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

Optimal. Leaf size=229 \[ \frac{20 i d^2 \text{PolyLog}\left (2,-e^{i (e+f x)}\right )}{3 a^2 f^3}-\frac{20 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac{d (c+d x) \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^2}-\frac{5 (c+d x)^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right ) \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{5 i (c+d x)^2}{3 a^2 f}+\frac{(c+d x)^3}{3 a^2 d}+\frac{2 d^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.498258, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4191, 3318, 4186, 3767, 8, 4184, 3719, 2190, 2279, 2391} \[ \frac{20 i d^2 \text{PolyLog}\left (2,-e^{i (e+f x)}\right )}{3 a^2 f^3}-\frac{20 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac{d (c+d x) \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^2}-\frac{5 (c+d x)^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right ) \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{6 a^2 f}+\frac{5 i (c+d x)^2}{3 a^2 f}+\frac{(c+d x)^3}{3 a^2 d}+\frac{2 d^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((5*I)/3)*(c + d*x)^2)/(a^2*f) + (c + d*x)^3/(3*a^2*d) - (20*d*(c + d*x)*Log[1 + E^(I*(e + f*x))])/(3*a^2*f^2
) + (((20*I)/3)*d^2*PolyLog[2, -E^(I*(e + f*x))])/(a^2*f^3) - (d*(c + d*x)*Sec[e/2 + (f*x)/2]^2)/(3*a^2*f^2) +
(2*d^2*Tan[e/2 + (f*x)/2])/(3*a^2*f^3) - (5*(c + d*x)^2*Tan[e/2 + (f*x)/2])/(3*a^2*f) + ((c + d*x)^2*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 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 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 2279

