3.275 \(\int \frac{(e+f x)^3 \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=475 \[ \frac{i f^2 (e+f x) \text{PolyLog}\left (2,-i e^{i (c+d x)}\right )}{a d^3}-\frac{i f^2 (e+f x) \text{PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac{2 i f^2 (e+f x) \text{PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{a d^3}-\frac{f^3 \text{PolyLog}\left (3,-i e^{i (c+d x)}\right )}{a d^4}+\frac{f^3 \text{PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}+\frac{f^3 \text{PolyLog}\left (3,-e^{2 i (c+d x)}\right )}{a d^4}+\frac{f^2 (e+f x) \tan (c+d x)}{a d^3}-\frac{f^2 (e+f x) \sec (c+d x)}{a d^3}+\frac{2 f (e+f x)^2 \log \left (1+e^{2 i (c+d x)}\right )}{a d^2}-\frac{i f (e+f x)^2 \tan ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}-\frac{f (e+f x)^2 \sec ^2(c+d x)}{2 a d^2}+\frac{f (e+f x)^2 \tan (c+d x) \sec (c+d x)}{2 a d^2}+\frac{f^3 \tanh ^{-1}(\sin (c+d x))}{a d^4}+\frac{f^3 \log (\cos (c+d x))}{a d^4}+\frac{2 (e+f x)^3 \tan (c+d x)}{3 a d}-\frac{(e+f x)^3 \sec ^3(c+d x)}{3 a d}+\frac{(e+f x)^3 \tan (c+d x) \sec ^2(c+d x)}{3 a d}-\frac{2 i (e+f x)^3}{3 a d} \]

[Out]

(((-2*I)/3)*(e + f*x)^3)/(a*d) - (I*f*(e + f*x)^2*ArcTan[E^(I*(c + d*x))])/(a*d^2) + (f^3*ArcTanh[Sin[c + d*x]
])/(a*d^4) + (2*f*(e + f*x)^2*Log[1 + E^((2*I)*(c + d*x))])/(a*d^2) + (f^3*Log[Cos[c + d*x]])/(a*d^4) + (I*f^2
*(e + f*x)*PolyLog[2, (-I)*E^(I*(c + d*x))])/(a*d^3) - (I*f^2*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3)
- ((2*I)*f^2*(e + f*x)*PolyLog[2, -E^((2*I)*(c + d*x))])/(a*d^3) - (f^3*PolyLog[3, (-I)*E^(I*(c + d*x))])/(a*
d^4) + (f^3*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^4) + (f^3*PolyLog[3, -E^((2*I)*(c + d*x))])/(a*d^4) - (f^2*(e
+ f*x)*Sec[c + d*x])/(a*d^3) - (f*(e + f*x)^2*Sec[c + d*x]^2)/(2*a*d^2) - ((e + f*x)^3*Sec[c + d*x]^3)/(3*a*d)
+ (f^2*(e + f*x)*Tan[c + d*x])/(a*d^3) + (2*(e + f*x)^3*Tan[c + d*x])/(3*a*d) + (f*(e + f*x)^2*Sec[c + d*x]*T
an[c + d*x])/(2*a*d^2) + ((e + f*x)^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.593911, antiderivative size = 475, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4531, 4186, 4184, 3475, 3719, 2190, 2531, 2282, 6589, 4409, 3770, 4181} \[ \frac{i f^2 (e+f x) \text{PolyLog}\left (2,-i e^{i (c+d x)}\right )}{a d^3}-\frac{i f^2 (e+f x) \text{PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac{2 i f^2 (e+f x) \text{PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{a d^3}-\frac{f^3 \text{PolyLog}\left (3,-i e^{i (c+d x)}\right )}{a d^4}+\frac{f^3 \text{PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}+\frac{f^3 \text{PolyLog}\left (3,-e^{2 i (c+d x)}\right )}{a d^4}+\frac{f^2 (e+f x) \tan (c+d x)}{a d^3}-\frac{f^2 (e+f x) \sec (c+d x)}{a d^3}+\frac{2 f (e+f x)^2 \log \left (1+e^{2 i (c+d x)}\right )}{a d^2}-\frac{i f (e+f x)^2 \tan ^{-1}\left (e^{i (c+d x)}\right )}{a d^2}-\frac{f (e+f x)^2 \sec ^2(c+d x)}{2 a d^2}+\frac{f (e+f x)^2 \tan (c+d x) \sec (c+d x)}{2 a d^2}+\frac{f^3 \tanh ^{-1}(\sin (c+d x))}{a d^4}+\frac{f^3 \log (\cos (c+d x))}{a d^4}+\frac{2 (e+f x)^3 \tan (c+d x)}{3 a d}-\frac{(e+f x)^3 \sec ^3(c+d x)}{3 a d}+\frac{(e+f x)^3 \tan (c+d x) \sec ^2(c+d x)}{3 a d}-\frac{2 i (e+f x)^3}{3 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-2*I)/3)*(e + f*x)^3)/(a*d) - (I*f*(e + f*x)^2*ArcTan[E^(I*(c + d*x))])/(a*d^2) + (f^3*ArcTanh[Sin[c + d*x]
