3.3.3 \(\int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx\) [203]

Optimal. Leaf size=463 \[ -\frac {2 i (e+f x)^3}{a d}+\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {12 i f^2 (e+f x) \text {Li}_2\left (i e^{i (c+d x)}\right )}{a d^3}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f^2 (e+f x) \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {12 f^3 \text {Li}_3\left (i e^{i (c+d x)}\right )}{a d^4}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\frac {3 f^3 \text {Li}_3\left (e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}-\frac {6 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a d^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.50, antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4631, 4269, 3798, 2221, 2611, 2320, 6724, 4268, 6744, 3399} \begin {gather*} \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 {6 i f^3 \text {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}-\frac {6 i f^3 \text {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4}-\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 {6 f^2 (e+f x) \text {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}-\frac {6 f^2 (e+f x) \text {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}-\frac {3 i f (e+f x)^2 \text {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \text {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}+\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {(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 {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac {2 i (e+f x)^3}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)*(e + f*x)^3)/(a*d) + (2*(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) + (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*(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*(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) + (6*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) - (6*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) + ((6*I)*f^3*PolyLog[4, -E^(I*(c + d*
x))])/(a*d^4) - ((6*I)*f^3*PolyLog[4, E^(I*(c + d*x))])/(a*d^4)

Rule 2221

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

Rule 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 2611

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

Rule 3399

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

Rule 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4268

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

Rule 4269

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

Rule 4631

Int[(Csc[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbo
l] :> Dist[1/a, Int[(e + f*x)^m*Csc[c + d*x]^n, x], x] - Dist[b/a, Int[(e + f*x)^m*(Csc[c + d*x]^(n - 1)/(a +
b*Sin[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 6724

