3.2.97 \(\int \frac {(e+f x)^3 \csc (c+d x)}{a+a \sin (c+d x)} \, dx\) [197]

Optimal. Leaf size=352 \[ \frac {i (e+f x)^3}{a d}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \text {Li}_2\left (i e^{i (c+d x)}\right )}{a d^3}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}-\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}-\frac {12 f^3 \text {Li}_3\left (i e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\frac {6 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {6 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a d^4} \]

[Out]

I*(f*x+e)^3/a/d-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-6*f*(f*x+e)^2*
ln(1-I*exp(I*(d*x+c)))/a/d^2+3*I*f*(f*x+e)^2*polylog(2,-exp(I*(d*x+c)))/a/d^2+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^2-6*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^3-6*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

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Rubi [A]
time = 0.33, antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {4631, 4268, 2611, 6744, 2320, 6724, 3399, 4269, 3798, 2221} \begin {gather*} -\frac {12 f^3 \text {PolyLog}\left (3,i 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 {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 {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 {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{a d}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {i (e+f x)^3}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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) - (6*f*(e + f*x)^2*Log[1 - I*E^(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) - (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) - ((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 (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int (e+f x)^3 \csc (c+d x) \, dx}{a}-\int \frac {(e+f x)^3}{a+a \sin (c+d x)} \, dx\\ &=-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{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 {(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 {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 {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 {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 {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 {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 (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 {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 {6 f (e+f x)^2 \log \left (1-i 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 (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}-\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 {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 {6 f (e+f x)^2 \log \left (1-i 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 {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 {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 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 {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 {6 f (e+f x)^2 \log \left (1-i 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 {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 {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 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 {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 {6 f (e+f x)^2 \log \left (1-i 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 {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 {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 {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 [A]
time = 4.03, size = 565, normalized size = 1.61 \begin {gather*} \frac {-2 d^3 e^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )+3 d^3 e^2 f x \log \left (1-e^{i (c+d x)}\right )+3 d^3 e 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^3 e^2 f x \log \left (1+e^{i (c+d x)}\right )-3 d^3 e 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 i d^2 f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )-3 i d^2 f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )-6 d e f^2 \text {Li}_3\left (-e^{i (c+d x)}\right )-6 d f^3 x \text {Li}_3\left (-e^{i (c+d x)}\right )+6 d e f^2 \text {Li}_3\left (e^{i (c+d x)}\right )+6 d f^3 x \text {Li}_3\left (e^{i (c+d x)}\right )-6 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )+6 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )+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 {i d^3 x \left (3 e^2+3 e f x+f^2 x^2\right ) (\cos (c)+i \sin (c))}{\cos (c)+i (1+\sin (c))}\right )-\frac {2 d^3 (e+f x)^3 \sin \left (\frac {d x}{2}\right )}{\left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}}{a d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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. 1150 vs. \(2 (317 ) = 634\).
time = 0.26, size = 1151, normalized size = 3.27

