3.209 \(\int \frac{(e+f x)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx\)
Optimal. Leaf size=600 \[ \frac{12 i f^2 (e+f x) \text{PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac{3 i f^2 (e+f x) \text{PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac{9 f^2 (e+f x) \text{PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac{9 f^2 (e+f x) \text{PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac{9 i f (e+f x)^2 \text{PolyLog}\left (2,-e^{i (c+d x)}\right )}{2 a d^2}-\frac{9 i f (e+f x)^2 \text{PolyLog}\left (2,e^{i (c+d x)}\right )}{2 a d^2}+\frac{3 i f^3 \text{PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^4}-\frac{3 i f^3 \text{PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^4}-\frac{12 f^3 \text{PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}-\frac{3 f^3 \text{PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}-\frac{9 i f^3 \text{PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}+\frac{9 i f^3 \text{PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac{6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac{3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac{3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}+\frac{(e+f x)^3 \cot \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right )}{a d}+\frac{(e+f x)^3 \cot (c+d x)}{a d}-\frac{3 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac{(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac{2 i (e+f x)^3}{a d} \]
[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)
________________________________________________________________________________________
Rubi [A] time = 1.10824, antiderivative size = 600, normalized size of antiderivative = 1.,
number of steps used = 40, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464, Rules
used = {4535, 4186, 4183, 2279, 2391, 2531, 6609, 2282, 6589, 4184, 3717, 2190, 3318}
\[ \frac{12 i f^2 (e+f x) \text{PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac{3 i f^2 (e+f x) \text{PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac{9 f^2 (e+f x) \text{PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac{9 f^2 (e+f x) \text{PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac{9 i f (e+f x)^2 \text{PolyLog}\left (2,-e^{i (c+d x)}\right )}{2 a d^2}-\frac{9 i f (e+f x)^2 \text{PolyLog}\left (2,e^{i (c+d x)}\right )}{2 a d^2}+\frac{3 i f^3 \text{PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^4}-\frac{3 i f^3 \text{PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^4}-\frac{12 f^3 \text{PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}-\frac{3 f^3 \text{PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}-\frac{9 i f^3 \text{PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}+\frac{9 i f^3 \text{PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac{6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac{3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac{3 f (e+f x)^2 \csc (c+d x)}{2 a d^2}+\frac{(e+f x)^3 \cot \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right )}{a d}+\frac{(e+f x)^3 \cot (c+d x)}{a d}-\frac{3 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac{(e+f x)^3 \cot (c+d x) \csc (c+d x)}{2 a d}+\frac{2 i (e+f x)^3}{a d} \]
Antiderivative was successfully verified.
[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 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 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 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 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 6609
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]