])/(a*d^4) + (2*f*(e + f*x)^2*Log[1 + E^((2*I)*(c + d*x))])/(a*d^2) + (f^3*Log[Cos[c + d*x]])/(a*d^4) + (I*f^2
*(e + f*x)*PolyLog[2, (-I)*E^(I*(c + d*x))])/(a*d^3) - (I*f^2*(e + f*x)*PolyLog[2, I*E^(I*(c + d*x))])/(a*d^3)
- ((2*I)*f^2*(e + f*x)*PolyLog[2, -E^((2*I)*(c + d*x))])/(a*d^3) - (f^3*PolyLog[3, (-I)*E^(I*(c + d*x))])/(a*
d^4) + (f^3*PolyLog[3, I*E^(I*(c + d*x))])/(a*d^4) + (f^3*PolyLog[3, -E^((2*I)*(c + d*x))])/(a*d^4) - (f^2*(e
+ f*x)*Sec[c + d*x])/(a*d^3) - (f*(e + f*x)^2*Sec[c + d*x]^2)/(2*a*d^2) - ((e + f*x)^3*Sec[c + d*x]^3)/(3*a*d)
+ (f^2*(e + f*x)*Tan[c + d*x])/(a*d^3) + (2*(e + f*x)^3*Tan[c + d*x])/(3*a*d) + (f*(e + f*x)^2*Sec[c + d*x]*T
an[c + d*x])/(2*a*d^2) + ((e + f*x)^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*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 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 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

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]

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 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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

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]

Rubi steps

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

Mathematica [B]  time = 8.75923, size = 1117, normalized size = 2.35 \[ \frac{\frac{d^3 (e+f x)^3}{-i+e^{i c}}+3 d^2 f \log \left (1-i e^{-i (c+d x)}\right ) (e+f x)^2+6 f^2 \left (i d (e+f x) \text{PolyLog}\left (2,i e^{-i (c+d x)}\right )+f \text{PolyLog}\left (3,i e^{-i (c+d x)}\right )\right )}{2 a d^4}-\frac{f (\cos (c)+i \sin (c)) \left (\frac{5}{3} d^2 f^2 (\cos (c)-i \sin (c)) x^3+5 d^2 e f \cos (c) x^2-5 i d^2 e f \sin (c) x^2-5 d f^2 \log (i \cos (c+d x)+\sin (c+d x)+1) (\cos (c)-i \sin (c)) (\cos (c)+i (\sin (c)+1)) x^2+\left (5 d^2 e^2+4 f^2\right ) (\cos (c)-i \sin (c)) x-10 d e f \log (i \cos (c+d x)+\sin (c+d x)+1) (\cos (c)-i \sin (c)) (\cos (c)+i (\sin (c)+1)) x+\frac{10 f^2 (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))) (\cos (c)-i \sin (c)) (-i \cos (c)+\sin (c)+1)}{d}+10 e f \text{PolyLog}(2,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i (\sin (c)+1))+\frac{\left (5 d^2 e^2+4 f^2\right ) (d x+i \log (\cos (c+d x)+i (\sin (c+d x)+1))) (i \cos (c)+\sin (c)) (\cos (c)+i (\sin (c)+1))}{d}\right )}{2 a d^3 (\cos (c)+i (\sin (c)+1))}+\frac{\sin \left (\frac{d x}{2}\right ) e^3+3 f x \sin \left (\frac{d x}{2}\right ) e^2+3 f^2 x^2 \sin \left (\frac{d x}{2}\right ) e+f^3 x^3 \sin \left (\frac{d x}{2}\right )}{2 a d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}+\frac{5 d^2 \sin \left (\frac{d x}{2}\right ) e^3+15 d^2 f x \sin \left (\frac{d x}{2}\right ) e^2+12 f^2 \sin \left (\frac{d x}{2}\right ) e+15 d^2 