3.24 \(\int (c+d x)^3 (a+b \sec (e+f x)) \, dx\)

Optimal. Leaf size=227 \[ -\frac{6 b d^2 (c+d x) \text{PolyLog}\left (3,-i e^{i (e+f x)}\right )}{f^3}+\frac{6 b d^2 (c+d x) \text{PolyLog}\left (3,i e^{i (e+f x)}\right )}{f^3}+\frac{3 i b d (c+d x)^2 \text{PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac{3 i b d (c+d x)^2 \text{PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}-\frac{6 i b d^3 \text{PolyLog}\left (4,-i e^{i (e+f x)}\right )}{f^4}+\frac{6 i b d^3 \text{PolyLog}\left (4,i e^{i (e+f x)}\right )}{f^4}+\frac{a (c+d x)^4}{4 d}-\frac{2 i b (c+d x)^3 \tan ^{-1}\left (e^{i (e+f x)}\right )}{f} \]

[Out]

(a*(c + d*x)^4)/(4*d) - ((2*I)*b*(c + d*x)^3*ArcTan[E^(I*(e + f*x))])/f + ((3*I)*b*d*(c + d*x)^2*PolyLog[2, (-
I)*E^(I*(e + f*x))])/f^2 - ((3*I)*b*d*(c + d*x)^2*PolyLog[2, I*E^(I*(e + f*x))])/f^2 - (6*b*d^2*(c + d*x)*Poly
Log[3, (-I)*E^(I*(e + f*x))])/f^3 + (6*b*d^2*(c + d*x)*PolyLog[3, I*E^(I*(e + f*x))])/f^3 - ((6*I)*b*d^3*PolyL
og[4, (-I)*E^(I*(e + f*x))])/f^4 + ((6*I)*b*d^3*PolyLog[4, I*E^(I*(e + f*x))])/f^4

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Rubi [A]  time = 0.212537, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4190, 4181, 2531, 6609, 2282, 6589} \[ -\frac{6 b d^2 (c+d x) \text{PolyLog}\left (3,-i e^{i (e+f x)}\right )}{f^3}+\frac{6 b d^2 (c+d x) \text{PolyLog}\left (3,i e^{i (e+f x)}\right )}{f^3}+\frac{3 i b d (c+d x)^2 \text{PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac{3 i b d (c+d x)^2 \text{PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}-\frac{6 i b d^3 \text{PolyLog}\left (4,-i e^{i (e+f x)}\right )}{f^4}+\frac{6 i b d^3 \text{PolyLog}\left (4,i e^{i (e+f x)}\right )}{f^4}+\frac{a (c+d x)^4}{4 d}-\frac{2 i b (c+d x)^3 \tan ^{-1}\left (e^{i (e+f x)}\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(c + d*x)^4)/(4*d) - ((2*I)*b*(c + d*x)^3*ArcTan[E^(I*(e + f*x))])/f + ((3*I)*b*d*(c + d*x)^2*PolyLog[2, (-
I)*E^(I*(e + f*x))])/f^2 - ((3*I)*b*d*(c + d*x)^2*PolyLog[2, I*E^(I*(e + f*x))])/f^2 - (6*b*d^2*(c + d*x)*Poly
Log[3, (-I)*E^(I*(e + f*x))])/f^3 + (6*b*d^2*(c + d*x)*PolyLog[3, I*E^(I*(e + f*x))])/f^3 - ((6*I)*b*d^3*PolyL
og[4, (-I)*E^(I*(e + f*x))])/f^4 + ((6*I)*b*d^3*PolyLog[4, I*E^(I*(e + f*x))])/f^4

Rule 4190

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 4181

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

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]

Rubi steps

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

Mathematica [A]  time = 0.485525, size = 365, normalized size = 1.61 \[ \frac{-24 b c d^2 f \text{PolyLog}\left (3,-i e^{i (e+f x)}\right )+24 b c d^2 f \text{PolyLog}\left (3,i e^{i (e+f x)}\right )+12 i b d f^2 (c+d x)^2 \text{PolyLog}\left (2,-i e^{i (e+f x)}\right )-12 i b d f^2 (c+d x)^2 \text{PolyLog}\left (2,i e^{i (e+f x)}\right )-24 b d^3 f x \text{PolyLog}\left (3,-i e^{i (e+f x)}\right )+24 b d^3 f x \text{PolyLog}\left (3,i e^{i (e+f x)}\right )-24 i b d^3 \text{PolyLog}\left (4,-i e^{i (e+f x)}\right )+24 i b d^3 \text{PolyLog}\left (4,i e^{i (e+f x)}\right )+6 a c^2 d f^4 x^2+4 a c^3 f^4 x+4 a c d^2 f^4 x^3+a d^3 f^4 x^4-24 i b c^2 d f^3 x \tan ^{-1}\left (e^{i (e+f x)}\right )+4 b c^3 f^3 \tanh ^{-1}(\sin (e+f x))-24 i b c d^2 f^3 x^2 \tan ^{-1}\left (e^{i (e+f x)}\right )-8 i b d^3 f^3 x^3 \tan ^{-1}\left (e^{i (e+f x)}\right )}{4 f^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*a*c^3*f^4*x + 6*a*c^2*d*f^4*x^2 + 4*a*c*d^2*f^4*x^3 + a*d^3*f^4*x^4 - (24*I)*b*c^2*d*f^3*x*ArcTan[E^(I*(e +
f*x))] - (24*I)*b*c*d^2*f^3*x^2*ArcTan[E^(I*(e + f*x))] - (8*I)*b*d^3*f^3*x^3*ArcTan[E^(I*(e + f*x))] + 4*b*c
^3*f^3*ArcTanh[Sin[e + f*x]] + (12*I)*b*d*f^2*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(e + f*x))] - (12*I)*b*d*f^2*(c
+ d*x)^2*PolyLog[2, I*E^(I*(e + f*x))] - 24*b*c*d^2*f*PolyLog[3, (-I)*E^(I*(e + f*x))] - 24*b*d^3*f*x*PolyLog
[3, (-I)*E^(I*(e + f*x))] + 24*b*c*d^2*f*PolyLog[3, I*E^(I*(e + f*x))] + 24*b*d^3*f*x*PolyLog[3, I*E^(I*(e + f
*x))] - (24*I)*b*d^3*PolyLog[4, (-I)*E^(I*(e + f*x))] + (24*I)*b*d^3*PolyLog[4, I*E^(I*(e + f*x))])/(4*f^4)

