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

Optimal. Leaf size=1523 \[ \text{result too large to display} \]

[Out]

((-I)*b^2*(c + d*x)^3)/(a^2*(a^2 - b^2)*f) + (c + d*x)^4/(4*a^2*d) + (3*b^2*d*(c + d*x)^2*Log[1 + (a*E^(I*(e +
f*x)))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*f^2) + (3*b^2*d*(c + d*x)^2*Log[1 + (a*E^(I*(e + f*x)))/(b
+ I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*f^2) - (I*b^3*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 +
b^2])])/(a^2*(-a^2 + b^2)^(3/2)*f) + ((2*I)*b*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2])]
)/(a^2*Sqrt[-a^2 + b^2]*f) + (I*b^3*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^
2 + b^2)^(3/2)*f) - ((2*I)*b*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 +
b^2]*f) - ((6*I)*b^2*d^2*(c + d*x)*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^
2)*f^3) - ((6*I)*b^2*d^2*(c + d*x)*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2
)*f^3) - (3*b^3*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/
2)*f^2) + (6*b*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*
f^2) + (3*b^3*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)
*f^2) - (6*b*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^
2) + (6*b^2*d^3*PolyLog[3, -((a*E^(I*(e + f*x)))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*f^4) + (6*b^2*d^3
*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*f^4) - ((6*I)*b^3*d^2*(c + d*x)*
PolyLog[3, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^3) + ((12*I)*b*d^2*(c + d
*x)*PolyLog[3, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^3) + ((6*I)*b^3*d^2*(c
+ d*x)*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^3) - ((12*I)*b*d^2
*(c + d*x)*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^3) + (6*b^3*d^3*
PolyLog[4, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^4) - (12*b*d^3*PolyLog[4,
-((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^4) - (6*b^3*d^3*PolyLog[4, -((a*E^(I*
(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^4) + (12*b*d^3*PolyLog[4, -((a*E^(I*(e + f*x))
)/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^4) + (b^2*(c + d*x)^3*Sin[e + f*x])/(a*(a^2 - b^2)*f*(b +
a*Cos[e + f*x]))

________________________________________________________________________________________

Rubi [A]  time = 2.80916, antiderivative size = 1523, normalized size of antiderivative = 1., number of steps used = 36, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4191, 3324, 3321, 2264, 2190, 2531, 6609, 2282, 6589, 4522} \[ \frac{(c+d x)^4}{4 a^2 d}+\frac{2 i b \log \left (\frac{e^{i (e+f x)} a}{b-\sqrt{b^2-a^2}}+1\right ) (c+d x)^3}{a^2 \sqrt{b^2-a^2} f}-\frac{i b^3 \log \left (\frac{e^{i (e+f x)} a}{b-\sqrt{b^2-a^2}}+1\right ) (c+d x)^3}{a^2 \left (b^2-a^2\right )^{3/2} f}-\frac{2 i b \log \left (\frac{e^{i (e+f x)} a}{b+\sqrt{b^2-a^2}}+1\right ) (c+d x)^3}{a^2 \sqrt{b^2-a^2} f}+\frac{i b^3 \log \left (\frac{e^{i (e+f x)} a}{b+\sqrt{b^2-a^2}}+1\right ) (c+d x)^3}{a^2 \left (b^2-a^2\right )^{3/2} f}+\frac{b^2 \sin (e+f x) (c+d x)^3}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}-\frac{i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac{3 b^2 d \log \left (\frac{e^{i (e+f x)} a}{b-i \sqrt{a^2-b^2}}+1\right ) (c+d x)^2}{a^2 \left (a^2-b^2\right ) f^2}+\frac{3 b^2 d \log \left (\frac{e^{i (e+f x)} a}{b+i \sqrt{a^2-b^2}}+1\right ) (c+d x)^2}{a^2 \left (a^2-b^2\right ) f^2}+\frac{6 b d \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right ) (c+d x)^2}{a^2 \sqrt{b^2-a^2} f^2}-\frac{3 b^3 d \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right ) (c+d x)^2}{a^2 \left (b^2-a^2\right )^{3/2} f^2}-\frac{6 b d \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right ) (c+d x)^2}{a^2 \sqrt{b^2-a^2} f^2}+\frac{3 b^3 d \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right ) (c+d x)^2}{a^2 \left (b^2-a^2\right )^{3/2} f^2}-\frac{6 i b^2 d^2 \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right ) (c+d x)}{a^2 \left (a^2-b^2\right ) f^3}-\frac{6 i b^2 d^2 \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right ) (c+d x)}{a^2 \left (a^2-b^2\right ) f^3}+\frac{12 i b d^2 \text{PolyLog}\left (3,-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right ) (c+d x)}{a^2 \sqrt{b^2-a^2} f^3}-\frac{6 i b^3 d^2 \text{PolyLog}\left (3,-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right ) (c+d x)}{a^2 \left (b^2-a^2\right )^{3/2} f^3}-\frac{12 i b d^2 \text{PolyLog}\left (3,-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right ) (c+d x)}{a^2 \sqrt{b^2-a^2} f^3}+\frac{6 i b^3 d^2 \text{PolyLog}\left (3,-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right ) (c+d x)}{a^2 \left (b^2-a^2\right )^{3/2} f^3}+\frac{6 b^2 d^3 \text{PolyLog}\left (3,-\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}+\frac{6 b^2 d^3 \text{PolyLog}\left (3,-\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}-\frac{12 b d^3 \text{PolyLog}\left (4,-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a^2 \sqrt{b^2-a^2} f^4}+\frac{6 b^3 d^3 \text{PolyLog}\left (4,-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} f^4}+\frac{12 b d^3 \text{PolyLog}\left (4,-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right )}{a^2 \sqrt{b^2-a^2} f^4}-\frac{6 b^3 d^3 \text{PolyLog}\left (4,-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*b^2*(c + d*x)^3)/(a^2*(a^2 - b^2)*f) + (c + d*x)^4/(4*a^2*d) + (3*b^2*d*(c + d*x)^2*Log[1 + (a*E^(I*(e +
f*x)))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*f^2) + (3*b^2*d*(c + d*x)^2*Log[1 + (a*E^(I*(e + f*x)))/(b
+ I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*f^2) - (I*b^3*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 +
b^2])])/(a^2*(-a^2 + b^2)^(3/2)*f) + ((2*I)*b*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2])]
)/(a^2*Sqrt[-a^2 + b^2]*f) + (I*b^3*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^
2 + b^2)^(3/2)*f) - ((2*I)*b*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 +
b^2]*f) - ((6*I)*b^2*d^2*(c + d*x)*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^
2)*f^3) - ((6*I)*b^2*d^2*(c + d*x)*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2
)*f^3) - (3*b^3*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/
2)*f^2) + (6*b*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*
f^2) + (3*b^3*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)
*f^2) - (6*b*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^
2) + (6*b^2*d^3*PolyLog[3, -((a*E^(I*(e + f*x)))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*f^4) + (6*b^2*d^3
*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*f^4) - ((6*I)*b^3*d^2*(c + d*x)*
PolyLog[3, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^3) + ((12*I)*b*d^2*(c + d
*x)*PolyLog[3, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^3) + ((6*I)*b^3*d^2*(c
+ d*x)*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^3) - ((12*I)*b*d^2
*(c + d*x)*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^3) + (6*b^3*d^3*
PolyLog[4, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^4) - (12*b*d^3*PolyLog[4,
-((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^4) - (6*b^3*d^3*PolyLog[4, -((a*E^(I*
(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^4) + (12*b*d^3*PolyLog[4, -((a*E^(I*(e + f*x))
)/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^4) + (b^2*(c + d*x)^3*Sin[e + f*x])/(a*(a^2 - b^2)*f*(b +
a*Cos[e + f*x]))

Rule 4191

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 3324

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*(c + d*x)^m*Cos[
e + f*x])/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[(b*d*m)/(f*(a^2 - b^2)), Int[((c + d*x)^(m - 1)*Cos[e + f*x])/(a + b*Sin[e + f*x]), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3321

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

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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 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 4522

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)])/(Cos[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :>
Simp[(I*(e + f*x)^(m + 1))/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a - Rt[-a^2 + b^2, 2] + I
