\(\int (c+d x)^3 (a+b \sec (e+f x))^2 \, dx\) [29]

Optimal result

Integrand size = 20, antiderivative size = 364 \[ \int (c+d x)^3 (a+b \sec (e+f x))^2 \, dx=-\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {4 i a b (c+d x)^3 \arctan \left (e^{i (e+f x)}\right )}{f}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {6 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac {6 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}-\frac {12 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )}{f^3}+\frac {12 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )}{f^3}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {12 i a b d^3 \operatorname {PolyLog}\left (4,-i e^{i (e+f x)}\right )}{f^4}+\frac {12 i a b d^3 \operatorname {PolyLog}\left (4,i e^{i (e+f x)}\right )}{f^4}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {4275, 4266, 2611, 6744, 2320, 6724, 4269, 3800, 2221} \[ \int (c+d x)^3 (a+b \sec (e+f x))^2 \, dx=\frac {a^2 (c+d x)^4}{4 d}-\frac {4 i a b (c+d x)^3 \arctan \left (e^{i (e+f x)}\right )}{f}-\frac {12 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )}{f^3}+\frac {12 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )}{f^3}+\frac {6 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac {6 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}-\frac {12 i a b d^3 \operatorname {PolyLog}\left (4,-i e^{i (e+f x)}\right )}{f^4}+\frac {12 i a b d^3 \operatorname {PolyLog}\left (4,i e^{i (e+f x)}\right )}{f^4}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {i b^2 (c+d x)^3}{f}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^4} \]

[In]

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

[Out]

((-I)*b^2*(c + d*x)^3)/f + (a^2*(c + d*x)^4)/(4*d) - ((4*I)*a*b*(c + d*x)^3*ArcTan[E^(I*(e + f*x))])/f + (3*b^
2*d*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])/f^2 + ((6*I)*a*b*d*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(e + f*x))])
/f^2 - ((6*I)*a*b*d*(c + d*x)^2*PolyLog[2, I*E^(I*(e + f*x))])/f^2 - ((3*I)*b^2*d^2*(c + d*x)*PolyLog[2, -E^((
2*I)*(e + f*x))])/f^3 - (12*a*b*d^2*(c + d*x)*PolyLog[3, (-I)*E^(I*(e + f*x))])/f^3 + (12*a*b*d^2*(c + d*x)*Po
lyLog[3, I*E^(I*(e + f*x))])/f^3 + (3*b^2*d^3*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^4) - ((12*I)*a*b*d^3*Poly
Log[4, (-I)*E^(I*(e + f*x))])/f^4 + ((12*I)*a*b*d^3*PolyLog[4, I*E^(I*(e + f*x))])/f^4 + (b^2*(c + d*x)^3*Tan[
e + f*x])/f

Rule 2221

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

Rule 2320

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

Rule 2611

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

Rule 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 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 4275

