3.269 \(\int \frac{(e+f x)^3 \sec (c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=502 \[ -\frac{3 f^2 (e+f x) \text{PolyLog}\left (3,-i e^{i (c+d x)}\right )}{a d^3}+\frac{3 f^2 (e+f x) \text{PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^3}+\frac{3 i f (e+f x)^2 \text{PolyLog}\left (2,-i e^{i (c+d x)}\right )}{2 a d^2}-\frac{3 i f (e+f x)^2 \text{PolyLog}\left (2,i e^{i (c+d x)}\right )}{2 a d^2}+\frac{3 i f^3 \text{PolyLog}\left (2,-i e^{i (c+d x)}\right )}{a d^4}-\frac{3 i f^3 \text{PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^4}-\frac{3 i f^3 \text{PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 a d^4}-\frac{3 i f^3 \text{PolyLog}\left (4,-i e^{i (c+d x)}\right )}{a d^4}+\frac{3 i f^3 \text{PolyLog}\left (4,i e^{i (c+d x)}\right )}{a d^4}+\frac{3 f^2 (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{a d^3}-\frac{6 i f^2 (e+f x) \tan ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}+\frac{3 f (e+f x)^2 \tan (c+d x)}{2 a d^2}-\frac{3 f (e+f x)^2 \sec (c+d x)}{2 a d^2}-\frac{i (e+f x)^3 \tan ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac{(e+f x)^3 \sec ^2(c+d x)}{2 a d}+\frac{(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{3 i f (e+f x)^2}{2 a d^2} \]

[Out]