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

Rule 6744

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

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int (e+f x)^3 \csc ^2(c+d x) \, dx}{a}-\int \frac {(e+f x)^3 \csc (c+d x)}{a+a \sin (c+d x)} \, dx\\ &=-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {\int (e+f x)^3 \csc (c+d x) \, dx}{a}+\frac {(3 f) \int (e+f x)^2 \cot (c+d x) \, dx}{a d}+\int \frac {(e+f x)^3}{a+a \sin (c+d x)} \, dx\\ &=-\frac {i (e+f x)^3}{a d}+\frac {2 (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 {\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 {i (e+f x)^3}{a d}+\frac {2 (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 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\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 {2 i (e+f x)^3}{a d}+\frac {2 (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 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{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 {6 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {6 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 ) \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 {2 (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 {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 (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{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 {6 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\frac {\left (12 f^2\right ) \int (e+f x) \log \left (1-i e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^2}+\frac {\left (6 i f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}-\frac {\left (6 i f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\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 {2 (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 {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 (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{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 (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{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 {6 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {6 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 {6 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}-\frac {6 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 {2 (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 {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 (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{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 (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{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 {6 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {6 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 {6 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}-\frac {6 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a d^4}+\frac {\left (12 f^3\right ) \text {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 {2 (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 {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 (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{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 (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{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 {6 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 {6 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 {6 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}-\frac {6 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a d^4}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1208\) vs. \(2(463)=926\).