f^2 x^2 \sin \left (\frac{d x}{2}\right ) e+5 d^2 f^3 x^3 \sin \left (\frac{d x}{2}\right )+12 f^3 x \sin \left (\frac{d x}{2}\right )}{6 a d^3 \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}+\frac{-d \cos \left (\frac{c}{2}\right ) e^3+d \sin \left (\frac{c}{2}\right ) e^3-3 f \cos \left (\frac{c}{2}\right ) e^2-3 d f x \cos \left (\frac{c}{2}\right ) e^2-3 f \sin \left (\frac{c}{2}\right ) e^2+3 d f x \sin \left (\frac{c}{2}\right ) e^2-3 d f^2 x^2 \cos \left (\frac{c}{2}\right ) e-6 f^2 x \cos \left (\frac{c}{2}\right ) e+3 d f^2 x^2 \sin \left (\frac{c}{2}\right ) e-6 f^2 x \sin \left (\frac{c}{2}\right ) e-d f^3 x^3 \cos \left (\frac{c}{2}\right )-3 f^3 x^2 \cos \left (\frac{c}{2}\right )+d f^3 x^3 \sin \left (\frac{c}{2}\right )-3 f^3 x^2 \sin \left (\frac{c}{2}\right )}{6 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 )^2}+\frac{\sin \left (\frac{d x}{2}\right ) e^3+3 f x \sin \left (\frac{d x}{2}\right ) e^2+3 f^2 x^2 \sin \left (\frac{d x}{2}\right ) e+f^3 x^3 \sin \left (\frac{d x}{2}\right )}{3 a d \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )+\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((d^3*(e + f*x)^3)/(-I + E^(I*c)) + 3*d^2*f*(e + f*x)^2*Log[1 - I/E^(I*(c + d*x))] + 6*f^2*(I*d*(e + f*x)*Poly
Log[2, I/E^(I*(c + d*x))] + f*PolyLog[3, I/E^(I*(c + d*x))]))/(2*a*d^4) - (f*(Cos[c] + I*Sin[c])*(5*d^2*e*f*x^
2*Cos[c] + (5*d^2*e^2 + 4*f^2)*x*(Cos[c] - I*Sin[c]) + (5*d^2*f^2*x^3*(Cos[c] - I*Sin[c]))/3 - (5*I)*d^2*e*f*x
^2*Sin[c] + (10*f^2*(d*x*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c
+ d*x]])*(Cos[c] - I*Sin[c])*(1 - I*Cos[c] + Sin[c]))/d + 10*e*f*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]]
*(Cos[c] - I*(1 + Sin[c])) - 10*d*e*f*x*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*(Cos[c] - I*Sin[c])*(Cos[c] + I
*(1 + Sin[c])) - 5*d*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
])) + ((5*d^2*e^2 + 4*f^2)*(d*x + I*Log[Cos[c + d*x] + I*(1 + Sin[c + d*x])])*(I*Cos[c] + Sin[c])*(Cos[c] + I*
(1 + Sin[c])))/d))/(2*a*d^3*(Cos[c] + I*(1 + Sin[c]))) + (e^3*Sin[(d*x)/2] + 3*e^2*f*x*Sin[(d*x)/2] + 3*e*f^2*
x^2*Sin[(d*x)/2] + f^3*x^3*Sin[(d*x)/2])/(2*a*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]
)) + (e^3*Sin[(d*x)/2] + 3*e^2*f*x*Sin[(d*x)/2] + 3*e*f^2*x^2*Sin[(d*x)/2] + f^3*x^3*Sin[(d*x)/2])/(3*a*d*(Cos
[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (-(d*e^3*Cos[c/2]) - 3*e^2*f*Cos[c/2] - 3*d*e
^2*f*x*Cos[c/2] - 6*e*f^2*x*Cos[c/2] - 3*d*e*f^2*x^2*Cos[c/2] - 3*f^3*x^2*Cos[c/2] - d*f^3*x^3*Cos[c/2] + d*e^
3*Sin[c/2] - 3*e^2*f*Sin[c/2] + 3*d*e^2*f*x*Sin[c/2] - 6*e*f^2*x*Sin[c/2] + 3*d*e*f^2*x^2*Sin[c/2] - 3*f^3*x^2
*Sin[c/2] + d*f^3*x^3*Sin[c/2])/(6*a*d^2*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) +
(5*d^2*e^3*Sin[(d*x)/2] + 12*e*f^2*Sin[(d*x)/2] + 15*d^2*e^2*f*x*Sin[(d*x)/2] + 12*f^3*x*Sin[(d*x)/2] + 15*d^2
*e*f^2*x^2*Sin[(d*x)/2] + 5*d^2*f^3*x^3*Sin[(d*x)/2])/(6*a*d^3*(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 = 0.338, size = 1124, normalized size = 2.4 \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)^2/(a+a*sin(d*x+c)),x)