(((-3*I)/2)*f*(e + f*x)^2)/(a*d^2) - ((6*I)*f^2*(e + f*x)*ArcTan[E^(I*(c + d*x))])/(a*d^3) - (I*(e + f*x)^3*Ar
cTan[E^(I*(c + d*x))])/(a*d) + (3*f^2*(e + f*x)*Log[1 + E^((2*I)*(c + d*x))])/(a*d^3) + ((3*I)*f^3*PolyLog[2,
(-I)*E^(I*(c + d*x))])/(a*d^4) + (((3*I)/2)*f*(e + f*x)^2*PolyLog[2, (-I)*E^(I*(c + d*x))])/(a*d^2) - ((3*I)*f
^3*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^4) - (((3*I)/2)*f*(e + f*x)^2*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^2) -
(((3*I)/2)*f^3*PolyLog[2, -E^((2*I)*(c + d*x))])/(a*d^4) - (3*f^2*(e + f*x)*PolyLog[3, (-I)*E^(I*(c + d*x))])/
(a*d^3) + (3*f^2*(e + f*x)*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^3) - ((3*I)*f^3*PolyLog[4, (-I)*E^(I*(c + d*x))
])/(a*d^4) + ((3*I)*f^3*PolyLog[4, I*E^(I*(c + d*x))])/(a*d^4) - (3*f*(e + f*x)^2*Sec[c + d*x])/(2*a*d^2) - ((
e + f*x)^3*Sec[c + d*x]^2)/(2*a*d) + (3*f*(e + f*x)^2*Tan[c + d*x])/(2*a*d^2) + ((e + f*x)^3*Sec[c + d*x]*Tan[
c + d*x])/(2*a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.488311, antiderivative size = 502, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 13, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4531, 4186, 4181, 2279, 2391, 2531, 6609, 2282, 6589, 4409, 4184, 3719, 2190} \[ -\frac{3 f^2 (e+f x) \text{PolyLog}\left (3,-i e^{i (c+d x)}\right )}{a d^3}+\frac{3 f^2 (e+f x) \text{PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^3}+\frac{3 i f (e+f x)^2 \text{PolyLog}\left (2,-i e^{i (c+d x)}\right )}{2 a d^2}-\frac{3 i f (e+f x)^2 \text{PolyLog}\left (2,i e^{i (c+d x)}\right )}{2 a d^2}+\frac{3 i f^3 \text{PolyLog}\left (2,-i e^{i (c+d x)}\right )}{a d^4}-\frac{3 i f^3 \text{PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^4}-\frac{3 i f^3 \text{PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 a d^4}-\frac{3 i f^3 \text{PolyLog}\left (4,-i e^{i (c+d x)}\right )}{a d^4}+\frac{3 i f^3 \text{PolyLog}\left (4,i e^{i (c+d x)}\right )}{a d^4}+\frac{3 f^2 (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{a d^3}-\frac{6 i f^2 (e+f x) \tan ^{-1}\left (e^{i (c+d x)}\right )}{a d^3}+\frac{3 f (e+f x)^2 \tan (c+d x)}{2 a d^2}-\frac{3 f (e+f x)^2 \sec (c+d x)}{2 a d^2}-\frac{i (e+f x)^3 \tan ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac{(e+f x)^3 \sec ^2(c+d x)}{2 a d}+\frac{(e+f x)^3 \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{3 i f (e+f x)^2}{2 a d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-3*I)/2)*f*(e + f*x)^2)/(a*d^2) - ((6*I)*f^2*(e + f*x)*ArcTan[E^(I*(c + d*x))])/(a*d^3) - (I*(e + f*x)^3*Ar
cTan[E^(I*(c + d*x))])/(a*d) + (3*f^2*(e + f*x)*Log[1 + E^((2*I)*(c + d*x))])/(a*d^3) + ((3*I)*f^3*PolyLog[2,
(-I)*E^(I*(c + d*x))])/(a*d^4) + (((3*I)/2)*f*(e + f*x)^2*PolyLog[2, (-I)*E^(I*(c + d*x))])/(a*d^2) - ((3*I)*f
^3*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^4) - (((3*I)/2)*f*(e + f*x)^2*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^2) -
(((3*I)/2)*f^3*PolyLog[2, -E^((2*I)*(c + d*x))])/(a*d^4) - (3*f^2*(e + f*x)*PolyLog[3, (-I)*E^(I*(c + d*x))])/
(a*d^3) + (3*f^2*(e + f*x)*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^3) - ((3*I)*f^3*PolyLog[4, (-I)*E^(I*(c + d*x))
])/(a*d^4) + ((3*I)*f^3*PolyLog[4, I*E^(I*(c + d*x))])/(a*d^4) - (3*f*(e + f*x)^2*Sec[c + d*x])/(2*a*d^2) - ((
e + f*x)^3*Sec[c + d*x]^2)/(2*a*d) + (3*f*(e + f*x)^2*Tan[c + d*x])/(2*a*d^2) + ((e + f*x)^3*Sec[c + d*x]*Tan[
c + d*x])/(2*a*d)

Rule 4531

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

Rule 4186

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

Rule 4181

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

Rule 2279

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2531

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

Rule 6609

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

Rule 2282

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

Rule 6589

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

Rule 4409

Int[((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.)*Tan[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Simp[
((c + d*x)^m*Sec[a + b*x]^n)/(b*n), x] - Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Sec[a + b*x]^n, x], x] /; Fre
eQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 4184

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

Rule 3719

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*(e + f*x)))/(1 + E^(2*I*(e + f*x))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 2190