time = 20.54, size = 1208, normalized size = 2.61 \begin {gather*} -\frac {e^3 \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}-\frac {3 e^2 f \left ((c+d x) \left (\log \left (1-e^{i (c+d x)}\right )-\log \left (1+e^{i (c+d x)}\right )\right )-c \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )+i \left (\text {Li}_2\left (-e^{i (c+d x)}\right )-\text {Li}_2\left (e^{i (c+d x)}\right )\right )\right )}{a d^2}-\frac {e^{-i c} f^3 \csc (c) \left (2 d^2 x^2 \left (2 d e^{2 i c} x+3 i \left (-1+e^{2 i c}\right ) \log \left (1-e^{2 i (c+d x)}\right )\right )+6 d \left (-1+e^{2 i c}\right ) x \text {Li}_2\left (e^{2 i (c+d x)}\right )+3 i \left (-1+e^{2 i c}\right ) \text {Li}_3\left (e^{2 i (c+d x)}\right )\right )}{4 a d^4}+\frac {6 e f^2 \left (d^2 x^2 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x))-i d x \text {Li}_2(-\cos (c+d x)-i \sin (c+d x))+i d x \text {Li}_2(\cos (c+d x)+i \sin (c+d x))+\text {Li}_3(-\cos (c+d x)-i \sin (c+d x))-\text {Li}_3(\cos (c+d x)+i \sin (c+d x))\right )}{a d^3}-\frac {f^3 \left (-2 d^3 x^3 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x))+3 i d^2 x^2 \text {Li}_2(-\cos (c+d x)-i \sin (c+d x))-3 i d^2 x^2 \text {Li}_2(\cos (c+d x)+i \sin (c+d x))-6 d x \text {Li}_3(-\cos (c+d x)-i \sin (c+d x))+6 d x \text {Li}_3(\cos (c+d x)+i \sin (c+d x))-6 i \text {Li}_4(-\cos (c+d x)-i \sin (c+d x))+6 i \text {Li}_4(\cos (c+d x)+i \sin (c+d x))\right )}{a d^4}+\frac {3 e^2 f \csc (c) (-d x \cos (c)+\log (\cos (d x) \sin (c)+\cos (c) \sin (d x)) \sin (c))}{a d^2 \left (\cos ^2(c)+\sin ^2(c)\right )}+\frac {2 f \left (3 d^2 (e+f x)^2 \log (1-i \cos (c+d x)+\sin (c+d x))-6 i d f (e+f x) \text {Li}_2(i \cos (c+d x)-\sin (c+d x))+6 f^2 \text {Li}_3(i \cos (c+d x)-\sin (c+d x))+\frac {d^3 x \left (3 e^2+3 e f x+f^2 x^2\right ) (-i \cos (c)+\sin (c))}{\cos (c)+i (1+\sin (c))}\right )}{a d^4}+\frac {\csc \left (\frac {c}{2}\right ) \csc \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^3 \sin \left (\frac {d x}{2}\right )+3 e^2 f x \sin \left (\frac {d x}{2}\right )+3 e f^2 x^2 \sin \left (\frac {d x}{2}\right )+f^3 x^3 \sin \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {\sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^3 \sin \left (\frac {d x}{2}\right )+3 e^2 f x \sin \left (\frac {d x}{2}\right )+3 e f^2 x^2 \sin \left (\frac {d x}{2}\right )+f^3 x^3 \sin \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {2 \left (e^3 \sin \left (\frac {d x}{2}\right )+3 e^2 f x \sin \left (\frac {d x}{2}\right )+3 e f^2 x^2 \sin \left (\frac {d x}{2}\right )+f^3 x^3 \sin \left (\frac {d x}{2}\right )\right )}{a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}-\frac {3 e f^2 \csc (c) \sec (c) \left (d^2 e^{i \tan ^{-1}(\tan (c))} x^2+\frac {\left (i d x \left (-\pi +2 \tan ^{-1}(\tan (c))\right )-\pi \log \left (1+e^{-2 i d x}\right )-2 \left (d x+\tan ^{-1}(\tan (c))\right ) \log \left (1-e^{2 i \left (d x+\tan ^{-1}(\tan (c))\right )}\right )+\pi \log (\cos (d x))+2 \tan ^{-1}(\tan (c)) \log \left (\sin \left (d x+\tan ^{-1}(\tan (c))\right )\right )+i \text {Li}_2\left (e^{2 i \left (d x+\tan ^{-1}(\tan (c))\right )}\right )\right ) \tan (c)}{\sqrt {1+\tan ^2(c)}}\right )}{a d^3 \sqrt {\sec ^2(c) \left (\cos ^2(c)+\sin ^2(c)\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((e^3*Log[Tan[(c + d*x)/2]])/(a*d)) - (3*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))] - PolyLog[2, E^(I*(c + d*x))])))/(a*d^2) - (f^3
*Csc[c]*(2*d^2*x^2*(2*d*E^((2*I)*c)*x + (3*I)*(-1 + E^((2*I)*c))*Log[1 - E^((2*I)*(c + d*x))]) + 6*d*(-1 + E^(
(2*I)*c))*x*PolyLog[2, E^((2*I)*(c + d*x))] + (3*I)*(-1 + E^((2*I)*c))*PolyLog[3, E^((2*I)*(c + d*x))]))/(4*a*
d^4*E^(I*c)) + (6*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*S
in[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) - (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]]))/(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)
) + (2*f*(3*d^2*(e + f*x)^2*Log[1 - I*Cos[c + d*x] + Sin[c + d*x]] - (6*I)*d*f*(e + f*x)*PolyLog[2, I*Cos[c +
d*x] - Sin[c + d*x]] + 6*f^2*PolyLog[3, I*Cos[c + d*x] - Sin[c + d*x]] + (d^3*x*(3*e^2 + 3*e*f*x + f^2*x^2)*((
-I)*Cos[c] + Sin[c]))/(Cos[c] + I*(1 + Sin[c]))))/(a*d^4) + (Csc[c/2]*Csc[c/2 + (d*x)/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]))/(2*a*d) + (Sec[c/2]*Sec[c/2 + (d*x)/
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]))/(2*a*d) + (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)])