*b*E^(I*(c + d*x))), x] + Int[((e + f*x)^m*E^(I*(c + d*x)))/(I*a + Rt[-a^2 + b^2, 2] + I*b*E^(I*(c + d*x))), x
]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]

Rubi steps

\begin{align*} \int \frac{(c+d x)^3}{(a+b \sec (e+f x))^2} \, dx &=\int \left (\frac{(c+d x)^3}{a^2}+\frac{b^2 (c+d x)^3}{a^2 (b+a \cos (e+f x))^2}-\frac{2 b (c+d x)^3}{a^2 (b+a \cos (e+f x))}\right ) \, dx\\ &=\frac{(c+d x)^4}{4 a^2 d}-\frac{(2 b) \int \frac{(c+d x)^3}{b+a \cos (e+f x)} \, dx}{a^2}+\frac{b^2 \int \frac{(c+d x)^3}{(b+a \cos (e+f x))^2} \, dx}{a^2}\\ &=\frac{(c+d x)^4}{4 a^2 d}+\frac{b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}-\frac{(4 b) \int \frac{e^{i (e+f x)} (c+d x)^3}{a+2 b e^{i (e+f x)}+a e^{2 i (e+f x)}} \, dx}{a^2}-\frac{b^3 \int \frac{(c+d x)^3}{b+a \cos (e+f x)} \, dx}{a^2 \left (a^2-b^2\right )}-\frac{\left (3 b^2 d\right ) \int \frac{(c+d x)^2 \sin (e+f x)}{b+a \cos (e+f x)} \, dx}{a \left (a^2-b^2\right ) f}\\ &=-\frac{i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac{(c+d x)^4}{4 a^2 d}+\frac{b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}-\frac{\left (2 b^3\right ) \int \frac{e^{i (e+f x)} (c+d x)^3}{a+2 b e^{i (e+f x)}+a e^{2 i (e+f x)}} \, dx}{a^2 \left (a^2-b^2\right )}-\frac{(4 b) \int \frac{e^{i (e+f x)} (c+d x)^3}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \sqrt{-a^2+b^2}}+\frac{(4 b) \int \frac{e^{i (e+f x)} (c+d x)^3}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \sqrt{-a^2+b^2}}-\frac{\left (3 b^2 d\right ) \int \frac{e^{i (e+f x)} (c+d x)^2}{i b-\sqrt{a^2-b^2}+i a e^{i (e+f x)}} \, dx}{a \left (a^2-b^2\right ) f}-\frac{\left (3 b^2 d\right ) \int \frac{e^{i (e+f x)} (c+d x)^2}{i b+\sqrt{a^2-b^2}+i a e^{i (e+f x)}} \, dx}{a \left (a^2-b^2\right ) f}\\ &=-\frac{i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac{(c+d x)^4}{4 a^2 d}+\frac{3 b^2 d (c+d x)^2 \log \left (1+\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac{3 b^2 d (c+d x)^2 \log \left (1+\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac{2 i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f}-\frac{2 i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f}+\frac{b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac{\left (2 b^3\right ) \int \frac{e^{i (e+f x)} (c+d x)^3}{2 b-2 \sqrt{-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \left (-a^2+b^2\right )^{3/2}}-\frac{\left (2 b^3\right ) \int \frac{e^{i (e+f x)} (c+d x)^3}{2 b+2 \sqrt{-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \left (-a^2+b^2\right )^{3/2}}-\frac{\left (6 b^2 d^2\right ) \int (c+d x) \log \left (1+\frac{i a e^{i (e+f x)}}{i b-\sqrt{a^2-b^2}}\right ) \, dx}{a^2 \left (a^2-b^2\right ) f^2}-\frac{\left (6 b^2 d^2\right ) \int (c+d x) \log \left (1+\frac{i a e^{i (e+f x)}}{i b+\sqrt{a^2-b^2}}\right ) \, dx}{a^2 \left (a^2-b^2\right ) f^2}-\frac{(6 i b d) \int (c+d x)^2 \log \left (1+\frac{2 a e^{i (e+f x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx}{a^2 \sqrt{-a^2+b^2} f}+\frac{(6 i b d) \int (c+d x)^2 \log \left (1+\frac{2 a e^{i (e+f x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx}{a^2 \sqrt{-a^2+b^2} f}\\ &=-\frac{i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac{(c+d x)^4}{4 a^2 d}+\frac{3 b^2 d (c+d x)^2 \log \left (1+\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac{3 b^2 d (c+d x)^2 \log \left (1+\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac{i b^3 (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac{2 i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f}+\frac{i b^3 (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac{2 i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f}-\frac{6 i b^2 d^2 (c+d x) \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac{6 i b^2 d^2 (c+d x) \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}+\frac{6 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^2}-\frac{6 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^2}+\frac{b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac{\left (6 i b^2 d^3\right ) \int \text{Li}_2\left (-\frac{i a e^{i (e+f x)}}{i b-\sqrt{a^2-b^2}}\right ) \, dx}{a^2 \left (a^2-b^2\right ) f^3}+\frac{\left (6 i b^2 d^3\right ) \int \text{Li}_2\left (-\frac{i a e^{i (e+f x)}}{i b+\sqrt{a^2-b^2}}\right ) \, dx}{a^2 \left (a^2-b^2\right ) f^3}-\frac{\left (12 b d^2\right ) \int (c+d x) \text{Li}_2\left (-\frac{2 a e^{i (e+f x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx}{a^2 \sqrt{-a^2+b^2} f^2}+\frac{\left (12 b d^2\right ) \int (c+d x) \text{Li}_2\left (-\frac{2 a e^{i (e+f x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx}{a^2 \sqrt{-a^2+b^2} f^2}+\frac{\left (3 i b^3 d\right ) \int (c+d x)^2 \log \left (1+\frac{2 a e^{i (e+f x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac{\left (3 i b^3 d\right ) \int (c+d x)^2 \log \left (1+\frac{2 a e^{i (e+f x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx}{a^2 \left (-a^2+b^2\right )^{3/2} f}\\ &=-\frac{i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac{(c+d x)^4}{4 a^2 d}+\frac{3 b^2 d (c+d x)^2 \log \left (1+\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac{3 b^2 d (c+d x)^2 \log \left (1+\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac{i b^3 (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac{2 i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f}+\frac{i b^3 (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac{2 i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f}-\frac{6 i b^2 d^2 (c+d x) \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac{6 i b^2 d^2 (c+d x) \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac{3 b^3 d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}+\frac{6 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^2}+\frac{3 b^3 d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}-\frac{6 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^2}+\frac{12 i b d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^3}-\frac{12 i b d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^3}+\frac{b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac{\left (6 b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i a x}{-i b+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \left (a^2-b^2\right ) f^4}+\frac{\left (6 b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{i a x}{i b+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \left (a^2-b^2\right ) f^4}-\frac{\left (12 i b d^3\right ) \int \text{Li}_3\left (-\frac{2 a e^{i (e+f x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx}{a^2 \sqrt{-a^2+b^2} f^3}+\frac{\left (12 i b d^3\right ) \int \text{Li}_3\left (-\frac{2 a e^{i (e+f x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx}{a^2 \sqrt{-a^2+b^2} f^3}+\frac{\left (6 b^3 d^2\right ) \int (c+d x) \text{Li}_2\left (-\frac{2 a e^{i (e+f x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}-\frac{\left (6 b^3 d^2\right ) \int (c+d x) \text{Li}_2\left (-\frac{2 a e^{i (e+f x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}\\ &=-\frac{i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac{(c+d x)^4}{4 a^2 d}+\frac{3 b^2 d (c+d x)^2 \log \left (1+\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac{3 b^2 d (c+d x)^2 \log \left (1+\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac{i b^3 (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac{2 i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f}+\frac{i b^3 (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac{2 i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f}-\frac{6 i b^2 d^2 (c+d x) \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac{6 i b^2 d^2 (c+d x) \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac{3 b^3 d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}+\frac{6 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^2}+\frac{3 b^3 d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}-\frac{6 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^2}+\frac{6 b^2 d^3 \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}+\frac{6 b^2 d^3 \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}-\frac{6 i b^3 d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^3}+\frac{12 i b d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^3}+\frac{6 