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 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)^3 \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\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) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac{(e+f x)^3 \tanh ^{-1}\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) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac{3 (e+f x)^3 \tanh ^{-1}\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 \text{Li}_2\left (-e^{i (c+d x)}\right )}{2 a d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (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) \text{Li}_2\left (-e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac{\left (3 i f^2\right ) \int (e+f x) \text{Li}_2\left (e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac{\left (3 i f^3\right ) \operatorname{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 ) \operatorname{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) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac{3 (e+f x)^3 \tanh ^{-1}\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 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^4}+\frac{9 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{2 a d^2}-\frac{3 i f^3 \text{Li}_2\left (e^{i (c+d x)}\right )}{a d^4}-\frac{9 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{2 a d^2}-\frac{3 f^2 (e+f x) \text{Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac{3 f^2 (e+f x) \text{Li}_3\left (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) \text{Li}_2\left (-e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac{\left (6 i f^2\right ) \int (e+f x) \text{Li}_2\left (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 \text{Li}_3\left (-e^{i (c+d x)}\right ) \, dx}{a d^3}-\frac{\left (3 f^3\right ) \int \text{Li}_3\left (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) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac{3 (e+f x)^3 \tanh ^{-1}\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 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^4}+\frac{9 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{2 a d^2}-\frac{3 i f^3 \text{Li}_2\left (e^{i (c+d x)}\right )}{a d^4}-\frac{9 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{2 a d^2}+\frac{3 i f^2 (e+f x) \text{Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}-\frac{9 f^2 (e+f x) \text{Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac{9 f^2 (e+f x) \text{Li}_3\left (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 ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac{\left (3 i f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}-\frac{\left (3 i f^3\right ) \int \text{Li}_2\left (e^{2 i (c+d x)}\right ) \, dx}{a d^3}+\frac{\left (6 f^3\right ) \int \text{Li}_3\left (-e^{i (c+d x)}\right ) \, dx}{a d^3}-\frac{\left (6 f^3\right ) \int \text{Li}_3\left (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) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac{3 (e+f x)^3 \tanh ^{-1}\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 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^4}+\frac{9 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{2 a d^2}-\frac{3 i f^3 \text{Li}_2\left (e^{i (c+d x)}\right )}{a d^4}-\frac{9 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{2 a d^2}+\frac{3 i f^2 (e+f x) \text{Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}-\frac{9 f^2 (e+f x) \text{Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac{9 f^2 (e+f x) \text{Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\frac{3 i f^3 \text{Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac{3 i f^3 \text{Li}_4\left (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 ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac{\left (6 i f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}-\frac{\left (3 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_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) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac{3 (e+f x)^3 \tanh ^{-1}\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 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^4}+\frac{9 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{2 a d^2}+\frac{12 i f^2 (e+f x) \text{Li}_2\left (i e^{i (c+d x)}\right )}{a d^3}-\frac{3 i f^3 \text{Li}_2\left (e^{i (c+d x)}\right )}{a d^4}-\frac{9 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{2 a d^2}+\frac{3 i f^2 (e+f x) \text{Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}-\frac{9 f^2 (e+f x) \text{Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac{9 f^2 (e+f x) \text{Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\frac{3 f^3 \text{Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^4}-\frac{9 i f^3 \text{Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac{9 i f^3 \text{Li}_4\left (e^{i (c+d x)}\right )}{a d^4}-\frac{\left (12 i f^3\right ) \int \text{Li}_2\left (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) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac{3 (e+f x)^3 \tanh ^{-1}\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 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^4}+\frac{9 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{2 a d^2}+\frac{12 i f^2 (e+f x) \text{Li}_2\left (i e^{i (c+d x)}\right )}{a d^3}-\frac{3 i f^3 \text{Li}_2\left (e^{i (c+d x)}\right )}{a d^4}-\frac{9 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{2 a d^2}+\frac{3 i f^2 (e+f x) \text{Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}-\frac{9 f^2 (e+f x) \text{Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac{9 f^2 (e+f x) \text{Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\frac{3 f^3 \text{Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^4}-\frac{9 i f^3 \text{Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac{9 i f^3 \text{Li}_4\left (e^{i (c+d x)}\right )}{a d^4}-\frac{\left (12 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_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) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}-\frac{3 (e+f x)^3 \tanh ^{-1}\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 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^4}+\frac{9 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{2 a d^2}+\frac{12 i f^2 (e+f x) \text{Li}_2\left (i e^{i (c+d x)}\right )}{a d^3}-\frac{3 i f^3 \text{Li}_2\left (e^{i (c+d x)}\right )}{a d^4}-\frac{9 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{2 a d^2}+\frac{3 i f^2 (e+f x) \text{Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}-\frac{9 f^2 (e+f x) \text{Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}-\frac{12 f^3 \text{Li}_3\left (i e^{i (c+d x)}\right )}{a d^4}+\frac{9 f^2 (e+f x) \text{Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\frac{3 f^3 \text{Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^4}-\frac{9 i f^3 \text{Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac{9 i f^3 \text{Li}_4\left (e^{i (c+d x)}\right )}{a d^4}\\ \end{align*}
Mathematica [B] time = 31.369, size = 1485, normalized size = 2.48 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[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*(-1 + E^((2*I)*c))*(d*x*PolyLog[2, -E^((-I)*(c +
