\(\int \frac {(c+d x)^2}{(a+a \sec (e+f x))^2} \, dx\) [17]

Optimal result

Integrand size = 20, antiderivative size = 229 \[ \int \frac {(c+d x)^2}{(a+a \sec (e+f x))^2} \, dx=\frac {5 i (c+d x)^2}{3 a^2 f}+\frac {(c+d x)^3}{3 a^2 d}-\frac {20 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}+\frac {20 i d^2 \operatorname {PolyLog}\left (2,-e^{i (e+f x)}\right )}{3 a^2 f^3}-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {2 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}-\frac {5 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4276, 3399, 4271, 3852, 8, 4269, 3800, 2221, 2317, 2438} \[ \int \frac {(c+d x)^2}{(a+a \sec (e+f x))^2} \, dx=-\frac {20 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {5 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {5 i (c+d x)^2}{3 a^2 f}+\frac {(c+d x)^3}{3 a^2 d}+\frac {20 i d^2 \operatorname {PolyLog}\left (2,-e^{i (e+f x)}\right )}{3 a^2 f^3}+\frac {2 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3} \]

[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 8

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

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 2317

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 2438

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]

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 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 3852

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 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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 \operatorname {PolyLog}\left (2,-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 ) \text {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 \operatorname {PolyLog}\left (2,-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] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(925\) vs. \(2(229)=458\).

Time = 7.17 (sec) , antiderivative size = 925, normalized size of antiderivative = 4.04 \[ \int \frac {(c+d x)^2}{(a+a \sec (e+f x))^2} \, dx=-\frac {80 c 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 )}{3 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 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 \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 \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}-\arctan \left (\cot \left (\frac {e}{2}\right )\right )\right ) \log \left (1-e^{2 i \left (\frac {f x}{2}-\arctan \left (\cot \left (\frac {e}{2}\right )\right )\right )}\right )+\pi \log \left (\cos \left (\frac {f x}{2}\right )\right )-2 \arctan \left (\cot \left (\frac {e}{2}\right )\right ) \log \left (\sin \left (\frac {f x}{2}-\arctan \left (\cot \left (\frac {e}{2}\right )\right )\right )\right )+i \operatorname {PolyLog}\left (2,e^{2 i \left (\frac {f x}{2}-\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)}{3 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 (-4 c d f \cos \left (\frac {f x}{2}\right )-4 d^2 f x \cos \left (\frac {f x}{2}\right )+9 c^2 f^3 x \cos \left (\frac {f x}{2}\right )+9 c d f^3 x^2 \cos \left (\frac {f x}{2}\right )+3 d^2 f^3 x^3 \cos \left (\frac {f x}{2}\right )-4 c d f \cos \left (e+\frac {f x}{2}\right )-4 d^2 f x \cos \left (e+\frac {f x}{2}\right )+9 c^2 f^3 x \cos \left (e+\frac {f x}{2}\right )+9 c d f^3 x^2 \cos \left (e+\frac {f x}{2}\right )+3 d^2 f^3 x^3 \cos \left (e+\frac {f x}{2}\right )+3 c^2 f^3 x \cos \left (e+\frac {3 f x}{2}\right )+3 c d f^3 x^2 \cos \left (e+\frac {3 f x}{2}\right )+d^2 f^3 x^3 \cos \left (e+\frac {3 f x}{2}\right )+3 c^2 f^3 x \cos \left (2 e+\frac {3 f x}{2}\right )+3 c d f^3 x^2 \cos \left (2 e+\frac {3 f x}{2}\right )+d^2 f^3 x^3 \cos \left (2 e+\frac {3 f x}{2}\right )+8 d^2 \sin \left (\frac {f x}{2}\right )-18 c^2 f^2 \sin \left (\frac {f x}{2}\right )-36 c d f^2 x \sin \left (\frac {f x}{2}\right )-18 d^2 f^2 x^2 \sin \left (\frac {f x}{2}\right )-4 d^2 \sin \left (e+\frac {f x}{2}\right )+12 c^2 f^2 \sin \left (e+\frac {f x}{2}\right )+24 c d f^2 x \sin \left (e+\frac {f x}{2}\right )+12 d^2 f^2 x^2 \sin \left (e+\frac {f x}{2}\right )+4 d^2 \sin \left (e+\frac {3 f x}{2}\right )-10 c^2 f^2 \sin \left (e+\frac {3 f x}{2}\right )-20 c d f^2 x \sin \left (e+\frac {3 f x}{2}\right )-10 d^2 f^2 x^2 \sin \left (e+\frac {3 f x}{2}\right )\right )}{6 f^3 (a+a \sec (e+f x))^2} \]