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

Rule 6724

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

Rule 6744

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

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 (c+d x)^3+2 a b (c+d x)^3 \sec (e+f x)+b^2 (c+d x)^3 \sec ^2(e+f x)\right ) \, dx \\ & = \frac {a^2 (c+d x)^4}{4 d}+(2 a b) \int (c+d x)^3 \sec (e+f x) \, dx+b^2 \int (c+d x)^3 \sec ^2(e+f x) \, dx \\ & = \frac {a^2 (c+d x)^4}{4 d}-\frac {4 i a b (c+d x)^3 \arctan \left (e^{i (e+f x)}\right )}{f}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {(6 a b d) \int (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right ) \, dx}{f}+\frac {(6 a b d) \int (c+d x)^2 \log \left (1+i e^{i (e+f x)}\right ) \, dx}{f}-\frac {\left (3 b^2 d\right ) \int (c+d x)^2 \tan (e+f x) \, dx}{f} \\ & = -\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {4 i a b (c+d x)^3 \arctan \left (e^{i (e+f x)}\right )}{f}+\frac {6 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac {6 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {\left (12 i a b d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right ) \, dx}{f^2}+\frac {\left (12 i a b d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right ) \, dx}{f^2}+\frac {\left (6 i b^2 d\right ) \int \frac {e^{2 i (e+f x)} (c+d x)^2}{1+e^{2 i (e+f x)}} \, dx}{f} \\ & = -\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {4 i a b (c+d x)^3 \arctan \left (e^{i (e+f x)}\right )}{f}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {6 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac {6 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}-\frac {12 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )}{f^3}+\frac {12 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )}{f^3}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}+\frac {\left (12 a b d^3\right ) \int \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right ) \, dx}{f^3}-\frac {\left (12 a b d^3\right ) \int \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right ) \, dx}{f^3}-\frac {\left (6 b^2 d^2\right ) \int (c+d x) \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f^2} \\ & = -\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {4 i a b (c+d x)^3 \arctan \left (e^{i (e+f x)}\right )}{f}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {6 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac {6 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}-\frac {12 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )}{f^3}+\frac {12 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )}{f^3}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {\left (12 i a b d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{f^4}+\frac {\left (12 i a b d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{f^4}+\frac {\left (3 i b^2 d^3\right ) \int \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right ) \, dx}{f^3} \\ & = -\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {4 i a b (c+d x)^3 \arctan \left (e^{i (e+f x)}\right )}{f}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {6 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac {6 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}-\frac {12 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )}{f^3}+\frac {12 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )}{f^3}-\frac {12 i a b d^3 \operatorname {PolyLog}\left (4,-i e^{i (e+f x)}\right )}{f^4}+\frac {12 i a b d^3 \operatorname {PolyLog}\left (4,i e^{i (e+f x)}\right )}{f^4}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}+\frac {\left (3 b^2 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^4} \\ & = -\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {4 i a b (c+d x)^3 \arctan \left (e^{i (e+f x)}\right )}{f}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {6 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac {6 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}-\frac {12 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )}{f^3}+\frac {12 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )}{f^3}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {12 i a b d^3 \operatorname {PolyLog}\left (4,-i e^{i (e+f x)}\right )}{f^4}+\frac {12 i a b d^3 \operatorname {PolyLog}\left (4,i e^{i (e+f x)}\right )}{f^4}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.59 (sec) , antiderivative size = 646, normalized size of antiderivative = 1.77 \[ \int (c+d x)^3 (a+b \sec (e+f x))^2 \, dx=\frac {4 a^2 c^3 f^4 x-12 i b^2 c d^2 f^3 x^2+6 a^2 c^2 d f^4 x^2-4 i b^2 d^3 f^3 x^3+4 a^2 c d^2 f^4 x^3+a^2 d^3 f^4 x^4-48 i a b c^2 d f^3 x \arctan \left (e^{i (e+f x)}\right )-48 i a b c d^2 f^3 x^2 \arctan \left (e^{i (e+f x)}\right )-16 i a b d^3 f^3 x^3 \arctan \left (e^{i (e+f x)}\right )+8 a b c^3 f^3 \text {arctanh}(\sin (e+f x))+24 b^2 c d^2 f^2 x \log \left (1+e^{2 i (e+f x)}\right )+12 b^2 d^3 f^2 x^2 \log \left (1+e^{2 i (e+f x)}\right )+12 b^2 c^2 d f^2 \log (\cos (e+f x))+24 i a b d f^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )-24 i a b d f^2 (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )-12 i b^2 c d^2 f \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )-12 i b^2 d^3 f x \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )-48 a b c d^2 f \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )-48 a b d^3 f x \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )+48 a b c d^2 f \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )+48 a b d^3 f x \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )+6 b^2 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )-48 i a b d^3 \operatorname {PolyLog}\left (4,-i e^{i (e+f x)}\right )+48 i a b d^3 \operatorname {PolyLog}\left (4,i e^{i (e+f x)}\right )+4 b^2 c^3 f^3 \tan (e+f x)+12 b^2 c^2 d f^3 x \tan (e+f x)+12 b^2 c d^2 f^3 x^2 \tan (e+f x)+4 b^2 d^3 f^3 x^3 \tan (e+f x)}{4 f^4} \]

[In]

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

[Out]