method	result	size
risch	\(\text {Expression too large to display}\)	\(1151\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2842 vs. \(2 (314) = 628\).
time = 0.82, size = 2842, normalized size = 8.07 \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)/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-(3*c*f*(2/(a*d + a*d*sin(d*x + c)/(cos(d*x + c) + 1)) + log(sin(d*x + c)/(cos(d*x + c) + 1))/(a*d))*e^2 - (lo
g(sin(d*x + c)/(cos(d*x + c) + 1))/a + 2/(a + a*sin(d*x + c)/(cos(d*x + c) + 1)))*e^3 + (-4*I*c^3*f^3 + 12*I*c
^2*d*f^2*e - 12*(-I*c^2*f^3 + 2*I*c*d*f^2*e - I*d^2*f*e^2 - (c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2)*cos(d*x + c) +
(-I*c^2*f^3 + 2*I*c*d*f^2*e - I*d^2*f*e^2)*sin(d*x + c))*arctan2(sin(d*x + c) + 1, cos(d*x + c)) - 12*(I*(d*x
+ c)^2*f^3 + 2*(-I*c*f^3 + I*d*f^2*e)*(d*x + c) + ((d*x + c)^2*f^3 - 2*(c*f^3 - d*f^2*e)*(d*x + c))*cos(d*x +
c) + (I*(d*x + c)^2*f^3 + 2*(-I*c*f^3 + I*d*f^2*e)*(d*x + c))*sin(d*x + c))*arctan2(cos(d*x + c), sin(d*x + c
) + 1) - 2*(-I*(d*x + c)^3*f^3 + I*c^3*f^3 - 3*I*c^2*d*f^2*e + 3*(I*c*f^3 - I*d*f^2*e)*(d*x + c)^2 + 3*(-I*c^2
*f^3 + 2*I*c*d*f^2*e - I*d^2*f*e^2)*(d*x + c) - ((d*x + c)^3*f^3 - c^3*f^3 + 3*c^2*d*f^2*e - 3*(c*f^3 - d*f^2*
e)*(d*x + c)^2 + 3*(c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2)*(d*x + c))*cos(d*x + c) + (-I*(d*x + c)^3*f^3 + I*c^3*f
^3 - 3*I*c^2*d*f^2*e + 3*(I*c*f^3 - I*d*f^2*e)*(d*x + c)^2 + 3*(-I*c^2*f^3 + 2*I*c*d*f^2*e - I*d^2*f*e^2)*(d*x
+ c))*sin(d*x + c))*arctan2(sin(d*x + c), cos(d*x + c) + 1) - 2*(-I*c^3*f^3 + 3*I*c^2*d*f^2*e - (c^3*f^3 - 3*
c^2*d*f^2*e)*cos(d*x + c) + (-I*c^3*f^3 + 3*I*c^2*d*f^2*e)*sin(d*x + c))*arctan2(sin(d*x + c), cos(d*x + c) -
1) - 2*(-I*(d*x + c)^3*f^3 + 3*(I*c*f^3 - I*d*f^2*e)*(d*x + c)^2 + 3*(-I*c^2*f^3 + 2*I*c*d*f^2*e - I*d^2*f*e^2
)*(d*x + c) - ((d*x + c)^3*f^3 - 3*(c*f^3 - d*f^2*e)*(d*x + c)^2 + 3*(c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2)*(d*x
+ c))*cos(d*x + c) + (-I*(d*x + c)^3*f^3 + 3*(I*c*f^3 - I*d*f^2*e)*(d*x + c)^2 + 3*(-I*c^2*f^3 + 2*I*c*d*f^2*e
- I*d^2*f*e^2)*(d*x + c))*sin(d*x + c))*arctan2(sin(d*x + c), -cos(d*x + c) + 1) - 4*((d*x + c)^3*f^3 - 3*(c*
f^3 - d*f^2*e)*(d*x + c)^2 + 3*(c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2)*(d*x + c))*cos(d*x + c) - 24*(I*(d*x + c)*f
^3 - I*c*f^3 + I*d*f^2*e + ((d*x + c)*f^3 - c*f^3 + d*f^2*e)*cos(d*x + c) + (I*(d*x + c)*f^3 - I*c*f^3 + I*d*f
^2*e)*sin(d*x + c))*dilog(I*e^(I*d*x + I*c)) - 6*(I*(d*x + c)^2*f^3 + I*c^2*f^3 - 2*I*c*d*f^2*e + I*d^2*f*e^2
+ 2*(-I*c*f^3 + I*d*f^2*e)*(d*x + c) + ((d*x + c)^2*f^3 + c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2 - 2*(c*f^3 - d*f^2
*e)*(d*x + c))*cos(d*x + c) + (I*(d*x + c)^2*f^3 + I*c^2*f^3 - 2*I*c*d*f^2*e + I*d^2*f*e^2 + 2*(-I*c*f^3 + I*d