d*x))] - I*PolyLog[3, -E^((-I)*(c + d*x))]))/E^((2*I)*c) - (6*(-1 + E^((2*I)*c))*(d*x*PolyLog[2, E^((-I)*(c +
d*x))] - I*PolyLog[3, E^((-I)*(c + d*x))]))/E^((2*I)*c)))/(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]) + Log[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]])*(C
os[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 + Sin[c]^2)])
________________________________________________________________________________________
Maple [B] time = 0.286, size = 2257, normalized size = 3.8 \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)^3*csc(d*x+c)^3/(a+a*sin(d*x+c)),x)
[Out]
-3/d^4/a*f^3*c*ln(exp(I*(d*x+c))-1)+3/d^3/a*e*f^2*ln(exp(I*(d*x+c))-1)-3/d^3/a*e*f^2*ln(exp(I*(d*x+c))+1)-3/d^
2/a*f^3*ln(exp(I*(d*x+c))+1)*x^2-3/d^2/a*f^3*ln(1-exp(I*(d*x+c)))*x^2+3/d^4/a*f^3*ln(1-exp(I*(d*x+c)))*c^2-3/d
^4/a*f^3*c^2*ln(exp(I*(d*x+c))-1)-3/d^2/a*e^2*f*ln(exp(I*(d*x+c))-1)-3/d^2/a*e^2*f*ln(exp(I*(d*x+c))+1)-12*f^2
/d^2/a*e*ln(1-I*exp(I*(d*x+c)))*x+24*I/d^2/a*c*e*f^2*x+3/2/d/a*e^3*ln(exp(I*(d*x+c))-1)-3/2/d/a*e^3*ln(exp(I*(
d*x+c))+1)-9/2*I/d^2/a*f^3*polylog(2,exp(I*(d*x+c)))*x^2+12*I/d^3/a*c^2*e*f^2-12*I/d^3/a*f^3*c^2*x+9/2*I/d^2/a
*f^3*polylog(2,-exp(I*(d*x+c)))*x^2+6*I/d^3/a*f^3*polylog(2,-exp(I*(d*x+c)))*x+6*I/d^3/a*f^3*polylog(2,exp(I*(
d*x+c)))*x+12*I/d/a*e*f^2*x^2-9/2*I/d^2/a*e^2*f*polylog(2,exp(I*(d*x+c)))+9/2*I/d^2/a*e^2*f*polylog(2,-exp(I*(
d*x+c)))+6*I/d^3/a*e*f^2*polylog(2,exp(I*(d*x+c)))+6*I/d^3/a*e*f^2*polylog(2,-exp(I*(d*x+c)))+6/d^3/a*e*f^2*c*
ln(exp(I*(d*x+c))-1)+3/d^3/a*f^3*ln(1-exp(I*(d*x+c)))*x+3/d^4/a*f^3*ln(1-exp(I*(d*x+c)))*c-3/d^3/a*f^3*ln(exp(
I*(d*x+c))+1)*x-8*I/d^4/a*f^3*c^3+4*I/d/a*f^3*x^3+12*f^3/d^4/a*c^2*ln(exp(I*(d*x+c)))-6*f/d^2/a*ln(exp(I*(d*x+
c))+I)*e^2-6*f^3/d^4/a*c^2*ln(exp(I*(d*x+c))+I)+12*f/d^2/a*ln(exp(I*(d*x+c)))*e^2-9/2/d^2/a*e^2*f*c*ln(exp(I*(
d*x+c))-1)+9/2/d^3/a*e*f^2*c^2*ln(exp(I*(d*x+c))-1)-3/2/d/a*f^3*ln(exp(I*(d*x+c))+1)*x^3-12*f^2/d^3/a*e*ln(1-I
*exp(I*(d*x+c)))*c-6*f^3/d^2/a*ln(1-I*exp(I*(d*x+c)))*x^2+6*f^3/d^4/a*ln(1-I*exp(I*(d*x+c)))*c^2-24*f^2/d^3/a*
e*c*ln(exp(I*(d*x+c)))+12*f^2/d^3/a*e*c*ln(exp(I*(d*x+c))+I)+12*I*f^3/d^3/a*polylog(2,I*exp(I*(d*x+c)))*x+12*I
*f^2/d^3/a*e*polylog(2,I*exp(I*(d*x+c)))+3*I*f^3*polylog(2,-exp(I*(d*x+c)))/a/d^4+9*I*f^3*polylog(4,exp(I*(d*x
+c)))/a/d^4-6*f^3*polylog(3,-exp(I*(d*x+c)))/a/d^4-6*f^3*polylog(3,exp(I*(d*x+c)))/a/d^4+3/2/d/a*f^3*ln(1-exp(
I*(d*x+c)))*x^3+3/2/d^4/a*f^3*ln(1-exp(I*(d*x+c)))*c^3-9/2/d/a*e*f^2*ln(exp(I*(d*x+c))+1)*x^2+9/2/d/a*ln(1-exp
(I*(d*x+c)))*e^2*f*x-9/2/d/a*ln(exp(I*(d*x+c))+1)*e^2*f*x+9/2/d/a*e*f^2*ln(1-exp(I*(d*x+c)))*x^2-9/2/d^3/a*e*f
^2*ln(1-exp(I*(d*x+c)))*c^2+9/2/d^2/a*ln(1-exp(I*(d*x+c)))*c*e^2*f-3*I*f^3*polylog(2,exp(I*(d*x+c)))/a/d^4+9*I
/d^2/a*polylog(2,-exp(I*(d*x+c)))*e*f^2*x-9*I/d^2/a*polylog(2,exp(I*(d*x+c)))*e*f^2*x-9*I*f^3*polylog(4,-exp(I
*(d*x+c)))/a/d^4-12*f^3*polylog(3,I*exp(I*(d*x+c)))/a/d^4+9/d^3/a*f^3*polylog(3,exp(I*(d*x+c)))*x-9/d^3/a*f^3*
polylog(3,-exp(I*(d*x+c)))*x-3/2/d^4/a*f^3*c^3*ln(exp(I*(d*x+c))-1)+9/d^3/a*e*f^2*polylog(3,exp(I*(d*x+c)))-9/
d^3/a*e*f^2*polylog(3,-exp(I*(d*x+c)))-6/d^2/a*e*f^2*ln(1-exp(I*(d*x+c)))*x-6/d^3/a*e*f^2*ln(1-exp(I*(d*x+c)))
*c-6/d^2/a*e*f^2*ln(exp(I*(d*x+c))+1)*x+(3*d*f^3*x^3*exp(4*I*(d*x+c))+6*e*f^2*x*exp(3*I*(d*x+c))+3*I*d*e^3*exp
(3*I*(d*x+c))+9*I*d*e*f^2*x^2*exp(3*I*(d*x+c))-3*I*f^3*x^2*exp(4*I*(d*x+c))-3*I*e^2*f*exp(4*I*(d*x+c))-5*d*f^3
*x^3*exp(2*I*(d*x+c))+3*I*e^2*f*exp(2*I*(d*x+c))+3*I*f^3*x^2*exp(2*I*(d*x+c))+4*d*e^3-5*d*e^3*exp(2*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*e^2*f*exp(3*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))+4*d*f^3*x^3-3*f^3*x^2*exp(I*(d*x+c))-3*exp(I*(d*x+c))*e^2*f-I*d*e^3*exp(I*(d*x+c))