(4*a^2*c^3*f^4*x - (12*I)*b^2*c*d^2*f^3*x^2 + 6*a^2*c^2*d*f^4*x^2 - (4*I)*b^2*d^3*f^3*x^3 + 4*a^2*c*d^2*f^4*x^
3 + a^2*d^3*f^4*x^4 - (48*I)*a*b*c^2*d*f^3*x*ArcTan[E^(I*(e + f*x))] - (48*I)*a*b*c*d^2*f^3*x^2*ArcTan[E^(I*(e
+ f*x))] - (16*I)*a*b*d^3*f^3*x^3*ArcTan[E^(I*(e + f*x))] + 8*a*b*c^3*f^3*ArcTanh[Sin[e + f*x]] + 24*b^2*c*d^
2*f^2*x*Log[1 + E^((2*I)*(e + f*x))] + 12*b^2*d^3*f^2*x^2*Log[1 + E^((2*I)*(e + f*x))] + 12*b^2*c^2*d*f^2*Log[
Cos[e + f*x]] + (24*I)*a*b*d*f^2*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(e + f*x))] - (24*I)*a*b*d*f^2*(c + d*x)^2*P
olyLog[2, I*E^(I*(e + f*x))] - (12*I)*b^2*c*d^2*f*PolyLog[2, -E^((2*I)*(e + f*x))] - (12*I)*b^2*d^3*f*x*PolyLo
g[2, -E^((2*I)*(e + f*x))] - 48*a*b*c*d^2*f*PolyLog[3, (-I)*E^(I*(e + f*x))] - 48*a*b*d^3*f*x*PolyLog[3, (-I)*
E^(I*(e + f*x))] + 48*a*b*c*d^2*f*PolyLog[3, I*E^(I*(e + f*x))] + 48*a*b*d^3*f*x*PolyLog[3, I*E^(I*(e + f*x))]
+ 6*b^2*d^3*PolyLog[3, -E^((2*I)*(e + f*x))] - (48*I)*a*b*d^3*PolyLog[4, (-I)*E^(I*(e + f*x))] + (48*I)*a*b*d
^3*PolyLog[4, I*E^(I*(e + f*x))] + 4*b^2*c^3*f^3*Tan[e + f*x] + 12*b^2*c^2*d*f^3*x*Tan[e + f*x] + 12*b^2*c*d^2
*f^3*x^2*Tan[e + f*x] + 4*b^2*d^3*f^3*x^3*Tan[e + f*x])/(4*f^4)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1488 vs. \(2 (327 ) = 654\).

Time = 1.65 (sec) , antiderivative size = 1489, normalized size of antiderivative = 4.09

method	result	size
risch	\(\text {Expression too large to display}\)	\(1489\)

[In]

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

[Out]