[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)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 452 vs. \(2 (187 ) = 374\).

Time = 0.58 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.98

method	result	size
risch	\(\frac {d^{2} x^{3}}{3 a^{2}}+\frac {d c \,x^{2}}{a^{2}}+\frac {c^{2} x}{a^{2}}+\frac {c^{3}}{3 a^{2} d}-\frac {2 i \left (6 d^{2} f^{2} x^{2} {\mathrm e}^{2 i \left (f x +e \right )}-2 i d^{2} f x \,{\mathrm e}^{i \left (f x +e \right )}+12 c d \,f^{2} x \,{\mathrm e}^{2 i \left (f x +e \right )}+9 d^{2} f^{2} x^{2} {\mathrm e}^{i \left (f x +e \right )}-2 i d^{2} f x \,{\mathrm e}^{2 i \left (f x +e \right )}-2 i c d f \,{\mathrm e}^{i \left (f x +e \right )}+6 c^{2} f^{2} {\mathrm e}^{2 i \left (f x +e \right )}+18 c d \,f^{2} x \,{\mathrm e}^{i \left (f x +e \right )}+5 d^{2} f^{2} x^{2}-2 i c d f \,{\mathrm e}^{2 i \left (f x +e \right )}+9 c^{2} f^{2} {\mathrm e}^{i \left (f x +e \right )}+10 c d \,f^{2} x +5 c^{2} f^{2}-2 d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-4 d^{2} {\mathrm e}^{i \left (f x +e \right )}-2 d^{2}\right )}{3 f^{3} a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}+\frac {20 d c \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{3 a^{2} f^{2}}-\frac {20 d c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{3 a^{2} f^{2}}+\frac {10 i d^{2} x^{2}}{3 a^{2} f}+\frac {20 i d^{2} e x}{3 a^{2} f^{2}}+\frac {10 i d^{2} e^{2}}{3 a^{2} f^{3}}-\frac {20 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) x}{3 a^{2} f^{2}}+\frac {20 i d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{i \left (f x +e \right )}\right )}{3 a^{2} f^{3}}-\frac {20 d^{2} e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{3 a^{2} f^{3}}\)	\(453\)

[In]

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

[Out]

1/3/a^2*d^2*x^3+1/a^2*d*c*x^2+1/a^2*c^2*x+1/3/a^2/d*c^3-2/3*I*(6*d^2*f^2*x^2*exp(2*I*(f*x+e))-2*I*d^2*f*x*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(2*I*(f*x+e))-2*I*c*d*f*e
xp(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*c*d*f*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/(ex
p(I*(f*x+e))+1)^3+20/3/a^2*d/f^2*c*ln(exp(I*(f*x+e)))-20/3/a^2*d/f^2*c*ln(exp(I*(f*x+e))+1)+10/3*I/a^2*d^2/f*x
^2+20/3*I/a^2*d^2/f^2*e*x+10/3*I/a^2*d^2/f^3*e^2-20/3/a^2*d^2/f^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/a^2*d^2/f^3*e*ln(exp(I*(f*x+e)))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (184) = 368\).