[Out]

3*f^3*polylog(3,-I*exp(I*(d*x+c)))/a/d^4+5*f^2/d^2/a*e*ln(1-I*exp(I*(d*x+c)))*x+3/2/d^4/a*f^3*c^2*ln(exp(I*(d*
x+c))-I)+3/2/d^2/a*e^2*f*ln(exp(I*(d*x+c))-I)-3/2/d^4/a*ln(1+I*exp(I*(d*x+c)))*c^2*f^3+8/3*I/d^4/a*c^3*f^3-4/3
*I/d/a*f^3*x^3-4*f^3/d^4/a*c^2*ln(exp(I*(d*x+c)))+5/2*f/d^2/a*ln(exp(I*(d*x+c))+I)*e^2+5/2*f^3/d^4/a*c^2*ln(ex
p(I*(d*x+c))+I)-4*f/d^2/a*ln(exp(I*(d*x+c)))*e^2+5*f^2/d^3/a*e*ln(1-I*exp(I*(d*x+c)))*c+5/2*f^3/d^2/a*ln(1-I*e
xp(I*(d*x+c)))*x^2-5/2*f^3/d^4/a*ln(1-I*exp(I*(d*x+c)))*c^2+8*f^2/d^3/a*e*c*ln(exp(I*(d*x+c)))-5*f^2/d^3/a*e*c
*ln(exp(I*(d*x+c))+I)-1/3*(12*I*d^2*e^2*f*x+6*I*d*e*f^2*x*exp(I*(d*x+c))+6*f^3*x*exp(I*(d*x+c))+6*e*f^2*exp(I*
(d*x+c))+3*I*d*f^3*x^2*exp(3*I*(d*x+c))+3*I*d*e^2*f*exp(3*I*(d*x+c))+6*I*e*f^2*exp(2*I*(d*x+c))+12*I*d^2*e*f^2
*x^2+6*I*e*f^2+6*I*d*e*f^2*x*exp(3*I*(d*x+c))+24*d^2*e*f^2*x^2*exp(I*(d*x+c))+24*d^2*e^2*f*x*exp(I*(d*x+c))+3*
I*d*e^2*f*exp(I*(d*x+c))+6*I*f^3*x*exp(2*I*(d*x+c))+8*d^2*f^3*x^3*exp(I*(d*x+c))+3*I*d*f^3*x^2*exp(I*(d*x+c))+
4*I*d^2*f^3*x^3+8*d^2*e^3*exp(I*(d*x+c))+4*I*d^2*e^3+6*f^3*x*exp(3*I*(d*x+c))+6*e*f^2*exp(3*I*(d*x+c))+6*I*f^3
*x)/(exp(I*(d*x+c))-I)/(exp(I*(d*x+c))+I)^3/d^3/a-8*I/d^2/a*c*e*f^2*x+2/d^4/a*f^3*ln(exp(I*(d*x+c))+I)-2/d^4/a
*f^3*ln(exp(I*(d*x+c)))-3/d^3/a*e*f^2*c*ln(exp(I*(d*x+c))-I)+3/d^2/a*ln(1+I*exp(I*(d*x+c)))*e*f^2*x+3/d^3/a*ln
(1+I*exp(I*(d*x+c)))*c*e*f^2+3/2/d^2/a*ln(1+I*exp(I*(d*x+c)))*f^3*x^2-4*I/d^3/a*c^2*e*f^2-3*I/d^3/a*polylog(2,
-I*exp(I*(d*x+c)))*f^3*x-5*I/d^3/a*polylog(2,I*exp(I*(d*x+c)))*f^3*x-3*I/d^3/a*e*f^2*polylog(2,-I*exp(I*(d*x+c
)))-5*I/d^3/a*e*f^2*polylog(2,I*exp(I*(d*x+c)))+4*I/d^3/a*c^2*f^3*x-4*I/d/a*e*f^2*x^2+5*f^3*polylog(3,I*exp(I*
(d*x+c)))/a/d^4

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Maxima [B]  time = 6.6732, size = 6893, normalized size = 14.51 \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)^2/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/12*(24*c^2*e*f^2*(sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*sin(d*x + c)^3
/(cos(d*x + c) + 1)^3 - 1)/(a*d^2 + 2*a*d^2*sin(d*x + c)/(cos(d*x + c) + 1) - 2*a*d^2*sin(d*x + c)^3/(cos(d*x
+ c) + 1)^3 - a*d^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + 6*(4*(8*(d*x + c)*cos(d*x + c) - sin(3*d*x + 3*c) -
sin(d*x + c))*cos(4*d*x + 4*c) + 16*(2*d*x + 4*(d*x + c)*sin(d*x + c) + 2*c + cos(d*x + c))*cos(3*d*x + 3*c)
+ 8*cos(3*d*x + 3*c)^2 + 8*cos(d*x + c)^2 + 5*(2*(2*sin(3*d*x + 3*c) + 2*sin(d*x + c) + 1)*cos(4*d*x + 4*c) -
cos(4*d*x + 4*c)^2 - 4*cos(3*d*x + 3*c)^2 - 8*cos(3*d*x + 3*c)*cos(d*x + c) - 4*cos(d*x + c)^2 - 4*(cos(3*d*x
+ 3*c) + cos(d*x + c))*sin(4*d*x + 4*c) - sin(4*d*x + 4*c)^2 - 4*(2*sin(d*x + c) + 1)*sin(3*d*x + 3*c) - 4*sin
(3*d*x + 3*c)^2 - 4*sin(d*x + c)^2 - 4*sin(d*x + c) - 1)*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c)
+ 1) + 3*(2*(2*sin(3*d*x + 3*c) + 2*sin(d*x + c) + 1)*cos(4*d*x + 4*c) - cos(4*d*x + 4*c)^2 - 4*cos(3*d*x + 3*
