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

Optimal. Leaf size=288 \[ \frac{20 i d^2 (c+d x) \text{PolyLog}\left (2,-e^{i (e+f x)}\right )}{a^2 f^3}-\frac{20 d^3 \text{PolyLog}\left (3,-e^{i (e+f x)}\right )}{a^2 f^4}+\frac{2 d^2 (c+d x) \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f^3}-\frac{10 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}-\frac{d (c+d x)^2 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}-\frac{5 (c+d x)^3 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^3 \tan \left (\frac{e}{2}+\frac{f x}{2}\right ) \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{5 i (c+d x)^3}{3 a^2 f}+\frac{(c+d x)^4}{4 a^2 d}+\frac{4 d^3 \log \left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{a^2 f^4} \]

[Out]

(((5*I)/3)*(c + d*x)^3)/(a^2*f) + (c + d*x)^4/(4*a^2*d) - (10*d*(c + d*x)^2*Log[1 + E^(I*(e + f*x))])/(a^2*f^2
) + (4*d^3*Log[Cos[e/2 + (f*x)/2]])/(a^2*f^4) + ((20*I)*d^2*(c + d*x)*PolyLog[2, -E^(I*(e + f*x))])/(a^2*f^3)
- (20*d^3*PolyLog[3, -E^(I*(e + f*x))])/(a^2*f^4) - (d*(c + d*x)^2*Sec[e/2 + (f*x)/2]^2)/(2*a^2*f^2) + (2*d^2*
(c + d*x)*Tan[e/2 + (f*x)/2])/(a^2*f^3) - (5*(c + d*x)^3*Tan[e/2 + (f*x)/2])/(3*a^2*f) + ((c + d*x)^3*Sec[e/2
+ (f*x)/2]^2*Tan[e/2 + (f*x)/2])/(6*a^2*f)

________________________________________________________________________________________

Rubi [A]  time = 0.719461, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4191, 3318, 4186, 4184, 3475, 3719, 2190, 2531, 2282, 6589} \[ \frac{20 i d^2 (c+d x) \text{PolyLog}\left (2,-e^{i (e+f x)}\right )}{a^2 f^3}-\frac{20 d^3 \text{PolyLog}\left (3,-e^{i (e+f x)}\right )}{a^2 f^4}+\frac{2 d^2 (c+d x) \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f^3}-\frac{10 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}-\frac{d (c+d x)^2 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}-\frac{5 (c+d x)^3 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^3 \tan \left (\frac{e}{2}+\frac{f x}{2}\right ) \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{5 i (c+d x)^3}{3 a^2 f}+\frac{(c+d x)^4}{4 a^2 d}+\frac{4 d^3 \log \left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{a^2 f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((5*I)/3)*(c + d*x)^3)/(a^2*f) + (c + d*x)^4/(4*a^2*d) - (10*d*(c + d*x)^2*Log[1 + E^(I*(e + f*x))])/(a^2*f^2
) + (4*d^3*Log[Cos[e/2 + (f*x)/2]])/(a^2*f^4) + ((20*I)*d^2*(c + d*x)*PolyLog[2, -E^(I*(e + f*x))])/(a^2*f^3)
- (20*d^3*PolyLog[3, -E^(I*(e + f*x))])/(a^2*f^4) - (d*(c + d*x)^2*Sec[e/2 + (f*x)/2]^2)/(2*a^2*f^2) + (2*d^2*
(c + d*x)*Tan[e/2 + (f*x)/2])/(a^2*f^3) - (5*(c + d*x)^3*Tan[e/2 + (f*x)/2])/(3*a^2*f) + ((c + d*x)^3*Sec[e/2
+ (f*x)/2]^2*Tan[e/2 + (f*x)/2])/(6*a^2*f)

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 3318

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

Rule 4186

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(b^2*(c + d*x)^m*Cot[e
+ f*x]*(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*d^2*m*(m - 1))/(f^2*(n - 1)*(n - 2)), Int[(c + d
*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)^m*(b*Csc[e + f*x])^(n
- 2), x], x] - Simp[(b^2*d*m*(c + d*x)^(m - 1)*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[
{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3475

