3.1.16 \(\int \frac {(c+d x)^3}{(a+a \sec (e+f x))^2} \, dx\) [16]

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

[Out]

5/3*I*(d*x+c)^3/a^2/f+1/4*(d*x+c)^4/a^2/d-10*d*(d*x+c)^2*ln(1+exp(I*(f*x+e)))/a^2/f^2+4*d^3*ln(cos(1/2*f*x+1/2
*e))/a^2/f^4+20*I*d^2*(d*x+c)*polylog(2,-exp(I*(f*x+e)))/a^2/f^3-20*d^3*polylog(3,-exp(I*(f*x+e)))/a^2/f^4-1/2
*d*(d*x+c)^2*sec(1/2*f*x+1/2*e)^2/a^2/f^2+2*d^2*(d*x+c)*tan(1/2*f*x+1/2*e)/a^2/f^3-5/3*(d*x+c)^3*tan(1/2*f*x+1
/2*e)/a^2/f+1/6*(d*x+c)^3*sec(1/2*f*x+1/2*e)^2*tan(1/2*f*x+1/2*e)/a^2/f

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Rubi [A]
time = 0.49, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4276, 3399, 4271, 4269, 3556, 3800, 2221, 2611, 2320, 6724} \begin {gather*} \frac {20 i d^2 (c+d x) \text {Li}_2\left (-e^{i (e+f x)}\right )}{a^2 f^3}+\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 {20 d^3 \text {Li}_3\left (-e^{i (e+f x)}\right )}{a^2 f^4}+\frac {4 d^3 \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4} \end {gather*}

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 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 3399

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + 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 3556

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

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 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 4271

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]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 4276

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 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]

