\(\int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [281]

Optimal result

Integrand size = 28, antiderivative size = 698 \[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {i f (e+f x)^2}{2 a d^2}-\frac {5 i f^2 (e+f x) \arctan \left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{4 a d}+\frac {f^2 (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{a d^3}+\frac {5 i f^3 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{2 a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{8 a d^2}-\frac {5 i f^3 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{2 a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{8 a d^2}-\frac {i f^3 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{4 a d^3}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{4 a d^3}-\frac {9 i f^3 \operatorname {PolyLog}\left (4,-i e^{i (c+d x)}\right )}{4 a d^4}+\frac {9 i f^3 \operatorname {PolyLog}\left (4,i e^{i (c+d x)}\right )}{4 a d^4}-\frac {f^3 \sec (c+d x)}{4 a d^4}-\frac {9 f (e+f x)^2 \sec (c+d x)}{8 a d^2}-\frac {f^2 (e+f x) \sec ^2(c+d x)}{4 a d^3}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 a d^2}-\frac {(e+f x)^3 \sec ^4(c+d x)}{4 a d}+\frac {f^3 \tan (c+d x)}{4 a d^4}+\frac {f (e+f x)^2 \tan (c+d x)}{2 a d^2}+\frac {f^2 (e+f x) \sec (c+d x) \tan (c+d x)}{4 a d^3}+\frac {3 (e+f x)^3 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac {f (e+f x)^2 \sec ^2(c+d x) \tan (c+d x)}{4 a d^2}+\frac {(e+f x)^3 \sec ^3(c+d x) \tan (c+d x)}{4 a d} \]

[Out]

-5/2*I*f^3*polylog(2,I*exp(I*(d*x+c)))/a/d^4-9/4*I*f^3*polylog(4,-I*exp(I*(d*x+c)))/a/d^4-1/2*I*f*(f*x+e)^2/a/
d^2+f^2*(f*x+e)*ln(1+exp(2*I*(d*x+c)))/a/d^3+5/2*I*f^3*polylog(2,-I*exp(I*(d*x+c)))/a/d^4-1/2*I*f^3*polylog(2,
-exp(2*I*(d*x+c)))/a/d^4-5*I*f^2*(f*x+e)*arctan(exp(I*(d*x+c)))/a/d^3+9/8*I*f*(f*x+e)^2*polylog(2,-I*exp(I*(d*
x+c)))/a/d^2-3/4*I*(f*x+e)^3*arctan(exp(I*(d*x+c)))/a/d-9/4*f^2*(f*x+e)*polylog(3,-I*exp(I*(d*x+c)))/a/d^3+9/4
*f^2*(f*x+e)*polylog(3,I*exp(I*(d*x+c)))/a/d^3-9/8*I*f*(f*x+e)^2*polylog(2,I*exp(I*(d*x+c)))/a/d^2+9/4*I*f^3*p
olylog(4,I*exp(I*(d*x+c)))/a/d^4-1/4*f^3*sec(d*x+c)/a/d^4-9/8*f*(f*x+e)^2*sec(d*x+c)/a/d^2-1/4*f^2*(f*x+e)*sec
(d*x+c)^2/a/d^3-1/4*f*(f*x+e)^2*sec(d*x+c)^3/a/d^2-1/4*(f*x+e)^3*sec(d*x+c)^4/a/d+1/4*f^3*tan(d*x+c)/a/d^4+1/2
*f*(f*x+e)^2*tan(d*x+c)/a/d^2+1/4*f^2*(f*x+e)*sec(d*x+c)*tan(d*x+c)/a/d^3+3/8*(f*x+e)^3*sec(d*x+c)*tan(d*x+c)/
a/d+1/4*f*(f*x+e)^2*sec(d*x+c)^2*tan(d*x+c)/a/d^2+1/4*(f*x+e)^3*sec(d*x+c)^3*tan(d*x+c)/a/d

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 698, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4627, 4271, 4270, 4266, 2317, 2438, 2611, 6744, 2320, 6724, 4494, 3852, 8, 4269, 3800, 2221} \[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5 i f^2 (e+f x) \arctan \left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{4 a d}+\frac {5 i f^3 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{2 a d^4}-\frac {5 i f^3 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{2 a d^4}-\frac {i f^3 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {9 i f^3 \operatorname {PolyLog}\left (4,-i e^{i (c+d x)}\right )}{4 a d^4}+\frac {9 i f^3 \operatorname {PolyLog}\left (4,i e^{i (c+d x)}\right )}{4 a d^4}+\frac {f^3 \tan (c+d x)}{4 a d^4}-\frac {f^3 \sec (c+d x)}{4 a d^4}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{4 a d^3}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{4 a d^3}+\frac {f^2 (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{a d^3}-\frac {f^2 (e+f x) \sec ^2(c+d x)}{4 a d^3}+\frac {f^2 (e+f x) \tan (c+d x) \sec (c+d x)}{4 a d^3}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{8 a d^2}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{8 a d^2}+\frac {f (e+f x)^2 \tan (c+d x)}{2 a d^2}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 a d^2}-\frac {9 f (e+f x)^2 \sec (c+d x)}{8 a d^2}+\frac {f (e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{4 a d^2}-\frac {(e+f x)^3 \sec ^4(c+d x)}{4 a d}+\frac {(e+f x)^3 \tan (c+d x) \sec ^3(c+d x)}{4 a d}+\frac {3 (e+f x)^3 \tan (c+d x) \sec (c+d x)}{8 a d}-\frac {i f (e+f x)^2}{2 a d^2} \]

