3.1.6 \(\int (c+d x)^3 (a+a \sec (e+f x))^2 \, dx\) [6]

Optimal. Leaf size=371 \[ -\frac {i a^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {4 i a^2 (c+d x)^3 \text {ArcTan}\left (e^{i (e+f x)}\right )}{f}+\frac {3 a^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {6 i a^2 d (c+d x)^2 \text {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac {6 i a^2 d (c+d x)^2 \text {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}-\frac {3 i a^2 d^2 (c+d x) \text {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}-\frac {12 a^2 d^2 (c+d x) \text {PolyLog}\left (3,-i e^{i (e+f x)}\right )}{f^3}+\frac {12 a^2 d^2 (c+d x) \text {PolyLog}\left (3,i e^{i (e+f x)}\right )}{f^3}+\frac {3 a^2 d^3 \text {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {12 i a^2 d^3 \text {PolyLog}\left (4,-i e^{i (e+f x)}\right )}{f^4}+\frac {12 i a^2 d^3 \text {PolyLog}\left (4,i e^{i (e+f x)}\right )}{f^4}+\frac {a^2 (c+d x)^3 \tan (e+f x)}{f} \]

[Out]

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

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Rubi [A]
time = 0.32, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {4275, 4266, 2611, 6744, 2320, 6724, 4269, 3800, 2221} \begin {gather*} -\frac {4 i a^2 (c+d x)^3 \text {ArcTan}\left (e^{i (e+f x)}\right )}{f}-\frac {3 i a^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}-\frac {12 a^2 d^2 (c+d x) \text {Li}_3\left (-i e^{i (e+f x)}\right )}{f^3}+\frac {12 a^2 d^2 (c+d x) \text {Li}_3\left (i e^{i (e+f x)}\right )}{f^3}+\frac {6 i a^2 d (c+d x)^2 \text {Li}_2\left (-i e^{i (e+f x)}\right )}{f^2}-\frac {6 i a^2 d (c+d x)^2 \text {Li}_2\left (i e^{i (e+f x)}\right )}{f^2}+\frac {3 a^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {a^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {i a^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}+\frac {3 a^2 d^3 \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {12 i a^2 d^3 \text {Li}_4\left (-i e^{i (e+f x)}\right )}{f^4}+\frac {12 i a^2 d^3 \text {Li}_4\left (i e^{i (e+f x)}\right )}{f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*a^2*(c + d*x)^3)/f + (a^2*(c + d*x)^4)/(4*d) - ((4*I)*a^2*(c + d*x)^3*ArcTan[E^(I*(e + f*x))])/f + (3*a^
2*d*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])/f^2 + ((6*I)*a^2*d*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(e + f*x))])
/f^2 - ((6*I)*a^2*d*(c + d*x)^2*PolyLog[2, I*E^(I*(e + f*x))])/f^2 - ((3*I)*a^2*d^2*(c + d*x)*PolyLog[2, -E^((
2*I)*(e + f*x))])/f^3 - (12*a^2*d^2*(c + d*x)*PolyLog[3, (-I)*E^(I*(e + f*x))])/f^3 + (12*a^2*d^2*(c + d*x)*Po
lyLog[3, I*E^(I*(e + f*x))])/f^3 + (3*a^2*d^3*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^4) - ((12*I)*a^2*d^3*Poly
Log[4, (-I)*E^(I*(e + f*x))])/f^4 + ((12*I)*a^2*d^3*PolyLog[4, I*E^(I*(e + f*x))])/f^4 + (a^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*} \int (c+d x)^3 (a+a \sec (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^3+2 a^2 (c+d x)^3 \sec (e+f x)+a^2 (c+d x)^3 \sec ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}+a^2 \int (c+d x)^3 \sec ^2(e+f x) \, dx+\left (2 a^2\right ) \int (c+d x)^3 \sec (e+f x) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}-\frac {4 i a^2 (c+d x)^3 \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {a^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {\left (3 a^2 d\right ) \int (c+d x)^2 \tan (e+f x) \, dx}{f}-\frac {\left (6 a^2 d\right ) \int (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right ) \, dx}{f}+\frac {\left (6 a^2 d\right ) \int (c+d x)^2 \log \left (1+i e^{i (e+f x)}\right ) \, dx}{f}\\ &=-\frac {i a^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {4 i a^2 (c+d x)^3 \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {6 i a^2 d (c+d x)^2 \text {Li}_2\left (-i e^{i (e+f x)}\right )}{f^2}-\frac {6 i a^2 