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

Rule 3719

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

Rule 2190

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

Rule 2531

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

Rule 2282

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

Rule 6589

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

Rubi steps

\begin{align*} \int \frac{(c+d x)^3}{(a+a \sec (e+f x))^2} \, dx &=\int \left (\frac{(c+d x)^3}{a^2}+\frac{(c+d x)^3}{a^2 (1+\cos (e+f x))^2}-\frac{2 (c+d x)^3}{a^2 (1+\cos (e+f x))}\right ) \, dx\\ &=\frac{(c+d x)^4}{4 a^2 d}+\frac{\int \frac{(c+d x)^3}{(1+\cos (e+f x))^2} \, dx}{a^2}-\frac{2 \int \frac{(c+d x)^3}{1+\cos (e+f x)} \, dx}{a^2}\\ &=\frac{(c+d x)^4}{4 a^2 d}+\frac{\int (c+d x)^3 \csc ^4\left (\frac{e+\pi }{2}+\frac{f x}{2}\right ) \, dx}{4 a^2}-\frac{\int (c+d x)^3 \csc ^2\left (\frac{e+\pi }{2}+\frac{f x}{2}\right ) \, dx}{a^2}\\ &=\frac{(c+d x)^4}{4 a^2 d}-\frac{d (c+d x)^2 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}-\frac{2 (c+d x)^3 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f}+\frac{(c+d x)^3 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{\int (c+d x)^3 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{6 a^2}+\frac{d^2 \int (c+d x) \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{a^2 f^2}+\frac{(6 d) \int (c+d x)^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{a^2 f}\\ &=\frac{2 i (c+d x)^3}{a^2 f}+\frac{(c+d x)^4}{4 a^2 d}-\frac{d (c+d x)^2 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}+\frac{2 d^2 (c+d x) \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f^3}-\frac{5 (c+d x)^3 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^3 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}-\frac{\left (2 d^3\right ) \int \tan \left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{a^2 f^3}-\frac{(12 i d) \int \frac{e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )} (c+d x)^2}{1+e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}} \, dx}{a^2 f}-\frac{d \int (c+d x)^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{a^2 f}\\ &=\frac{5 i (c+d x)^3}{3 a^2 f}+\frac{(c+d x)^4}{4 a^2 d}-\frac{12 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}+\frac{4 d^3 \log \left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{a^2 f^4}-\frac{d (c+d x)^2 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}+\frac{2 d^2 (c+d x) \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f^3}-\frac{5 (c+d x)^3 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^3 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{\left (24 d^2\right ) \int (c+d x) \log \left (1+e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}\right ) \, dx}{a^2 f^2}+\frac{(2 i d) \int \frac{e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )} (c+d x)^2}{1+e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}} \, dx}{a^2 f}\\ &=\frac{5 i (c+d x)^3}{3 a^2 f}+\frac{(c+d x)^4}{4 a^2 d}-\frac{10 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}+\frac{4 d^3 \log \left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{a^2 f^4}+\frac{24 i d^2 (c+d x) \text{Li}_2\left (-e^{i (e+f x)}\right )}{a^2 f^3}-\frac{d (c+d x)^2 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}+\frac{2 d^2 (c+d x) \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f^3}-\frac{5 (c+d x)^3 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^3 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}-\frac{\left (24 i d^3\right ) \int \text{Li}_2\left (-e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}\right ) \, dx}{a^2 f^3}-\frac{\left (4 d^2\right ) \int (c+d x) \log \left (1+e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}\right ) \, dx}{a^2 f^2}\\ &=\frac{5 i (c+d x)^3}{3 a^2 f}+\frac{(c+d x)^4}{4 a^2 d}-\frac{10 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}+\frac{4 d^3 \log \left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{a^2 f^4}+\frac{20 i d^2 (c+d x) \text{Li}_2\left (-e^{i (e+f x)}\right )}{a^2 f^3}-\frac{d (c+d x)^2 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}+\frac{2 d^2 (c+d x) \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f^3}-\frac{5 (c+d x)^3 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^3 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}-\frac{\left (24 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}\right )}{a^2 f^4}+\frac{\left (4 i d^3\right ) \int \text{Li}_2\left (-e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}\right ) \, dx}{a^2 f^3}\\ &=\frac{5 i (c+d x)^3}{3 a^2 f}+\frac{(c+d x)^4}{4 a^2 d}-\frac{10 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}+\frac{4 d^3 \log \left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{a^2 f^4}+\frac{20 i d^2 (c+d x) \text{Li}_2\left (-e^{i (e+f x)}\right )}{a^2 f^3}-\frac{24 d^3 \text{Li}_3\left (-e^{i (e+f x)}\right )}{a^2 f^4}-\frac{d (c+d x)^2 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}+\frac{2 d^2 (c+d x) \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f^3}-\frac{5 (c+d x)^3 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^3 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{\left (4 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}\right )}{a^2 f^4}\\ &=\frac{5 i (c+d x)^3}{3 a^2 f}+\frac{(c+d x)^4}{4 a^2 d}-\frac{10 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}+\frac{4 d^3 \log \left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{a^2 f^4}+\frac{20 i d^2 (c+d x) \text{Li}_2\left (-e^{i (e+f x)}\right )}{a^2 f^3}-\frac{20 d^3 \text{Li}_3\left (-e^{i (e+f x)}\right )}{a^2 f^4}-\frac{d (c+d x)^2 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{2 a^2 f^2}+\frac{2 d^2 (c+d x) \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f^3}-\frac{5 (c+d x)^3 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^3 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}\\ \end{align*}