[In]

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

[Out]

((-1/2*I)*f*(e + f*x)^2)/(a*d^2) - ((5*I)*f^2*(e + f*x)*ArcTan[E^(I*(c + d*x))])/(a*d^3) - (((3*I)/4)*(e + f*x
)^3*ArcTan[E^(I*(c + d*x))])/(a*d) + (f^2*(e + f*x)*Log[1 + E^((2*I)*(c + d*x))])/(a*d^3) + (((5*I)/2)*f^3*Pol
yLog[2, (-I)*E^(I*(c + d*x))])/(a*d^4) + (((9*I)/8)*f*(e + f*x)^2*PolyLog[2, (-I)*E^(I*(c + d*x))])/(a*d^2) -
(((5*I)/2)*f^3*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^4) - (((9*I)/8)*f*(e + f*x)^2*PolyLog[2, I*E^(I*(c + d*x))]
)/(a*d^2) - ((I/2)*f^3*PolyLog[2, -E^((2*I)*(c + d*x))])/(a*d^4) - (9*f^2*(e + f*x)*PolyLog[3, (-I)*E^(I*(c +
d*x))])/(4*a*d^3) + (9*f^2*(e + f*x)*PolyLog[3, I*E^(I*(c + d*x))])/(4*a*d^3) - (((9*I)/4)*f^3*PolyLog[4, (-I)
*E^(I*(c + d*x))])/(a*d^4) + (((9*I)/4)*f^3*PolyLog[4, I*E^(I*(c + d*x))])/(a*d^4) - (f^3*Sec[c + d*x])/(4*a*d
^4) - (9*f*(e + f*x)^2*Sec[c + d*x])/(8*a*d^2) - (f^2*(e + f*x)*Sec[c + d*x]^2)/(4*a*d^3) - (f*(e + f*x)^2*Sec
[c + d*x]^3)/(4*a*d^2) - ((e + f*x)^3*Sec[c + d*x]^4)/(4*a*d) + (f^3*Tan[c + d*x])/(4*a*d^4) + (f*(e + f*x)^2*
Tan[c + d*x])/(2*a*d^2) + (f^2*(e + f*x)*Sec[c + d*x]*Tan[c + d*x])/(4*a*d^3) + (3*(e + f*x)^3*Sec[c + d*x]*Ta
n[c + d*x])/(8*a*d) + (f*(e + f*x)^2*Sec[c + d*x]^2*Tan[c + d*x])/(4*a*d^2) + ((e + f*x)^3*Sec[c + d*x]^3*Tan[
c + d*x])/(4*a*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 2317

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 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 2438

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 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 3800

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 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 4266

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 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 4270

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

Rule 4271

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]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 4494