*f^2*e)*(d*x + c))*sin(d*x + c))*dilog(-e^(I*d*x + I*c)) - 6*(-I*(d*x + c)^2*f^3 - I*c^2*f^3 + 2*I*c*d*f^2*e -
I*d^2*f*e^2 + 2*(I*c*f^3 - I*d*f^2*e)*(d*x + c) - ((d*x + c)^2*f^3 + c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2 - 2*(c
*f^3 - d*f^2*e)*(d*x + c))*cos(d*x + c) + (-I*(d*x + c)^2*f^3 - I*c^2*f^3 + 2*I*c*d*f^2*e - I*d^2*f*e^2 + 2*(I
*c*f^3 - I*d*f^2*e)*(d*x + c))*sin(d*x + c))*dilog(e^(I*d*x + I*c)) + ((d*x + c)^3*f^3 - c^3*f^3 + 3*c^2*d*f^2
*e - 3*(c*f^3 - d*f^2*e)*(d*x + c)^2 + 3*(c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2)*(d*x + c) + (-I*(d*x + c)^3*f^3 +
I*c^3*f^3 - 3*I*c^2*d*f^2*e - 3*(-I*c*f^3 + I*d*f^2*e)*(d*x + c)^2 - 3*(I*c^2*f^3 - 2*I*c*d*f^2*e + I*d^2*f*e
^2)*(d*x + c))*cos(d*x + c) + ((d*x + c)^3*f^3 - c^3*f^3 + 3*c^2*d*f^2*e - 3*(c*f^3 - d*f^2*e)*(d*x + c)^2 + 3
*(c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2)*(d*x + c))*sin(d*x + c))*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x
+ c) + 1) - ((d*x + c)^3*f^3 - c^3*f^3 + 3*c^2*d*f^2*e - 3*(c*f^3 - d*f^2*e)*(d*x + c)^2 + 3*(c^2*f^3 - 2*c*d*
f^2*e + d^2*f*e^2)*(d*x + c) - (I*(d*x + c)^3*f^3 - I*c^3*f^3 + 3*I*c^2*d*f^2*e - 3*(I*c*f^3 - I*d*f^2*e)*(d*x
+ c)^2 - 3*(-I*c^2*f^3 + 2*I*c*d*f^2*e - I*d^2*f*e^2)*(d*x + c))*cos(d*x + c) + ((d*x + c)^3*f^3 - c^3*f^3 +
3*c^2*d*f^2*e - 3*(c*f^3 - d*f^2*e)*(d*x + c)^2 + 3*(c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2)*(d*x + c))*sin(d*x + c
))*log(cos(d*x + c)^2 + sin(d*x + c)^2 - 2*cos(d*x + c) + 1) + 6*((d*x + c)^2*f^3 + c^2*f^3 - 2*c*d*f^2*e + d^
2*f*e^2 - 2*(c*f^3 - d*f^2*e)*(d*x + c) - (I*(d*x + c)^2*f^3 + I*c^2*f^3 - 2*I*c*d*f^2*e + I*d^2*f*e^2 + 2*(-I
*c*f^3 + I*d*f^2*e)*(d*x + c))*cos(d*x + c) + ((d*x + c)^2*f^3 + c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2 - 2*(c*f^3
- d*f^2*e)*(d*x + c))*sin(d*x + c))*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1) + 12*(f^3*cos(d*
x + c) + I*f^3*sin(d*x + c) + I*f^3)*polylog(4, -e^(I*d*x + I*c)) - 12*(f^3*cos(d*x + c) + I*f^3*sin(d*x + c)
+ I*f^3)*polylog(4, e^(I*d*x + I*c)) - 24*(I*f^3*cos(d*x + c) - f^3*sin(d*x + c) - f^3)*polylog(3, I*e^(I*d*x
+ I*c)) + 12*((d*x + c)*f^3 - c*f^3 + d*f^2*e - (I*(d*x + c)*f^3 - I*c*f^3 + I*d*f^2*e)*cos(d*x + c) + ((d*x +
c)*f^3 - c*f^3 + d*f^2*e)*sin(d*x + c))*polylog(3, -e^(I*d*x + I*c)) - 12*((d*x + c)*f^3 - c*f^3 + d*f^2*e +
(-I*(d*x + c)*f^3 + I*c*f^3 - I*d*f^2*e)*cos(d*x + c) + ((d*x + c)*f^3 - c*f^3 + d*f^2*e)*sin(d*x + c))*polylo
g(3, e^(I*d*x + I*c)) - 4*(I*(d*x + c)^3*f^3 + 3*(-I*c*f^3 + I*d*f^2*e)*(d*x + c)^2 + 3*(I*c^2*f^3 - 2*I*c*d*f
^2*e + I*d^2*f*e^2)*(d*x + c))*sin(d*x + c))/(-2*I*a*d^3*cos(d*x + c) + 2*a*d^3*sin(d*x + c) + 2*a*d^3))/d