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

Rubi steps

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

Mathematica [A]  time = 8.75896, size = 865, normalized size = 1.72 \[ -\frac{(e+f x)^3}{2 a d \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^2}-\frac{(\cos (c)+i \sin (c)) \left (\frac{(\cos (c)-i \sin (c)) (e+f x)^4}{4 f}+\frac{\log (-i \cos (c+d x)-\sin (c+d x)+1) (-i \cos (c)-\sin (c)+1) (e+f x)^3}{d}+\frac{3 f \left (-2 \text{PolyLog}(4,i \cos (c+d x)+\sin (c+d x)) f^2-2 i d (e+f x) \text{PolyLog}(3,i \cos (c+d x)+\sin (c+d x)) f+d^2 (e+f x)^2 \text{PolyLog}(2,i \cos (c+d x)+\sin (c+d x))\right ) (\cos (c)+i (\sin (c)-1)) (i \cos (c)+\sin (c))}{d^4}\right )}{2 a (\cos (c)+i (\sin (c)-1))}-\frac{12 \left (d^2 \text{PolyLog}(2,-i \cos (c+d x)-\sin (c+d x)) x^2-2 i d \text{PolyLog}(3,-i \cos (c+d x)-\sin (c+d x)) x-2 \text{PolyLog}(4,-i \cos (c+d x)-\sin (c+d x))\right ) (-i \cos (c)+\sin (c)+1) f^4-4 d^3 x^3 \log (i \cos (c+d x)+\sin (c+d x)+1) (\cos (c)+i (\sin (c)+1)) f^4+24 d e (d x \text{PolyLog}(2,-i \cos (c+d x)-\sin (c+d x))-i \text{PolyLog}(3,-i \cos (c+d x)-\sin (c+d x))) (-i \cos (c)+\sin (c)+1) f^3-12 d^3 e x^2 \log (i \cos (c+d x)+\sin (c+d x)+1) (\cos (c)+i (\sin (c)+1)) f^3+12 \left (d^2 e^2+4 f^2\right ) \text{PolyLog}(2,-i \cos (c+d x)-\sin (c+d x)) (-i \cos (c)+\sin (c)+1) f^2-12 d \left (d^2 e^2+4 f^2\right ) x \log (i \cos (c+d x)+\sin (c+d x)+1) (\cos (c)+i (\sin (c)+1)) f^2+4 i d e \left (d^2 e^2+12 f^2\right ) (d x+i \log (\cos (c+d x)+i (\sin (c+d x)+1))) (\cos (c)+i (\sin (c)+1)) f+\left (12 f^2+d^2 (e+f x)^2\right )^2}{8 a d^4 f (\cos (c)+i (\sin (c)+1))}+\frac{x \left (4 e^3+6 f x e^2+4 f^2 x^2 e+f^3 x^3\right )}{8 a \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right )}+\frac{3 \left (x^2 \sin \left (\frac{d x}{2}\right ) f^3+2 e x \sin \left (\frac{d x}{2}\right ) f^2+e^2 \sin \left (\frac{d x}{2}\right ) f\right )}{a d^2 \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 )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3))/(8*a*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])) - ((Cos[c] +
I*Sin[c])*(((e + f*x)^3*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(1 - I*Cos[c] - Sin[c]))/d + ((e + f*x)^4*(Cos
[c] - I*Sin[c]))/(4*f) + (3*f*(d^2*(e + f*x)^2*PolyLog[2, I*Cos[c + d*x] + Sin[c + d*x]] - (2*I)*d*f*(e + f*x)
*PolyLog[3, I*Cos[c + d*x] + Sin[c + d*x]] - 2*f^2*PolyLog[4, I*Cos[c + d*x] + Sin[c + d*x]])*(Cos[c] + I*(-1
+ Sin[c]))*(I*Cos[c] + Sin[c]))/d^4))/(2*a*(Cos[c] + I*(-1 + Sin[c]))) - ((12*f^2 + d^2*(e + f*x)^2)^2 + 12*f^
2*(d^2*e^2 + 4*f^2)*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(1 - I*Cos[c] + Sin[c]) + 24*d*e*f^3*(d*x*Pol
yLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(1 - I*Cos[c] + Si
n[c]) + 12*f^4*(d^2*x^2*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - (2*I)*d*x*PolyLog[3, (-I)*Cos[c + d*x]
- Sin[c + d*x]] - 2*PolyLog[4, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(1 - I*Cos[c] + Sin[c]) - 12*d*f^2*(d^2*e^2
+ 4*f^2)*x*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] + I*(1 + Sin[c])) - 12*d^3*e*f^3*x^2*Log[1 + I*Cos[c
+ d*x] + Sin[c + d*x]]*(Cos[c] + I*(1 + Sin[c])) - 4*d^3*f^4*x^3*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[
c] + I*(1 + Sin[c])) + (4*I)*d*e*f*(d^2*e^2 + 12*f^2)*(d*x + I*Log[Cos[c + d*x] + I*(1 + Sin[c + d*x])])*(Cos[
c] + I*(1 + Sin[c])))/(8*a*d^4*f*(Cos[c] + I*(1 + Sin[c]))) - (e + f*x)^3/(2*a*d*(Cos[c/2 + (d*x)/2] + Sin[c/2
+ (d*x)/2])^2) + (3*(e^2*f*Sin[(d*x)/2] + 2*e*f^2*x*Sin[(d*x)/2] + f^3*x^2*Sin[(d*x)/2]))/(a*d^2*(Cos[c/2] +
Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))