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 4627

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 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 \sec ^5(c+d x) \, dx}{a}-\frac {\int (e+f x)^3 \sec ^4(c+d x) \tan (c+d x) \, dx}{a} \\ & = -\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 a d^2}-\frac {(e+f x)^3 \sec ^4(c+d x)}{4 a d}+\frac {(e+f x)^3 \sec ^3(c+d x) \tan (c+d x)}{4 a d}+\frac {3 \int (e+f x)^3 \sec ^3(c+d x) \, dx}{4 a}+\frac {(3 f) \int (e+f x)^2 \sec ^4(c+d x) \, dx}{4 a d}+\frac {f^2 \int (e+f x) \sec ^3(c+d x) \, dx}{2 a d^2} \\ & = -\frac {f^3 \sec (c+d x)}{4 a d^4}-\frac {9 f (e+f x)^2 \sec (c+d x)}{8 a d^2}-\frac {f^2 (e+f x) \sec ^2(c+d x)}{4 a d^3}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 a d^2}-\frac {(e+f x)^3 \sec ^4(c+d x)}{4 a d}+\frac {f^2 (e+f x) \sec (c+d x) \tan (c+d x)}{4 a d^3}+\frac {3 (e+f x)^3 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac {f (e+f x)^2 \sec ^2(c+d x) \tan (c+d x)}{4 a d^2}+\frac {(e+f x)^3 \sec ^3(c+d x) \tan (c+d x)}{4 a d}+\frac {3 \int (e+f x)^3 \sec (c+d x) \, dx}{8 a}+\frac {f \int (e+f x)^2 \sec ^2(c+d x) \, dx}{2 a d}+\frac {f^2 \int (e+f x) \sec (c+d x) \, dx}{4 a d^2}+\frac {\left (9 f^2\right ) \int (e+f x) \sec (c+d x) \, dx}{4 a d^2}+\frac {f^3 \int \sec ^2(c+d x) \, dx}{4 a d^3} \\ & = -\frac {5 i f^2 (e+f x) \arctan \left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{4 a d}-\frac {f^3 \sec (c+d x)}{4 a d^4}-\frac {9 f (e+f x)^2 \sec (c+d x)}{8 a d^2}-\frac {f^2 (e+f x) \sec ^2(c+d x)}{4 a d^3}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 a d^2}-\frac {(e+f x)^3 \sec ^4(c+d x)}{4 a d}+\frac {f (e+f x)^2 \tan (c+d x)}{2 a d^2}+\frac {f^2 (e+f x) \sec (c+d x) \tan (c+d x)}{4 a d^3}+\frac {3 (e+f x)^3 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac {f (e+f x)^2 \sec ^2(c+d x) \tan (c+d x)}{4 a d^2}+\frac {(e+f x)^3 \sec ^3(c+d x) \tan (c+d x)}{4 a d}-\frac {(9 f) \int (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right ) \, dx}{8 a d}+\frac {(9 f) \int (e+f x)^2 \log \left (1+i e^{i (c+d x)}\right ) \, dx}{8 a d}-\frac {f^2 \int (e+f x) \tan (c+d x) \, dx}{a d^2}-\frac {f^3 \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{4 a d^4}-\frac {f^3 \int \log \left (1-i e^{i (c+d x)}\right ) \, dx}{4 a d^3}+\frac {f^3 \int \log \left (1+i e^{i (c+d x)}\right ) \, dx}{4 a d^3}-\frac {\left (9 f^3\right ) \int \log \left (1-i e^{i (c+d x)}\right ) \, dx}{4 a d^3}+\frac {\left (9 f^3\right ) \int \log \left (1+i e^{i (c+d x)}\right ) \, dx}{4 a d^3} \\ & = -\frac {i f (e+f x)^2}{2 a d^2}-\frac {5 i f^2 (e+f x) \arctan \left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{4 a d}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{8 a d^2}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{8 a d^2}-\frac {f^3 \sec (c+d x)}{4 a d^4}-\frac {9 f (e+f x)^2 \sec (c+d x)}{8 a d^2}-\frac {f^2 (e+f x) \sec ^2(c+d x)}{4 a d^3}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 a d^2}-\frac {(e+f x)^3 \sec ^4(c+d x)}{4 a d}+\frac {f^3 \tan (c+d x)}{4 a d^4}+\frac {f (e+f x)^2 \tan (c+d x)}{2 a d^2}+\frac {f^2 (e+f x) \sec (c+d x) \tan (c+d x)}{4 a d^3}+\frac {3 (e+f x)^3 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac {f (e+f x)^2 \sec ^2(c+d x) \tan (c+d x)}{4 a d^2}+\frac {(e+f x)^3 \sec ^3(c+d x) \tan (c+d x)}{4 a d}+\frac {\left (2 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 (9 i f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right ) \, dx}{4 a d^2}+\frac {\left (9 i f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right ) \, dx}{4 a d^2}+\frac {\left (i f^3\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{4 a d^4}-\frac {\left (i f^3\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{4 a d^4}+\frac {\left (9 i f^3\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{4 a d^4}-\frac {\left (9 i f^3\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{4 a d^4} \\ & = -\frac {i f (e+f x)^2}{2 a d^2}-\frac {5 i f^2 (e+f x) \arctan \left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{4 a d}+\frac {f^2 (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{a