i b^3 d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^3}-\frac{12 i b d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^3}+\frac{b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}-\frac{\left (12 b d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{a x}{-b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \sqrt{-a^2+b^2} f^4}+\frac{\left (12 b d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \sqrt{-a^2+b^2} f^4}+\frac{\left (6 i b^3 d^3\right ) \int \text{Li}_3\left (-\frac{2 a e^{i (e+f x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx}{a^2 \left (-a^2+b^2\right )^{3/2} f^3}-\frac{\left (6 i b^3 d^3\right ) \int \text{Li}_3\left (-\frac{2 a e^{i (e+f x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx}{a^2 \left (-a^2+b^2\right )^{3/2} f^3}\\ &=-\frac{i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac{(c+d x)^4}{4 a^2 d}+\frac{3 b^2 d (c+d x)^2 \log \left (1+\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac{3 b^2 d (c+d x)^2 \log \left (1+\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac{i b^3 (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac{2 i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f}+\frac{i b^3 (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac{2 i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f}-\frac{6 i b^2 d^2 (c+d x) \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac{6 i b^2 d^2 (c+d x) \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac{3 b^3 d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}+\frac{6 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^2}+\frac{3 b^3 d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}-\frac{6 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^2}+\frac{6 b^2 d^3 \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}+\frac{6 b^2 d^3 \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}-\frac{6 i b^3 d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^3}+\frac{12 i b d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^3}+\frac{6 i b^3 d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^3}-\frac{12 i b d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^3}-\frac{12 b d^3 \text{Li}_4\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^4}+\frac{12 b d^3 \text{Li}_4\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^4}+\frac{b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac{\left (6 b^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{a x}{-b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^4}-\frac{\left (6 b^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^4}\\ &=-\frac{i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac{(c+d x)^4}{4 a^2 d}+\frac{3 b^2 d (c+d x)^2 \log \left (1+\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac{3 b^2 d (c+d x)^2 \log \left (1+\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac{i b^3 (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac{2 i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f}+\frac{i b^3 (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac{2 i b (c+d x)^3 \log \left (1+\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f}-\frac{6 i b^2 d^2 (c+d x) \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac{6 i b^2 d^2 (c+d x) \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac{3 b^3 d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}+\frac{6 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^2}+\frac{3 b^3 d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}-\frac{6 b d (c+d x)^2 \text{Li}_2\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^2}+\frac{6 b^2 d^3 \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}+\frac{6 b^2 d^3 \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}-\frac{6 i b^3 d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^3}+\frac{12 i b d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^3}+\frac{6 i b^3 d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^3}-\frac{12 i b d^2 (c+d x) \text{Li}_3\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^3}+\frac{6 b^3 d^3 \text{Li}_4\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^4}-\frac{12 b d^3 \text{Li}_4\left (-\frac{a e^{i (e+f x)}}{b-\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^4}-\frac{6 b^3 d^3 \text{Li}_4\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^4}+\frac{12 b d^3 \text{Li}_4\left (-\frac{a e^{i (e+f x)}}{b+\sqrt{-a^2+b^2}}\right )}{a^2 \sqrt{-a^2+b^2} f^4}+\frac{b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}\\ \end{align*}