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 ) \text {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 ) \text {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*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1455\) vs. \(2(288)=576\).
time = 7.19, size = 1455, normalized size = 5.05 \begin {gather*} \frac {20 d^3 e^{-\frac {i e}{2}} \cos ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) \left (i f^2 x^2 \left (e^{i e} f x+3 i \left (1+e^{i e}\right ) \log \left (1+e^{i (e+f x)}\right )\right )+6 i \left (1+e^{i e}\right ) f x \text {PolyLog}\left (2,-e^{i (e+f x)}\right )-6 \left (1+e^{i e}\right ) \text {PolyLog}\left (3,-e^{i (e+f x)}\right )\right ) \sec \left (\frac {e}{2}\right ) \sec ^2(e+f x)}{3 f^4 (a+a \sec (e+f x))^2}+\frac {16 d^3 \cos ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) \sec \left (\frac {e}{2}\right ) \sec ^2(e+f x) \left (\cos \left (\frac {e}{2}\right ) \log \left (\cos \left (\frac {e}{2}\right ) \cos \left (\frac {f x}{2}\right )-\sin \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )\right )+\frac {1}{2} f x \sin \left (\frac {e}{2}\right )\right )}{f^4 (a+a \sec (e+f x))^2 \left (\cos ^2\left (\frac {e}{2}\right )+\sin ^2\left (\frac {e}{2}\right )\right )}-\frac {40 c^2 d \cos ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) \sec \left (\frac {e}{2}\right ) \sec ^2(e+f x) \left (\cos \left (\frac {e}{2}\right ) \log \left (\cos \left (\frac {e}{2}\right ) \cos \left (\frac {f x}{2}\right )-\sin \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )\right )+\frac {1}{2} f x \sin \left (\frac {e}{2}\right )\right )}{f^2 (a+a \sec (e+f x))^2 \left (\cos ^2\left (\frac {e}{2}\right )+\sin ^2\left (\frac {e}{2}\right )\right )}-\frac {80 c d^2 \cos ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) \csc \left (\frac {e}{2}\right ) \left (\frac {1}{4} e^{-i \text {ArcTan}\left (\cot \left (\frac {e}{2}\right )\right )} f^2 x^2-\frac {\cot \left (\frac {e}{2}\right ) \left (\frac {1}{2} i f x \left (-\pi -2 \text {ArcTan}\left (\cot \left (\frac {e}{2}\right )\right )\right )-\pi \log \left (1+e^{-i f x}\right )-2 \left (\frac {f x}{2}-\text {ArcTan}\left (\cot \left (\frac {e}{2}\right )\right )\right ) \log \left (1-e^{2 i \left (\frac {f x}{2}-\text {ArcTan}\left (\cot \left (\frac {e}{2}\right )\right )\right )}\right )+\pi \log \left (\cos \left (\frac {f x}{2}\right )\right )-2 \text {ArcTan}\left (\cot \left (\frac {e}{2}\right )\right ) \log \left (\sin \left (\frac {f x}{2}-\text {ArcTan}\left (\cot \left (\frac {e}{2}\right )\right )\right )\right )+i \text {PolyLog}\left (2,e^{2 i \left (\frac {f x}{2}-\text {ArcTan}\left (\cot \left (\frac {e}{2}\right )\right )\right )}\right )\right )}{\sqrt {1+\cot ^2\left (\frac {e}{2}\right )}}\right ) \sec \left (\frac {e}{2}\right ) \sec ^2(e+f x)}{f^3 (a+a \sec (e+f x))^2 \sqrt {\csc ^2\left (\frac {e}{2}\right ) \left (\cos ^2\left (\frac {e}{2}\right )+\sin ^2\left (\frac {e}{2}\right )\right )}}+\frac {\cos \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec \left (\frac {e}{2}\right ) \sec ^2(e+f x) \left (-24 c^2 d f \cos \left (\frac {f x}{2}\right )-48 c d^2 f x \cos \left (\frac {f x}{2}\right )+36 c^3 f^3 x \cos \left (\frac {f x}{2}\right )-24 d^3 f x^2 \cos \left (\frac {f x}{2}\right )+54 c^2 d f^3 x^2 \cos \left (\frac {f x}{2}\right )+36 c d^2 f^3 x^3 \cos \left (\frac {f x}{2}\right )+9 d^3 f^3 x^4 \cos \left (\frac {f x}{2}\right )-24 c^2 d f \cos \left (e+\frac {f x}{2}\right )-48 c d^2 f x \cos \left (e+\frac {f x}{2}\right )+36 c^3 f^3 x \cos \left (e+\frac {f x}{2}\right )-24 d^3 f x^2 \cos \left (e+\frac {f x}{2}\right )+54 c^2 d f^3 x^2 \cos \left (e+\frac {f x}{2}\right )+36 c d^2 f^3 x^3 \cos \left (e+\frac {f x}{2}\right )+9 d^3 f^3 x^4 \cos \left (e+\frac {f x}{2}\right )+12 c^3 f^3 x \cos \left (e+\frac {3 f x}{2}\right )+18 c^2 d f^3 x^2 \cos \left (e+\frac {3 f x}{2}\right )+12 c d^2 f^3 x^3 \cos \left (e+\frac {3 f x}{2}\right )+3 d^3 f^3 x^4 \cos \left (e+\frac {3 f x}{2}\right )+12 c^3 f^3 x \cos \left (2 e+\frac {3 f x}{2}\right )+18 c^2 d f^3 x^2 \cos \left (2 e+\frac {3 f x}{2}\right )+12 c d^2 f^3 x^3 \cos \left (2 e+\frac {3 f x}{2}\right )+3 d^3 f^3 x^4 \cos \left (2 e+\frac {3 f x}{2}\right )+96 c d^2 \sin \left (\frac {f x}{2}\right )-72 c^3 f^2 \sin \left (\frac {f x}{2}\right )+96 d^3 x \sin \left (\frac {f x}{2}\right )-216 c^2 d f^2 x \sin \left (\frac {f x}{2}\right )-216 c d^2 f^2 x^2 \sin \left (\frac {f x}{2}\right )-72 d^3 f^2 x^3 \sin \left (\frac {f x}{2}\right )-48 c d^2 \sin \left (e+\frac {f x}{2}\right )+48 c^3 f^2 \sin \left (e+\frac {f x}{2}\right )-48 d^3 x \sin \left (e+\frac {f x}{2}\right )+144 c^2 d f^2 x \sin \left (e+\frac {f x}{2}\right )+144 c d^2 f^2 x^2 \sin \left (e+\frac {f x}{2}\right )+48 d^3 f^2 x^3 \sin \left (e+\frac {f x}{2}\right )+48 c d^2 \sin \left (e+\frac {3 f x}{2}\right )-40 c^3 f^2 \sin \left (e+\frac {3 f x}{2}\right )+48 d^3 x \sin \left (e+\frac {3 f x}{2}\right )-120 c^2 d f^2 x \sin \left (e+\frac {3 f x}{2}\right )-120 c d^2 f^2 x^2 \sin \left (e+\frac {3 f x}{2}\right )-40 d^3 f^2 x^3 \sin \left (e+\frac {3 f x}{2}\right )\right )}{24 f^3 (a+a \sec (e+f x))^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(20*d^3*Cos[e/2 + (f*x)/2]^4*(I*f^2*x^2*(E^(I*e)*f*x + (3*I)*(1 + E^(I*e))*Log[1 + E^(I*(e + f*x))]) + (6*I)*(
1 + E^(I*e))*f*x*PolyLog[2, -E^(I*(e + f*x))] - 6*(1 + E^(I*e))*PolyLog[3, -E^(I*(e + f*x))])*Sec[e/2]*Sec[e +
f*x]^2)/(3*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*(Co
s[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*(Co
s[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*(-Pi - 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 - ArcT
an[Cot[e/2]]))] + Pi*Log[Cos[(f*x)/2]] - 2*ArcTan[Cot[e/2]]*Log[Sin[(f*x)/2 - ArcTan[Cot[e/2]]]] + I*PolyLog[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*Sec[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*Co
s[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*C
os[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*Sin[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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 809 vs. \(2 (246 ) = 492\).
time = 0.80, size = 810, normalized size = 2.81