d^3}+\frac {5 i f^3 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{2 a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{8 a d^2}-\frac {5 i f^3 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{2 a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{8 a d^2}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{4 a d^3}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{4 a d^3}-\frac {f^3 \sec (c+d x)}{4 a d^4}-\frac {9 f (e+f x)^2 \sec (c+d x)}{8 a d^2}-\frac {f^2 (e+f x) \sec ^2(c+d x)}{4 a d^3}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 a d^2}-\frac {(e+f x)^3 \sec ^4(c+d x)}{4 a d}+\frac {f^3 \tan (c+d x)}{4 a d^4}+\frac {f (e+f x)^2 \tan (c+d x)}{2 a d^2}+\frac {f^2 (e+f x) \sec (c+d x) \tan (c+d x)}{4 a d^3}+\frac {3 (e+f x)^3 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac {f (e+f x)^2 \sec ^2(c+d x) \tan (c+d x)}{4 a d^2}+\frac {(e+f x)^3 \sec ^3(c+d x) \tan (c+d x)}{4 a d}-\frac {f^3 \int \log \left (1+e^{2 i (c+d x)}\right ) \, dx}{a d^3}+\frac {\left (9 f^3\right ) \int \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right ) \, dx}{4 a d^3}-\frac {\left (9 f^3\right ) \int \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right ) \, dx}{4 a d^3} \\ & = -\frac {i f (e+f x)^2}{2 a d^2}-\frac {5 i f^2 (e+f x) \arctan \left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{4 a d}+\frac {f^2 (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{a d^3}+\frac {5 i f^3 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{2 a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{8 a d^2}-\frac {5 i f^3 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{2 a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{8 a d^2}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{4 a d^3}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{4 a d^3}-\frac {f^3 \sec (c+d x)}{4 a d^4}-\frac {9 f (e+f x)^2 \sec (c+d x)}{8 a d^2}-\frac {f^2 (e+f x) \sec ^2(c+d x)}{4 a d^3}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 a d^2}-\frac {(e+f x)^3 \sec ^4(c+d x)}{4 a d}+\frac {f^3 \tan (c+d x)}{4 a d^4}+\frac {f (e+f x)^2 \tan (c+d x)}{2 a d^2}+\frac {f^2 (e+f x) \sec (c+d x) \tan (c+d x)}{4 a d^3}+\frac {3 (e+f x)^3 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac {f (e+f x)^2 \sec ^2(c+d x) \tan (c+d x)}{4 a d^2}+\frac {(e+f x)^3 \sec ^3(c+d x) \tan (c+d x)}{4 a d}+\frac {\left (i f^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {\left (9 i f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{4 a d^4}+\frac {\left (9 i f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{4 a d^4} \\ & = -\frac {i f (e+f x)^2}{2 a d^2}-\frac {5 i f^2 (e+f x) \arctan \left (e^{i (c+d x)}\right )}{a d^3}-\frac {3 i (e+f x)^3 \arctan \left (e^{i (c+d x)}\right )}{4 a d}+\frac {f^2 (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{a d^3}+\frac {5 i f^3 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{2 a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{8 a d^2}-\frac {5 i f^3 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{2 a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{8 a d^2}-\frac {i f^3 \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 a d^4}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right )}{4 a d^3}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{4 a d^3}-\frac {9 i f^3 \operatorname {PolyLog}\left (4,-i e^{i (c+d x)}\right )}{4 a d^4}+\frac {9 i f^3 \operatorname {PolyLog}\left (4,i e^{i (c+d x)}\right )}{4 a d^4}-\frac {f^3 \sec (c+d x)}{4 a d^4}-\frac {9 f (e+f x)^2 \sec (c+d x)}{8 a d^2}-\frac {f^2 (e+f x) \sec ^2(c+d x)}{4 a d^3}-\frac {f (e+f x)^2 \sec ^3(c+d x)}{4 a d^2}-\frac {(e+f x)^3 \sec ^4(c+d x)}{4 a d}+\frac {f^3 \tan (c+d x)}{4 a d^4}+\frac {f (e+f x)^2 \tan (c+d x)}{2 a d^2}+\frac {f^2 (e+f x) \sec (c+d x) \tan (c+d x)}{4 a d^3}+\frac {3 (e+f x)^3 \sec (c+d x) \tan (c+d x)}{8 a d}+\frac {f (e+f x)^2 \sec ^2(c+d x) \tan (c+d x)}{4 a d^2}+\frac {(e+f x)^3 \sec ^3(c+d x) \tan (c+d x)}{4 a d} \\ \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. \(2278\) vs. \(2(698)=1396\).