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{(c+d x)^2}{(a+a \sec (e+f x))^2} \, dx &=\int \left (\frac{(c+d x)^2}{a^2}+\frac{(c+d x)^2}{a^2 (1+\cos (e+f x))^2}-\frac{2 (c+d x)^2}{a^2 (1+\cos (e+f x))}\right ) \, dx\\ &=\frac{(c+d x)^3}{3 a^2 d}+\frac{\int \frac{(c+d x)^2}{(1+\cos (e+f x))^2} \, dx}{a^2}-\frac{2 \int \frac{(c+d x)^2}{1+\cos (e+f x)} \, dx}{a^2}\\ &=\frac{(c+d x)^3}{3 a^2 d}+\frac{\int (c+d x)^2 \csc ^4\left (\frac{e+\pi }{2}+\frac{f x}{2}\right ) \, dx}{4 a^2}-\frac{\int (c+d x)^2 \csc ^2\left (\frac{e+\pi }{2}+\frac{f x}{2}\right ) \, dx}{a^2}\\ &=\frac{(c+d x)^3}{3 a^2 d}-\frac{d (c+d x) \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^2}-\frac{2 (c+d x)^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{a^2 f}+\frac{(c+d x)^2 \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)^2 \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{6 a^2}+\frac{d^2 \int \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{3 a^2 f^2}+\frac{(4 d) \int (c+d x) \tan \left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{a^2 f}\\ &=\frac{2 i (c+d x)^2}{a^2 f}+\frac{(c+d x)^3}{3 a^2 d}-\frac{d (c+d x) \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^2}-\frac{5 (c+d x)^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^2 \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^2\right ) \operatorname{Subst}\left (\int 1 \, dx,x,-\tan \left (\frac{e}{2}+\frac{f x}{2}\right )\right )}{3 a^2 f^3}-\frac{(8 i d) \int \frac{e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )} (c+d x)}{1+e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}} \, dx}{a^2 f}-\frac{(2 d) \int (c+d x) \tan \left (\frac{e}{2}+\frac{f x}{2}\right ) \, dx}{3 a^2 f}\\ &=\frac{5 i (c+d x)^2}{3 a^2 f}+\frac{(c+d x)^3}{3 a^2 d}-\frac{8 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}-\frac{d (c+d x) \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^2}+\frac{2 d^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^3}-\frac{5 (c+d x)^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^2 \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 (8 d^2\right ) \int \log \left (1+e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}\right ) \, dx}{a^2 f^2}+\frac{(4 i d) \int \frac{e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )} (c+d x)}{1+e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}} \, dx}{3 a^2 f}\\ &=\frac{5 i (c+d x)^2}{3 a^2 f}+\frac{(c+d x)^3}{3 a^2 d}-\frac{20 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac{d (c+d x) \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^2}+\frac{2 d^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^3}-\frac{5 (c+d x)^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^2 \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 (8 i d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}\right )}{a^2 f^3}-\frac{\left (4 d^2\right ) \int \log \left (1+e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}\right ) \, dx}{3 a^2 f^2}\\ &=\frac{5 i (c+d x)^2}{3 a^2 f}+\frac{(c+d x)^3}{3 a^2 d}-\frac{20 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}+\frac{8 i d^2 \text{Li}_2\left (-e^{i (e+f x)}\right )}{a^2 f^3}-\frac{d (c+d x) \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^2}+\frac{2 d^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^3}-\frac{5 (c+d x)^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^2 \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 i d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}\right )}{3 a^2 f^3}\\ &=\frac{5 i (c+d x)^2}{3 a^2 f}+\frac{(c+d x)^3}{3 a^2 d}-\frac{20 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}+\frac{20 i d^2 \text{Li}_2\left (-e^{i (e+f x)}\right )}{3 a^2 f^3}-\frac{d (c+d x) \sec ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^2}+\frac{2 d^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f^3}-\frac{5 (c+d x)^2 \tan \left (\frac{e}{2}+\frac{f x}{2}\right )}{3 a^2 f}+\frac{(c+d x)^2 \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 = 6.84881, size = 925, normalized size = 4.04 \[ -\frac{80 d^2 \csc \left (\frac{e}{2}\right ) \left (\frac{1}{4} e^{-i \tan ^{-1}\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 (-2 \tan ^{-1}\left (\cot \left (\frac{e}{2}\right )\right )-\pi \right )-\pi \log \left (1+e^{-i f x}\right )-2 \left (\frac{f x}{2}-\tan ^{-1}\left (\cot \left (\frac{e}{2}\right )\right )\right ) \log \left (1-e^{2 i \left (\frac{f x}{2}-\tan ^{-1}\left (\cot \left (\frac{e}{2}\right )\right )\right )}\right )+\pi \log \left (\cos \left (\frac{f x}{2}\right )\right )-2 \tan ^{-1}\left (\cot \left (\frac{e}{2}\right )\right ) \log \left (\sin \left (\frac{f x}{2}-\tan ^{-1}\left (\cot \left (\frac{e}{2}\right )\right )\right )\right )+i \text{PolyLog}\left (2,e^{2 i \left (\frac{f x}{2}-\tan ^{-1}\left (\cot \left (\frac{e}{2}\right )\right )\right )}\right )\right )}{\sqrt{\cot ^2\left (\frac{e}{2}\right )+1}}\right ) \sec \left (\frac{e}{2}\right ) \sec ^2(e+f x) \cos ^4\left (\frac{e}{2}+\frac{f x}{2}\right )}{3 f^3 (\sec (e+f x) a+a)^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{80 c d \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 ) \cos ^4\left (\frac{e}{2}+\frac{f x}{2}\right )}{3 f^2 (\sec (e+f x) a+a)^2 \left (\cos ^2\left (\frac{e}{2}\right )+\sin ^2\left (\frac{e}{2}\right )\right )}+\frac{\sec \left (\frac{e}{2}\right ) \sec ^2(e+f x) \left (3 d^2 x^3 \cos \left (\frac{f x}{2}\right ) f^3+9 c d x^2 \cos \left (\frac{f x}{2}\right ) f^3+9 c^2 x \cos \left (\frac{f x}{2}\right ) f^3+3 d^2 x^3 \cos \left (e+\frac{f x}{2}\right ) f^3+9 c d x^2 \cos \left (e+\frac{f x}{2}\right ) f^3+9 c^2 x \cos \left (e+\frac{f x}{2}\right ) f^3+d^2 x^3 \cos \left (e+\frac{3 f x}{2}\right ) f^3+3 c d x^2 \cos \left (e+\frac{3 f x}{2}\right ) f^3+3 c^2 x \cos \left (e+\frac{3 f x}{2}\right ) f^3+d^2 x^3 \cos \left (2 e+\frac{3 f x}{2}\right ) f^3+3 c d x^2 \cos \left (2 e+\frac{3 f x}{2}\right ) f^3+3 c^2 x \cos \left (2 e+\frac{3 f x}{2}\right ) f^3-18 c^2 \sin \left (\frac{f x}{2}\right ) f^2-18 d^2 x^2 \sin \left (\frac{f x}{2}\right ) f^2-36 c d x \sin \left (\frac{f x}{2}\right ) f^2+12 c^2 \sin \left (e+\frac{f x}{2}\right ) f^2+12 d^2 x^2 \sin \left (e+\frac{f x}{2}\right ) f^2+24 c d x \sin \left (e+\frac{f x}{2}\right ) f^2-10 c^2 \sin \left (e+\frac{3 f x}{2}\right ) f^2-10 d^2 x^2 \sin \left (e+\frac{3 f x}{2}\right ) f^2-20 c d x \sin \left (e+\frac{3 f x}{2}\right ) f^2-4 c d \cos \left (\frac{f x}{2}\right ) f-4 d^2 x \cos \left (\frac{f x}{2}\right ) f-4 c d \cos \left (e+\frac{f x}{2}\right ) f-4 d^2 x \cos \left (e+\frac{f x}{2}\right ) f+8 d^2 \sin \left (\frac{f x}{2}\right )-4 d^2 \sin \left (e+\frac{f x}{2}\right )+4 d^2 \sin \left (e+\frac{3 f x}{2}\right )\right ) \cos \left (\frac{e}{2}+\frac{f x}{2}\right )}{6 f^3 (\sec (e+f x) a+a)^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-80*c*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))/(3*f^2*(a + a*Sec[e + f*x])^2*(Cos[e/2]^2 + Sin[e/2]^2)) - (80*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 - ArcTan[Cot[e/2]]))] + Pi*Log[Co
s[(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 - ArcTa
n[Cot[e/2]]))]))/Sqrt[1 + Cot[e/2]^2])*Sec[e/2]*Sec[e + f*x]^2)/(3*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*(-4*c*d*f*Cos[(f*x)/2] - 4*d^2*f*x*C
os[(f*x)/2] + 9*c^2*f^3*x*Cos[(f*x)/2] + 9*c*d*f^3*x^2*Cos[(f*x)/2] + 3*d^2*f^3*x^3*Cos[(f*x)/2] - 4*c*d*f*Cos
[e + (f*x)/2] - 4*d^2*f*x*Cos[e + (f*x)/2] + 9*c^2*f^3*x*Cos[e + (f*x)/2] + 9*c*d*f^3*x^2*Cos[e + (f*x)/2] + 3
*d^2*f^3*x^3*Cos[e + (f*x)/2] + 3*c^2*f^3*x*Cos[e + (3*f*x)/2] + 3*c*d*f^3*x^2*Cos[e + (3*f*x)/2] + d^2*f^3*x^
3*Cos[e + (3*f*x)/2] + 3*c^2*f^3*x*Cos[2*e + (3*f*x)/2] + 3*c*d*f^3*x^2*Cos[2*e + (3*f*x)/2] + d^2*f^3*x^3*Cos
[2*e + (3*f*x)/2] + 8*d^2*Sin[(f*x)/2] - 18*c^2*f^2*Sin[(f*x)/2] - 36*c*d*f^2*x*Sin[(f*x)/2] - 18*d^2*f^2*x^2*
Sin[(f*x)/2] - 4*d^2*Sin[e + (f*x)/2] + 12*c^2*f^2*Sin[e + (f*x)/2] + 24*c*d*f^2*x*Sin[e + (f*x)/2] + 12*d^2*f
^2*x^2*Sin[e + (f*x)/2] + 4*d^2*Sin[e + (3*f*x)/2] - 10*c^2*f^2*Sin[e + (3*f*x)/2] - 20*c*d*f^2*x*Sin[e + (3*f
*x)/2] - 10*d^2*f^2*x^2*Sin[e + (3*f*x)/2]))/(6*f^3*(a + a*Sec[e + f*x])^2)