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Maple [B]  time = 0.26, size = 1265, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 5.55909, size = 5164, normalized size = 10.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(3*c*e^2*f*(2/(a*d*sin(d*x + c) + a*d) - log(sin(d*x + c) + 1)/(a*d) + log(sin(d*x + c) - 1)/(a*d)) + e^3*
(log(sin(d*x + c) + 1)/a - log(sin(d*x + c) - 1)/a - 2/(a*sin(d*x + c) + a)) - 4*(12*d^2*e^2*f - 24*c*d*e*f^2
+ 12*c^2*f^3 + (6*(c^2 + 4)*d*e*f^2 - 2*(c^3 + 12*c)*f^3 - 2*(3*(c^2 + 4)*d*e*f^2 - (c^3 + 12*c)*f^3)*cos(2*d*
x + 2*c) - ((12*I*c^2 + 48*I)*d*e*f^2 + (-4*I*c^3 - 48*I*c)*f^3)*cos(d*x + c) - ((6*I*c^2 + 24*I)*d*e*f^2 + (-
2*I*c^3 - 24*I*c)*f^3)*sin(2*d*x + 2*c) + 4*(3*(c^2 + 4)*d*e*f^2 - (c^3 + 12*c)*f^3)*sin(d*x + c))*arctan2(sin
(d*x + c) + 1, cos(d*x + c)) - (6*c^2*d*e*f^2 - 2*c^3*f^3 - 2*(3*c^2*d*e*f^2 - c^3*f^3)*cos(2*d*x + 2*c) + (-1
2*I*c^2*d*e*f^2 + 4*I*c^3*f^3)*cos(d*x + c) + (-6*I*c^2*d*e*f^2 + 2*I*c^3*f^3)*sin(2*d*x + 2*c) + 4*(3*c^2*d*e
*f^2 - c^3*f^3)*sin(d*x + c))*arctan2(sin(d*x + c) - 1, cos(d*x + c)) - (2*(d*x + c)^3*f^3 + 6*(d*e*f^2 - c*f^
3)*(d*x + c)^2 + 6*(d^2*e^2*f - 2*c*d*e*f^2 + (c^2 + 4)*f^3)*(d*x + c) - 2*((d*x + c)^3*f^3 + 3*(d*e*f^2 - c*f
^3)*(d*x + c)^2 + 3*(d^2*e^2*f - 2*c*d*e*f^2 + (c^2 + 4)*f^3)*(d*x + c))*cos(2*d*x + 2*c) + (-4*I*(d*x + c)^3*
f^3 + (-12*I*d*e*f^2 + 12*I*c*f^3)*(d*x + c)^2 + (-12*I*d^2*e^2*f + 24*I*c*d*e*f^2 + (-12*I*c^2 - 48*I)*f^3)*(
d*x + c))*cos(d*x + c) + (-2*I*(d*x + c)^3*f^3 + (-6*I*d*e*f^2 + 6*I*c*f^3)*(d*x + c)^2 + (-6*I*d^2*e^2*f + 12
*I*c*d*e*f^2 + (-6*I*c^2 - 24*I)*f^3)*(d*x + c))*sin(2*d*x + 2*c) + 4*((d*x + c)^3*f^3 + 3*(d*e*f^2 - c*f^3)*(
d*x + c)^2 + 3*(d^2*e^2*f - 2*c*d*e*f^2 + (c^2 + 4)*f^3)*(d*x + c))*sin(d*x + c))*arctan2(cos(d*x + c), sin(d*
x + c) + 1) - (2*(d*x + c)^3*f^3 + 6*(d*e*f^2 - c*f^3)*(d*x + c)^2 + 6*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*(d*