Mathematica [B]  time = 25.9251, size = 8174, normalized size = 5.37 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [F]  time = 0.794, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{3}}{ \left ( a+b\sec \left ( fx+e \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+b*sec(f*x+e))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 7.24652, size = 14575, normalized size = 9.57 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((a^4*b - 2*a^2*b^3 + b^5)*d^3*f^4*x^4 + 4*(a^4*b - 2*a^2*b^3 + b^5)*c*d^2*f^4*x^3 + 6*(a^4*b - 2*a^2*b^3
+ b^5)*c^2*d*f^4*x^2 + 4*(a^4*b - 2*a^2*b^3 + b^5)*c^3*f^4*x + 12*((2*a^4*b - a^2*b^3)*d^3*cos(f*x + e) + (2*a
^3*b^2 - a*b^4)*d^3)*sqrt(-(a^2 - b^2)/a^2)*polylog(4, -(b*cos(f*x + e) + I*b*sin(f*x + e) + (a*cos(f*x + e) +
I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/a^2))/a) - 12*((2*a^4*b - a^2*b^3)*d^3*cos(f*x + e) + (2*a^3*b^2 - a*b^4)
*d^3)*sqrt(-(a^2 - b^2)/a^2)*polylog(4, -(b*cos(f*x + e) + I*b*sin(f*x + e) - (a*cos(f*x + e) + I*a*sin(f*x +
e))*sqrt(-(a^2 - b^2)/a^2))/a) + 12*((2*a^4*b - a^2*b^3)*d^3*cos(f*x + e) + (2*a^3*b^2 - a*b^4)*d^3)*sqrt(-(a^
2 - b^2)/a^2)*polylog(4, -(b*cos(f*x + e) - I*b*sin(f*x + e) + (a*cos(f*x + e) - I*a*sin(f*x + e))*sqrt(-(a^2
- b^2)/a^2))/a) - 12*((2*a^4*b - a^2*b^3)*d^3*cos(f*x + e) + (2*a^3*b^2 - a*b^4)*d^3)*sqrt(-(a^2 - b^2)/a^2)*p
olylog(4, -(b*cos(f*x + e) - I*b*sin(f*x + e) - (a*cos(f*x + e) - I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/a^2))/a)
+ ((a^5 - 2*a^3*b^2 + a*b^4)*d^3*f^4*x^4 + 4*(a^5 - 2*a^3*b^2 + a*b^4)*c*d^2*f^4*x^3 + 6*(a^5 - 2*a^3*b^2 + a
*b^4)*c^2*d*f^4*x^2 + 4*(a^5 - 2*a^3*b^2 + a*b^4)*c^3*f^4*x)*cos(f*x + e) + (-12*I*(a^2*b^3 - b^5)*d^3*f*x - 1
2*I*(a^2*b^3 - b^5)*c*d^2*f + (-12*I*(a^3*b^2 - a*b^4)*d^3*f*x - 12*I*(a^3*b^2 - a*b^4)*c*d^2*f)*cos(f*x + e)
- 6*((2*a^3*b^2 - a*b^4)*d^3*f^2*x^2 + 2*(2*a^3*b^2 - a*b^4)*c*d^2*f^2*x + (2*a^3*b^2 - a*b^4)*c^2*d*f^2 + ((2
*a^4*b - a^2*b^3)*d^3*f^2*x^2 + 2*(2*a^4*b - a^2*b^3)*c*d^2*f^2*x + (2*a^4*b - a^2*b^3)*c^2*d*f^2)*cos(f*x + e
))*sqrt(-(a^2 - b^2)/a^2))*dilog(-1/2*(2*b*cos(f*x + e) + 2*I*b*sin(f*x + e) + 2*(a*cos(f*x + e) + I*a*sin(f*x
+ e))*sqrt(-(a^2 - b^2)/a^2) + 2*a)/a + 1) + (-12*I*(a^2*b^3 - b^5)*d^3*f*x - 12*I*(a^2*b^3 - b^5)*c*d^2*f +
(-12*I*(a^3*b^2 - a*b^4)*d^3*f*x - 12*I*(a^3*b^2 - a*b^4)*c*d^2*f)*cos(f*x + e) + 6*((2*a^3*b^2 - a*b^4)*d^3*f
^2*x^2 + 2*(2*a^3*b^2 - a*b^4)*c*d^2*f^2*x + (2*a^3*b^2 - a*b^4)*c^2*d*f^2 + ((2*a^4*b - a^2*b^3)*d^3*f^2*x^2
+ 2*(2*a^4*b - a^2*b^3)*c*d^2*f^2*x + (2*a^4*b - a^2*b^3)*c^2*d*f^2)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2))*dil
og(-1/2*(2*b*cos(f*x + e) + 2*I*b*sin(f*x + e) - 2*(a*cos(f*x + e) + I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/a^2)
+ 2*a)/a + 1) + (12*I*(a^2*b^3 - b^5)*d^3*f*x + 12*I*(a^2*b^3 - b^5)*c*d^2*f + (12*I*(a^3*b^2 - a*b^4)*d^3*f*x
+ 12*I*(a^3*b^2 - a*b^4)*c*d^2*f)*cos(f*x + e) - 6*((2*a^3*b^2 - a*b^4)*d^3*f^2*x^2 + 2*(2*a^3*b^2 - a*b^4)*c
*d^2*f^2*x + (2*a^3*b^2 - a*b^4)*c^2*d*f^2 + ((2*a^4*b - a^2*b^3)*d^3*f^2*x^2 + 2*(2*a^4*b - a^2*b^3)*c*d^2*f^
2*x + (2*a^4*b - a^2*b^3)*c^2*d*f^2)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2))*dilog(-1/2*(2*b*cos(f*x + e) - 2*I*
b*sin(f*x + e) + 2*(a*cos(f*x + e) - I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/a^2) + 2*a)/a + 1) + (12*I*(a^2*b^3 -
b^5)*d^3*f*x + 12*I*(a^2*b^3 - b^5)*c*d^2*f + (12*I*(a^3*b^2 - a*b^4)*d^3*f*x + 12*I*(a^3*b^2 - a*b^4)*c*d^2*
f)*cos(f*x + e) + 6*((2*a^3*b^2 - a*b^4)*d^3*f^2*x^2 + 2*(2*a^3*b^2 - a*b^4)*c*d^2*f^2*x + (2*a^3*b^2 - a*b^4)
*c^2*d*f^2 + ((2*a^4*b - a^2*b^3)*d^3*f^2*x^2 + 2*(2*a^4*b - a^2*b^3)*c*d^2*f^2*x + (2*a^4*b - a^2*b^3)*c^2*d*