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Maple [B]  time = 0.136, size = 442, normalized size = 1.9 \begin{align*}{\frac{{d}^{2}{x}^{3}}{3\,{a}^{2}}}+{\frac{cd{x}^{2}}{{a}^{2}}}+{\frac{{c}^{2}x}{{a}^{2}}}-{\frac{{\frac{2\,i}{3}} \left ( 6\,{d}^{2}{f}^{2}{x}^{2}{{\rm e}^{2\,i \left ( fx+e \right ) }}-2\,icdf{{\rm e}^{i \left ( fx+e \right ) }}+12\,cd{f}^{2}x{{\rm e}^{2\,i \left ( fx+e \right ) }}+9\,{d}^{2}{f}^{2}{x}^{2}{{\rm e}^{i \left ( fx+e \right ) }}-2\,i{d}^{2}fx{{\rm e}^{i \left ( fx+e \right ) }}-2\,icdf{{\rm e}^{2\,i \left ( fx+e \right ) }}+6\,{c}^{2}{f}^{2}{{\rm e}^{2\,i \left ( fx+e \right ) }}+18\,cd{f}^{2}x{{\rm e}^{i \left ( fx+e \right ) }}+5\,{d}^{2}{f}^{2}{x}^{2}-2\,i{d}^{2}fx{{\rm e}^{2\,i \left ( fx+e \right ) }}+9\,{c}^{2}{f}^{2}{{\rm e}^{i \left ( fx+e \right ) }}+10\,cd{f}^{2}x+5\,{c}^{2}{f}^{2}-2\,{d}^{2}{{\rm e}^{2\,i \left ( fx+e \right ) }}-4\,{d}^{2}{{\rm e}^{i \left ( fx+e \right ) }}-2\,{d}^{2} \right ) }{{a}^{2}{f}^{3} \left ({{\rm e}^{i \left ( fx+e \right ) }}+1 \right ) ^{3}}}-{\frac{20\,cd\ln \left ({{\rm e}^{i \left ( fx+e \right ) }}+1 \right ) }{3\,{a}^{2}{f}^{2}}}+{\frac{20\,cd\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{3\,{a}^{2}{f}^{2}}}+{\frac{{\frac{10\,i}{3}}{d}^{2}{x}^{2}}{f{a}^{2}}}+{\frac{{\frac{20\,i}{3}}{d}^{2}ex}{{a}^{2}{f}^{2}}}+{\frac{{\frac{10\,i}{3}}{d}^{2}{e}^{2}}{{a}^{2}{f}^{3}}}-{\frac{20\,{d}^{2}\ln \left ({{\rm e}^{i \left ( fx+e \right ) }}+1 \right ) x}{3\,{a}^{2}{f}^{2}}}+{\frac{{\frac{20\,i}{3}}{d}^{2}{\it polylog} \left ( 2,-{{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{a}^{2}{f}^{3}}}-{\frac{20\,{d}^{2}e\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{3\,{a}^{2}{f}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 4.54975, size = 1395, normalized size = 6.09 \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)^2/(a+a*sec(f*x+e))^2,x, algorithm="maxima")