x + c) - 2*((d*x + c)^3*f^3 + 3*(d*e*f^2 - c*f^3)*(d*x + c)^2 + 3*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*(d*x + c
))*cos(2*d*x + 2*c) + (-4*I*(d*x + c)^3*f^3 + (-12*I*d*e*f^2 + 12*I*c*f^3)*(d*x + c)^2 + (-12*I*d^2*e^2*f + 24
*I*c*d*e*f^2 - 12*I*c^2*f^3)*(d*x + c))*cos(d*x + c) + (-2*I*(d*x + c)^3*f^3 + (-6*I*d*e*f^2 + 6*I*c*f^3)*(d*x
+ c)^2 + (-6*I*d^2*e^2*f + 12*I*c*d*e*f^2 - 6*I*c^2*f^3)*(d*x + c))*sin(2*d*x + 2*c) + 4*((d*x + c)^3*f^3 + 3
*(d*e*f^2 - c*f^3)*(d*x + c)^2 + 3*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*(d*x + c))*sin(d*x + c))*arctan2(cos(d*
x + c), -sin(d*x + c) + 1) + 12*((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*cos(2*d*x + 2*c) + (4*(d*x +
c)^3*f^3 - 12*I*d^2*e^2*f + 12*(c^2 + 2*I*c)*d*e*f^2 - 4*(c^3 + 3*I*c^2)*f^3 + (12*d*e*f^2 - (12*c - 12*I)*f^
3)*(d*x + c)^2 + (12*d^2*e^2*f - (24*c - 24*I)*d*e*f^2 + 12*(c^2 - 2*I*c)*f^3)*(d*x + c))*cos(d*x + c) - (6*d^
2*e^2*f - 12*c*d*e*f^2 + 6*(d*x + c)^2*f^3 + 6*(c^2 + 4)*f^3 + 12*(d*e*f^2 - c*f^3)*(d*x + c) - 6*(d^2*e^2*f -
2*c*d*e*f^2 + (d*x + c)^2*f^3 + (c^2 + 4)*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*cos(2*d*x + 2*c) + (-12*I*d^2*
e^2*f + 24*I*c*d*e*f^2 - 12*I*(d*x + c)^2*f^3 + (-12*I*c^2 - 48*I)*f^3 + (-24*I*d*e*f^2 + 24*I*c*f^3)*(d*x + c
))*cos(d*x + c) + (-6*I*d^2*e^2*f + 12*I*c*d*e*f^2 - 6*I*(d*x + c)^2*f^3 + (-6*I*c^2 - 24*I)*f^3 + (-12*I*d*e*
f^2 + 12*I*c*f^3)*(d*x + c))*sin(2*d*x + 2*c) + 12*(d^2*e^2*f - 2*c*d*e*f^2 + (d*x + c)^2*f^3 + (c^2 + 4)*f^3
+ 2*(d*e*f^2 - c*f^3)*(d*x + c))*sin(d*x + c))*dilog(I*e^(I*d*x + I*c)) + (6*d^2*e^2*f - 12*c*d*e*f^2 + 6*(d*x
+ c)^2*f^3 + 6*c^2*f^3 + 12*(d*e*f^2 - c*f^3)*(d*x + c) - 6*(d^2*e^2*f - 2*c*d*e*f^2 + (d*x + c)^2*f^3 + c^2*
f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*cos(2*d*x + 2*c) - (12*I*d^2*e^2*f - 24*I*c*d*e*f^2 + 12*I*(d*x + c)^2*f^
3 + 12*I*c^2*f^3 + (24*I*d*e*f^2 - 24*I*c*f^3)*(d*x + c))*cos(d*x + c) - (6*I*d^2*e^2*f - 12*I*c*d*e*f^2 + 6*I