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1704 vs. \(2 (419 ) = 838\).
time = 0.29, size = 1705, normalized size = 3.68

method	result	size
risch	\(\text {Expression too large to display}\)	\(1705\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4792 vs. \(2 (417) = 834\).
time = 0.53, size = 4792, normalized size = 10.35 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*d^3*f^3*x^3 + 6*d^3*f^2*x^2*e + 6*d^3*f*x*e^2 + 2*d^3*e^3 - 4*(d^3*f^3*x^3 + 3*d^3*f^2*x^2*e + 3*d^3*f
*x*e^2 + d^3*e^3)*cos(d*x + c)^2 - 2*(d^3*f^3*x^3 + 3*d^3*f^2*x^2*e + 3*d^3*f*x*e^2 + d^3*e^3)*cos(d*x + c) -
3*(-I*d^2*f^3*x^2 + 2*I*d*f^3*x - I*d^2*f*e^2 + (I*d^2*f^3*x^2 - 2*I*d*f^3*x + I*d^2*f*e^2 + 2*I*(d^2*f^2*x -
d*f^2)*e)*cos(d*x + c)^2 - 2*I*(d^2*f^2*x - d*f^2)*e + (-I*d^2*f^3*x^2 + 2*I*d*f^3*x - I*d^2*f*e^2 + (-I*d^2*f
^3*x^2 + 2*I*d*f^3*x - I*d^2*f*e^2 - 2*I*(d^2*f^2*x - d*f^2)*e)*cos(d*x + c) - 2*I*(d^2*f^2*x - d*f^2)*e)*sin(
d*x + c))*dilog(cos(d*x + c) + I*sin(d*x + c)) - 3*(I*d^2*f^3*x^2 - 2*I*d*f^3*x + I*d^2*f*e^2 + (-I*d^2*f^3*x^
2 + 2*I*d*f^3*x - I*d^2*f*e^2 - 2*I*(d^2*f^2*x - d*f^2)*e)*cos(d*x + c)^2 + 2*I*(d^2*f^2*x - d*f^2)*e + (I*d^2
*f^3*x^2 - 2*I*d*f^3*x + I*d^2*f*e^2 + (I*d^2*f^3*x^2 - 2*I*d*f^3*x + I*d^2*f*e^2 + 2*I*(d^2*f^2*x - d*f^2)*e)
*cos(d*x + c) + 2*I*(d^2*f^2*x - d*f^2)*e)*sin(d*x + c))*dilog(cos(d*x + c) - I*sin(d*x + c)) - 12*(I*d*f^3*x
+ I*d*f^2*e + (-I*d*f^3*x - I*d*f^2*e)*cos(d*x + c)^2 + (I*d*f^3*x + I*d*f^2*e + (I*d*f^3*x + I*d*f^2*e)*cos(d
*x + c))*sin(d*x + c))*dilog(I*cos(d*x + c) - sin(d*x + c)) - 12*(-I*d*f^3*x - I*d*f^2*e + (I*d*f^3*x + I*d*f^
2*e)*cos(d*x + c)^2 + (-I*d*f^3*x - I*d*f^2*e + (-I*d*f^3*x - I*d*f^2*e)*cos(d*x + c))*sin(d*x + c))*dilog(-I*
cos(d*x + c) - sin(d*x + c)) - 3*(-I*d^2*f^3*x^2 - 2*I*d*f^3*x - I*d^2*f*e^2 + (I*d^2*f^3*x^2 + 2*I*d*f^3*x +
I*d^2*f*e^2 + 2*I*(d^2*f^2*x + d*f^2)*e)*cos(d*x + c)^2 - 2*I*(d^2*f^2*x + d*f^2)*e + (-I*d^2*f^3*x^2 - 2*I*d*
f^3*x - I*d^2*f*e^2 + (-I*d^2*f^3*x^2 - 2*I*d*f^3*x - I*d^2*f*e^2 - 2*I*(d^2*f^2*x + d*f^2)*e)*cos(d*x + c) -
2*I*(d^2*f^2*x + d*f^2)*e)*sin(d*x + c))*dilog(-cos(d*x + c) + I*sin(d*x + c)) - 3*(I*d^2*f^3*x^2 + 2*I*d*f^3*
x + I*d^2*f*e^2 + (-I*d^2*f^3*x^2 - 2*I*d*f^3*x - I*d^2*f*e^2 - 2*I*(d^2*f^2*x + d*f^2)*e)*cos(d*x + c)^2 + 2*
I*(d^2*f^2*x + d*f^2)*e + (I*d^2*f^3*x^2 + 2*I*d*f^3*x + I*d^2*f*e^2 + (I*d^2*f^3*x^2 + 2*I*d*f^3*x + I*d^2*f*
e^2 + 2*I*(d^2*f^2*x + d*f^2)*e)*cos(d*x + c) + 2*I*(d^2*f^2*x + d*f^2)*e)*sin(d*x + c))*dilog(-cos(d*x + c) -
I*sin(d*x + c)) + (d^3*f^3*x^3 + 3*d^2*f^3*x^2 + d^3*e^3 - (d^3*f^3*x^3 + 3*d^2*f^3*x^2 + d^3*e^3 + 3*(d^3*f*
x + d^2*f)*e^2 + 3*(d^3*f^2*x^2 + 2*d^2*f^2*x)*e)*cos(d*x + c)^2 + 3*(d^3*f*x + d^2*f)*e^2 + 3*(d^3*f^2*x^2 +
2*d^2*f^2*x)*e + (d^3*f^3*x^3 + 3*d^2*f^3*x^2 + d^3*e^3 + (d^3*f^3*x^3 + 3*d^2*f^3*x^2 + d^3*e^3 + 3*(d^3*f*x
+ d^2*f)*e^2 + 3*(d^3*f^2*x^2 + 2*d^2*f^2*x)*e)*cos(d*x + c) + 3*(d^3*f*x + d^2*f)*e^2 + 3*(d^3*f^2*x^2 + 2*d^
