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

Optimal result

Integrand size = 28, antiderivative size = 463 \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2 i (e+f x)^3}{a d}+\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {12 f^3 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}-\frac {6 i f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4} \]

[Out]

-6*I*f^3*polylog(4,exp(I*(d*x+c)))/a/d^4+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-exp(2*I*(d*x
+c)))/a/d^2+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^2-3*I*f^2*(f*x+e)*polylog(2,exp(2*I*(d*x+c)))/a/d^3+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^4-12*I*f^2*(f*x+e)*polylog(2,I*exp(I*(d*x+c)))/a/d^3-3*I*f*(f*x+e)^2*polylog(2,-exp(I*(d*x+c)))/a
/d^2

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4631, 4269, 3798, 2221, 2611, 2320, 6724, 4268, 6744, 3399} \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {12 f^3 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}-\frac {6 i f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4}-\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}+\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{a d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}-\frac {2 i (e+f x)^3}{a d} \]

[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*} \text {integral}& = \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 \text {arctanh}\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 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,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) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right ) \, dx}{a d^2}-\frac {\left (6 i f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right ) \, dx}{a d^2}-\frac {\left (6 f^2\right ) \int (e+f x) \log \left (1-e^{2 i (c+d x)}\right ) \, dx}{a d^2} \\ & = -\frac {2 i (e+f x)^3}{a d}+\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac {(6 f) \int \frac {e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )} (e+f x)^2}{1-i e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}} \, dx}{a d}+\frac {\left (3 i f^3\right ) \int \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right ) \, dx}{a d^3}-\frac {\left (6 f^3\right ) \int \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right ) \, dx}{a d^3}+\frac {\left (6 f^3\right ) \int \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right ) \, dx}{a d^3} \\ & = -\frac {2 i (e+f x)^3}{a d}+\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}+\frac {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 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,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 {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}-\frac {\left (6 i f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{2 a d^4} \\ & = -\frac {2 i (e+f x)^3}{a d}+\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}+\frac {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 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}-\frac {6 i f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4}+\frac {\left (12 i f^3\right ) \int \operatorname {PolyLog}\left (2,i e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^3} \\ & = -\frac {2 i (e+f x)^3}{a d}+\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}+\frac {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 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}-\frac {6 i f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4}+\frac {\left (12 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{a d^4} \\ & = -\frac {2 i (e+f x)^3}{a d}+\frac {2 (e+f x)^3 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(e+f x)^3 \cot (c+d x)}{a d}+\frac {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 \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {12 f^3 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}-\frac {6 i f^3 \operatorname {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1052\) vs. \(2(463)=926\).

Time = 8.61 (sec) , antiderivative size = 1052, normalized size of antiderivative = 2.27 \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {i d^3 e^2 (d e-3 f) x-i d^3 e^2 (d e+3 f) x-\frac {2 i d^3 (e+f x)^3}{-1+e^{2 i c}}-3 d^2 e (d e-2 f) f x \log \left (1-e^{-i (c+d x)}\right )-3 d^2 (d e-f) f^2 x^2 \log \left (1-e^{-i (c+d x)}\right )-d^3 f^3 x^3 \log \left (1-e^{-i (c+d x)}\right )+3 d^2 e f (d e+2 f) x \log \left (1+e^{-i (c+d x)}\right )+3 d^2 f^2 (d e+f) x^2 \log \left (1+e^{-i (c+d x)}\right )+d^3 f^3 x^3 \log \left (1+e^{-i (c+d x)}\right )-d^2 e^2 (d e-3 f) \log \left (1-e^{i (c+d x)}\right )+d^2 e^2 (d e+3 f) \log \left (1+e^{i (c+d x)}\right )+3 i d e f (d e+2 f) \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )+6 i d f^2 (d e+f) x \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )+3 i d^2 f^3 x^2 \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )-3 i d e (d e-2 f) f \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )-6 i d (d e-f) f^2 x \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )-3 i d^2 f^3 x^2 \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )+6 f^2 (d e+f) \operatorname {PolyLog}\left (3,-e^{-i (c+d x)}\right )+6 d f^3 x \operatorname {PolyLog}\left (3,-e^{-i (c+d x)}\right )-6 (d e-f) f^2 \operatorname {PolyLog}\left (3,e^{-i (c+d x)}\right )-6 d f^3 x \operatorname {PolyLog}\left (3,e^{-i (c+d x)}\right )-6 i f^3 \operatorname {PolyLog}\left (4,-e^{-i (c+d x)}\right )+6 i f^3 \operatorname {PolyLog}\left (4,e^{-i (c+d x)}\right )}{a d^4}-\frac {6 f (\cos (c)+i \sin (c)) \left (\frac {(e+f x)^3 (\cos (c)-i \sin (c))}{3 f}-\frac {(e+f x)^2 \log (1+i \cos (c+d x)+\sin (c+d x)) (1+i \cos (c)+\sin (c))}{d}+\frac {2 f (d (e+f x) \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x))-i f \operatorname {PolyLog}(3,-i \cos (c+d x)-\sin (c+d x))) (\cos (c)-i (1+\sin (c)))}{d^3}\right )}{a d (\cos (c)+i (1+\sin (c)))}+\frac {\csc \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 )} \]

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1773 vs. \(2 (419 ) = 838\).