*(d*x + c)^2*f^3 + 6*I*c^2*f^3 + (12*I*d*e*f^2 - 12*I*c*f^3)*(d*x + c))*sin(2*d*x + 2*c) + 12*(d^2*e^2*f - 2*c
*d*e*f^2 + (d*x + c)^2*f^3 + c^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*sin(d*x + c))*dilog(-I*e^(I*d*x + I*c))
- (I*(d*x + c)^3*f^3 + (3*I*c^2 + 12*I)*d*e*f^2 + (-I*c^3 - 12*I*c)*f^3 + (3*I*d*e*f^2 - 3*I*c*f^3)*(d*x + c)^
2 + (3*I*d^2*e^2*f - 6*I*c*d*e*f^2 + (3*I*c^2 + 12*I)*f^3)*(d*x + c) + (-I*(d*x + c)^3*f^3 + (-3*I*c^2 - 12*I)
*d*e*f^2 + (I*c^3 + 12*I*c)*f^3 + (-3*I*d*e*f^2 + 3*I*c*f^3)*(d*x + c)^2 + (-3*I*d^2*e^2*f + 6*I*c*d*e*f^2 + (
-3*I*c^2 - 12*I)*f^3)*(d*x + c))*cos(2*d*x + 2*c) + 2*((d*x + c)^3*f^3 + 3*(c^2 + 4)*d*e*f^2 - (c^3 + 12*c)*f^
3 + 3*(d*e*f^2 - c*f^3)*(d*x + c)^2 + 3*(d^2*e^2*f - 2*c*d*e*f^2 + (c^2 + 4)*f^3)*(d*x + c))*cos(d*x + c) + ((
d*x + c)^3*f^3 + 3*(c^2 + 4)*d*e*f^2 - (c^3 + 12*c)*f^3 + 3*(d*e*f^2 - c*f^3)*(d*x + c)^2 + 3*(d^2*e^2*f - 2*c
*d*e*f^2 + (c^2 + 4)*f^3)*(d*x + c))*sin(2*d*x + 2*c) + (2*I*(d*x + c)^3*f^3 + (6*I*c^2 + 24*I)*d*e*f^2 + (-2*
I*c^3 - 24*I*c)*f^3 + (6*I*d*e*f^2 - 6*I*c*f^3)*(d*x + c)^2 + (6*I*d^2*e^2*f - 12*I*c*d*e*f^2 + (6*I*c^2 + 24*
I)*f^3)*(d*x + c))*sin(d*x + c))*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1) - (-3*I*c^2*d*e*f^2
- I*(d*x + c)^3*f^3 + I*c^3*f^3 + (-3*I*d*e*f^2 + 3*I*c*f^3)*(d*x + c)^2 + (-3*I*d^2*e^2*f + 6*I*c*d*e*f^2 -
3*I*c^2*f^3)*(d*x + c) + (3*I*c^2*d*e*f^2 + I*(d*x + c)^3*f^3 - I*c^3*f^3 + (3*I*d*e*f^2 - 3*I*c*f^3)*(d*x + c
)^2 + (3*I*d^2*e^2*f - 6*I*c*d*e*f^2 + 3*I*c^2*f^3)*(d*x + c))*cos(2*d*x + 2*c) - 2*(3*c^2*d*e*f^2 + (d*x + c)
^3*f^3 - c^3*f^3 + 3*(d*e*f^2 - c*f^3)*(d*x + c)^2 + 3*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*(d*x + c))*cos(d*x
+ c) - (3*c^2*d*e*f^2 + (d*x + c)^3*f^3 - c^3*f^3 + 3*(d*e*f^2 - c*f^3)*(d*x + c)^2 + 3*(d^2*e^2*f - 2*c*d*e*f