[Out]

-(I*d^2*f^3*x^3 + 3*I*c*d*f^3*x^2 + 3*I*c^2*f^3*x + 10*c^2*f^2 - 4*d^2 + (20*d^2*f*x + 20*c*d*f + 20*(d^2*f*x
+ c*d*f)*cos(3*f*x + 3*e) + 60*(d^2*f*x + c*d*f)*cos(2*f*x + 2*e) + 60*(d^2*f*x + c*d*f)*cos(f*x + e) + (20*I*
d^2*f*x + 20*I*c*d*f)*sin(3*f*x + 3*e) + (60*I*d^2*f*x + 60*I*c*d*f)*sin(2*f*x + 2*e) + (60*I*d^2*f*x + 60*I*c
*d*f)*sin(f*x + e))*arctan2(sin(f*x + e), cos(f*x + e) + 1) + (I*d^2*f^3*x^3 + (3*I*c*d*f^3 - 10*d^2*f^2)*x^2
+ (3*I*c^2*f^3 - 20*c*d*f^2)*x)*cos(3*f*x + 3*e) + (3*I*d^2*f^3*x^3 + 12*c^2*f^2 - 4*I*c*d*f - 9*(-I*c*d*f^3 +
2*d^2*f^2)*x^2 - 4*d^2 + (9*I*c^2*f^3 - 36*c*d*f^2 - 4*I*d^2*f)*x)*cos(2*f*x + 2*e) + (3*I*d^2*f^3*x^3 + 18*c
^2*f^2 - 4*I*c*d*f - 3*(-3*I*c*d*f^3 + 4*d^2*f^2)*x^2 - 8*d^2 + (9*I*c^2*f^3 - 24*c*d*f^2 - 4*I*d^2*f)*x)*cos(
f*x + e) - (20*d^2*cos(3*f*x + 3*e) + 60*d^2*cos(2*f*x + 2*e) + 60*d^2*cos(f*x + e) + 20*I*d^2*sin(3*f*x + 3*e
) + 60*I*d^2*sin(2*f*x + 2*e) + 60*I*d^2*sin(f*x + e) + 20*d^2)*dilog(-e^(I*f*x + I*e)) + (-10*I*d^2*f*x - 10*
I*c*d*f + (-10*I*d^2*f*x - 10*I*c*d*f)*cos(3*f*x + 3*e) + (-30*I*d^2*f*x - 30*I*c*d*f)*cos(2*f*x + 2*e) + (-30
*I*d^2*f*x - 30*I*c*d*f)*cos(f*x + e) + 10*(d^2*f*x + c*d*f)*sin(3*f*x + 3*e) + 30*(d^2*f*x + c*d*f)*sin(2*f*x
+ 2*e) + 30*(d^2*f*x + c*d*f)*sin(f*x + e))*log(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*cos(f*x + e) + 1) - (d^2*
f^3*x^3 + (3*c*d*f^3 + 10*I*d^2*f^2)*x^2 + (3*c^2*f^3 + 20*I*c*d*f^2)*x)*sin(3*f*x + 3*e) - (3*d^2*f^3*x^3 - 1
2*I*c^2*f^2 - 4*c*d*f + (9*c*d*f^3 + 18*I*d^2*f^2)*x^2 + 4*I*d^2 + (9*c^2*f^3 + 36*I*c*d*f^2 - 4*d^2*f)*x)*sin
(2*f*x + 2*e) - (3*d^2*f^3*x^3 - 18*I*c^2*f^2 - 4*c*d*f + (9*c*d*f^3 + 12*I*d^2*f^2)*x^2 + 8*I*d^2 + (9*c^2*f^
3 + 24*I*c*d*f^2 - 4*d^2*f)*x)*sin(f*x + e))/(-3*I*a^2*f^3*cos(3*f*x + 3*e) - 9*I*a^2*f^3*cos(2*f*x + 2*e) - 9
*I*a^2*f^3*cos(f*x + e) + 3*a^2*f^3*sin(3*f*x + 3*e) + 9*a^2*f^3*sin(2*f*x + 2*e) + 9*a^2*f^3*sin(f*x + e) - 3
*I*a^2*f^3)