c)^2 - 8*cos(3*d*x + 3*c)*cos(d*x + c) - 4*cos(d*x + c)^2 - 4*(cos(3*d*x + 3*c) + cos(d*x + c))*sin(4*d*x + 4*
c) - sin(4*d*x + 4*c)^2 - 4*(2*sin(d*x + c) + 1)*sin(3*d*x + 3*c) - 4*sin(3*d*x + 3*c)^2 - 4*sin(d*x + c)^2 -
4*sin(d*x + c) - 1)*log(cos(d*x + c)^2 + sin(d*x + c)^2 - 2*sin(d*x + c) + 1) + 4*(4*d*x + 8*(d*x + c)*sin(d*x
+ c) + 4*c + cos(3*d*x + 3*c) + cos(d*x + c))*sin(4*d*x + 4*c) - 4*(16*(d*x + c)*cos(d*x + c) - 4*sin(d*x + c
) - 1)*sin(3*d*x + 3*c) + 8*sin(3*d*x + 3*c)^2 + 8*sin(d*x + c)^2 + 4*sin(d*x + c))*c*e*f^2/(a*d^2*cos(4*d*x +
4*c)^2 + 4*a*d^2*cos(3*d*x + 3*c)^2 + 8*a*d^2*cos(3*d*x + 3*c)*cos(d*x + c) + 4*a*d^2*cos(d*x + c)^2 + a*d^2*
sin(4*d*x + 4*c)^2 + 4*a*d^2*sin(3*d*x + 3*c)^2 + 4*a*d^2*sin(d*x + c)^2 + 4*a*d^2*sin(d*x + c) + a*d^2 - 2*(2
*a*d^2*sin(3*d*x + 3*c) + 2*a*d^2*sin(d*x + c) + a*d^2)*cos(4*d*x + 4*c) + 4*(a*d^2*cos(3*d*x + 3*c) + a*d^2*c
os(d*x + c))*sin(4*d*x + 4*c) + 4*(2*a*d^2*sin(d*x + c) + a*d^2)*sin(3*d*x + 3*c)) - 24*c*e^2*f*(sin(d*x + c)/
(cos(d*x + c) + 1) + 3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 1)/(a*d +
2*a*d*sin(d*x + c)/(cos(d*x + c) + 1) - 2*a*d*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - a*d*sin(d*x + c)^4/(cos(d
*x + c) + 1)^4) - 3*(4*(8*(d*x + c)*cos(d*x + c) - sin(3*d*x + 3*c) - sin(d*x + c))*cos(4*d*x + 4*c) + 16*(2*d
*x + 4*(d*x + c)*sin(d*x + c) + 2*c + cos(d*x + c))*cos(3*d*x + 3*c) + 8*cos(3*d*x + 3*c)^2 + 8*cos(d*x + c)^2
+ 5*(2*(2*sin(3*d*x + 3*c) + 2*sin(d*x + c) + 1)*cos(4*d*x + 4*c) - cos(4*d*x + 4*c)^2 - 4*cos(3*d*x + 3*c)^2
- 8*cos(3*d*x + 3*c)*cos(d*x + c) - 4*cos(d*x + c)^2 - 4*(cos(3*d*x + 3*c) + cos(d*x + c))*sin(4*d*x + 4*c) -
sin(4*d*x + 4*c)^2 - 4*(2*sin(d*x + c) + 1)*sin(3*d*x + 3*c) - 4*sin(3*d*x + 3*c)^2 - 4*sin(d*x + c)^2 - 4*si
n(d*x + c) - 1)*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1) + 3*(2*(2*sin(3*d*x + 3*c) + 2*sin(d
*x + c) + 1)*cos(4*d*x + 4*c) - cos(4*d*x + 4*c)^2 - 4*cos(3*d*x + 3*c)^2 - 8*cos(3*d*x + 3*c)*cos(d*x + c) -
4*cos(d*x + c)^2 - 4*(cos(3*d*x + 3*c) + cos(d*x + c))*sin(4*d*x + 4*c) - sin(4*d*x + 4*c)^2 - 4*(2*sin(d*x +
c) + 1)*sin(3*d*x + 3*c) - 4*sin(3*d*x + 3*c)^2 - 4*sin(d*x + c)^2 - 4*sin(d*x + c) - 1)*log(cos(d*x + c)^2 +
sin(d*x + c)^2 - 2*sin(d*x + c) + 1) + 4*(4*d*x + 8*(d*x + c)*sin(d*x + c) + 4*c + cos(3*d*x + 3*c) + cos(d*x
+ c))*sin(4*d*x + 4*c) - 4*(16*(d*x + c)*cos(d*x + c) - 4*sin(d*x + c) - 1)*sin(3*d*x + 3*c) + 8*sin(3*d*x + 3
*c)^2 + 8*sin(d*x + c)^2 + 4*sin(d*x + c))*e^2*f/(a*d*cos(4*d*x + 4*c)^2 + 4*a*d*cos(3*d*x + 3*c)^2 + 8*a*d*co
s(3*d*x + 3*c)*cos(d*x + c) + 4*a*d*cos(d*x + c)^2 + a*d*sin(4*d*x + 4*c)^2 + 4*a*d*sin(3*d*x + 3*c)^2 + 4*a*d
*sin(d*x + c)^2 + 4*a*d*sin(d*x + c) + a*d - 2*(2*a*d*sin(3*d*x + 3*c) + 2*a*d*sin(d*x + c) + a*d)*cos(4*d*x +
4*c) + 4*(a*d*cos(3*d*x + 3*c) + a*d*cos(d*x + c))*sin(4*d*x + 4*c) + 4*(2*a*d*sin(d*x + c) + a*d)*sin(3*d*x
+ 3*c)) + 8*e^3*(sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*sin(d*x + c)^3/(c
os(d*x + c) + 1)^3 - 1)/(a + 2*a*sin(d*x + c)/(cos(d*x + c) + 1) - 2*a*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - a