Mathematica [B]  time = 7.49013, size = 1447, normalized size = 5.02 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(((-20*I)/3)*d^3*Cos[e/2 + (f*x)/2]^4*(f^2*x^2*(f*x - (3*I)*(1 + E^(I*e))*Log[1 + E^((-I)*(e + f*x))]) + 6*(1
+ E^(I*e))*f*x*PolyLog[2, -E^((-I)*(e + f*x))] - (6*I)*(1 + E^(I*e))*PolyLog[3, -E^((-I)*(e + f*x))])*Sec[e/2]
*Sec[e + f*x]^2)/(E^((I/2)*e)*f^4*(a + a*Sec[e + f*x])^2) + (16*d^3*Cos[e/2 + (f*x)/2]^4*Sec[e/2]*Sec[e + f*x]
^2*(Cos[e/2]*Log[Cos[e/2]*Cos[(f*x)/2] - Sin[e/2]*Sin[(f*x)/2]] + (f*x*Sin[e/2])/2))/(f^4*(a + a*Sec[e + f*x])
^2*(Cos[e/2]^2 + Sin[e/2]^2)) - (40*c^2*d*Cos[e/2 + (f*x)/2]^4*Sec[e/2]*Sec[e + f*x]^2*(Cos[e/2]*Log[Cos[e/2]*
Cos[(f*x)/2] - Sin[e/2]*Sin[(f*x)/2]] + (f*x*Sin[e/2])/2))/(f^2*(a + a*Sec[e + f*x])^2*(Cos[e/2]^2 + Sin[e/2]^
2)) - (80*c*d^2*Cos[e/2 + (f*x)/2]^4*Csc[e/2]*((f^2*x^2)/(4*E^(I*ArcTan[Cot[e/2]])) - (Cot[e/2]*((I/2)*f*x*(-P
i - 2*ArcTan[Cot[e/2]]) - Pi*Log[1 + E^((-I)*f*x)] - 2*((f*x)/2 - ArcTan[Cot[e/2]])*Log[1 - E^((2*I)*((f*x)/2
- ArcTan[Cot[e/2]]))] + Pi*Log[Cos[(f*x)/2]] - 2*ArcTan[Cot[e/2]]*Log[Sin[(f*x)/2 - ArcTan[Cot[e/2]]]] + I*Pol
yLog[2, E^((2*I)*((f*x)/2 - ArcTan[Cot[e/2]]))]))/Sqrt[1 + Cot[e/2]^2])*Sec[e/2]*Sec[e + f*x]^2)/(f^3*(a + a*S
ec[e + f*x])^2*Sqrt[Csc[e/2]^2*(Cos[e/2]^2 + Sin[e/2]^2)]) + (Cos[e/2 + (f*x)/2]*Sec[e/2]*Sec[e + f*x]^2*(-24*
c^2*d*f*Cos[(f*x)/2] - 48*c*d^2*f*x*Cos[(f*x)/2] + 36*c^3*f^3*x*Cos[(f*x)/2] - 24*d^3*f*x^2*Cos[(f*x)/2] + 54*
c^2*d*f^3*x^2*Cos[(f*x)/2] + 36*c*d^2*f^3*x^3*Cos[(f*x)/2] + 9*d^3*f^3*x^4*Cos[(f*x)/2] - 24*c^2*d*f*Cos[e + (
f*x)/2] - 48*c*d^2*f*x*Cos[e + (f*x)/2] + 36*c^3*f^3*x*Cos[e + (f*x)/2] - 24*d^3*f*x^2*Cos[e + (f*x)/2] + 54*c
^2*d*f^3*x^2*Cos[e + (f*x)/2] + 36*c*d^2*f^3*x^3*Cos[e + (f*x)/2] + 9*d^3*f^3*x^4*Cos[e + (f*x)/2] + 12*c^3*f^
3*x*Cos[e + (3*f*x)/2] + 18*c^2*d*f^3*x^2*Cos[e + (3*f*x)/2] + 12*c*d^2*f^3*x^3*Cos[e + (3*f*x)/2] + 3*d^3*f^3
*x^4*Cos[e + (3*f*x)/2] + 12*c^3*f^3*x*Cos[2*e + (3*f*x)/2] + 18*c^2*d*f^3*x^2*Cos[2*e + (3*f*x)/2] + 12*c*d^2
*f^3*x^3*Cos[2*e + (3*f*x)/2] + 3*d^3*f^3*x^4*Cos[2*e + (3*f*x)/2] + 96*c*d^2*Sin[(f*x)/2] - 72*c^3*f^2*Sin[(f
*x)/2] + 96*d^3*x*Sin[(f*x)/2] - 216*c^2*d*f^2*x*Sin[(f*x)/2] - 216*c*d^2*f^2*x^2*Sin[(f*x)/2] - 72*d^3*f^2*x^
3*Sin[(f*x)/2] - 48*c*d^2*Sin[e + (f*x)/2] + 48*c^3*f^2*Sin[e + (f*x)/2] - 48*d^3*x*Sin[e + (f*x)/2] + 144*c^2
*d*f^2*x*Sin[e + (f*x)/2] + 144*c*d^2*f^2*x^2*Sin[e + (f*x)/2] + 48*d^3*f^2*x^3*Sin[e + (f*x)/2] + 48*c*d^2*Si
n[e + (3*f*x)/2] - 40*c^3*f^2*Sin[e + (3*f*x)/2] + 48*d^3*x*Sin[e + (3*f*x)/2] - 120*c^2*d*f^2*x*Sin[e + (3*f*
x)/2] - 120*c*d^2*f^2*x^2*Sin[e + (3*f*x)/2] - 40*d^3*f^2*x^3*Sin[e + (3*f*x)/2]))/(24*f^3*(a + a*Sec[e + f*x]
)^2)