^2 + c^2*f^3)*(d*x + c))*sin(2*d*x + 2*c) + (-6*I*c^2*d*e*f^2 - 2*I*(d*x + c)^3*f^3 + 2*I*c^3*f^3 + (-6*I*d*e*
f^2 + 6*I*c*f^3)*(d*x + c)^2 + (-6*I*d^2*e^2*f + 12*I*c*d*e*f^2 - 6*I*c^2*f^3)*(d*x + c))*sin(d*x + c))*log(co
s(d*x + c)^2 + sin(d*x + c)^2 - 2*sin(d*x + c) + 1) - (12*f^3*cos(2*d*x + 2*c) + 24*I*f^3*cos(d*x + c) + 12*I*
f^3*sin(2*d*x + 2*c) - 24*f^3*sin(d*x + c) - 12*f^3)*polylog(4, I*e^(I*d*x + I*c)) + (12*f^3*cos(2*d*x + 2*c)
+ 24*I*f^3*cos(d*x + c) + 12*I*f^3*sin(2*d*x + 2*c) - 24*f^3*sin(d*x + c) - 12*f^3)*polylog(4, -I*e^(I*d*x + I
*c)) - (12*I*d*e*f^2 + 12*I*(d*x + c)*f^3 - 12*I*c*f^3 + (-12*I*d*e*f^2 - 12*I*(d*x + c)*f^3 + 12*I*c*f^3)*cos
(2*d*x + 2*c) + 24*(d*e*f^2 + (d*x + c)*f^3 - c*f^3)*cos(d*x + c) + 12*(d*e*f^2 + (d*x + c)*f^3 - c*f^3)*sin(2
*d*x + 2*c) + (24*I*d*e*f^2 + 24*I*(d*x + c)*f^3 - 24*I*c*f^3)*sin(d*x + c))*polylog(3, I*e^(I*d*x + I*c)) - (
-12*I*d*e*f^2 - 12*I*(d*x + c)*f^3 + 12*I*c*f^3 + (12*I*d*e*f^2 + 12*I*(d*x + c)*f^3 - 12*I*c*f^3)*cos(2*d*x +
2*c) - 24*(d*e*f^2 + (d*x + c)*f^3 - c*f^3)*cos(d*x + c) - 12*(d*e*f^2 + (d*x + c)*f^3 - c*f^3)*sin(2*d*x + 2
*c) + (-24*I*d*e*f^2 - 24*I*(d*x + c)*f^3 + 24*I*c*f^3)*sin(d*x + c))*polylog(3, -I*e^(I*d*x + I*c)) - (-12*I*
(d*x + c)^2*f^3 + (-24*I*d*e*f^2 + 24*I*c*f^3)*(d*x + c))*sin(2*d*x + 2*c) - (-4*I*(d*x + c)^3*f^3 - 12*d^2*e^
2*f + (-12*I*c^2 + 24*c)*d*e*f^2 + (4*I*c^3 - 12*c^2)*f^3 - 12*(I*d*e*f^2 + (-I*c - 1)*f^3)*(d*x + c)^2 + (-12
*I*d^2*e^2*f - 24*(-I*c - 1)*d*e*f^2 + (-12*I*c^2 - 24*c)*f^3)*(d*x + c))*sin(d*x + c))/(-4*I*a*d^3*cos(2*d*x
+ 2*c) + 8*a*d^3*cos(d*x + c) + 4*a*d^3*sin(2*d*x + 2*c) + 8*I*a*d^3*sin(d*x + c) + 4*I*a*d^3))/d

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Fricas [C]  time = 3.15735, size = 4439, normalized size = 8.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{3} \sec \left (d x + c\right )}{a \sin \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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