d (c+d x)^2 \text {Li}_2\left (i e^{i (e+f x)}\right )}{f^2}+\frac {a^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {\left (12 i a^2 d^2\right ) \int (c+d x) \text {Li}_2\left (-i e^{i (e+f x)}\right ) \, dx}{f^2}+\frac {\left (12 i a^2 d^2\right ) \int (c+d x) \text {Li}_2\left (i e^{i (e+f x)}\right ) \, dx}{f^2}+\frac {\left (6 i a^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 a^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {4 i a^2 (c+d x)^3 \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {3 a^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {6 i a^2 d (c+d x)^2 \text {Li}_2\left (-i e^{i (e+f x)}\right )}{f^2}-\frac {6 i a^2 d (c+d x)^2 \text {Li}_2\left (i e^{i (e+f x)}\right )}{f^2}-\frac {12 a^2 d^2 (c+d x) \text {Li}_3\left (-i e^{i (e+f x)}\right )}{f^3}+\frac {12 a^2 d^2 (c+d x) \text {Li}_3\left (i e^{i (e+f x)}\right )}{f^3}+\frac {a^2 (c+d x)^3 \tan (e+f x)}{f}+\frac {\left (12 a^2 d^3\right ) \int \text {Li}_3\left (-i e^{i (e+f x)}\right ) \, dx}{f^3}-\frac {\left (12 a^2 d^3\right ) \int \text {Li}_3\left (i e^{i (e+f x)}\right ) \, dx}{f^3}-\frac {\left (6 a^2 d^2\right ) \int (c+d x) \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f^2}\\ &=-\frac {i a^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {4 i a^2 (c+d x)^3 \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {3 a^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {6 i a^2 d (c+d x)^2 \text {Li}_2\left (-i e^{i (e+f x)}\right )}{f^2}-\frac {6 i a^2 d (c+d x)^2 \text {Li}_2\left (i e^{i (e+f x)}\right )}{f^2}-\frac {3 i a^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}-\frac {12 a^2 d^2 (c+d x) \text {Li}_3\left (-i e^{i (e+f x)}\right )}{f^3}+\frac {12 a^2 d^2 (c+d x) \text {Li}_3\left (i e^{i (e+f x)}\right )}{f^3}+\frac {a^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {\left (12 i a^2 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{f^4}+\frac {\left (12 i a^2 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{f^4}+\frac {\left (3 i a^2 d^3\right ) \int \text {Li}_2\left (-e^{2 i (e+f x)}\right ) \, dx}{f^3}\\ &=-\frac {i a^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {4 i a^2 (c+d x)^3 \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {3 a^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {6 i a^2 d (c+d x)^2 \text {Li}_2\left (-i e^{i (e+f x)}\right )}{f^2}-\frac {6 i a^2 d (c+d x)^2 \text {Li}_2\left (i e^{i (e+f x)}\right )}{f^2}-\frac {3 i a^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}-\frac {12 a^2 d^2 (c+d x) \text {Li}_3\left (-i e^{i (e+f x)}\right )}{f^3}+\frac {12 a^2 d^2 (c+d x) \text {Li}_3\left (i e^{i (e+f x)}\right )}{f^3}-\frac {12 i a^2 d^3 \text {Li}_4\left (-i e^{i (e+f x)}\right )}{f^4}+\frac {12 i a^2 d^3 \text {Li}_4\left (i e^{i (e+f x)}\right )}{f^4}+\frac {a^2 (c+d x)^3 \tan (e+f x)}{f}+\frac {\left (3 a^2 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^4}\\ &=-\frac {i a^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {4 i a^2 (c+d x)^3 \tan ^{-1}\left (e^{i (e+f x)}\right )}{f}+\frac {3 a^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {6 i a^2 d (c+d x)^2 \text {Li}_2\left (-i e^{i (e+f x)}\right )}{f^2}-\frac {6 i a^2 d (c+d x)^2 \text {Li}_2\left (i e^{i (e+f x)}\right )}{f^2}-\frac {3 i a^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}-\frac {12 a^2 d^2 (c+d x) \text {Li}_3\left (-i e^{i (e+f x)}\right )}{f^3}+\frac {12 a^2 d^2 (c+d x) \text {Li}_3\left (i e^{i (e+f x)}\right )}{f^3}+\frac {3 a^2 d^3 \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {12 i a^2 d^3 \text {Li}_4\left (-i e^{i (e+f x)}\right )}{f^4}+\frac {12 i a^2 d^3 \text {Li}_4\left (i e^{i (e+f x)}\right )}{f^4}+\frac {a^2 (c+d x)^3 \tan (e+f x)}{f}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(811\) vs. \(2(371)=742\).