*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) - 12*(24*d*e*f^2 - 8*(2*c^3 + 3*c)*f^3 - (6*(5*c^2 + 4)*f^3*cos(4*d*x +
4*c) + (60*I*c^2 + 48*I)*f^3*cos(3*d*x + 3*c) + (60*I*c^2 + 48*I)*f^3*cos(d*x + c) + (30*I*c^2 + 24*I)*f^3*sin
(4*d*x + 4*c) - 12*(5*c^2 + 4)*f^3*sin(3*d*x + 3*c) - 12*(5*c^2 + 4)*f^3*sin(d*x + c) - 6*(5*c^2 + 4)*f^3)*arc
tan2(sin(d*x + c) + 1, cos(d*x + c)) - (18*c^2*f^3*cos(4*d*x + 4*c) + 36*I*c^2*f^3*cos(3*d*x + 3*c) + 36*I*c^2
*f^3*cos(d*x + c) + 18*I*c^2*f^3*sin(4*d*x + 4*c) - 36*c^2*f^3*sin(3*d*x + 3*c) - 36*c^2*f^3*sin(d*x + c) - 18
*c^2*f^3)*arctan2(sin(d*x + c) - 1, cos(d*x + c)) - (30*(d*x + c)^2*f^3 + 60*(d*e*f^2 - c*f^3)*(d*x + c) - 30*
((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*cos(4*d*x + 4*c) + (-60*I*(d*x + c)^2*f^3 + (-120*I*d*e*f^2
+ 120*I*c*f^3)*(d*x + c))*cos(3*d*x + 3*c) + (-60*I*(d*x + c)^2*f^3 + (-120*I*d*e*f^2 + 120*I*c*f^3)*(d*x + c)
)*cos(d*x + c) + (-30*I*(d*x + c)^2*f^3 + (-60*I*d*e*f^2 + 60*I*c*f^3)*(d*x + c))*sin(4*d*x + 4*c) + 60*((d*x
+ c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*sin(3*d*x + 3*c) + 60*((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x
+ c))*sin(d*x + c))*arctan2(cos(d*x + c), sin(d*x + c) + 1) + (18*(d*x + c)^2*f^3 + 36*(d*e*f^2 - c*f^3)*(d*x
+ c) - 18*((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*cos(4*d*x + 4*c) - (36*I*(d*x + c)^2*f^3 + (72*I*d
*e*f^2 - 72*I*c*f^3)*(d*x + c))*cos(3*d*x + 3*c) - (36*I*(d*x + c)^2*f^3 + (72*I*d*e*f^2 - 72*I*c*f^3)*(d*x +
c))*cos(d*x + c) - (18*I*(d*x + c)^2*f^3 + (36*I*d*e*f^2 - 36*I*c*f^3)*(d*x + c))*sin(4*d*x + 4*c) + 36*((d*x
+ c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*sin(3*d*x + 3*c) + 36*((d*x + c)^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x
+ c))*sin(d*x + c))*arctan2(cos(d*x + c), -sin(d*x + c) + 1) + 8*(2*(d*x + c)^3*f^3 + 3*(2*c^2 + 1)*(d*x + c)*
f^3 + 6*(d*e*f^2 - c*f^3)*(d*x + c)^2)*cos(4*d*x + 4*c) - (-32*I*(d*x + c)^3*f^3 + 24*I*d*e*f^2 - 12*(c^2 + 2*
I*c)*f^3 - 12*(8*I*d*e*f^2 + (-8*I*c + 1)*f^3)*(d*x + c)^2 - (24*d*e*f^2 - (-96*I*c^2 + 24*c - 24*I)*f^3)*(d*x
+ c))*cos(3*d*x + 3*c) + 24*(d*e*f^2 + (d*x + c)*f^3 - c*f^3)*cos(2*d*x + 2*c) + (12*(d*x + c)^2*f^3 - 24*I*d
*e*f^2 - (-32*I*c^3 - 12*c^2 - 24*I*c)*f^3 + (24*d*e*f^2 - (24*c - 24*I)*f^3)*(d*x + c))*cos(d*x + c) - (60*d*
e*f^2 + 60*(d*x + c)*f^3 - 60*c*f^3 - 60*(d*e*f^2 + (d*x + c)*f^3 - c*f^3)*cos(4*d*x + 4*c) + (-120*I*d*e*f^2
- 120*I*(d*x + c)*f^3 + 120*I*c*f^3)*cos(3*d*x + 3*c) + (-120*I*d*e*f^2 - 120*I*(d*x + c)*f^3 + 120*I*c*f^3)*c
os(d*x + c) + (-60*I*d*e*f^2 - 60*I*(d*x + c)*f^3 + 60*I*c*f^3)*sin(4*d*x + 4*c) + 120*(d*e*f^2 + (d*x + c)*f^
3 - c*f^3)*sin(3*d*x + 3*c) + 120*(d*e*f^2 + (d*x + c)*f^3 - c*f^3)*sin(d*x + c))*dilog(I*e^(I*d*x + I*c)) - (
36*d*e*f^2 + 36*(d*x + c)*f^3 - 36*c*f^3 - 36*(d*e*f^2 + (d*x + c)*f^3 - c*f^3)*cos(4*d*x + 4*c) + (-72*I*d*e*
f^2 - 72*I*(d*x + c)*f^3 + 72*I*c*f^3)*cos(3*d*x + 3*c) + (-72*I*d*e*f^2 - 72*I*(d*x + c)*f^3 + 72*I*c*f^3)*co