Time = 10.14 (sec) , antiderivative size = 2278, normalized size of antiderivative = 3.26 \[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(-3*(6*d^4*e^2*f*x^2 + 8*d^2*f^3*x^2 + 4*d^4*e*f^2*x^3 + d^4*f^3*x^4 - (4*I)*d^4*e^3*x*Cos[c] - (16*I)*d^2*e*f
^2*x*Cos[c] - (4*I)*d^3*e^3*Log[-Cos[c + d*x] - I*(-1 + Sin[c + d*x])] - (16*I)*d*e*f^2*Log[-Cos[c + d*x] - I*
(-1 + Sin[c + d*x])] - (12*I)*d^3*e^2*f*x*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]] - (16*I)*d*f^3*x*Log[1 - I*Co
s[c + d*x] - Sin[c + d*x]] - (12*I)*d^3*e*f^2*x^2*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]] - (4*I)*d^3*f^3*x^3*L
og[1 - I*Cos[c + d*x] - Sin[c + d*x]] - 24*f^3*PolyLog[4, I*Cos[c + d*x] + Sin[c + d*x]] + 24*d*f^2*(e + f*x)*
PolyLog[3, I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] + I*(-1 + Sin[c])) + 4*f*(4*f^2 + 3*d^2*(e + f*x)^2)*PolyLog
[2, I*Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] - Sin[c]) + 4*d^3*e^3*Log[-Cos[c + d*x] - I*(-1 + Sin[c + d*x
])]*(Cos[c] + I*Sin[c]) + 16*d*e*f^2*Log[-Cos[c + d*x] - I*(-1 + Sin[c + d*x])]*(Cos[c] + I*Sin[c]) + 12*d^3*e
^2*f*x*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] + I*Sin[c]) + 16*d*f^3*x*Log[1 - I*Cos[c + d*x] - Sin[c
+ d*x]]*(Cos[c] + I*Sin[c]) + 12*d^3*e*f^2*x^2*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] + I*Sin[c]) + 4*
d^3*f^3*x^3*Log[1 - I*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] + I*Sin[c]) + 4*d^4*e^3*x*Sin[c] + 16*d^2*e*f^2*x*S
in[c] + 24*f^3*PolyLog[4, I*Cos[c + d*x] + Sin[c + d*x]]*((-I)*Cos[c] + Sin[c])))/(32*a*d^4*(Cos[c] + I*(-1 +
Sin[c]))) - ((Cos[c] + I*Sin[c])*(((28*f^2 + 3*d^2*(e + f*x)^2)^2*(Cos[c] - I*Sin[c]))/(12*d^2*f) + (f*(9*d^2*
e^2 + 28*f^2)*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*Sin[c])*(1 - I*Cos[c] + Sin[c]))/d^2 +
18*e*f^2*x*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*Sin[c])*(1 - I*Cos[c] + Sin[c]) + 9*f^3*x^
2*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])) - (18*f^3*PolyLog[4, (-I)*Cos[c + d*x
] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])))/d^2 - (f*(9*d^2*e^2 + 28*f^2)*x*Log[1 + I*Cos[c + d*x] + Sin[c +
d*x]]*(Cos[c] - I*Sin[c])*(Cos[c] + I*(1 + Sin[c])))/d - 9*d*e*f^2*x^2*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*
(Cos[c] - I*Sin[c])*(Cos[c] + I*(1 + Sin[c])) - 3*d*f^3*x^3*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I
*Sin[c])*(Cos[c] + I*(1 + Sin[c])) - (e*(3*d^2*e^2 + 28*f^2)*Log[Cos[c + d*x] + I*(1 + Sin[c + d*x])]*(Cos[c]
- I*Sin[c])*(Cos[c] + I*(1 + Sin[c])))/d - (18*e*f^2*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*
Sin[c])*(Cos[c] + I*(1 + Sin[c])))/d - (18*f^3*x*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*Sin[
c])*(Cos[c] + I*(1 + Sin[c])))/d + e*(3*d^2*e^2 + 28*f^2)*x*(I*Cos[c] + Sin[c])*(Cos[c] + I*(1 + Sin[c]))))/(8
*a*d^2*(Cos[c] + I*(1 + Sin[c]))) + ((3*e^3*x*Cos[c])/(4*a) + (((3*I)/4)*e^3*x*Sin[c])/a)/(1 + Cos[2*c] + I*Si
n[2*c]) + ((9*e^2*f*x^2*Cos[c])/(8*a) + (((9*I)/8)*e^2*f*x^2*Sin[c])/a)/(1 + Cos[2*c] + I*Sin[2*c]) + ((3*e*f^
2*x^3*Cos[c])/(4*a) + (((3*I)/4)*e*f^2*x^3*Sin[c])/a)/(1 + Cos[2*c] + I*Sin[2*c]) + ((3*f^3*x^4*Cos[c])/(16*a)
+ (((3*I)/16)*f^3*x^4*Sin[c])/a)/(1 + Cos[2*c] + I*Sin[2*c]) + (e^3 + 3*e^2*f*x + 3*e*f^2*x^2 + f^3*x^3)/(8*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*Si
n[(d*x)/2]))/(4*a*d^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (-e^3 - 3*e^2*f*x - 3
*e*f^2*x^2 - f^3*x^3)/(8*a*d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) + (e^2*f*Sin[(d*x)/2] + 2*e*f^2*x*Si
n[(d*x)/2] + f^3*x^2*Sin[(d*x)/2])/(4*a*d^2*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3)
+ (-2*d^2*e^3*Cos[c/2] - d*e^2*f*Cos[c/2] - 2*e*f^2*Cos[c/2] - 6*d^2*e^2*f*x*Cos[c/2] - 2*d*e*f^2*x*Cos[c/2]
- 2*f^3*x*Cos[c/2] - 6*d^2*e*f^2*x^2*Cos[c/2] - d*f^3*x^2*Cos[c/2] - 2*d^2*f^3*x^3*Cos[c/2] - 2*d^2*e^3*Sin[c/
2] + d*e^2*f*Sin[c/2] - 2*e*f^2*Sin[c/2] - 6*d^2*e^2*f*x*Sin[c/2] + 2*d*e*f^2*x*Sin[c/2] - 2*f^3*x*Sin[c/2] -
6*d^2*e*f^2*x^2*Sin[c/2] + d*f^3*x^2*Sin[c/2] - 2*d^2*f^3*x^3*Sin[c/2])/(8*a*d^3*(Cos[c/2] + Sin[c/2])*(Cos[c/
2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (7*d^2*e^2*f*Sin[(d*x)/2] + 2*f^3*Sin[(d*x)/2] + 14*d^2*e*f^2*x*Sin[(d
*x)/2] + 7*d^2*f^3*x^2*Sin[(d*x)/2])/(4*a*d^4*(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. 2040 vs. \(2 (612 ) = 1224\).