f^2)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2))*dilog(-1/2*(2*b*cos(f*x + e) - 2*I*b*sin(f*x + e) - 2*(a*cos(f*x +
e) - I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/a^2) + 2*a)/a + 1) + 2*(3*(a^2*b^3 - b^5)*d^3*e^2 - 6*(a^2*b^3 - b^5)
*c*d^2*e*f + 3*(a^2*b^3 - b^5)*c^2*d*f^2 + 3*((a^3*b^2 - a*b^4)*d^3*e^2 - 2*(a^3*b^2 - a*b^4)*c*d^2*e*f + (a^3
*b^2 - a*b^4)*c^2*d*f^2)*cos(f*x + e) + (-I*(2*a^3*b^2 - a*b^4)*d^3*e^3 + 3*I*(2*a^3*b^2 - a*b^4)*c*d^2*e^2*f
- 3*I*(2*a^3*b^2 - a*b^4)*c^2*d*e*f^2 + I*(2*a^3*b^2 - a*b^4)*c^3*f^3 + (-I*(2*a^4*b - a^2*b^3)*d^3*e^3 + 3*I*
(2*a^4*b - a^2*b^3)*c*d^2*e^2*f - 3*I*(2*a^4*b - a^2*b^3)*c^2*d*e*f^2 + I*(2*a^4*b - a^2*b^3)*c^3*f^3)*cos(f*x
+ e))*sqrt(-(a^2 - b^2)/a^2))*log(2*a*cos(f*x + e) + 2*I*a*sin(f*x + e) + 2*a*sqrt(-(a^2 - b^2)/a^2) + 2*b) +
2*(3*(a^2*b^3 - b^5)*d^3*e^2 - 6*(a^2*b^3 - b^5)*c*d^2*e*f + 3*(a^2*b^3 - b^5)*c^2*d*f^2 + 3*((a^3*b^2 - a*b^
4)*d^3*e^2 - 2*(a^3*b^2 - a*b^4)*c*d^2*e*f + (a^3*b^2 - a*b^4)*c^2*d*f^2)*cos(f*x + e) + (I*(2*a^3*b^2 - a*b^4
)*d^3*e^3 - 3*I*(2*a^3*b^2 - a*b^4)*c*d^2*e^2*f + 3*I*(2*a^3*b^2 - a*b^4)*c^2*d*e*f^2 - I*(2*a^3*b^2 - a*b^4)*
c^3*f^3 + (I*(2*a^4*b - a^2*b^3)*d^3*e^3 - 3*I*(2*a^4*b - a^2*b^3)*c*d^2*e^2*f + 3*I*(2*a^4*b - a^2*b^3)*c^2*d
*e*f^2 - I*(2*a^4*b - a^2*b^3)*c^3*f^3)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2))*log(2*a*cos(f*x + e) - 2*I*a*sin
(f*x + e) + 2*a*sqrt(-(a^2 - b^2)/a^2) + 2*b) + 2*(3*(a^2*b^3 - b^5)*d^3*e^2 - 6*(a^2*b^3 - b^5)*c*d^2*e*f + 3
*(a^2*b^3 - b^5)*c^2*d*f^2 + 3*((a^3*b^2 - a*b^4)*d^3*e^2 - 2*(a^3*b^2 - a*b^4)*c*d^2*e*f + (a^3*b^2 - a*b^4)*
c^2*d*f^2)*cos(f*x + e) + (-I*(2*a^3*b^2 - a*b^4)*d^3*e^3 + 3*I*(2*a^3*b^2 - a*b^4)*c*d^2*e^2*f - 3*I*(2*a^3*b
^2 - a*b^4)*c^2*d*e*f^2 + I*(2*a^3*b^2 - a*b^4)*c^3*f^3 + (-I*(2*a^4*b - a^2*b^3)*d^3*e^3 + 3*I*(2*a^4*b - a^2
*b^3)*c*d^2*e^2*f - 3*I*(2*a^4*b - a^2*b^3)*c^2*d*e*f^2 + I*(2*a^4*b - a^2*b^3)*c^3*f^3)*cos(f*x + e))*sqrt(-(
a^2 - b^2)/a^2))*log(-2*a*cos(f*x + e) + 2*I*a*sin(f*x + e) + 2*a*sqrt(-(a^2 - b^2)/a^2) - 2*b) + 2*(3*(a^2*b^
3 - b^5)*d^3*e^2 - 6*(a^2*b^3 - b^5)*c*d^2*e*f + 3*(a^2*b^3 - b^5)*c^2*d*f^2 + 3*((a^3*b^2 - a*b^4)*d^3*e^2 -
2*(a^3*b^2 - a*b^4)*c*d^2*e*f + (a^3*b^2 - a*b^4)*c^2*d*f^2)*cos(f*x + e) + (I*(2*a^3*b^2 - a*b^4)*d^3*e^3 - 3
*I*(2*a^3*b^2 - a*b^4)*c*d^2*e^2*f + 3*I*(2*a^3*b^2 - a*b^4)*c^2*d*e*f^2 - I*(2*a^3*b^2 - a*b^4)*c^3*f^3 + (I*
(2*a^4*b - a^2*b^3)*d^3*e^3 - 3*I*(2*a^4*b - a^2*b^3)*c*d^2*e^2*f + 3*I*(2*a^4*b - a^2*b^3)*c^2*d*e*f^2 - I*(2
*a^4*b - a^2*b^3)*c^3*f^3)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2))*log(-2*a*cos(f*x + e) - 2*I*a*sin(f*x + e) +
2*a*sqrt(-(a^2 - b^2)/a^2) - 2*b) + 2*(3*(a^2*b^3 - b^5)*d^3*f^2*x^2 + 6*(a^2*b^3 - b^5)*c*d^2*f^2*x - 3*(a^2*
b^3 - b^5)*d^3*e^2 + 6*(a^2*b^3 - b^5)*c*d^2*e*f + 3*((a^3*b^2 - a*b^4)*d^3*f^2*x^2 + 2*(a^3*b^2 - a*b^4)*c*d^
2*f^2*x - (a^3*b^2 - a*b^4)*d^3*e^2 + 2*(a^3*b^2 - a*b^4)*c*d^2*e*f)*cos(f*x + e) + (-I*(2*a^3*b^2 - a*b^4)*d^
3*f^3*x^3 - 3*I*(2*a^3*b^2 - a*b^4)*c*d^2*f^3*x^2 - 3*I*(2*a^3*b^2 - a*b^4)*c^2*d*f^3*x - I*(2*a^3*b^2 - a*b^4
)*d^3*e^3 + 3*I*(2*a^3*b^2 - a*b^4)*c*d^2*e^2*f - 3*I*(2*a^3*b^2 - a*b^4)*c^2*d*e*f^2 + (-I*(2*a^4*b - a^2*b^3
)*d^3*f^3*x^3 - 3*I*(2*a^4*b - a^2*b^3)*c*d^2*f^3*x^2 - 3*I*(2*a^4*b - a^2*b^3)*c^2*d*f^3*x - I*(2*a^4*b - a^2
*b^3)*d^3*e^3 + 3*I*(2*a^4*b - a^2*b^3)*c*d^2*e^2*f - 3*I*(2*a^4*b - a^2*b^3)*c^2*d*e*f^2)*cos(f*x + e))*sqrt(
-(a^2 - b^2)/a^2))*log(1/2*(2*b*cos(f*x + e) + 2*I*b*sin(f*x + e) + 2*(a*cos(f*x + e) + I*a*sin(f*x + e))*sqrt
(-(a^2 - b^2)/a^2) + 2*a)/a) + 2*(3*(a^2*b^3 - b^5)*d^3*f^2*x^2 + 6*(a^2*b^3 - b^5)*c*d^2*f^2*x - 3*(a^2*b^3 -
b^5)*d^3*e^2 + 6*(a^2*b^3 - b^5)*c*d^2*e*f + 3*((a^3*b^2 - a*b^4)*d^3*f^2*x^2 + 2*(a^3*b^2 - a*b^4)*c*d^2*f^2
*x - (a^3*b^2 - a*b^4)*d^3*e^2 + 2*(a^3*b^2 - a*b^4)*c*d^2*e*f)*cos(f*x + e) + (I*(2*a^3*b^2 - a*b^4)*d^3*f^3*
x^3 + 3*I*(2*a^3*b^2 - a*b^4)*c*d^2*f^3*x^2 + 3*I*(2*a^3*b^2 - a*b^4)*c^2*d*f^3*x + I*(2*a^3*b^2 - a*b^4)*d^3*
e^3 - 3*I*(2*a^3*b^2 - a*b^4)*c*d^2*e^2*f + 3*I*(2*a^3*b^2 - a*b^4)*c^2*d*e*f^2 + (I*(2*a^4*b - a^2*b^3)*d^3*f