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Fricas [B]  time = 1.84353, size = 1173, normalized size = 5.12 \begin{align*} \frac{d^{2} f^{3} x^{3} + 3 \, c d f^{3} x^{2} - 2 \, c d f +{\left (d^{2} f^{3} x^{3} + 3 \, c d f^{3} x^{2} + 3 \, c^{2} f^{3} x\right )} \cos \left (f x + e\right )^{2} +{\left (3 \, c^{2} f^{3} - 2 \, d^{2} f\right )} x + 2 \,{\left (d^{2} f^{3} x^{3} + 3 \, c d f^{3} x^{2} - c d f +{\left (3 \, c^{2} f^{3} - d^{2} f\right )} x\right )} \cos \left (f x + e\right ) +{\left (-10 i \, d^{2} \cos \left (f x + e\right )^{2} - 20 i \, d^{2} \cos \left (f x + e\right ) - 10 i \, d^{2}\right )}{\rm Li}_2\left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) +{\left (10 i \, d^{2} \cos \left (f x + e\right )^{2} + 20 i \, d^{2} \cos \left (f x + e\right ) + 10 i \, d^{2}\right )}{\rm Li}_2\left (-\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 10 \,{\left (d^{2} f x + c d f +{\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )^{2} + 2 \,{\left (d^{2} f x + c d f\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 ) - 10 \,{\left (d^{2} f x + c d f +{\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )^{2} + 2 \,{\left (d^{2} f x + c d f\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 ) -{\left (4 \, d^{2} f^{2} x^{2} + 8 \, c d f^{2} x + 4 \, c^{2} f^{2} - 2 \, d^{2} +{\left (5 \, d^{2} f^{2} x^{2} + 10 \, c d f^{2} x + 5 \, c^{2} f^{2} - 2 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{3 \,{\left (a^{2} f^{3} \cos \left (f x + e\right )^{2} + 2 \, a^{2} f^{3} \cos \left (f x + e\right ) + a^{2} f^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(d^2*f^3*x^3 + 3*c*d*f^3*x^2 - 2*c*d*f + (d^2*f^3*x^3 + 3*c*d*f^3*x^2 + 3*c^2*f^3*x)*cos(f*x + e)^2 + (3*c
^2*f^3 - 2*d^2*f)*x + 2*(d^2*f^3*x^3 + 3*c*d*f^3*x^2 - c*d*f + (3*c^2*f^3 - d^2*f)*x)*cos(f*x + e) + (-10*I*d^
2*cos(f*x + e)^2 - 20*I*d^2*cos(f*x + e) - 10*I*d^2)*dilog(-cos(f*x + e) + I*sin(f*x + e)) + (10*I*d^2*cos(f*x
+ e)^2 + 20*I*d^2*cos(f*x + e) + 10*I*d^2)*dilog(-cos(f*x + e) - I*sin(f*x + e)) - 10*(d^2*f*x + c*d*f + (d^2
*f*x + c*d*f)*cos(f*x + e)^2 + 2*(d^2*f*x + c*d*f)*cos(f*x + e))*log(cos(f*x + e) + I*sin(f*x + e) + 1) - 10*(
d^2*f*x + c*d*f + (d^2*f*x + c*d*f)*cos(f*x + e)^2 + 2*(d^2*f*x + c*d*f)*cos(f*x + e))*log(cos(f*x + e) - I*si
n(f*x + e) + 1) - (4*d^2*f^2*x^2 + 8*c*d*f^2*x + 4*c^2*f^2 - 2*d^2 + (5*d^2*f^2*x^2 + 10*c*d*f^2*x + 5*c^2*f^2
- 2*d^2)*cos(f*x + e))*sin(f*x + e))/(a^2*f^3*cos(f*x + e)^2 + 2*a^2*f^3*cos(f*x + e) + a^2*f^3)

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

[Out]

(Integral(c**2/(sec(e + f*x)**2 + 2*sec(e + f*x) + 1), x) + Integral(d**2*x**2/(sec(e + f*x)**2 + 2*sec(e + f*
x) + 1), x) + Integral(2*c*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 )}^{2}}{{\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)^2/(a+a*sec(f*x+e))^2,x, algorithm="giac")

[Out]

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