Time = 1.18 (sec) , antiderivative size = 2041, normalized size of antiderivative = 2.92

method	result	size
risch	\(\text {Expression too large to display}\)	\(2041\)

[In]

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

[Out]

9/8/a/d*e^2*f*ln(1-I*exp(I*(d*x+c)))*x+9/8/a/d^2*e^2*f*ln(1-I*exp(I*(d*x+c)))*c-9/8/a/d*e^2*f*ln(1+I*exp(I*(d*
x+c)))*x-9/8/a/d^2*e^2*f*ln(1+I*exp(I*(d*x+c)))*c+9/8/a/d*e*f^2*ln(1-I*exp(I*(d*x+c)))*x^2-9/8/a/d*e*f^2*ln(1+
I*exp(I*(d*x+c)))*x^2-9/8/a/d^3*c^2*e*f^2*ln(1-I*exp(I*(d*x+c)))-9/4*I/a/d^3*e*f^2*c^2*arctan(exp(I*(d*x+c)))+
9/4*I/a/d^2*e^2*f*c*arctan(exp(I*(d*x+c)))-9/4*I/a/d^2*e*f^2*polylog(2,I*exp(I*(d*x+c)))*x+9/4*I/a/d^2*e*f^2*p
olylog(2,-I*exp(I*(d*x+c)))*x+9/4/a/d^3*e*f^2*polylog(3,I*exp(I*(d*x+c)))-9/4/a/d^3*e*f^2*polylog(3,-I*exp(I*(
d*x+c)))+2/a/d^4*f^3*c*ln(exp(I*(d*x+c)))-1/a/d^4*f^3*c*ln(1+exp(2*I*(d*x+c)))+3/8/a/d^4*c^3*f^3*ln(1-I*exp(I*
(d*x+c)))-3/8/a/d^4*c^3*f^3*ln(1+I*exp(I*(d*x+c)))+7/2/a/d^3*f^3*ln(1-I*exp(I*(d*x+c)))*x+7/2/a/d^4*f^3*ln(1-I
*exp(I*(d*x+c)))*c-2/a/d^3*e*f^2*ln(exp(I*(d*x+c)))+1/a/d^3*e*f^2*ln(1+exp(2*I*(d*x+c)))+3/8/a/d*f^3*ln(1-I*ex
p(I*(d*x+c)))*x^3+9/4/a/d^3*f^3*polylog(3,I*exp(I*(d*x+c)))*x-3/8/a/d*f^3*ln(1+I*exp(I*(d*x+c)))*x^3-9/4/a/d^3
*f^3*polylog(3,-I*exp(I*(d*x+c)))*x+9/4*I*f^3*polylog(4,I*exp(I*(d*x+c)))/a/d^4+5*I/a/d^4*f^3*c*arctan(exp(I*(
d*x+c)))-5*I/a/d^3*e*f^2*arctan(exp(I*(d*x+c)))-1/4*I*(-6*I*d^3*f^3*x^3*exp(2*I*(d*x+c))-8*I*d^2*f^3*x^2*exp(3
*I*(d*x+c))-8*I*d^2*e^2*f*exp(3*I*(d*x+c))-9*I*d^2*f^3*x^2*exp(5*I*(d*x+c))+9*d^3*e*f^2*x^2*exp(I*(d*x+c))+9*d
^3*e^2*f*x*exp(I*(d*x+c))+I*d^2*f^3*x^2*exp(I*(d*x+c))-2*I*f^3*exp(5*I*(d*x+c))-4*I*f^3*exp(3*I*(d*x+c))+3*d^3
*e^3*exp(5*I*(d*x+c))+2*d^3*e^3*exp(3*I*(d*x+c))+6*I*d^3*e^3*exp(4*I*(d*x+c))-6*I*d^3*e^3*exp(2*I*(d*x+c))+4*d
*f^3*x*exp(3*I*(d*x+c))+4*d*e*f^2*exp(3*I*(d*x+c))+18*d^2*f^3*x^2*exp(4*I*(d*x+c))-18*I*d^2*e*f^2*x*exp(5*I*(d
*x+c))-18*I*d^3*e*f^2*x^2*exp(2*I*(d*x+c))-18*I*d^3*e^2*f*x*exp(2*I*(d*x+c))-16*I*d^2*e*f^2*x*exp(3*I*(d*x+c))
+18*I*d^3*e^2*f*x*exp(4*I*(d*x+c))+18*I*d^3*e*f^2*x^2*exp(4*I*(d*x+c))+3*d^3*e^3*exp(I*(d*x+c))-2*I*f^3*exp(I*
(d*x+c))+2*f^3+2*d*f^3*x*exp(I*(d*x+c))+2*d*e*f^2*exp(I*(d*x+c))+3*d^3*f^3*x^3*exp(I*(d*x+c))+4*d^2*e^2*f+2*I*
d^2*e*f^2*x*exp(I*(d*x+c))+4*d^2*f^3*x^2+18*d^2*e^2*f*exp(4*I*(d*x+c))+2*d^3*f^3*x^3*exp(3*I*(d*x+c))+22*d^2*f
^3*x^2*exp(2*I*(d*x+c))+22*d^2*e^2*f*exp(2*I*(d*x+c))+8*d^2*e*f^2*x+3*d^3*f^3*x^3*exp(5*I*(d*x+c))+2*d*f^3*x*e
xp(5*I*(d*x+c))+2*d*e*f^2*exp(5*I*(d*x+c))+2*f^3*exp(4*I*(d*x+c))+4*f^3*exp(2*I*(d*x+c))+6*I*d^3*f^3*x^3*exp(4
*I*(d*x+c))-9*I*d^2*e^2*f*exp(5*I*(d*x+c))+6*d^3*e*f^2*x^2*exp(3*I*(d*x+c))+6*d^3*e^2*f*x*exp(3*I*(d*x+c))+44*
d^2*e*f^2*x*exp(2*I*(d*x+c))+36*d^2*e*f^2*x*exp(4*I*(d*x+c))+I*d^2*e^2*f*exp(I*(d*x+c))+9*d^3*e^2*f*x*exp(5*I*
(d*x+c))+9*d^3*e*f^2*x^2*exp(5*I*(d*x+c)))/(exp(I*(d*x+c))+I)^4/d^4/(-I+exp(I*(d*x+c)))^2/a+9/8/a/d^3*c^2*e*f^
2*ln(1+I*exp(I*(d*x+c)))-3/2/a/d^3*f^3*ln(1+I*exp(I*(d*x+c)))*x-3/2/a/d^4*f^3*ln(1+I*exp(I*(d*x+c)))*c-I/a/d^2
*f^3*x^2-3/4*I/a/d*e^3*arctan(exp(I*(d*x+c)))-I/a/d^4*f^3*c^2-7/2*I/a/d^4*f^3*polylog(2,I*exp(I*(d*x+c)))+3/2*
I/a/d^4*f^3*polylog(2,-I*exp(I*(d*x+c)))+9/8*I/a/d^2*f^3*polylog(2,-I*exp(I*(d*x+c)))*x^2-9/8*I/a/d^2*f^3*poly
log(2,I*exp(I*(d*x+c)))*x^2-9/8*I/a/d^2*e^2*f*polylog(2,I*exp(I*(d*x+c)))+9/8*I/a/d^2*e^2*f*polylog(2,-I*exp(I
*(d*x+c)))-2*I/a/d^3*f^3*c*x+3/4*I/a/d^4*f^3*c^3*arctan(exp(I*(d*x+c)))-9/4*I*f^3*polylog(4,-I*exp(I*(d*x+c)))
/a/d^4