-6*e*f^2*x*exp(I*(d*x+c))+12*d*e*f^2*x^2+12*d*e^2*f*x-I*d*f^3*x^3*exp(I*(d*x+c))+9*d*e^2*f*x*exp(4*I*(d*x+c))-
6*I*e*f^2*x*exp(4*I*(d*x+c))+3*I*d*f^3*x^3*exp(3*I*(d*x+c))+6*I*e*f^2*x*exp(2*I*(d*x+c))-15*d*e*f^2*x^2*exp(2*
I*(d*x+c))-15*d*e^2*f*x*exp(2*I*(d*x+c))+9*d*e*f^2*x^2*exp(4*I*(d*x+c))+9*I*d*e^2*f*x*exp(3*I*(d*x+c)))/(exp(2
*I*(d*x+c))-1)^2/d^2/(exp(I*(d*x+c))+I)/a
________________________________________________________________________________________
Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^3*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")
[Out]
Timed out
________________________________________________________________________________________
Fricas [C] time = 5.57135, size = 17747, normalized size = 29.58 \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)^3*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")
[Out]
-1/4*(4*d^3*f^3*x^3 + 4*d^3*e^3 - 6*d^2*e^2*f - 8*(d^3*f^3*x^3 + 3*d^3*e*f^2*x^2 + 3*d^3*e^2*f*x + d^3*e^3)*co
s(d*x + c)^3 + 6*(2*d^3*e*f^2 - d^2*f^3)*x^2 - 6*(d^3*f^3*x^3 + d^3*e^3 - d^2*e^2*f + (3*d^3*e*f^2 - d^2*f^3)*
x^2 + (3*d^3*e^2*f - 2*d^2*e*f^2)*x)*cos(d*x + c)^2 + 12*(d^3*e^2*f - d^2*e*f^2)*x + 6*(d^3*f^3*x^3 + 3*d^3*e*
f^2*x^2 + 3*d^3*e^2*f*x + d^3*e^3)*cos(d*x + c) - (9*I*d^2*f^3*x^2 + 9*I*d^2*e^2*f - 12*I*d*e*f^2 + (-9*I*d^2*
f^3*x^2 - 9*I*d^2*e^2*f + 12*I*d*e*f^2 - 6*I*f^3 - 6*I*(3*d^2*e*f^2 - 2*d*f^3)*x)*cos(d*x + c)^3 + 6*I*f^3 + (
-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f + 12*I*d*e*f^2 - 6*I*f^3 - 6*I*(3*d^2*e*f^2 - 2*d*f^3)*x)*cos(d*x + c)^2 + 6*
I*(3*d^2*e*f^2 - 2*d*f^3)*x + (9*I*d^2*f^3*x^2 + 9*I*d^2*e^2*f - 12*I*d*e*f^2 + 6*I*f^3 + 6*I*(3*d^2*e*f^2 - 2
*d*f^3)*x)*cos(d*x + c) + (9*I*d^2*f^3*x^2 + 9*I*d^2*e^2*f - 12*I*d*e*f^2 + 6*I*f^3 + (-9*I*d^2*f^3*x^2 - 9*I*
d^2*e^2*f + 12*I*d*e*f^2 - 6*I*f^3 - 6*I*(3*d^2*e*f^2 - 2*d*f^3)*x)*cos(d*x + c)^2 + 6*I*(3*d^2*e*f^2 - 2*d*f^
3)*x)*sin(d*x + c))*dilog(cos(d*x + c) + I*sin(d*x + c)) - (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f + 12*I*d*e*f^2 +
(9*I*d^2*f^3*x^2 + 9*I*d^2*e^2*f - 12*I*d*e*f^2 + 6*I*f^3 + 6*I*(3*d^2*e*f^2 - 2*d*f^3)*x)*cos(d*x + c)^3 - 6*
I*f^3 + (9*I*d^2*f^3*x^2 + 9*I*d^2*e^2*f - 12*I*d*e*f^2 + 6*I*f^3 + 6*I*(3*d^2*e*f^2 - 2*d*f^3)*x)*cos(d*x + c
)^2 - 6*I*(3*d^2*e*f^2 - 2*d*f^3)*x + (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f + 12*I*d*e*f^2 - 6*I*f^3 - 6*I*(3*d^2*
e*f^2 - 2*d*f^3)*x)*cos(d*x + c) + (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f + 12*I*d*e*f^2 - 6*I*f^3 + (9*I*d^2*f^3*x
^2 + 9*I*d^2*e^2*f - 12*I*d*e*f^2 + 6*I*f^3 + 6*I*(3*d^2*e*f^2 - 2*d*f^3)*x)*cos(d*x + c)^2 - 6*I*(3*d^2*e*f^2
- 2*d*f^3)*x)*sin(d*x + c))*dilog(cos(d*x + c) - I*sin(d*x + c)) - (-24*I*d*f^3*x - 24*I*d*e*f^2 + (24*I*d*f^
3*x + 24*I*d*e*f^2)*cos(d*x + c)^3 + (24*I*d*f^3*x + 24*I*d*e*f^2)*cos(d*x + c)^2 + (-24*I*d*f^3*x - 24*I*d*e*
f^2)*cos(d*x + c) + (-24*I*d*f^3*x - 24*I*d*e*f^2 + (24*I*d*f^3*x + 24*I*d*e*f^2)*cos(d*x + c)^2)*sin(d*x + c)
)*dilog(I*cos(d*x + c) - sin(d*x + c)) - (24*I*d*f^3*x + 24*I*d*e*f^2 + (-24*I*d*f^3*x - 24*I*d*e*f^2)*cos(d*x
+ c)^3 + (-24*I*d*f^3*x - 24*I*d*e*f^2)*cos(d*x + c)^2 + (24*I*d*f^3*x + 24*I*d*e*f^2)*cos(d*x + c) + (24*I*d
*f^3*x + 24*I*d*e*f^2 + (-24*I*d*f^3*x - 24*I*d*e*f^2)*cos(d*x + c)^2)*sin(d*x + c))*dilog(-I*cos(d*x + c) - s
in(d*x + c)) - (9*I*d^2*f^3*x^2 + 9*I*d^2*e^2*f + 12*I*d*e*f^2 + (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f - 12*I*d*e*
f^2 - 6*I*f^3 - 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^3 + 6*I*f^3 + (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f -