method	result	size
risch	\(\frac {d^{3} x^{4}}{4 a^{2}}+\frac {d^{2} c \,x^{3}}{a^{2}}+\frac {3 d \,c^{2} x^{2}}{2 a^{2}}+\frac {c^{3} x}{a^{2}}+\frac {c^{4}}{4 a^{2} d}+\frac {10 i d^{2} c \,x^{2}}{a^{2} f}-\frac {20 i d^{3} e^{3}}{3 a^{2} f^{4}}-\frac {2 i \left (6 d^{3} f^{2} x^{3} {\mathrm e}^{2 i \left (f x +e \right )}-3 i c^{2} d f \,{\mathrm e}^{i \left (f x +e \right )}+18 c \,d^{2} f^{2} x^{2} {\mathrm e}^{2 i \left (f x +e \right )}+9 d^{3} f^{2} x^{3} {\mathrm e}^{i \left (f x +e \right )}-6 i c \,d^{2} f x \,{\mathrm e}^{i \left (f x +e \right )}-3 i d^{3} f \,x^{2} {\mathrm e}^{2 i \left (f x +e \right )}+18 c^{2} d \,f^{2} x \,{\mathrm e}^{2 i \left (f x +e \right )}+27 c \,d^{2} f^{2} x^{2} {\mathrm e}^{i \left (f x +e \right )}+5 d^{3} f^{2} x^{3}-6 i c \,d^{2} f x \,{\mathrm e}^{2 i \left (f x +e \right )}-3 i c^{2} d f \,{\mathrm e}^{2 i \left (f x +e \right )}+6 c^{3} f^{2} {\mathrm e}^{2 i \left (f x +e \right )}+27 c^{2} d \,f^{2} x \,{\mathrm e}^{i \left (f x +e \right )}+15 c \,d^{2} f^{2} x^{2}-3 i d^{3} f \,x^{2} {\mathrm e}^{i \left (f x +e \right )}+9 c^{3} f^{2} {\mathrm e}^{i \left (f x +e \right )}+15 c^{2} d \,f^{2} x -6 d^{3} x \,{\mathrm e}^{2 i \left (f x +e \right )}+5 c^{3} f^{2}-6 d^{2} c \,{\mathrm e}^{2 i \left (f x +e \right )}-12 d^{3} x \,{\mathrm e}^{i \left (f x +e \right )}-12 d^{2} c \,{\mathrm e}^{i \left (f x +e \right )}-6 d^{3} x -6 d^{2} c \right )}{3 f^{3} a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}+\frac {20 i d^{2} c e x}{a^{2} f^{2}}+\frac {10 i d^{2} c \,e^{2}}{a^{2} f^{3}}-\frac {10 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) x^{2}}{a^{2} f^{2}}+\frac {10 i d^{3} x^{3}}{3 a^{2} f}-\frac {10 i d^{3} e^{2} x}{a^{2} f^{3}}-\frac {20 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) c x}{a^{2} f^{2}}+\frac {20 i d^{2} c \polylog \left (2, -{\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{3}}+\frac {20 i d^{3} \polylog \left (2, -{\mathrm e}^{i \left (f x +e \right )}\right ) x}{a^{2} f^{3}}+\frac {4 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{a^{2} f^{4}}-\frac {4 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{4}}-\frac {20 d^{2} c e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{3}}-\frac {20 d^{3} \polylog \left (3, -{\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{4}}-\frac {10 d \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{a^{2} f^{2}}+\frac {10 d \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{2}}+\frac {10 d^{3} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{4}}\)	\(810\)