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. 2572 vs. \(2 (589) = 1178\).

Time = 0.43 (sec) , antiderivative size = 2572, normalized size of antiderivative = 3.68 \[ \int \frac {(e+f x)^3 \sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/16*(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*(2*d^2*f^3*x^2 + 4*d^2*e*f^2*x + 2*d^2*e
^2*f + f^3)*cos(d*x + c)^3 - 2*(3*d^3*f^3*x^3 + 9*d^3*e*f^2*x^2 + 3*d^3*e^3 + 2*d*e*f^2 + (9*d^3*e^2*f + 2*d*f
^3)*x)*cos(d*x + c)^2 - 14*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + d^2*e^2*f)*cos(d*x + c) - 3*((3*I*d^2*f^3*x^2 + 6*I*
d^2*e*f^2*x + 3*I*d^2*e^2*f + 4*I*f^3)*cos(d*x + c)^2*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 + 4*I*f^3)*cos(d*x + c)^2)*dilog(I*cos(d*x + c) + sin(d*x + c)) + ((-9*I*d^2*f^3*x^2 - 18*I*d^2*e*f^
2*x - 9*I*d^2*e^2*f - 28*I*f^3)*cos(d*x + c)^2*sin(d*x + c) + (-9*I*d^2*f^3*x^2 - 18*I*d^2*e*f^2*x - 9*I*d^2*e
^2*f - 28*I*f^3)*cos(d*x + c)^2)*dilog(I*cos(d*x + c) - sin(d*x + c)) - 3*((-3*I*d^2*f^3*x^2 - 6*I*d^2*e*f^2*x
- 3*I*d^2*e^2*f - 4*I*f^3)*cos(d*x + c)^2*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
- 4*I*f^3)*cos(d*x + c)^2)*dilog(-I*cos(d*x + c) + sin(d*x + c)) + ((9*I*d^2*f^3*x^2 + 18*I*d^2*e*f^2*x + 9*I*
d^2*e^2*f + 28*I*f^3)*cos(d*x + c)^2*sin(d*x + c) + (9*I*d^2*f^3*x^2 + 18*I*d^2*e*f^2*x + 9*I*d^2*e^2*f + 28*I
*f^3)*cos(d*x + c)^2)*dilog(-I*cos(d*x + c) - sin(d*x + c)) + ((3*d^3*e^3 - 9*c*d^2*e^2*f + (9*c^2 + 28)*d*e*f
^2 - (3*c^3 + 28*c)*f^3)*cos(d*x + c)^2*sin(d*x + c) + (3*d^3*e^3 - 9*c*d^2*e^2*f + (9*c^2 + 28)*d*e*f^2 - (3*
c^3 + 28*c)*f^3)*cos(d*x + c)^2)*log(cos(d*x + c) + I*sin(d*x + c) + I) - 3*((d^3*e^3 - 3*c*d^2*e^2*f + (3*c^2
+ 4)*d*e*f^2 - (c^3 + 4*c)*f^3)*cos(d*x + c)^2*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 + 4*c)*f^3)*cos(d*x + c)^2)*log(cos(d*x + c) - I*sin(d*x + c) + I) + ((3*d^3*f^3*x^3 + 9*d^3*e*f^2*x^2
+ 9*c*d^2*e^2*f - 9*c^2*d*e*f^2 + (3*c^3 + 28*c)*f^3 + (9*d^3*e^2*f + 28*d*f^3)*x)*cos(d*x + c)^2*sin(d*x + c)
+ (3*d^3*f^3*x^3 + 9*d^3*e*f^2*x^2 + 9*c*d^2*e^2*f - 9*c^2*d*e*f^2 + (3*c^3 + 28*c)*f^3 + (9*d^3*e^2*f + 28*d
*f^3)*x)*cos(d*x + c)^2)*log(I*cos(d*x + c) + sin(d*x + c) + 1) - 3*((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 + 4*c)*f^3 + (3*d^3*e^2*f + 4*d*f^3)*x)*cos(d*x + c)^2*sin(d*x + c) + (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 + 4*c)*f^3 + (3*d^3*e^2*f + 4*d*f^3)*x)*cos(d*x + c