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Maple [B]  time = 0.2, size = 799, normalized size = 2.8 \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+a*sec(f*x+e))^2,x)

[Out]

1/4/a^2*d^3*x^4+1/a^2*c*d^2*x^3+3/2/a^2*c^2*d*x^2+1/a^2*c^3*x+20*I*d^2/f^2/a^2*c*e*x-10*I*d^3/f^3/a^2*e^2*x+10
*I*d^2/f/a^2*c*x^2-20*d^2/f^3/a^2*c*e*ln(exp(I*(f*x+e)))+4*d^3/f^4/a^2*ln(exp(I*(f*x+e))+1)-4*d^3/f^4/a^2*ln(e
xp(I*(f*x+e)))-20*d^3*polylog(3,-exp(I*(f*x+e)))/a^2/f^4+20*I*d^3/f^3/a^2*polylog(2,-exp(I*(f*x+e)))*x-20*d^2/
f^2/a^2*ln(exp(I*(f*x+e))+1)*c*x-20/3*I*d^3/f^4/a^2*e^3-10*d/f^2/a^2*c^2*ln(exp(I*(f*x+e))+1)+10*d/f^2/a^2*c^2
*ln(exp(I*(f*x+e)))+10*d^3/f^4/a^2*e^2*ln(exp(I*(f*x+e)))+10/3*I*d^3/f/a^2*x^3+10*I*d^2/f^3/a^2*c*e^2-2/3*I*(6
*d^3*f^2*x^3*exp(2*I*(f*x+e))-3*I*c^2*d*f*exp(2*I*(f*x+e))+18*c*d^2*f^2*x^2*exp(2*I*(f*x+e))+9*d^3*f^2*x^3*exp
(I*(f*x+e))-6*I*c*d^2*f*x*exp(2*I*(f*x+e))-6*I*c*d^2*f*x*exp(I*(f*x+e))+18*c^2*d*f^2*x*exp(2*I*(f*x+e))+27*c*d
^2*f^2*x^2*exp(I*(f*x+e))+5*d^3*f^2*x^3-3*I*d^3*f*x^2*exp(I*(f*x+e))-3*I*c^2*d*f*exp(I*(f*x+e))+6*c^3*f^2*exp(
2*I*(f*x+e))+27*c^2*d*f^2*x*exp(I*(f*x+e))+15*c*d^2*f^2*x^2-3*I*d^3*f*x^2*exp(2*I*(f*x+e))+9*c^3*f^2*exp(I*(f*
x+e))+15*c^2*d*f^2*x-6*d^3*x*exp(2*I*(f*x+e))+5*c^3*f^2-6*c*d^2*exp(2*I*(f*x+e))-12*d^3*x*exp(I*(f*x+e))-12*c*
d^2*exp(I*(f*x+e))-6*d^3*x-6*d^2*c)/f^3/a^2/(exp(I*(f*x+e))+1)^3+20*I*d^2/f^3/a^2*c*polylog(2,-exp(I*(f*x+e)))
-10*d^3/f^2/a^2*ln(exp(I*(f*x+e))+1)*x^2

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Maxima [B]  time = 8.52382, size = 5750, normalized size = 19.97 \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+a*sec(f*x+e))^2,x, algorithm="maxima")

[Out]