12*I*d*e*f^2 - 6*I*f^3 - 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x + (9*I*
d^2*f^3*x^2 + 9*I*d^2*e^2*f + 12*I*d*e*f^2 + 6*I*f^3 + 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c) + (9*I*d^2*
f^3*x^2 + 9*I*d^2*e^2*f + 12*I*d*e*f^2 + 6*I*f^3 + (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f - 12*I*d*e*f^2 - 6*I*f^3
- 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*sin(d*x + c))*dilog(-cos(d*x
+ c) + I*sin(d*x + c)) - (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f - 12*I*d*e*f^2 + (9*I*d^2*f^3*x^2 + 9*I*d^2*e^2*f +
12*I*d*e*f^2 + 6*I*f^3 + 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^3 - 6*I*f^3 + (9*I*d^2*f^3*x^2 + 9*I*d^2
*e^2*f + 12*I*d*e*f^2 + 6*I*f^3 + 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 - 6*I*(3*d^2*e*f^2 + 2*d*f^3)*
x + (-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f - 12*I*d*e*f^2 - 6*I*f^3 - 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c) +
(-9*I*d^2*f^3*x^2 - 9*I*d^2*e^2*f - 12*I*d*e*f^2 - 6*I*f^3 + (9*I*d^2*f^3*x^2 + 9*I*d^2*e^2*f + 12*I*d*e*f^2
+ 6*I*f^3 + 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 - 6*I*(3*d^2*e*f^2 + 2*d*f^3)*x)*sin(d*x + c))*dilog
(-cos(d*x + c) - I*sin(d*x + c)) - 3*(d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 - (d^3*f^3*x^3 + d^3*e^3
+ 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^2*f^3)*x^2 + (3*d^3*e^2*f + 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x
+ c)^3 + (3*d^3*e*f^2 + 2*d^2*f^3)*x^2 - (d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d
^2*f^3)*x^2 + (3*d^3*e^2*f + 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + (3*d^3*e^2*f + 4*d^2*e*f^2 + 2*d*f^3)*
x + (d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^2*f^3)*x^2 + (3*d^3*e^2*f + 4*d^2*e*
f^2 + 2*d*f^3)*x)*cos(d*x + c) + (d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^2*f^3)*
x^2 - (d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^2*f^3)*x^2 + (3*d^3*e^2*f + 4*d^2*
e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + (3*d^3*e^2*f + 4*d^2*e*f^2 + 2*d*f^3)*x)*sin(d*x + c))*log(cos(d*x + c) +
I*sin(d*x + c) + 1) - 12*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3 - (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x + c
)^3 - (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x + c)^2 + (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x + c) +
(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3 - (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x + c)^2)*sin(d*x + c))*log(cos
(d*x + c) + I*sin(d*x + c) + I) - 3*(d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 - (d^3*f^3*x^3 + d^3*e^3
+ 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^2*f^3)*x^2 + (3*d^3*e^2*f + 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x +
c)^3 + (3*d^3*e*f^2 + 2*d^2*f^3)*x^2 - (d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^
2*f^3)*x^2 + (3*d^3*e^2*f + 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + (3*d^3*e^2*f + 4*d^2*e*f^2 + 2*d*f^3)*x
+ (d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^2*f^3)*x^2 + (3*d^3*e^2*f + 4*d^2*e*f
^2 + 2*d*f^3)*x)*cos(d*x + c) + (d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^2*f^3)*x
^2 - (d^3*f^3*x^3 + d^3*e^3 + 2*d^2*e^2*f + 2*d*e*f^2 + (3*d^3*e*f^2 + 2*d^2*f^3)*x^2 + (3*d^3*e^2*f + 4*d^2*e
*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + (3*d^3*e^2*f + 4*d^2*e*f^2 + 2*d*f^3)*x)*sin(d*x + c))*log(cos(d*x + c) -
I*sin(d*x + c) + 1) - 12*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3 - (d^2*f^3*x^2 + 2*d^2*e*f^2*x +