)^2)*log(I*cos(d*x + c) - sin(d*x + c) + 1) + ((3*d^3*f^3*x^3 + 9*d^3*e*f^2*x^2 + 9*c*d^2*e^2*f - 9*c^2*d*e*f^
2 + (3*c^3 + 28*c)*f^3 + (9*d^3*e^2*f + 28*d*f^3)*x)*cos(d*x + c)^2*sin(d*x + c) + (3*d^3*f^3*x^3 + 9*d^3*e*f^
2*x^2 + 9*c*d^2*e^2*f - 9*c^2*d*e*f^2 + (3*c^3 + 28*c)*f^3 + (9*d^3*e^2*f + 28*d*f^3)*x)*cos(d*x + c)^2)*log(-
I*cos(d*x + c) + sin(d*x + c) + 1) - 3*((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
+ 4*c)*f^3 + (3*d^3*e^2*f + 4*d*f^3)*x)*cos(d*x + c)^2*sin(d*x + c) + (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 + 4*c)*f^3 + (3*d^3*e^2*f + 4*d*f^3)*x)*cos(d*x + c)^2)*log(-I*cos(d*x + c) - si
n(d*x + c) + 1) + ((3*d^3*e^3 - 9*c*d^2*e^2*f + (9*c^2 + 28)*d*e*f^2 - (3*c^3 + 28*c)*f^3)*cos(d*x + c)^2*sin(
d*x + c) + (3*d^3*e^3 - 9*c*d^2*e^2*f + (9*c^2 + 28)*d*e*f^2 - (3*c^3 + 28*c)*f^3)*cos(d*x + c)^2)*log(-cos(d*
x + c) + I*sin(d*x + c) + I) - 3*((d^3*e^3 - 3*c*d^2*e^2*f + (3*c^2 + 4)*d*e*f^2 - (c^3 + 4*c)*f^3)*cos(d*x +
c)^2*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 + 4*c)*f^3)*cos(d*x + c)^2)*log(-cos
(d*x + c) - I*sin(d*x + c) + I) - 18*(-I*f^3*cos(d*x + c)^2*sin(d*x + c) - I*f^3*cos(d*x + c)^2)*polylog(4, I*
cos(d*x + c) + sin(d*x + c)) - 18*(-I*f^3*cos(d*x + c)^2*sin(d*x + c) - I*f^3*cos(d*x + c)^2)*polylog(4, I*cos
(d*x + c) - sin(d*x + c)) - 18*(I*f^3*cos(d*x + c)^2*sin(d*x + c) + I*f^3*cos(d*x + c)^2)*polylog(4, -I*cos(d*
x + c) + sin(d*x + c)) - 18*(I*f^3*cos(d*x + c)^2*sin(d*x + c) + I*f^3*cos(d*x + c)^2)*polylog(4, -I*cos(d*x +
c) - sin(d*x + c)) - 18*((d*f^3*x + d*e*f^2)*cos(d*x + c)^2*sin(d*x + c) + (d*f^3*x + d*e*f^2)*cos(d*x + c)^2
)*polylog(3, I*cos(d*x + c) + sin(d*x + c)) + 18*((d*f^3*x + d*e*f^2)*cos(d*x + c)^2*sin(d*x + c) + (d*f^3*x +
d*e*f^2)*cos(d*x + c)^2)*polylog(3, I*cos(d*x + c) - sin(d*x + c)) - 18*((d*f^3*x + d*e*f^2)*cos(d*x + c)^2*s
in(d*x + c) + (d*f^3*x + d*e*f^2)*cos(d*x + c)^2)*polylog(3, -I*cos(d*x + c) + sin(d*x + c)) + 18*((d*f^3*x +
d*e*f^2)*cos(d*x + c)^2*sin(d*x + c) + (d*f^3*x + d*e*f^2)*cos(d*x + c)^2)*polylog(3, -I*cos(d*x + c) - sin(d*
x + c)) + 2*(3*d^3*f^3*x^3 + 9*d^3*e*f^2*x^2 + 9*d^3*e^2*f*x + 3*d^3*e^3 - 5*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + d^
2*e^2*f)*cos(d*x + c))*sin(d*x + c))/(a*d^4*cos(d*x + c)^2*sin(d*x + c) + a*d^4*cos(d*x + c)^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}