-1/6*(3*c*d^2*e^2*((9*sin(f*x + e)/(cos(f*x + e) + 1) - sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/(a^2*f^2) - 12*ar
ctan(sin(f*x + e)/(cos(f*x + e) + 1))/(a^2*f^2)) - 3*c^2*d*e*((9*sin(f*x + e)/(cos(f*x + e) + 1) - sin(f*x + e
)^3/(cos(f*x + e) + 1)^3)/(a^2*f) - 12*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/(a^2*f)) + 6*(3*(f*x + e)^2*cos
(3*f*x + 3*e)^2 + 3*(f*x + e)^2*sin(3*f*x + 3*e)^2 + 3*(9*(f*x + e)^2 - 4)*cos(2*f*x + 2*e)^2 + 3*(9*(f*x + e)
^2 - 4)*cos(f*x + e)^2 + 3*(9*(f*x + e)^2 - 4)*sin(2*f*x + 2*e)^2 + 3*(9*(f*x + e)^2 - 4)*sin(f*x + e)^2 + 3*(
f*x + e)^2 + 2*(3*(f*x + e)^2 + (9*(f*x + e)^2 - 2)*cos(2*f*x + 2*e) + (9*(f*x + e)^2 - 2)*cos(f*x + e) + 12*(
f*x + e)*sin(2*f*x + 2*e) + 18*(f*x + e)*sin(f*x + e))*cos(3*f*x + 3*e) + 2*(9*(f*x + e)^2 + 3*(9*(f*x + e)^2
- 4)*cos(f*x + e) + 18*(f*x + e)*sin(f*x + e) - 2)*cos(2*f*x + 2*e) + 2*(9*(f*x + e)^2 - 2)*cos(f*x + e) - 10*
(2*(3*cos(2*f*x + 2*e) + 3*cos(f*x + e) + 1)*cos(3*f*x + 3*e) + cos(3*f*x + 3*e)^2 + 6*(3*cos(f*x + e) + 1)*co
s(2*f*x + 2*e) + 9*cos(2*f*x + 2*e)^2 + 9*cos(f*x + e)^2 + 6*(sin(2*f*x + 2*e) + sin(f*x + e))*sin(3*f*x + 3*e
) + sin(3*f*x + 3*e)^2 + 9*sin(2*f*x + 2*e)^2 + 18*sin(2*f*x + 2*e)*sin(f*x + e) + 9*sin(f*x + e)^2 + 6*cos(f*
x + e) + 1)*log(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*cos(f*x + e) + 1) - 2*(10*f*x + 12*(f*x + e)*cos(2*f*x + 2
*e) + 18*(f*x + e)*cos(f*x + e) - (9*(f*x + e)^2 - 2)*sin(2*f*x + 2*e) - (9*(f*x + e)^2 - 2)*sin(f*x + e) + 10
*e)*sin(3*f*x + 3*e) - 6*(6*f*x + 6*(f*x + e)*cos(f*x + e) - (9*(f*x + e)^2 - 4)*sin(f*x + e) + 6*e)*sin(2*f*x
+ 2*e) - 24*(f*x + e)*sin(f*x + e))*c*d^2*e/(a^2*f^2*cos(3*f*x + 3*e)^2 + 9*a^2*f^2*cos(2*f*x + 2*e)^2 + 9*a^
2*f^2*cos(f*x + e)^2 + a^2*f^2*sin(3*f*x + 3*e)^2 + 9*a^2*f^2*sin(2*f*x + 2*e)^2 + 18*a^2*f^2*sin(2*f*x + 2*e)
*sin(f*x + e) + 9*a^2*f^2*sin(f*x + e)^2 + 6*a^2*f^2*cos(f*x + e) + a^2*f^2 + 2*(3*a^2*f^2*cos(2*f*x + 2*e) +
3*a^2*f^2*cos(f*x + e) + a^2*f^2)*cos(3*f*x + 3*e) + 6*(3*a^2*f^2*cos(f*x + e) + a^2*f^2)*cos(2*f*x + 2*e) + 6
*(a^2*f^2*sin(2*f*x + 2*e) + a^2*f^2*sin(f*x + e))*sin(3*f*x + 3*e)) + c^3*((9*sin(f*x + e)/(cos(f*x + e) + 1)
- sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/a^2 - 12*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) - 3*(3*(f*x + e)
^2*cos(3*f*x + 3*e)^2 + 3*(f*x + e)^2*sin(3*f*x + 3*e)^2 + 3*(9*(f*x + e)^2 - 4)*cos(2*f*x + 2*e)^2 + 3*(9*(f*
x + e)^2 - 4)*cos(f*x + e)^2 + 3*(9*(f*x + e)^2 - 4)*sin(2*f*x + 2*e)^2 + 3*(9*(f*x + e)^2 - 4)*sin(f*x + e)^2
+ 3*(f*x + e)^2 + 2*(3*(f*x + e)^2 + (9*(f*x + e)^2 - 2)*cos(2*f*x + 2*e) + (9*(f*x + e)^2 - 2)*cos(f*x + e)
+ 12*(f*x + e)*sin(2*f*x + 2*e) + 18*(f*x + e)*sin(f*x + e))*cos(3*f*x + 3*e) + 2*(9*(f*x + e)^2 + 3*(9*(f*x +