2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c)^3 - (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c)^2
+ (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c) + (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*
f^2 - c^2*f^3 - (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c)^2)*sin(d*x + c))*log(I*cos(
d*x + c) + sin(d*x + c) + 1) - 12*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3 - (d^2*f^3*x^2 + 2*d^2*
e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c)^3 - (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*
x + c)^2 + (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c) + (d^2*f^3*x^2 + 2*d^2*e*f^2*x +
2*c*d*e*f^2 - c^2*f^3 - (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c)^2)*sin(d*x + c))*l
og(-I*cos(d*x + c) + sin(d*x + c) + 1) + 3*(d^3*e^3 - (3*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 - (d^3*e^3 - (3*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)^3 - (d^3*e^3 - (3*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 + (d^3*e^3 - (3*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) + (d^
3*e^3 - (3*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 - (d^3*e^3 - (3*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)*sin(d*x + c))*log(-1/2*cos(d*x + c
) + 1/2*I*sin(d*x + c) + 1/2) + 3*(d^3*e^3 - (3*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 - (d^3*e^3 - (3*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)^
3 - (d^3*e^3 - (3*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 + (d^
3*e^3 - (3*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) + (d^3*e^3 - (
3*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 - (d^3*e^3 - (3*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)*sin(d*x + c))*log(-1/2*cos(d*x + c) - 1/2*I
*sin(d*x + c) + 1/2) + 3*(d^3*f^3*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3 - (d^3
*f^3*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (
3*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 - (d^3*f^3*x^3 + 3*c*d^
2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (3*d^3*e^2*f - 4*d
^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + (3*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x + (d^3*f^3*x^3 + 3*c*d^2*e^2*f
- (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (3*d^3*e^2*f - 4*d^2*e*f^
2 + 2*d*f^3)*x)*cos(d*x + c) + (d^3*f^3*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3
+ (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 - (d^3*f^3*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f
^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (3*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + (3*d^3*e^2*f -
4*d^2*e*f^2 + 2*d*f^3)*x)*sin(d*x + c))*log(-cos(d*x + c) + I*sin(d*x + c) + 1) - 12*(d^2*e^2*f - 2*c*d*e*f^2
+ c^2*f^3 - (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x + c)^3 - (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x +
c)^2 + (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x + c) + (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3 - (d^2*e^2*f - 2
*c*d*e*f^2 + c^2*f^3)*cos(d*x + c)^2)*sin(d*x + c))*log(-cos(d*x + c) + I*sin(d*x + c) + I) + 3*(d^3*f^3*x^3 +
3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3 - (d^3*f^3*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c
)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (3*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x)
*cos(d*x + c)^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 - (d^3*f^3*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3
+ 2*c^2 + 2*c)*f^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (3*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 +