s(d*x + c) + (-36*I*d*e*f^2 - 36*I*(d*x + c)*f^3 + 36*I*c*f^3)*sin(4*d*x + 4*c) + 72*(d*e*f^2 + (d*x + c)*f^3
- c*f^3)*sin(3*d*x + 3*c) + 72*(d*e*f^2 + (d*x + c)*f^3 - c*f^3)*sin(d*x + c))*dilog(-I*e^(I*d*x + I*c)) - (15
*I*(d*x + c)^2*f^3 + (15*I*c^2 + 12*I)*f^3 + (30*I*d*e*f^2 - 30*I*c*f^3)*(d*x + c) + (-15*I*(d*x + c)^2*f^3 +
(-15*I*c^2 - 12*I)*f^3 + (-30*I*d*e*f^2 + 30*I*c*f^3)*(d*x + c))*cos(4*d*x + 4*c) + 6*(5*(d*x + c)^2*f^3 + (5*
c^2 + 4)*f^3 + 10*(d*e*f^2 - c*f^3)*(d*x + c))*cos(3*d*x + 3*c) + 6*(5*(d*x + c)^2*f^3 + (5*c^2 + 4)*f^3 + 10*
(d*e*f^2 - c*f^3)*(d*x + c))*cos(d*x + c) + 3*(5*(d*x + c)^2*f^3 + (5*c^2 + 4)*f^3 + 10*(d*e*f^2 - c*f^3)*(d*x
+ c))*sin(4*d*x + 4*c) + (30*I*(d*x + c)^2*f^3 + (30*I*c^2 + 24*I)*f^3 + (60*I*d*e*f^2 - 60*I*c*f^3)*(d*x + c
))*sin(3*d*x + 3*c) + (30*I*(d*x + c)^2*f^3 + (30*I*c^2 + 24*I)*f^3 + (60*I*d*e*f^2 - 60*I*c*f^3)*(d*x + c))*s
in(d*x + c))*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1) - (9*I*(d*x + c)^2*f^3 + 9*I*c^2*f^3 +
(18*I*d*e*f^2 - 18*I*c*f^3)*(d*x + c) + (-9*I*(d*x + c)^2*f^3 - 9*I*c^2*f^3 + (-18*I*d*e*f^2 + 18*I*c*f^3)*(d*
x + c))*cos(4*d*x + 4*c) + 18*((d*x + c)^2*f^3 + c^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*cos(3*d*x + 3*c) + 1
8*((d*x + c)^2*f^3 + c^2*f^3 + 2*(d*e*f^2 - c*f^3)*(d*x + c))*cos(d*x + c) + 9*((d*x + c)^2*f^3 + c^2*f^3 + 2*
(d*e*f^2 - c*f^3)*(d*x + c))*sin(4*d*x + 4*c) + (18*I*(d*x + c)^2*f^3 + 18*I*c^2*f^3 + (36*I*d*e*f^2 - 36*I*c*
f^3)*(d*x + c))*sin(3*d*x + 3*c) + (18*I*(d*x + c)^2*f^3 + 18*I*c^2*f^3 + (36*I*d*e*f^2 - 36*I*c*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) - (-60*I*f^3*cos(4*d*x + 4*c) + 120
*f^3*cos(3*d*x + 3*c) + 120*f^3*cos(d*x + c) + 60*f^3*sin(4*d*x + 4*c) + 120*I*f^3*sin(3*d*x + 3*c) + 120*I*f^
3*sin(d*x + c) + 60*I*f^3)*polylog(3, I*e^(I*d*x + I*c)) - (-36*I*f^3*cos(4*d*x + 4*c) + 72*f^3*cos(3*d*x + 3*
c) + 72*f^3*cos(d*x + c) + 36*f^3*sin(4*d*x + 4*c) + 72*I*f^3*sin(3*d*x + 3*c) + 72*I*f^3*sin(d*x + c) + 36*I*
f^3)*polylog(3, -I*e^(I*d*x + I*c)) - (-16*I*(d*x + c)^3*f^3 + (-48*I*c^2 - 24*I)*(d*x + c)*f^3 + (-48*I*d*e*f
^2 + 48*I*c*f^3)*(d*x + c)^2)*sin(4*d*x + 4*c) - (32*(d*x + c)^3*f^3 - 24*d*e*f^2 + (-12*I*c^2 + 24*c)*f^3 + (
96*d*e*f^2 - (96*c + 12*I)*f^3)*(d*x + c)^2 + (-24*I*d*e*f^2 + 24*(4*c^2 + I*c + 1)*f^3)*(d*x + c))*sin(3*d*x
+ 3*c) - (-24*I*d*e*f^2 - 24*I*(d*x + c)*f^3 + 24*I*c*f^3)*sin(2*d*x + 2*c) - (-12*I*(d*x + c)^2*f^3 - 24*d*e*
f^2 + (32*c^3 - 12*I*c^2 + 24*c)*f^3 - 24*(I*d*e*f^2 + (-I*c - 1)*f^3)*(d*x + c))*sin(d*x + c))/(-12*I*a*d^3*c
os(4*d*x + 4*c) + 24*a*d^3*cos(3*d*x + 3*c) + 24*a*d^3*cos(d*x + c) + 12*a*d^3*sin(4*d*x + 4*c) + 24*I*a*d^3*s
in(3*d*x + 3*c) + 24*I*a*d^3*sin(d*x + c) + 12*I*a*d^3))/d

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Fricas [C]  time = 3.11652, size = 3767, normalized size = 7.93 \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)^2/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/12*(4*d^3*f^3*x^3 + 12*d^3*e*f^2*x^2 + 12*d^3*e^2*f*x + 4*d^3*e^3 - 4*(2*d^3*f^3*x^3 + 6*d^3*e*f^2*x^2 + 2*d