e)^2 - 4)*cos(f*x + e) + 18*(f*x + e)*sin(f*x + e) - 2)*cos(2*f*x + 2*e) + 2*(9*(f*x + e)^2 - 2)*cos(f*x + e)
- 10*(2*(3*cos(2*f*x + 2*e) + 3*cos(f*x + e) + 1)*cos(3*f*x + 3*e) + cos(3*f*x + 3*e)^2 + 6*(3*cos(f*x + e) +
1)*cos(2*f*x + 2*e) + 9*cos(2*f*x + 2*e)^2 + 9*cos(f*x + e)^2 + 6*(sin(2*f*x + 2*e) + sin(f*x + e))*sin(3*f*x
+ 3*e) + sin(3*f*x + 3*e)^2 + 9*sin(2*f*x + 2*e)^2 + 18*sin(2*f*x + 2*e)*sin(f*x + e) + 9*sin(f*x + e)^2 + 6*
cos(f*x + e) + 1)*log(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*cos(f*x + e) + 1) - 2*(10*f*x + 12*(f*x + e)*cos(2*f
*x + 2*e) + 18*(f*x + e)*cos(f*x + e) - (9*(f*x + e)^2 - 2)*sin(2*f*x + 2*e) - (9*(f*x + e)^2 - 2)*sin(f*x + e
) + 10*e)*sin(3*f*x + 3*e) - 6*(6*f*x + 6*(f*x + e)*cos(f*x + e) - (9*(f*x + e)^2 - 4)*sin(f*x + e) + 6*e)*sin
(2*f*x + 2*e) - 24*(f*x + e)*sin(f*x + e))*c^2*d/(a^2*f*cos(3*f*x + 3*e)^2 + 9*a^2*f*cos(2*f*x + 2*e)^2 + 9*a^
2*f*cos(f*x + e)^2 + a^2*f*sin(3*f*x + 3*e)^2 + 9*a^2*f*sin(2*f*x + 2*e)^2 + 18*a^2*f*sin(2*f*x + 2*e)*sin(f*x
+ e) + 9*a^2*f*sin(f*x + e)^2 + 6*a^2*f*cos(f*x + e) + a^2*f + 2*(3*a^2*f*cos(2*f*x + 2*e) + 3*a^2*f*cos(f*x
+ e) + a^2*f)*cos(3*f*x + 3*e) + 6*(3*a^2*f*cos(f*x + e) + a^2*f)*cos(2*f*x + 2*e) + 6*(a^2*f*sin(2*f*x + 2*e)
+ a^2*f*sin(f*x + e))*sin(3*f*x + 3*e)) + 6*(3*I*(f*x + e)^4*d^3 + 18*I*(f*x + e)^2*d^3*e^2 - 12*I*(f*x + e)*
d^3*e^3 - 40*d^3*e^3 + (-12*I*d^3*e + 12*I*c*d^2*f)*(f*x + e)^3 + 48*d^3*e - 48*c*d^2*f + (120*(f*x + e)^2*d^3
+ 120*d^3*e^2 - 48*d^3 - 240*(d^3*e - c*d^2*f)*(f*x + e) + 24*(5*(f*x + e)^2*d^3 + 5*d^3*e^2 - 2*d^3 - 10*(d^
3*e - c*d^2*f)*(f*x + e))*cos(3*f*x + 3*e) + 72*(5*(f*x + e)^2*d^3 + 5*d^3*e^2 - 2*d^3 - 10*(d^3*e - c*d^2*f)*
(f*x + e))*cos(2*f*x + 2*e) + 72*(5*(f*x + e)^2*d^3 + 5*d^3*e^2 - 2*d^3 - 10*(d^3*e - c*d^2*f)*(f*x + e))*cos(
f*x + e) + (120*I*(f*x + e)^2*d^3 + 120*I*d^3*e^2 - 48*I*d^3 + (-240*I*d^3*e + 240*I*c*d^2*f)*(f*x + e))*sin(3
*f*x + 3*e) + (360*I*(f*x + e)^2*d^3 + 360*I*d^3*e^2 - 144*I*d^3 + (-720*I*d^3*e + 720*I*c*d^2*f)*(f*x + e))*s
in(2*f*x + 2*e) + (360*I*(f*x + e)^2*d^3 + 360*I*d^3*e^2 - 144*I*d^3 + (-720*I*d^3*e + 720*I*c*d^2*f)*(f*x + e
))*sin(f*x + e))*arctan2(sin(f*x + e), cos(f*x + e) + 1) + (3*I*(f*x + e)^4*d^3 + (-12*I*d^3*e + 12*I*c*d^2*f
- 40*d^3)*(f*x + e)^3 + (18*I*d^3*e^2 + 120*d^3*e - 120*c*d^2*f)*(f*x + e)^2 + (-12*I*d^3*e^3 - 120*d^3*e^2 +
48*d^3)*(f*x + e))*cos(3*f*x + 3*e) + (9*I*(f*x + e)^4*d^3 - 48*d^3*e^3 - 24*I*d^3*e^2 + (-36*I*d^3*e + 36*I*c
*d^2*f - 72*d^3)*(f*x + e)^3 + 48*d^3*e - 48*c*d^2*f + (54*I*d^3*e^2 + 216*d^3*e - 216*c*d^2*f - 24*I*d^3)*(f*
x + e)^2 + (-36*I*d^3*e^3 - 216*d^3*e^2 + 48*I*d^3*e - 48*I*c*d^2*f + 96*d^3)*(f*x + e))*cos(2*f*x + 2*e) + (9