^3*x^3 + 3*I*(2*a^4*b - a^2*b^3)*c*d^2*f^3*x^2 + 3*I*(2*a^4*b - a^2*b^3)*c^2*d*f^3*x + I*(2*a^4*b - a^2*b^3)*d
^3*e^3 - 3*I*(2*a^4*b - a^2*b^3)*c*d^2*e^2*f + 3*I*(2*a^4*b - a^2*b^3)*c^2*d*e*f^2)*cos(f*x + e))*sqrt(-(a^2 -
b^2)/a^2))*log(1/2*(2*b*cos(f*x + e) + 2*I*b*sin(f*x + e) - 2*(a*cos(f*x + e) + I*a*sin(f*x + e))*sqrt(-(a^2
- b^2)/a^2) + 2*a)/a) + 2*(3*(a^2*b^3 - b^5)*d^3*f^2*x^2 + 6*(a^2*b^3 - b^5)*c*d^2*f^2*x - 3*(a^2*b^3 - b^5)*d
^3*e^2 + 6*(a^2*b^3 - b^5)*c*d^2*e*f + 3*((a^3*b^2 - a*b^4)*d^3*f^2*x^2 + 2*(a^3*b^2 - a*b^4)*c*d^2*f^2*x - (a
^3*b^2 - a*b^4)*d^3*e^2 + 2*(a^3*b^2 - a*b^4)*c*d^2*e*f)*cos(f*x + e) + (I*(2*a^3*b^2 - a*b^4)*d^3*f^3*x^3 + 3
*I*(2*a^3*b^2 - a*b^4)*c*d^2*f^3*x^2 + 3*I*(2*a^3*b^2 - a*b^4)*c^2*d*f^3*x + I*(2*a^3*b^2 - a*b^4)*d^3*e^3 - 3
*I*(2*a^3*b^2 - a*b^4)*c*d^2*e^2*f + 3*I*(2*a^3*b^2 - a*b^4)*c^2*d*e*f^2 + (I*(2*a^4*b - a^2*b^3)*d^3*f^3*x^3
+ 3*I*(2*a^4*b - a^2*b^3)*c*d^2*f^3*x^2 + 3*I*(2*a^4*b - a^2*b^3)*c^2*d*f^3*x + I*(2*a^4*b - a^2*b^3)*d^3*e^3
- 3*I*(2*a^4*b - a^2*b^3)*c*d^2*e^2*f + 3*I*(2*a^4*b - a^2*b^3)*c^2*d*e*f^2)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a
^2))*log(1/2*(2*b*cos(f*x + e) - 2*I*b*sin(f*x + e) + 2*(a*cos(f*x + e) - I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/
a^2) + 2*a)/a) + 2*(3*(a^2*b^3 - b^5)*d^3*f^2*x^2 + 6*(a^2*b^3 - b^5)*c*d^2*f^2*x - 3*(a^2*b^3 - b^5)*d^3*e^2
+ 6*(a^2*b^3 - b^5)*c*d^2*e*f + 3*((a^3*b^2 - a*b^4)*d^3*f^2*x^2 + 2*(a^3*b^2 - a*b^4)*c*d^2*f^2*x - (a^3*b^2
- a*b^4)*d^3*e^2 + 2*(a^3*b^2 - a*b^4)*c*d^2*e*f)*cos(f*x + e) + (-I*(2*a^3*b^2 - a*b^4)*d^3*f^3*x^3 - 3*I*(2*
a^3*b^2 - a*b^4)*c*d^2*f^3*x^2 - 3*I*(2*a^3*b^2 - a*b^4)*c^2*d*f^3*x - I*(2*a^3*b^2 - a*b^4)*d^3*e^3 + 3*I*(2*
a^3*b^2 - a*b^4)*c*d^2*e^2*f - 3*I*(2*a^3*b^2 - a*b^4)*c^2*d*e*f^2 + (-I*(2*a^4*b - a^2*b^3)*d^3*f^3*x^3 - 3*I
*(2*a^4*b - a^2*b^3)*c*d^2*f^3*x^2 - 3*I*(2*a^4*b - a^2*b^3)*c^2*d*f^3*x - I*(2*a^4*b - a^2*b^3)*d^3*e^3 + 3*I
*(2*a^4*b - a^2*b^3)*c*d^2*e^2*f - 3*I*(2*a^4*b - a^2*b^3)*c^2*d*e*f^2)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2))*
log(1/2*(2*b*cos(f*x + e) - 2*I*b*sin(f*x + e) - 2*(a*cos(f*x + e) - I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/a^2)
+ 2*a)/a) + 2*(6*(a^3*b^2 - a*b^4)*d^3*cos(f*x + e) + 6*(a^2*b^3 - b^5)*d^3 + (-6*I*(2*a^3*b^2 - a*b^4)*d^3*f*
x - 6*I*(2*a^3*b^2 - a*b^4)*c*d^2*f + (-6*I*(2*a^4*b - a^2*b^3)*d^3*f*x - 6*I*(2*a^4*b - a^2*b^3)*c*d^2*f)*cos
(f*x + e))*sqrt(-(a^2 - b^2)/a^2))*polylog(3, -(b*cos(f*x + e) + I*b*sin(f*x + e) + (a*cos(f*x + e) + I*a*sin(
f*x + e))*sqrt(-(a^2 - b^2)/a^2))/a) + 2*(6*(a^3*b^2 - a*b^4)*d^3*cos(f*x + e) + 6*(a^2*b^3 - b^5)*d^3 + (6*I*
(2*a^3*b^2 - a*b^4)*d^3*f*x + 6*I*(2*a^3*b^2 - a*b^4)*c*d^2*f + (6*I*(2*a^4*b - a^2*b^3)*d^3*f*x + 6*I*(2*a^4*
b - a^2*b^3)*c*d^2*f)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2))*polylog(3, -(b*cos(f*x + e) + I*b*sin(f*x + e) - (
a*cos(f*x + e) + I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/a^2))/a) + 2*(6*(a^3*b^2 - a*b^4)*d^3*cos(f*x + e) + 6*(a
^2*b^3 - b^5)*d^3 + (6*I*(2*a^3*b^2 - a*b^4)*d^3*f*x + 6*I*(2*a^3*b^2 - a*b^4)*c*d^2*f + (6*I*(2*a^4*b - a^2*b
^3)*d^3*f*x + 6*I*(2*a^4*b - a^2*b^3)*c*d^2*f)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2))*polylog(3, -(b*cos(f*x +
e) - I*b*sin(f*x + e) + (a*cos(f*x + e) - I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/a^2))/a) + 2*(6*(a^3*b^2 - a*b^4
)*d^3*cos(f*x + e) + 6*(a^2*b^3 - b^5)*d^3 + (-6*I*(2*a^3*b^2 - a*b^4)*d^3*f*x - 6*I*(2*a^3*b^2 - a*b^4)*c*d^2
*f + (-6*I*(2*a^4*b - a^2*b^3)*d^3*f*x - 6*I*(2*a^4*b - a^2*b^3)*c*d^2*f)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2)
)*polylog(3, -(b*cos(f*x + e) - I*b*sin(f*x + e) - (a*cos(f*x + e) - I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/a^2))
/a) + 4*((a^3*b^2 - a*b^4)*d^3*f^3*x^3 + 3*(a^3*b^2 - a*b^4)*c*d^2*f^3*x^2 + 3*(a^3*b^2 - a*b^4)*c^2*d*f^3*x +
(a^3*b^2 - a*b^4)*c^3*f^3)*sin(f*x + e))/((a^7 - 2*a^5*b^2 + a^3*b^4)*f^4*cos(f*x + e) + (a^6*b - 2*a^4*b^3 +
a^2*b^5)*f^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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