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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. 2929 vs. \(2 (314) = 628\).
time = 0.51, size = 2929, normalized size = 8.32 \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)/(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 + 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^2*f^2*x*e + I*d^2*f*e^2 + (I*d^2*f^3*x^2 + 2*I*d^2*f^
2*x*e + I*d^2*f*e^2)*cos(d*x + c) + (I*d^2*f^3*x^2 + 2*I*d^2*f^2*x*e + I*d^2*f*e^2)*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^2*f^2*x*e - I*d^2*f*e^2 + (-I*d^2*f^3*x^2 - 2*I*d^2*f^2*x
*e - I*d^2*f*e^2)*cos(d*x + c) + (-I*d^2*f^3*x^2 - 2*I*d^2*f^2*x*e - I*d^2*f*e^2)*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) + (-I*d*f^3*x - I*
d*f^2*e)*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) + (I*d*f^3*x + I*d*f^2*e)*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^2*f^2*x*e + I*d^2*f*e^2 + (I*d^2*f^3*x^2 + 2*I*d^2*f^2*x*e + I*d^2*f*e^2)*cos(d*x + c) + (I*d^2*f^
3*x^2 + 2*I*d^2*f^2*x*e + I*d^2*f*e^2)*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^2*f^2*x*e - I*d^2*f*e^2 + (-I*d^2*f^3*x^2 - 2*I*d^2*f^2*x*e - I*d^2*f*e^2)*cos(d*x + c) + (-I*d^2*f^3
*x^2 - 2*I*d^2*f^2*x*e - I*d^2*f*e^2)*sin(d*x + c))*dilog(-cos(d*x + c) - I*sin(d*x + c)) - (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 + (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
) + (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)*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) + (c^2*f^3 - 2*
c*d*f^2*e + d^2*f*e^2)*sin(d*x + c))*log(cos(d*x + c) + I*sin(d*x + c) + I) - (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 + (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) + (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)*sin(d*x + c))*log(cos(d*x + c) - I*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*(d^2*f^2*x + c*d*f^2)
*e + (d^2*f^3*x^2 - c^2*f^3 + 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*(d^2*f^2*x +
c*d*f^2)*e + (d^2*f^3*x^2 - c^2*f^3 + 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) - (c^3*f^3 - 3*c^2*d*f^2*e + 3*c*d^2*f*e^2 - d^3*e^3 + (c^3*f^3 - 3*c^2*d*f^2*e + 3*c*d^2*f*e^2 - d^
3*e^3)*cos(d*x + c) + (c^3*f^3 - 3*c^2*d*f^2*e + 3*c*d^2*f*e^2 - d^3*e^3)*sin(d*x + c))*log(-1/2*cos(d*x + c)