*I*(f*x + e)^4*d^3 - 72*d^3*e^3 - 24*I*d^3*e^2 + (-36*I*d^3*e + 36*I*c*d^2*f - 48*d^3)*(f*x + e)^3 + 96*d^3*e
- 96*c*d^2*f + (54*I*d^3*e^2 + 144*d^3*e - 144*c*d^2*f - 24*I*d^3)*(f*x + e)^2 + (-36*I*d^3*e^3 - 144*d^3*e^2
+ 48*I*d^3*e - 48*I*c*d^2*f + 48*d^3)*(f*x + e))*cos(f*x + e) - (240*(f*x + e)*d^3 - 240*d^3*e + 240*c*d^2*f +
240*((f*x + e)*d^3 - d^3*e + c*d^2*f)*cos(3*f*x + 3*e) + 720*((f*x + e)*d^3 - d^3*e + c*d^2*f)*cos(2*f*x + 2*
e) + 720*((f*x + e)*d^3 - d^3*e + c*d^2*f)*cos(f*x + e) - (-240*I*(f*x + e)*d^3 + 240*I*d^3*e - 240*I*c*d^2*f)
*sin(3*f*x + 3*e) - (-720*I*(f*x + e)*d^3 + 720*I*d^3*e - 720*I*c*d^2*f)*sin(2*f*x + 2*e) - (-720*I*(f*x + e)*
d^3 + 720*I*d^3*e - 720*I*c*d^2*f)*sin(f*x + e))*dilog(-e^(I*f*x + I*e)) + (-60*I*(f*x + e)^2*d^3 - 60*I*d^3*e
^2 + 24*I*d^3 + (120*I*d^3*e - 120*I*c*d^2*f)*(f*x + e) + (-60*I*(f*x + e)^2*d^3 - 60*I*d^3*e^2 + 24*I*d^3 + (
120*I*d^3*e - 120*I*c*d^2*f)*(f*x + e))*cos(3*f*x + 3*e) + (-180*I*(f*x + e)^2*d^3 - 180*I*d^3*e^2 + 72*I*d^3
+ (360*I*d^3*e - 360*I*c*d^2*f)*(f*x + e))*cos(2*f*x + 2*e) + (-180*I*(f*x + e)^2*d^3 - 180*I*d^3*e^2 + 72*I*d
^3 + (360*I*d^3*e - 360*I*c*d^2*f)*(f*x + e))*cos(f*x + e) + 12*(5*(f*x + e)^2*d^3 + 5*d^3*e^2 - 2*d^3 - 10*(d
^3*e - c*d^2*f)*(f*x + e))*sin(3*f*x + 3*e) + 36*(5*(f*x + e)^2*d^3 + 5*d^3*e^2 - 2*d^3 - 10*(d^3*e - c*d^2*f)
*(f*x + e))*sin(2*f*x + 2*e) + 36*(5*(f*x + e)^2*d^3 + 5*d^3*e^2 - 2*d^3 - 10*(d^3*e - c*d^2*f)*(f*x + e))*sin
(f*x + e))*log(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*cos(f*x + e) + 1) + (-240*I*d^3*cos(3*f*x + 3*e) - 720*I*d^
3*cos(2*f*x + 2*e) - 720*I*d^3*cos(f*x + e) + 240*d^3*sin(3*f*x + 3*e) + 720*d^3*sin(2*f*x + 2*e) + 720*d^3*si
n(f*x + e) - 240*I*d^3)*polylog(3, -e^(I*f*x + I*e)) - (3*(f*x + e)^4*d^3 - 4*(3*d^3*e - 3*c*d^2*f - 10*I*d^3)
*(f*x + e)^3 + (18*d^3*e^2 - 120*I*d^3*e + 120*I*c*d^2*f)*(f*x + e)^2 - (12*d^3*e^3 - 120*I*d^3*e^2 + 48*I*d^3
)*(f*x + e))*sin(3*f*x + 3*e) - (9*(f*x + e)^4*d^3 + 48*I*d^3*e^3 - 24*d^3*e^2 - 36*(d^3*e - c*d^2*f - 2*I*d^3
)*(f*x + e)^3 - 48*I*d^3*e + 48*I*c*d^2*f + (54*d^3*e^2 - 216*I*d^3*e + 216*I*c*d^2*f - 24*d^3)*(f*x + e)^2 -
(36*d^3*e^3 - 216*I*d^3*e^2 - 48*d^3*e + 48*c*d^2*f + 96*I*d^3)*(f*x + e))*sin(2*f*x + 2*e) - (9*(f*x + e)^4*d
^3 + 72*I*d^3*e^3 - 24*d^3*e^2 - 12*(3*d^3*e - 3*c*d^2*f - 4*I*d^3)*(f*x + e)^3 - 96*I*d^3*e + 96*I*c*d^2*f +
(54*d^3*e^2 - 144*I*d^3*e + 144*I*c*d^2*f - 24*d^3)*(f*x + e)^2 - (36*d^3*e^3 - 144*I*d^3*e^2 - 48*d^3*e + 48*
c*d^2*f + 48*I*d^3)*(f*x + e))*sin(f*x + e))/(-12*I*a^2*f^3*cos(3*f*x + 3*e) - 36*I*a^2*f^3*cos(2*f*x + 2*e) -
36*I*a^2*f^3*cos(f*x + e) + 12*a^2*f^3*sin(3*f*x + 3*e) + 36*a^2*f^3*sin(2*f*x + 2*e) + 36*a^2*f^3*sin(f*x +
e) - 12*I*a^2*f^3))/f