6/f^2*b^2*c*d^2*ln(1+exp(2*I*(f*x+e)))*x-12/f^3*b*d^3*a*polylog(3,-I*exp(I*(f*x+e)))*x+12/f^3*b*d^2*c*a*polylo
g(3,I*exp(I*(f*x+e)))+2/f^4*b*a*e^3*d^3*ln(1-I*exp(I*(f*x+e)))-2/f^4*b*a*e^3*d^3*ln(1+I*exp(I*(f*x+e)))-4*I/f*
b*a*c^3*arctan(exp(I*(f*x+e)))+6*I/f^3*b^2*d^3*e^2*x-6*I/f^3*b^2*d^3*polylog(2,I*exp(I*(f*x+e)))*x-6*I/f^4*b^2
*d^3*polylog(2,I*exp(I*(f*x+e)))*e+6/f^4*b^2*d^3*polylog(3,I*exp(I*(f*x+e)))+6/f^4*b^2*d^3*polylog(3,-I*exp(I*
(f*x+e)))+6/f*b*d^2*c*a*ln(1-I*exp(I*(f*x+e)))*x^2-6/f^3*b*e^2*a*c*d^2*ln(1-I*exp(I*(f*x+e)))-6/f*b*a*c^2*d*ln
(1+I*exp(I*(f*x+e)))*x+6/f*b*a*c^2*d*ln(1-I*exp(I*(f*x+e)))*x+2*I*b^2*(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3)/f/(1
+exp(2*I*(f*x+e)))+12/f^3*b^2*c*d^2*e*ln(exp(I*(f*x+e)))-12*I*a*b*d^3*polylog(4,-I*exp(I*(f*x+e)))/f^4+12*I*a*
b*d^3*polylog(4,I*exp(I*(f*x+e)))/f^4-6*I/f^4*b^2*d^3*polylog(2,-I*exp(I*(f*x+e)))*e-6*I/f*b^2*c*d^2*x^2-6*I/f
^3*b^2*c*d^2*e^2+a^2*d^2*c*x^3+3/2*a^2*d*c^2*x^2+a^2*c^3*x+6/f^2*b*a*c^2*d*ln(1-I*exp(I*(f*x+e)))*e+6/f^3*b*e^
2*a*c*d^2*ln(1+I*exp(I*(f*x+e)))-6/f^2*b*a*c^2*d*ln(1+I*exp(I*(f*x+e)))*e-6/f*b*d^2*c*a*ln(1+I*exp(I*(f*x+e)))
*x^2-12*I/f^2*b^2*c*d^2*e*x+4*I/f^4*b*a*d^3*e^3*arctan(exp(I*(f*x+e)))+6*I/f^2*b*a*c^2*d*polylog(2,-I*exp(I*(f
*x+e)))-6*I/f^2*b*a*c^2*d*polylog(2,I*exp(I*(f*x+e)))-6*I/f^2*b*d^3*a*polylog(2,I*exp(I*(f*x+e)))*x^2+6*I/f^2*
b*d^3*a*polylog(2,-I*exp(I*(f*x+e)))*x^2+12/f^3*b*d^3*a*polylog(3,I*exp(I*(f*x+e)))*x+6/f^3*b^2*d^3*ln(1-I*exp
(I*(f*x+e)))*e*x+6/f^3*b^2*d^3*ln(1+I*exp(I*(f*x+e)))*e*x-2/f*b*d^3*a*ln(1+I*exp(I*(f*x+e)))*x^3+2/f*b*d^3*a*l
n(1-I*exp(I*(f*x+e)))*x^3-12*I/f^3*b*a*c*d^2*e^2*arctan(exp(I*(f*x+e)))+12*I/f^2*b*d^2*c*a*polylog(2,-I*exp(I*
(f*x+e)))*x-12*I/f^2*b*d^2*c*a*polylog(2,I*exp(I*(f*x+e)))*x+12*I/f^2*b*a*c^2*d*e*arctan(exp(I*(f*x+e)))-6/f^3
*b^2*e*d^3*ln(1+exp(2*I*(f*x+e)))*x-12/f^3*b*d^2*c*a*polylog(3,-I*exp(I*(f*x+e)))+3/f^4*b^2*d^3*ln(1-I*exp(I*(
f*x+e)))*e^2-3/f^4*b^2*e^2*d^3*ln(1+exp(2*I*(f*x+e)))+3/f^2*b^2*c^2*d*ln(1+exp(2*I*(f*x+e)))+3/f^4*b^2*d^3*ln(
1+I*exp(I*(f*x+e)))*e^2+3/f^2*b^2*d^3*ln(1-I*exp(I*(f*x+e)))*x^2-6/f^2*b^2*c^2*d*ln(exp(I*(f*x+e)))+3/f^2*b^2*
d^3*ln(1+I*exp(I*(f*x+e)))*x^2-6/f^4*b^2*d^3*e^2*ln(exp(I*(f*x+e)))-2*I/f*b^2*d^3*x^3+4*I/f^4*b^2*d^3*e^3-3*I/
f^3*b^2*c*d^2*polylog(2,-exp(2*I*(f*x+e)))+1/4*a^2*d^3*x^4+1/4*a^2/d*c^4+3*I/f^4*b^2*e*d^3*polylog(2,-exp(2*I*
(f*x+e)))-6*I/f^3*b^2*d^3*polylog(2,-I*exp(I*(f*x+e)))*x

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. 1823 vs. \(2 (311) = 622\).

Time = 0.38 (sec) , antiderivative size = 1823, normalized size of antiderivative = 5.01 \[ \int (c+d x)^3 (a+b \sec (e+f x))^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3267 vs. \(2 (311) = 622\).

Time = 0.72 (sec) , antiderivative size = 3267, normalized size of antiderivative = 8.98 \[ \int (c+d x)^3 (a+b \sec (e+f x))^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^3 (a+b \sec (e+f x))^2 \, dx=\int {\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^2\,{\left (c+d\,x\right )}^3 \,d x \]

[In]

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

[Out]

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