time = 8.57, size = 811, normalized size = 2.19 \begin {gather*} \frac {1}{16} a^2 (1+\cos (e+f x))^2 \sec ^4\left (\frac {1}{2} (e+f x)\right ) \left (x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+\frac {4 (c+d x)^3 \sin \left (\frac {f x}{2}\right )}{f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {4 (c+d x)^3 \sin \left (\frac {f x}{2}\right )}{f \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {2 i \left (6 c^2 d f^3 x+6 c d^2 f^3 x^2+2 d^3 f^3 x^3+8 c^3 f^3 \text {ArcTan}(\cos (e+f x)+i \sin (e+f x))+24 c^2 d f^3 x \text {ArcTan}(\cos (e+f x)+i \sin (e+f x))+24 c d^2 f^3 x^2 \text {ArcTan}(\cos (e+f x)+i \sin (e+f x))+8 d^3 f^3 x^3 \text {ArcTan}(\cos (e+f x)+i \sin (e+f x))+6 i c^2 d f^2 \log (1+\cos (2 (e+f x))+i \sin (2 (e+f x)))+12 i c d^2 f^2 x \log (1+\cos (2 (e+f x))+i \sin (2 (e+f x)))+6 i d^3 f^2 x^2 \log (1+\cos (2 (e+f x))+i \sin (2 (e+f x)))+12 d f^2 (c+d x)^2 \text {PolyLog}(2,i \cos (e+f x)-\sin (e+f x))-12 d f^2 (c+d x)^2 \text {PolyLog}(2,-i \cos (e+f x)+\sin (e+f x))+6 c d^2 f \text {PolyLog}(2,-\cos (2 (e+f x))-i \sin (2 (e+f x)))+6 d^3 f x \text {PolyLog}(2,-\cos (2 (e+f x))-i \sin (2 (e+f x)))+24 i c d^2 f \text {PolyLog}(3,i \cos (e+f x)-\sin (e+f x))+24 i d^3 f x \text {PolyLog}(3,i \cos (e+f x)-\sin (e+f x))-24 i c d^2 f \text {PolyLog}(3,-i \cos (e+f x)+\sin (e+f x))-24 i d^3 f x \text {PolyLog}(3,-i \cos (e+f x)+\sin (e+f x))+3 i d^3 \text {PolyLog}(3,-\cos (2 (e+f x))-i \sin (2 (e+f x)))-24 d^3 \text {PolyLog}(4,i \cos (e+f x)-\sin (e+f x))+24 d^3 \text {PolyLog}(4,-i \cos (e+f x)+\sin (e+f x))+6 i c^2 d f^3 x \tan (e)+6 i c d^2 f^3 x^2 \tan (e)+2 i d^3 f^3 x^3 \tan (e)\right )}{f^4}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*(1 + Cos[e + f*x])^2*Sec[(e + f*x)/2]^4*(x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + (4*(c + d*x)^3*S
in[(f*x)/2])/(f*(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])) + (4*(c + d*x)^3*Sin[(f*x)/2])/(f
*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) - ((2*I)*(6*c^2*d*f^3*x + 6*c*d^2*f^3*x^2 + 2*d^
3*f^3*x^3 + 8*c^3*f^3*ArcTan[Cos[e + f*x] + I*Sin[e + f*x]] + 24*c^2*d*f^3*x*ArcTan[Cos[e + f*x] + I*Sin[e + f
*x]] + 24*c*d^2*f^3*x^2*ArcTan[Cos[e + f*x] + I*Sin[e + f*x]] + 8*d^3*f^3*x^3*ArcTan[Cos[e + f*x] + I*Sin[e +
f*x]] + (6*I)*c^2*d*f^2*Log[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]] + (12*I)*c*d^2*f^2*x*Log[1 + Cos[2*(e +
f*x)] + I*Sin[2*(e + f*x)]] + (6*I)*d^3*f^2*x^2*Log[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]] + 12*d*f^2*(c