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Maple [B]  time = 0.195, size = 747, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 2.55481, size = 1250, normalized size = 5.51 \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)),x, algorithm="maxima")

[Out]

1/4*(4*(f*x + e)*a*c^3 + (f*x + e)^4*a*d^3/f^3 - 4*(f*x + e)^3*a*d^3*e/f^3 + 6*(f*x + e)^2*a*d^3*e^2/f^3 - 4*(
f*x + e)*a*d^3*e^3/f^3 + 4*(f*x + e)^3*a*c*d^2/f^2 - 12*(f*x + e)^2*a*c*d^2*e/f^2 + 12*(f*x + e)*a*c*d^2*e^2/f
^2 + 6*(f*x + e)^2*a*c^2*d/f - 12*(f*x + e)*a*c^2*d*e/f + 4*b*c^3*log(sec(f*x + e) + tan(f*x + e)) - 4*b*d^3*e
^3*log(sec(f*x + e) + tan(f*x + e))/f^3 + 12*b*c*d^2*e^2*log(sec(f*x + e) + tan(f*x + e))/f^2 - 12*b*c^2*d*e*l
og(sec(f*x + e) + tan(f*x + e))/f + 2*(12*I*b*d^3*polylog(4, I*e^(I*f*x + I*e)) - 12*I*b*d^3*polylog(4, -I*e^(
I*f*x + I*e)) + (-2*I*(f*x + e)^3*b*d^3 + (6*I*b*d^3*e - 6*I*b*c*d^2*f)*(f*x + e)^2 + (-6*I*b*d^3*e^2 + 12*I*b
*c*d^2*e*f - 6*I*b*c^2*d*f^2)*(f*x + e))*arctan2(cos(f*x + e), sin(f*x + e) + 1) + (-2*I*(f*x + e)^3*b*d^3 + (
6*I*b*d^3*e - 6*I*b*c*d^2*f)*(f*x + e)^2 + (-6*I*b*d^3*e^2 + 12*I*b*c*d^2*e*f - 6*I*b*c^2*d*f^2)*(f*x + e))*ar
ctan2(cos(f*x + e), -sin(f*x + e) + 1) + (-6*I*(f*x + e)^2*b*d^3 - 6*I*b*d^3*e^2 + 12*I*b*c*d^2*e*f - 6*I*b*c^
2*d*f^2 + (12*I*b*d^3*e - 12*I*b*c*d^2*f)*(f*x + e))*dilog(I*e^(I*f*x + I*e)) + (6*I*(f*x + e)^2*b*d^3 + 6*I*b
*d^3*e^2 - 12*I*b*c*d^2*e*f + 6*I*b*c^2*d*f^2 + (-12*I*b*d^3*e + 12*I*b*c*d^2*f)*(f*x + e))*dilog(-I*e^(I*f*x
+ I*e)) + ((f*x + e)^3*b*d^3 - 3*(b*d^3*e - b*c*d^2*f)*(f*x + e)^2 + 3*(b*d^3*e^2 - 2*b*c*d^2*e*f + b*c^2*d*f^
2)*(f*x + e))*log(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*sin(f*x + e) + 1) - ((f*x + e)^3*b*d^3 - 3*(b*d^3*e - b*
c*d^2*f)*(f*x + e)^2 + 3*(b*d^3*e^2 - 2*b*c*d^2*e*f + b*c^2*d*f^2)*(f*x + e))*log(cos(f*x + e)^2 + sin(f*x + e
)^2 - 2*sin(f*x + e) + 1) + 12*((f*x + e)*b*d^3 - b*d^3*e + b*c*d^2*f)*polylog(3, I*e^(I*f*x + I*e)) - 12*((f*
x + e)*b*d^3 - b*d^3*e + b*c*d^2*f)*polylog(3, -I*e^(I*f*x + I*e)))/f^3)/f

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Fricas [C]  time = 2.30645, size = 2674, normalized size = 11.78 \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)),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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

[Out]

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