Time = 0.60 (sec) , antiderivative size = 1774, normalized size of antiderivative = 3.83

method	result	size
risch	\(\text {Expression too large to display}\)	\(1774\)

[In]

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

[Out]

12*I/a/d^3*e*f^2*c*arctan(exp(I*(d*x+c)))-24*I/a/d^2*f^2*e*c*x+6*I/a/d^2*f^2*e*polylog(2,exp(I*(d*x+c)))*x-6*I
/a/d^2*f^2*e*polylog(2,-exp(I*(d*x+c)))*x-3/a/d*f^2*e*ln(1-exp(I*(d*x+c)))*x^2+6/a/d^3*f^2*e*ln(1-exp(I*(d*x+c
)))*c+12/a/d^3*f^2*e*ln(1-I*exp(I*(d*x+c)))*c+6/a/d^2*f^2*e*ln(1-exp(I*(d*x+c)))*x+12/a/d^2*f^2*e*ln(1-I*exp(I
*(d*x+c)))*x+6/a/d^2*f^2*e*ln(exp(I*(d*x+c))+1)*x+3/a/d^2*c*e^2*f*ln(exp(I*(d*x+c))-1)-6*I/a/d^2*e^2*f*arctan(
exp(I*(d*x+c)))+3*I/a/d^2*e^2*f*polylog(2,exp(I*(d*x+c)))-3*I/a/d^2*e^2*f*polylog(2,-exp(I*(d*x+c)))-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-12*I/a/d^3*f^3*polylog(2,I*exp(I*(
d*x+c)))*x-6*I/a/d^4*c^2*f^3*arctan(exp(I*(d*x+c)))-12*I/a/d^3*e*f^2*c^2+12*I/a/d^3*c^2*f^3*x-3*I/a/d^2*f^3*po
lylog(2,-exp(I*(d*x+c)))*x^2+3*I/a/d^2*f^3*polylog(2,exp(I*(d*x+c)))*x^2-6*I/a/d^3*f^2*e*polylog(2,exp(I*(d*x+
c)))-12*I/a/d^3*f^2*e*polylog(2,I*exp(I*(d*x+c)))-6*I/a/d^3*f^2*e*polylog(2,-exp(I*(d*x+c)))-12*I/a/d*e*f^2*x^
2+24/a/d^3*e*f^2*c*ln(exp(I*(d*x+c)))-3/a/d^2*e^2*f*ln(1-exp(I*(d*x+c)))*c+3/a/d*e^2*f*ln(exp(I*(d*x+c))+1)*x-
3/a/d*e^2*f*ln(1-exp(I*(d*x+c)))*x+3/a/d^3*c^2*f^2*e*ln(1-exp(I*(d*x+c)))-3/a/d^3*c^2*f^2*e*ln(exp(I*(d*x+c))-
1)-6/a/d^3*c*f^2*e*ln(exp(I*(d*x+c))-1)-6/a/d^3*c*f^2*e*ln(1+exp(2*I*(d*x+c)))+3/a/d*f^2*e*ln(exp(I*(d*x+c))+1
)*x^2-2*(-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-1/a/d*e^3*ln(exp(I*(d*x+c))-1)+1/
a/d*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^4-12/a
/d^4*f^3*c^2*ln(exp(I*(d*x+c)))+3/a/d^2*e^2*f*ln(exp(I*(d*x+c))-1)+3/a/d^2*e^2*f*ln(1+exp(2*I*(d*x+c)))+3/a/d^
2*e^2*f*ln(exp(I*(d*x+c))+1)+6*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
-6*I*f^3*polylog(4,exp(I*(d*x+c)))/a/d^4+3/a/d^2*f^3*ln(exp(I*(d*x+c))+1)*x^2+3/a/d^2*f^3*ln(1-exp(I*(d*x+c)))
*x^2-3/a/d^4*c^2*f^3*ln(1-exp(I*(d*x+c)))-6/a/d^4*c^2*f^3*ln(1-I*exp(I*(d*x+c)))+3/a/d^4*c^2*f^3*ln(exp(I*(d*x
+c))-1)+3/a/d^4*c^2*f^3*ln(1+exp(2*I*(d*x+c)))+6/a/d^2*f^3*ln(1-I*exp(I*(d*x+c)))*x^2-12/a/d^2*e^2*f*ln(exp(I*
(d*x+c)))-6/a/d^3*f^2*e*polylog(3,exp(I*(d*x+c)))+6/a/d^3*f^2*e*polylog(3,-exp(I*(d*x+c)))+1/a/d*f^3*ln(exp(I*
(d*x+c))+1)*x^3+6/a/d^3*f^3*polylog(3,-exp(I*(d*x+c)))*x-1/a/d*f^3*ln(1-exp(I*(d*x+c)))*x^3-6/a/d^3*f^3*polylo
g(3,exp(I*(d*x+c)))*x+1/a/d^4*c^3*f^3*ln(exp(I*(d*x+c))-1)-1/a/d^4*c^3*f^3*ln(1-exp(I*(d*x+c)))-4*I/a/d*f^3*x^
3+8*I/a/d^4*f^3*c^3

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4799 vs. \(2 (405) = 810\).

Time = 0.45 (sec) , antiderivative size = 4799, normalized size of antiderivative = 10.37 \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

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

[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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^3 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]

[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.

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}