*f^2 - c*f^2)*cos(4*
d*x + 4*c) + (16*I*d*e*f + 16*I*(d*x + c)*f^2 - 16*I*c*f^2)*cos(3*d*x + 3*c) - 16*(d*e*f + (d*x + c)*f^2 - c*f
^2)*cos(2*d*x + 2*c) + (-8*I*d*e*f - 8*I*(d*x + c)*f^2 + 8*I*c*f^2)*cos(d*x + c) + 8*(d*e*f + (d*x + c)*f^2 -
c*f^2)*sin(5*d*x + 5*c) + (8*I*d*e*f + 8*I*(d*x + c)*f^2 - 8*I*c*f^2)*sin(4*d*x + 4*c) - 16*(d*e*f + (d*x + c)
*f^2 - c*f^2)*sin(3*d*x + 3*c) + (-16*I*d*e*f - 16*I*(d*x + c)*f^2 + 16*I*c*f^2)*sin(2*d*x + 2*c) + 8*(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*c
os(5*d*x + 5*c) + 12*f^2*cos(4*d*x + 4*c) + 24*I*f^2*cos(3*d*x + 3*c) - 24*f^2*cos(2*d*x + 2*c) - 12*I*f^2*cos
(d*x + c) + 12*f^2*sin(5*d*x + 5*c) + 12*I*f^2*sin(4*d*x + 4*c) - 24*f^2*sin(3*d*x + 3*c) - 24*I*f^2*sin(2*d*x
+ 2*c) + 12*f^2*sin(d*x + c) + 12*f^2)*polylog(3, -e^(I*d*x + I*c)) + (12*I*f^2*cos(5*d*x + 5*c) - 12*f^2*cos
(4*d*x + 4*c) - 24*I*f^2*cos(3*d*x + 3*c) + 24*f^2*cos(2*d*x + 2*c) + 12*I*f^2*cos(d*x + c) - 12*f^2*sin(5*d*x
+ 5*c) - 12*I*f^2*sin(4*d*x + 4*c) + 24*f^2*sin(3*d*x + 3*c) + 24*I*f^2*sin(2*d*x + 2*c) - 12*f^2*sin(d*x + c
) - 12*f^2)*polylog(3, e^(I*d*x + I*c)) + (-16*I*(d*x + c)^2*f^2 + (-32*I*d*e*f + 32*I*c*f^2)*(d*x + c))*sin(5
*d*x + 5*c) + (4*(d*x + c)^2*f^2 + 8*I*d*e*f - 4*(3*c^2 + 2*I*c)*f^2 + (8*d*e*f - (8*c - 8*I)*f^2)*(d*x + c))*
sin(4*d*x + 4*c) + (20*I*(d*x + c)^2*f^2 - 8*d*e*f + (-12*I*c^2 + 8*c)*f^2 - 8*(-5*I*d*e*f + (5*I*c + 1)*f^2)*
(d*x + c))*sin(3*d*x + 3*c) - (12*(d*x + c)^2*f^2 + 8*I*d*e*f - 4*(5*c^2 + 2*I*c)*f^2 + (24*d*e*f - (24*c - 8*
I)*f^2)*(d*x + c))*sin(2*d*x + 2*c) + (-12*I*(d*x + c)^2*f^2 + 8*d*e*f + (4*I*c^2 - 8*c)*f^2 - 8*(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*s
in(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

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Fricas [C]  time = 3.54559, size = 9528, normalized size = 24.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^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) - (6*I*d*f^2*x + (-6*I*d*f^2*x - 6*I*d*e*f + 4*I*f^2)*cos(d*x + c)^
3 + 6*I*d*e*f + (-6*I*d*f^2*x - 6*I*d*e*f + 4*I*f^2)*cos(d*x + c)^2 - 4*I*f^2 + (6*I*d*f^2*x + 6*I*d*e*f - 4*I
*f^2)*cos(d*x + c) + (6*I*d*f^2*x + 6*I*d*e*f + (-6*I*d*f^2*x - 6*I*d*e*f + 4*I*f^2)*cos(d*x + c)^2 - 4*I*f^2)
*sin(d*x + c))*dilog(cos(d*x + c) + I*sin(d*x + c)) - (-6*I*d*f^2*x + (6*I*d*f^2*x + 6*I*d*e*f - 4*I*f^2)*cos(
d*x + c)^3 - 6*I*d*e*f + (6*I*d*f^2*x + 6*I*d*e*f - 4*I*f^2)*cos(d*x + c)^2 + 4*I*f^2 + (-6*I*d*f^2*x - 6*I*d*
e*f + 4*I*f^2)*cos(d*x + c) + (-6*I*d*f^2*x - 6*I*d*e*f + (6*I*d*f^2*x + 6*I*d*e*f - 4*I*f^2)*cos(d*x + c)^2 +
4*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 + 8*I*f^2*cos(d*x + c)^
2 - 8*I*f^2*cos(d*x + c) - 8*I*f^2 + (8*I*f^2*cos(d*x + c)^2 - 8*I*f^2)*sin(d*x + c))*dilog(I*cos(d*x + c) - s
in(d*x + c)) - (-8*I*f^2*cos(d*x + c)^3 - 8*I*f^2*cos(d*x + c)^2 + 8*I*f^2*cos(d*x + c) + 8*I*f^2 + (-8*I*f^2*
cos(d*x + c)^2 + 8*I*f^2)*sin(d*x + c))*dilog(-I*cos(d*x + c) - sin(d*x + c)) - (6*I*d*f^2*x + (-6*I*d*f^2*x -
6*I*d*e*f - 4*I*f^2)*cos(d*x + c)^3 + 6*I*d*e*f + (-6*I*d*f^2*x - 6*I*d*e*f - 4*I*f^2)*cos(d*x + c)^2 + 4*I*f
^2 + (6*I*d*f^2*x + 6*I*d*e*f + 4*I*f^2)*cos(d*x + c) + (6*I*d*f^2*x + 6*I*d*e*f + (-6*I*d*f^2*x - 6*I*d*e*f -
4*I*f^2)*cos(d*x + c)^2 + 4*I*f^2)*sin(d*x + c))*dilog(-cos(d*x + c) + I*sin(d*x + c)) - (-6*I*d*f^2*x + (6*I
*d*f^2*x + 6*I*d*e*f + 4*I*f^2)*cos(d*x + c)^3 - 6*I*d*e*f + (6*I*d*f^2*x + 6*I*d*e*f + 4*I*f^2)*cos(d*x + c)^
2 - 4*I*f^2 + (-6*I*d*f^2*x - 6*I*d*e*f - 4*I*f^2)*cos(d*x + c) + (-6*I*d*f^2*x - 6*I*d*e*f + (6*I*d*f^2*x + 6
*I*d*e*f + 4*I*f^2)*cos(d*x + c)^2 - 4*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)*c
os(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*c
os(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)*co
s(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)*sin(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)*si
n(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*c
os(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*co
s(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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)