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Fricas [C]  time = 1.9916, size = 2179, normalized size = 7.57 \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+a*sec(f*x+e))^2,x, algorithm="fricas")

[Out]

1/12*(3*d^3*f^4*x^4 + 12*c*d^2*f^4*x^3 - 12*c^2*d*f^2 + 6*(3*c^2*d*f^4 - 2*d^3*f^2)*x^2 + 3*(d^3*f^4*x^4 + 4*c
*d^2*f^4*x^3 + 6*c^2*d*f^4*x^2 + 4*c^3*f^4*x)*cos(f*x + e)^2 + 12*(c^3*f^4 - 2*c*d^2*f^2)*x + 6*(d^3*f^4*x^4 +
4*c*d^2*f^4*x^3 - 2*c^2*d*f^2 + 2*(3*c^2*d*f^4 - d^3*f^2)*x^2 + 4*(c^3*f^4 - c*d^2*f^2)*x)*cos(f*x + e) + (-1
20*I*d^3*f*x - 120*I*c*d^2*f + (-120*I*d^3*f*x - 120*I*c*d^2*f)*cos(f*x + e)^2 + (-240*I*d^3*f*x - 240*I*c*d^2
*f)*cos(f*x + e))*dilog(-cos(f*x + e) + I*sin(f*x + e)) + (120*I*d^3*f*x + 120*I*c*d^2*f + (120*I*d^3*f*x + 12
0*I*c*d^2*f)*cos(f*x + e)^2 + (240*I*d^3*f*x + 240*I*c*d^2*f)*cos(f*x + e))*dilog(-cos(f*x + e) - I*sin(f*x +
e)) - 12*(5*d^3*f^2*x^2 + 10*c*d^2*f^2*x + 5*c^2*d*f^2 - 2*d^3 + (5*d^3*f^2*x^2 + 10*c*d^2*f^2*x + 5*c^2*d*f^2
- 2*d^3)*cos(f*x + e)^2 + 2*(5*d^3*f^2*x^2 + 10*c*d^2*f^2*x + 5*c^2*d*f^2 - 2*d^3)*cos(f*x + e))*log(cos(f*x
+ e) + I*sin(f*x + e) + 1) - 12*(5*d^3*f^2*x^2 + 10*c*d^2*f^2*x + 5*c^2*d*f^2 - 2*d^3 + (5*d^3*f^2*x^2 + 10*c*
d^2*f^2*x + 5*c^2*d*f^2 - 2*d^3)*cos(f*x + e)^2 + 2*(5*d^3*f^2*x^2 + 10*c*d^2*f^2*x + 5*c^2*d*f^2 - 2*d^3)*cos
(f*x + e))*log(cos(f*x + e) - I*sin(f*x + e) + 1) - 120*(d^3*cos(f*x + e)^2 + 2*d^3*cos(f*x + e) + d^3)*polylo
g(3, -cos(f*x + e) + I*sin(f*x + e)) - 120*(d^3*cos(f*x + e)^2 + 2*d^3*cos(f*x + e) + d^3)*polylog(3, -cos(f*x
+ e) - I*sin(f*x + e)) - 4*(4*d^3*f^3*x^3 + 12*c*d^2*f^3*x^2 + 4*c^3*f^3 - 6*c*d^2*f + 6*(2*c^2*d*f^3 - d^3*f
)*x + (5*d^3*f^3*x^3 + 15*c*d^2*f^3*x^2 + 5*c^3*f^3 - 6*c*d^2*f + 3*(5*c^2*d*f^3 - 2*d^3*f)*x)*cos(f*x + e))*s
in(f*x + e))/(a^2*f^4*cos(f*x + e)^2 + 2*a^2*f^4*cos(f*x + e) + a^2*f^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{c^{3}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx + \int \frac{d^{3} x^{3}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx + \int \frac{3 c d^{2} x^{2}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx + \int \frac{3 c^{2} d x}{\sec ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx}{a^{2}} \end{align*}

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]

(Integral(c**3/(sec(e + f*x)**2 + 2*sec(e + f*x) + 1), x) + Integral(d**3*x**3/(sec(e + f*x)**2 + 2*sec(e + f*
x) + 1), x) + Integral(3*c*d**2*x**2/(sec(e + f*x)**2 + 2*sec(e + f*x) + 1), x) + Integral(3*c**2*d*x/(sec(e +
f*x)**2 + 2*sec(e + f*x) + 1), x))/a**2

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

[Out]

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