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]

1/4/a^2*d^3*x^4+1/a^2*d^2*c*x^3+3/2/a^2*d*c^2*x^2+1/a^2*c^3*x+1/4/a^2/d*c^4+10*I/a^2/f*d^2*c*x^2+20*I/a^2/f^3*
d^3*polylog(2,-exp(I*(f*x+e)))*x-20/3*I/a^2/f^4*d^3*e^3-2/3*I*(6*d^3*f^2*x^3*exp(2*I*(f*x+e))-3*I*c^2*d*f*exp(
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(I*(f*x+e))-3*I*d^3
*f*x^2*exp(2*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^3-6*I*c*d^
2*f*x*exp(2*I*(f*x+e))-3*I*c^2*d*f*exp(2*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))+1
5*c*d^2*f^2*x^2-3*I*d^3*f*x^2*exp(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*d^2*c*exp(2*I*(f*x+e))-12*d^3*x*exp(I*(f*x+e))-12*d^2*c*exp(I*(f*x+e))-6*d^3*x-6*d^2*c)/f^3/a^2/(e
xp(I*(f*x+e))+1)^3+20*I/a^2/f^2*d^2*c*e*x-10/a^2/f^2*d^3*ln(exp(I*(f*x+e))+1)*x^2+10*I/a^2/f^3*d^2*c*e^2+10/3*
I/a^2/f*d^3*x^3-20/a^2/f^2*d^2*ln(exp(I*(f*x+e))+1)*c*x-10*I/a^2/f^3*d^3*e^2*x+20*I/a^2/f^3*d^2*c*polylog(2,-e
xp(I*(f*x+e)))+4/a^2/f^4*d^3*ln(exp(I*(f*x+e))+1)-4/a^2/f^4*d^3*ln(exp(I*(f*x+e)))-20/a^2/f^3*d^2*c*e*ln(exp(I
*(f*x+e)))-20*d^3*polylog(3,-exp(I*(f*x+e)))/a^2/f^4-10/a^2/f^2*d*c^2*ln(exp(I*(f*x+e))+1)+10/a^2/f^2*d*c^2*ln
(exp(I*(f*x+e)))+10/a^2/f^4*d^3*e^2*ln(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. 4460 vs. \(2 (252) = 504\).
time = 1.79, size = 4460, normalized size = 15.49 \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]

-1/6*(3*c*d^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*arctan
(sin(f*x + e)/(cos(f*x + e) + 1))/(a^2*f^2))*e^2 - 3*c^2*d*((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))*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) + 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)*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 + 1
2*(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)) - 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 - 12*(-I*c*d^2*f + I*d^3*e)*(f*x + e)^3 - 48*c*d^2*f - 8*d^3*(5*e^3 - 6*e) + 24*(5*(f*x + e)^2*d^3 + d
^3*(5*e^2 - 2) + 10*(c*d^2*f - d^3*e)*(f*x + e) + (5*(f*x + e)^2*d^3 + d^3*(5*e^2 - 2) + 10*(c*d^2*f - d^3*e)*
(f*x + e))*cos(3*f*x + 3*e) + 3*(5*(f*x + e)^2*d^3 + d^3*(5*e^2 - 2) + 10*(c*d^2*f - d^3*e)*(f*x + e))*cos(2*f
*x + 2*e) + 3*(5*(f*x + e)^2*d^3 + d^3*(5*e^2 - 2) + 10*(c*d^2*f - d^3*e)*(f*x + e))*cos(f*x + e) - (-5*I*(f*x
+ e)^2*d^3 + d^3*(-5*I*e^2 + 2*I) + 10*(-I*c*d^2*f + I*d^3*e)*(f*x + e))*sin(3*f*x + 3*e) - 3*(-5*I*(f*x + e)
^2*d^3 + d^3*(-5*I*e^2 + 2*I) + 10*(-I*c*d^2*f + I*d^3*e)*(f*x + e))*sin(2*f*x + 2*e) - 3*(-5*I*(f*x + e)^2*d^
3 + d^3*(-5*I*e^2 + 2*I) + 10*(-I*c*d^2*f + I*d^3*e)*(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*(f*x + e)*d^3*(I*e^3 + 10*e^2 - 4) - 4*(-3*I*c*d^2*f + d^3*(3*I*e + 10))*(
f*x + e)^3 - 6*(20*c*d^2*f + d^3*(-3*I*e^2 - 20*e))*(f*x + e)^2)*cos(3*f*x + 3*e) - 3*(-3*I*(f*x + e)^4*d^3 +
12*(-I*c*d^2*f + d^3*(I*e + 2))*(f*x + e)^3 + 1...