2*f^2*x)*e)*sin(d*x + c))*log(cos(d*x + c) + I*sin(d*x + c) + 1) + 6*(c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2 - (c^2
*f^3 - 2*c*d*f^2*e + d^2*f*e^2)*cos(d*x + c)^2 + (c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2 + (c^2*f^3 - 2*c*d*f^2*e +
d^2*f*e^2)*cos(d*x + c))*sin(d*x + c))*log(cos(d*x + c) + I*sin(d*x + c) + I) + (d^3*f^3*x^3 + 3*d^2*f^3*x^2
+ d^3*e^3 - (d^3*f^3*x^3 + 3*d^2*f^3*x^2 + d^3*e^3 + 3*(d^3*f*x + d^2*f)*e^2 + 3*(d^3*f^2*x^2 + 2*d^2*f^2*x)*e
)*cos(d*x + c)^2 + 3*(d^3*f*x + d^2*f)*e^2 + 3*(d^3*f^2*x^2 + 2*d^2*f^2*x)*e + (d^3*f^3*x^3 + 3*d^2*f^3*x^2 +
d^3*e^3 + (d^3*f^3*x^3 + 3*d^2*f^3*x^2 + d^3*e^3 + 3*(d^3*f*x + d^2*f)*e^2 + 3*(d^3*f^2*x^2 + 2*d^2*f^2*x)*e)*
cos(d*x + c) + 3*(d^3*f*x + d^2*f)*e^2 + 3*(d^3*f^2*x^2 + 2*d^2*f^2*x)*e)*sin(d*x + c))*log(cos(d*x + c) - I*s
in(d*x + c) + 1) + 6*(d^2*f^3*x^2 - c^2*f^3 - (d^2*f^3*x^2 - c^2*f^3 + 2*(d^2*f^2*x + c*d*f^2)*e)*cos(d*x + c)
^2 + 2*(d^2*f^2*x + c*d*f^2)*e + (d^2*f^3*x^2 - c^2*f^3 + (d^2*f^3*x^2 - c^2*f^3 + 2*(d^2*f^2*x + c*d*f^2)*e)*
cos(d*x + c) + 2*(d^2*f^2*x + c*d*f^2)*e)*sin(d*x + c))*log(I*cos(d*x + c) + sin(d*x + c) + 1) + 6*(d^2*f^3*x^
2 - c^2*f^3 - (d^2*f^3*x^2 - c^2*f^3 + 2*(d^2*f^2*x + c*d*f^2)*e)*cos(d*x + c)^2 + 2*(d^2*f^2*x + c*d*f^2)*e +
(d^2*f^3*x^2 - c^2*f^3 + (d^2*f^3*x^2 - c^2*f^3 + 2*(d^2*f^2*x + c*d*f^2)*e)*cos(d*x + c) + 2*(d^2*f^2*x + c*
d*f^2)*e)*sin(d*x + c))*log(-I*cos(d*x + c) + sin(d*x + c) + 1) + (3*(c + 1)*d^2*f*e^2 - 3*(c^2 + 2*c)*d*f^2*e
+ (c^3 + 3*c^2)*f^3 - d^3*e^3 - (3*(c + 1)*d^2*f*e^2 - 3*(c^2 + 2*c)*d*f^2*e + (c^3 + 3*c^2)*f^3 - d^3*e^3)*c
os(d*x + c)^2 + (3*(c + 1)*d^2*f*e^2 - 3*(c^2 + 2*c)*d*f^2*e + (c^3 + 3*c^2)*f^3 - d^3*e^3 + (3*(c + 1)*d^2*f*
e^2 - 3*(c^2 + 2*c)*d*f^2*e + (c^3 + 3*c^2)*f^3 - d^3*e^3)*cos(d*x + c))*sin(d*x + c))*log(-1/2*cos(d*x + c) +
1/2*I*sin(d*x + c) + 1/2) + (3*(c + 1)*d^2*f*e^2 - 3*(c^2 + 2*c)*d*f^2*e + (c^3 + 3*c^2)*f^3 - d^3*e^3 - (3*(
c + 1)*d^2*f*e^2 - 3*(c^2 + 2*c)*d*f^2*e + (c^3 + 3*c^2)*f^3 - d^3*e^3)*cos(d*x + c)^2 + (3*(c + 1)*d^2*f*e^2
- 3*(c^2 + 2*c)*d*f^2*e + (c^3 + 3*c^2)*f^3 - d^3*e^3 + (3*(c + 1)*d^2*f*e^2 - 3*(c^2 + 2*c)*d*f^2*e + (c^3 +
3*c^2)*f^3 - d^3*e^3)*cos(d*x + c))*sin(d*x + c))*log(-1/2*cos(d*x + c) - 1/2*I*sin(d*x + c) + 1/2) - (d^3*f^3
*x^3 - 3*d^2*f^3*x^2 + (c^3 + 3*c^2)*f^3 - (d^3*f^3*x^3 - 3*d^2*f^3*x^2 + (c^3 + 3*c^2)*f^3 + 3*(d^3*f*x + c*d
^2*f)*e^2 + 3*(d^3*f^2*x^2 - 2*d^2*f^2*x - (c^2 + 2*c)*d*f^2)*e)*cos(d*x + c)^2 + 3*(d^3*f*x + c*d^2*f)*e^2 +
3*(d^3*f^2*x^2 - 2*d^2*f^2*x - (c^2 + 2*c)*d*f^...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {e^{3} \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{3} x^{3} \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e f^{2} x^{2} \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e^{2} f x \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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