(3*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x + (d^3*f^3*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2
+ 2*c)*f^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (3*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c) + (d^3*f^3
*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 - (d^3*
f^3*x^3 + 3*c*d^2*e^2*f - (3*c^2 + 4*c)*d*e*f^2 + (c^3 + 2*c^2 + 2*c)*f^3 + (3*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (3
*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x)*cos(d*x + c)^2 + (3*d^3*e^2*f - 4*d^2*e*f^2 + 2*d*f^3)*x)*sin(d*x + c))
*log(-cos(d*x + c) - I*sin(d*x + c) + 1) - (18*I*f^3*cos(d*x + c)^3 + 18*I*f^3*cos(d*x + c)^2 - 18*I*f^3*cos(d
*x + c) - 18*I*f^3 + (18*I*f^3*cos(d*x + c)^2 - 18*I*f^3)*sin(d*x + c))*polylog(4, cos(d*x + c) + I*sin(d*x +
c)) - (-18*I*f^3*cos(d*x + c)^3 - 18*I*f^3*cos(d*x + c)^2 + 18*I*f^3*cos(d*x + c) + 18*I*f^3 + (-18*I*f^3*cos(
d*x + c)^2 + 18*I*f^3)*sin(d*x + c))*polylog(4, cos(d*x + c) - I*sin(d*x + c)) - (18*I*f^3*cos(d*x + c)^3 + 18
*I*f^3*cos(d*x + c)^2 - 18*I*f^3*cos(d*x + c) - 18*I*f^3 + (18*I*f^3*cos(d*x + c)^2 - 18*I*f^3)*sin(d*x + c))*
polylog(4, -cos(d*x + c) + I*sin(d*x + c)) - (-18*I*f^3*cos(d*x + c)^3 - 18*I*f^3*cos(d*x + c)^2 + 18*I*f^3*co
s(d*x + c) + 18*I*f^3 + (-18*I*f^3*cos(d*x + c)^2 + 18*I*f^3)*sin(d*x + c))*polylog(4, -cos(d*x + c) - I*sin(d
*x + c)) + 6*(3*d*f^3*x + 3*d*e*f^2 - (3*d*f^3*x + 3*d*e*f^2 - 2*f^3)*cos(d*x + c)^3 - 2*f^3 - (3*d*f^3*x + 3*
d*e*f^2 - 2*f^3)*cos(d*x + c)^2 + (3*d*f^3*x + 3*d*e*f^2 - 2*f^3)*cos(d*x + c) + (3*d*f^3*x + 3*d*e*f^2 - 2*f^
3 - (3*d*f^3*x + 3*d*e*f^2 - 2*f^3)*cos(d*x + c)^2)*sin(d*x + c))*polylog(3, cos(d*x + c) + I*sin(d*x + c)) +
6*(3*d*f^3*x + 3*d*e*f^2 - (3*d*f^3*x + 3*d*e*f^2 - 2*f^3)*cos(d*x + c)^3 - 2*f^3 - (3*d*f^3*x + 3*d*e*f^2 - 2
*f^3)*cos(d*x + c)^2 + (3*d*f^3*x + 3*d*e*f^2 - 2*f^3)*cos(d*x + c) + (3*d*f^3*x + 3*d*e*f^2 - 2*f^3 - (3*d*f^
3*x + 3*d*e*f^2 - 2*f^3)*cos(d*x + c)^2)*sin(d*x + c))*polylog(3, cos(d*x + c) - I*sin(d*x + c)) + 24*(f^3*cos
(d*x + c)^3 + f^3*cos(d*x + c)^2 - f^3*cos(d*x + c) - f^3 + (f^3*cos(d*x + c)^2 - f^3)*sin(d*x + c))*polylog(3
, I*cos(d*x + c) - sin(d*x + c)) + 24*(f^3*cos(d*x + c)^3 + f^3*cos(d*x + c)^2 - f^3*cos(d*x + c) - f^3 + (f^3
*cos(d*x + c)^2 - f^3)*sin(d*x + c))*polylog(3, -I*cos(d*x + c) - sin(d*x + c)) - 6*(3*d*f^3*x + 3*d*e*f^2 - (
3*d*f^3*x + 3*d*e*f^2 + 2*f^3)*cos(d*x + c)^3 + 2*f^3 - (3*d*f^3*x + 3*d*e*f^2 + 2*f^3)*cos(d*x + c)^2 + (3*d*
f^3*x + 3*d*e*f^2 + 2*f^3)*cos(d*x + c) + (3*d*f^3*x + 3*d*e*f^2 + 2*f^3 - (3*d*f^3*x + 3*d*e*f^2 + 2*f^3)*cos
(d*x + c)^2)*sin(d*x + c))*polylog(3, -cos(d*x + c) + I*sin(d*x + c)) - 6*(3*d*f^3*x + 3*d*e*f^2 - (3*d*f^3*x
+ 3*d*e*f^2 + 2*f^3)*cos(d*x + c)^3 + 2*f^3 - (3*d*f^3*x + 3*d*e*f^2 + 2*f^3)*cos(d*x + c)^2 + (3*d*f^3*x + 3*
d*e*f^2 + 2*f^3)*cos(d*x + c) + (3*d*f^3*x + 3*d*e*f^2 + 2*f^3 - (3*d*f^3*x + 3*d*e*f^2 + 2*f^3)*cos(d*x + c)^
2)*sin(d*x + c))*polylog(3, -cos(d*x + c) - I*sin(d*x + c)) - 2*(2*d^3*f^3*x^3 + 2*d^3*e^3 + 3*d^2*e^2*f + 3*(
2*d^3*e*f^2 + d^2*f^3)*x^2 - 4*(d^3*f^3*x^3 + 3*d^3*e*f^2*x^2 + 3*d^3*e^2*f*x + d^3*e^3)*cos(d*x + c)^2 + 6*(d
^3*e^2*f + d^2*e*f^2)*x - (d^3*f^3*x^3 + d^3*e^3 - 3*d^2*e^2*f + 3*(d^3*e*f^2 - d^2*f^3)*x^2 + 3*(d^3*e^2*f -
2*d^2*e*f^2)*x)*cos(d*x + c))*sin(d*x + c))/(a*d^4*cos(d*x + c)^3 + a*d^4*cos(d*x + c)^2 - a*d^4*cos(d*x + c)
- a*d^4 + (a*d^4*cos(d*x + c)^2 - a*d^4)*sin(d*x + c))
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)**3*csc(d*x+c)**3/(a+a*sin(d*x+c)),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^3*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")
[Out]
Timed out