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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. 963 vs. \(2 (252) = 504\).
time = 3.93, size = 963, normalized size = 3.34 \begin {gather*} \frac {3 \, d^{3} f^{4} x^{4} + 12 \, c d^{2} f^{4} x^{3} - 12 \, c^{2} d f^{2} + 6 \, {\left (3 \, c^{2} d f^{4} - 2 \, d^{3} f^{2}\right )} x^{2} + 3 \, {\left (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\right )} \cos \left (f x + e\right )^{2} + 12 \, {\left (c^{3} f^{4} - 2 \, c d^{2} f^{2}\right )} x + 6 \, {\left (d^{3} f^{4} x^{4} + 4 \, c d^{2} f^{4} x^{3} - 2 \, c^{2} d f^{2} + 2 \, {\left (3 \, c^{2} d f^{4} - d^{3} f^{2}\right )} x^{2} + 4 \, {\left (c^{3} f^{4} - c d^{2} f^{2}\right )} x\right )} \cos \left (f x + e\right ) - 120 \, {\left (i \, d^{3} f x + i \, c d^{2} f + {\left (i \, d^{3} f x + i \, c d^{2} f\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (i \, d^{3} f x + i \, c d^{2} f\right )} \cos \left (f x + e\right )\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - 120 \, {\left (-i \, d^{3} f x - i \, c d^{2} f + {\left (-i \, d^{3} f x - i \, c d^{2} f\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (-i \, d^{3} f x - i \, c d^{2} f\right )} \cos \left (f x + e\right )\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 12 \, {\left (5 \, d^{3} f^{2} x^{2} + 10 \, c d^{2} f^{2} x + 5 \, c^{2} d f^{2} - 2 \, d^{3} + {\left (5 \, d^{3} f^{2} x^{2} + 10 \, c d^{2} f^{2} x + 5 \, c^{2} d f^{2} - 2 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (5 \, d^{3} f^{2} x^{2} + 10 \, c d^{2} f^{2} x + 5 \, c^{2} d f^{2} - 2 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + 1\right ) - 12 \, {\left (5 \, d^{3} f^{2} x^{2} + 10 \, c d^{2} f^{2} x + 5 \, c^{2} d f^{2} - 2 \, d^{3} + {\left (5 \, d^{3} f^{2} x^{2} + 10 \, c d^{2} f^{2} x + 5 \, c^{2} d f^{2} - 2 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (5 \, d^{3} f^{2} x^{2} + 10 \, c d^{2} f^{2} x + 5 \, c^{2} d f^{2} - 2 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + 1\right ) - 120 \, {\left (d^{3} \cos \left (f x + e\right )^{2} + 2 \, d^{3} \cos \left (f x + e\right ) + d^{3}\right )} {\rm polylog}\left (3, -\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - 120 \, {\left (d^{3} \cos \left (f x + e\right )^{2} + 2 \, d^{3} \cos \left (f x + e\right ) + d^{3}\right )} {\rm polylog}\left (3, -\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 4 \, {\left (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 \, {\left (2 \, c^{2} d f^{3} - d^{3} f\right )} x + {\left (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 \, {\left (5 \, c^{2} d f^{3} - 2 \, d^{3} f\right )} x\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{12 \, {\left (a^{2} f^{4} \cos \left (f x + e\right )^{2} + 2 \, a^{2} f^{4} \cos \left (f x + e\right ) + a^{2} f^{4}\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, 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) - 120
*(I*d^3*f*x + I*c*d^2*f + (I*d^3*f*x + I*c*d^2*f)*cos(f*x + e)^2 + 2*(I*d^3*f*x + I*c*d^2*f)*cos(f*x + e))*dil
og(-cos(f*x + e) + I*sin(f*x + e)) - 120*(-I*d^3*f*x - I*c*d^2*f + (-I*d^3*f*x - I*c*d^2*f)*cos(f*x + e)^2 + 2
*(-I*d^3*f*x - 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)*polylog(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))*sin(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {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]

(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 = 0.00, 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(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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