+ 1/2*I*sin(d*x + c) + 1/2) - (c^3*f^3 - 3*c^2*d*f^2*e + 3*c*d^2*f*e^2 - d^3*e^3 + (c^3*f^3 - 3*c^2*d*f^2*e +
3*c*d^2*f*e^2 - d^3*e^3)*cos(d*x + c) + (c^3*f^3 - 3*c^2*d*f^2*e + 3*c*d^2*f*e^2 - d^3*e^3)*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 + c^3*f^3 + (d^3*f^3*x^3 + c^3*f^3 + 3*(d^3*f*x +
c*d^2*f)*e^2 + 3*(d^3*f^2*x^2 - c^2*d*f^2)*e)*cos(d*x + c) + 3*(d^3*f*x + c*d^2*f)*e^2 + 3*(d^3*f^2*x^2 - c^2
*d*f^2)*e + (d^3*f^3*x^3 + c^3*f^3 + 3*(d^3*f*x + c*d^2*f)*e^2 + 3*(d^3*f^2*x^2 - c^2*d*f^2)*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) + (c^2*f^3 - 2*c*d*f^2*e + d^2*f*e^2)*sin(d*x + c))*log(-cos(d*x + c) + I*sin(d*x + c) + I
) + (d^3*f^3*x^3 + c^3*f^3 + (d^3*f^3*x^3 + c^3*f^3 + 3*(d^3*f*x + c*d^2*f)*e^2 + 3*(d^3*f^2*x^2 - c^2*d*f^2)*
e)*cos(d*x + c) + 3*(d^3*f*x + c*d^2*f)*e^2 + 3*(d^3*f^2*x^2 - c^2*d*f^2)*e + (d^3*f^3*x^3 + c^3*f^3 + 3*(d^3*
f*x + c*d^2*f)*e^2 + 3*(d^3*f^2*x^2 - c^2*d*f^2)*e)*sin(d*x + c))*log(-cos(d*x + c) - I*sin(d*x + c) + 1) - 6*
(-I*f^3*cos(d*x + c) - I*f^3*sin(d*x + c) - I*f^3)*polylog(4, cos(d*x + c) + I*sin(d*x + c)) - 6*(I*f^3*cos(d*
x + c) + I*f^3*sin(d*x + c) + I*f^3)*polylog(4, cos(d*x + c) - I*sin(d*x + c)) - 6*(-I*f^3*cos(d*x + c) - I*f^
3*sin(d*x + c) - I*f^3)*polylog(4, -cos(d*x + c) + I*sin(d*x + c)) - 6*(I*f^3*cos(d*x + c) + I*f^3*sin(d*x + c
) + I*f^3)*polylog(4, -cos(d*x + c) - I*sin(d*x + c)) + 6*(d*f^3*x + d*f^2*e + (d*f^3*x + d*f^2*e)*cos(d*x + c
) + (d*f^3*x + d*f^2*e)*sin(d*x + c))*polylog(3, cos(d*x + c) + I*sin(d*x + c)) + 6*(d*f^3*x + d*f^2*e + (d*f^
3*x + d*f^2*e)*cos(d*x + c) + (d*f^3*x + d*f^2*e)*sin(d*x + c))*polylog(3, cos(d*x + c) - I*sin(d*x + c)) - 12
*(f^3*cos(d*x + c) + f^3*sin(d*x + c) + f^3)*polylog(3, I*cos(d*x + c) - sin(d*x + c)) - 12*(f^3*cos(d*x + c)
+ f^3*sin(d*x + c) + f^3)*polylog(3, -I*cos(d*x + c) - sin(d*x + c)) - 6*(d*f^3*x + d*f^2*e + (d*f^3*x + d*f^2
*e)*cos(d*x + c) + (d*f^3*x + d*f^2*e)*sin(d*x + c))*polylog(3, -cos(d*x + c) + I*sin(d*x + c)) - 6*(d*f^3*x +
d*f^2*e + (d*f^3*x + d*f^2*e)*cos(d*x + c) + (d*f^3*x + d*f^2*e)*sin(d*x + c))*polylog(3, -cos(d*x + c) - I*s
in(d*x + c)) - 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)*sin(d*x + c))/(a*d^4*cos(d*x + c) +
a*d^4*sin(d*x + c) + a*d^4)

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)*(a + a*sin(c + d*x))),x)

[Out]

\text{Hanged}

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