Time = 0.30 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.15 \[ \int \frac {(c+d x)^2}{(a+a \sec (e+f x))^2} \, dx=\frac {d^{2} f^{3} x^{3} + 3 \, c d f^{3} x^{2} - 2 \, c d f + {\left (d^{2} f^{3} x^{3} + 3 \, c d f^{3} x^{2} + 3 \, c^{2} f^{3} x\right )} \cos \left (f x + e\right )^{2} + {\left (3 \, c^{2} f^{3} - 2 \, d^{2} f\right )} x + 2 \, {\left (d^{2} f^{3} x^{3} + 3 \, c d f^{3} x^{2} - c d f + {\left (3 \, c^{2} f^{3} - d^{2} f\right )} x\right )} \cos \left (f x + e\right ) - 10 \, {\left (i \, d^{2} \cos \left (f x + e\right )^{2} + 2 i \, d^{2} \cos \left (f x + e\right ) + i \, d^{2}\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - 10 \, {\left (-i \, d^{2} \cos \left (f x + e\right )^{2} - 2 i \, d^{2} \cos \left (f x + e\right ) - i \, d^{2}\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 10 \, {\left (d^{2} f x + c d f + {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + 1\right ) - 10 \, {\left (d^{2} f x + c d f + {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + 1\right ) - {\left (4 \, d^{2} f^{2} x^{2} + 8 \, c d f^{2} x + 4 \, c^{2} f^{2} - 2 \, d^{2} + {\left (5 \, d^{2} f^{2} x^{2} + 10 \, c d f^{2} x + 5 \, c^{2} f^{2} - 2 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{3 \, {\left (a^{2} f^{3} \cos \left (f x + e\right )^{2} + 2 \, a^{2} f^{3} \cos \left (f x + e\right ) + a^{2} f^{3}\right )}} \]

[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 + 2*I*d^2*cos(f*x + e) + I*d^2)*dilog(-cos(f*x + e) + I*sin(f*x + e)) - 10*(-I*d^2*cos(f*x + e
)^2 - 2*I*d^2*cos(f*x + e) - 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*sin(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)

Sympy [F]

\[ \int \frac {(c+d x)^2}{(a+a \sec (e+f x))^2} \, dx=\frac {\int \frac {c^{2}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{2} x^{2}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {2 c d x}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]

[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

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1035 vs. \(2 (184) = 368\).

Time = 0.78 (sec) , antiderivative size = 1035, normalized size of antiderivative = 4.52 \[ \int \frac {(c+d x)^2}{(a+a \sec (e+f x))^2} \, dx=\text {Too large to display} \]

[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 + c*d*f + (d^2*f*x + c*d*
f)*cos(3*f*x + 3*e) + 3*(d^2*f*x + c*d*f)*cos(2*f*x + 2*e) + 3*(d^2*f*x + c*d*f)*cos(f*x + e) - (-I*d^2*f*x -
I*c*d*f)*sin(3*f*x + 3*e) - 3*(-I*d^2*f*x - I*c*d*f)*sin(2*f*x + 2*e) - 3*(-I*d^2*f*x - 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) + 3*d^2*cos(2*f*x + 2*e) + 3*d^2*cos(f*x + e) + I*d^2*sin(3*f*x + 3*e) + 3*I*d^2*sin(2*f*x +
2*e) + 3*I*d^2*sin(f*x + e) + d^2)*dilog(-e^(I*f*x + I*e)) - 10*(I*d^2*f*x + I*c*d*f + (I*d^2*f*x + I*c*d*f)*c
os(3*f*x + 3*e) + 3*(I*d^2*f*x + I*c*d*f)*cos(2*f*x + 2*e) + 3*(I*d^2*f*x + I*c*d*f)*cos(f*x + e) - (d^2*f*x +
c*d*f)*sin(3*f*x + 3*e) - 3*(d^2*f*x + c*d*f)*sin(2*f*x + 2*e) - 3*(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 - 12*I*c^2*f^2 - 4*c*d*f + 9*(c*d*f^3 + 2*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
+ 3*(3*c*d*f^3 + 4*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)

Giac [F]

\[ \int \frac {(c+d x)^2}{(a+a \sec (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (a \sec \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2}{(a+a \sec (e+f x))^2} \, dx=\text {Hanged} \]

[In]

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

[Out]

\text{Hanged}