+ d*x)^2*PolyLog[2, I*Cos[e + f*x] - Sin[e + f*x]] - 12*d*f^2*(c + d*x)^2*PolyLog[2, (-I)*Cos[e + f*x] + Sin[e
+ f*x]] + 6*c*d^2*f*PolyLog[2, -Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)]] + 6*d^3*f*x*PolyLog[2, -Cos[2*(e + f*x
)] - I*Sin[2*(e + f*x)]] + (24*I)*c*d^2*f*PolyLog[3, I*Cos[e + f*x] - Sin[e + f*x]] + (24*I)*d^3*f*x*PolyLog[3
, I*Cos[e + f*x] - Sin[e + f*x]] - (24*I)*c*d^2*f*PolyLog[3, (-I)*Cos[e + f*x] + Sin[e + f*x]] - (24*I)*d^3*f*
x*PolyLog[3, (-I)*Cos[e + f*x] + Sin[e + f*x]] + (3*I)*d^3*PolyLog[3, -Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)]]
- 24*d^3*PolyLog[4, I*Cos[e + f*x] - Sin[e + f*x]] + 24*d^3*PolyLog[4, (-I)*Cos[e + f*x] + Sin[e + f*x]] + (6*
I)*c^2*d*f^3*x*Tan[e] + (6*I)*c*d^2*f^3*x^2*Tan[e] + (2*I)*d^3*f^3*x^3*Tan[e]))/f^4))/16

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1516 vs. \(2 (334 ) = 668\).
time = 1.10, size = 1517, normalized size = 4.09

method	result	size
risch	\(\text {Expression too large to display}\)	\(1517\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3584 vs. \(2 (329) = 658\).
time = 0.95, size = 3584, normalized size = 9.66 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1940 vs. \(2 (329) = 658\).
time = 3.44, size = 1940, normalized size = 5.23 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{2} \left (\int c^{3}\, dx + \int 2 c^{3} \sec {\left (e + f x \right )}\, dx + \int c^{3} \sec ^{2}{\left (e + f x \right )}\, dx + \int d^{3} x^{3}\, dx + \int 3 c d^{2} x^{2}\, dx + \int 3 c^{2} d x\, dx + \int 2 d^{3} x^{3} \sec {\left (e + f x \right )}\, dx + \int d^{3} x^{3} \sec ^{2}{\left (e + f x \right )}\, dx + \int 6 c d^{2} x^{2} \sec {\left (e + f x \right )}\, dx + \int 3 c d^{2} x^{2} \sec ^{2}{\left (e + f x \right )}\, dx + \int 6 c^{2} d x \sec {\left (e + f x \right )}\, dx + \int 3 c^{2} d x \sec ^{2}{\left (e + f x \right )}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*(Integral(c**3, x) + Integral(2*c**3*sec(e + f*x), x) + Integral(c**3*sec(e + f*x)**2, x) + Integral(d**3
*x**3, x) + Integral(3*c*d**2*x**2, x) + Integral(3*c**2*d*x, x) + Integral(2*d**3*x**3*sec(e + f*x), x) + Int
egral(d**3*x**3*sec(e + f*x)**2, x) + Integral(6*c*d**2*x**2*sec(e + f*x), x) + Integral(3*c*d**2*x**2*sec(e +
f*x)**2, x) + Integral(6*c**2*d*x*sec(e + f*x), x) + Integral(3*c**2*d*x*sec(e + f*x)**2, x))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^2\,{\left (c+d\,x\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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