^3*e^3 + 3*d*e*f^2 + 3*(2*d^3*e^2*f + d*f^3)*x)*cos(d*x + c)^2 - 6*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + d^2*e^2*f)*c
os(d*x + c) + ((18*I*d*f^3*x + 18*I*d*e*f^2)*cos(d*x + c)*sin(d*x + c) + (18*I*d*f^3*x + 18*I*d*e*f^2)*cos(d*x
+ c))*dilog(I*cos(d*x + c) + sin(d*x + c)) + ((-30*I*d*f^3*x - 30*I*d*e*f^2)*cos(d*x + c)*sin(d*x + c) + (-30
*I*d*f^3*x - 30*I*d*e*f^2)*cos(d*x + c))*dilog(I*cos(d*x + c) - sin(d*x + c)) + ((-18*I*d*f^3*x - 18*I*d*e*f^2
)*cos(d*x + c)*sin(d*x + c) + (-18*I*d*f^3*x - 18*I*d*e*f^2)*cos(d*x + c))*dilog(-I*cos(d*x + c) + sin(d*x + c
)) + ((30*I*d*f^3*x + 30*I*d*e*f^2)*cos(d*x + c)*sin(d*x + c) + (30*I*d*f^3*x + 30*I*d*e*f^2)*cos(d*x + c))*di
log(-I*cos(d*x + c) - sin(d*x + c)) + 3*((5*d^2*e^2*f - 10*c*d*e*f^2 + (5*c^2 + 4)*f^3)*cos(d*x + c)*sin(d*x +
c) + (5*d^2*e^2*f - 10*c*d*e*f^2 + (5*c^2 + 4)*f^3)*cos(d*x + c))*log(cos(d*x + c) + I*sin(d*x + c) + I) + 9*
((d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x + c)*sin(d*x + c) + (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x +
c))*log(cos(d*x + c) - I*sin(d*x + c) + I) + 15*((d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*
x + c)*sin(d*x + c) + (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c))*log(I*cos(d*x + c) +
sin(d*x + c) + 1) + 9*((d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c)*sin(d*x + c) + (d^2
*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c))*log(I*cos(d*x + c) - sin(d*x + c) + 1) + 15*((
d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c)*sin(d*x + c) + (d^2*f^3*x^2 + 2*d^2*e*f^2*x
+ 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c))*log(-I*cos(d*x + c) + sin(d*x + c) + 1) + 9*((d^2*f^3*x^2 + 2*d^2*e*f^2
*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c)*sin(d*x + c) + (d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*
cos(d*x + c))*log(-I*cos(d*x + c) - sin(d*x + c) + 1) + 3*((5*d^2*e^2*f - 10*c*d*e*f^2 + (5*c^2 + 4)*f^3)*cos(
d*x + c)*sin(d*x + c) + (5*d^2*e^2*f - 10*c*d*e*f^2 + (5*c^2 + 4)*f^3)*cos(d*x + c))*log(-cos(d*x + c) + I*sin
(d*x + c) + I) + 9*((d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x + c)*sin(d*x + c) + (d^2*e^2*f - 2*c*d*e*f^2 +
c^2*f^3)*cos(d*x + c))*log(-cos(d*x + c) - I*sin(d*x + c) + I) + 18*(f^3*cos(d*x + c)*sin(d*x + c) + f^3*cos(
d*x + c))*polylog(3, I*cos(d*x + c) + sin(d*x + c)) + 30*(f^3*cos(d*x + c)*sin(d*x + c) + f^3*cos(d*x + c))*po
lylog(3, I*cos(d*x + c) - sin(d*x + c)) + 18*(f^3*cos(d*x + c)*sin(d*x + c) + f^3*cos(d*x + c))*polylog(3, -I*
cos(d*x + c) + sin(d*x + c)) + 30*(f^3*cos(d*x + c)*sin(d*x + c) + f^3*cos(d*x + c))*polylog(3, -I*cos(d*x + c
) - sin(d*x + c)) + 8*(d^3*f^3*x^3 + 3*d^3*e*f^2*x^2 + 3*d^3*e^2*f*x + d^3*e^3)*sin(d*x + c))/(a*d^4*cos(d*x +
c)*sin(d*x + c) + a*d^4*cos(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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 )^{2}}{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)^2/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

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