\(\int \frac {(c+d x)^2}{(a+b \sec (e+f x))^2} \, dx\) [40]

Optimal result

Integrand size = 20, antiderivative size = 1117 \[ \int \frac {(c+d x)^2}{(a+b \sec (e+f x))^2} \, dx=-\frac {i b^2 (c+d x)^2}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^3}{3 a^2 d}+\frac {2 b^2 d (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {2 b^2 d (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac {i b^3 (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac {2 i b (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {i b^3 (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac {2 i b (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}-\frac {2 i b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {2 i b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {2 b^3 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}+\frac {4 b d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {2 b^3 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}-\frac {4 b d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}-\frac {2 i b^3 d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^3}+\frac {4 i b d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {2 i b^3 d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^3}-\frac {4 i b d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {b^2 (c+d x)^2 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))} \]

[Out]

I*b^3*(d*x+c)^2*ln(1+a*exp(I*(f*x+e))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/f+1/3*(d*x+c)^3/a^2/d+2*b^2*d
*(d*x+c)*ln(1+a*exp(I*(f*x+e))/(b-I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/f^2+2*b^2*d*(d*x+c)*ln(1+a*exp(I*(f*x+e))/
(b+I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/f^2-2*I*b*(d*x+c)^2*ln(1+a*exp(I*(f*x+e))/(b+(-a^2+b^2)^(1/2)))/a^2/f/(-a
^2+b^2)^(1/2)-2*I*b^3*d^2*polylog(3,-a*exp(I*(f*x+e))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/f^3+4*I*b*d^2
*polylog(3,-a*exp(I*(f*x+e))/(b-(-a^2+b^2)^(1/2)))/a^2/f^3/(-a^2+b^2)^(1/2)-4*I*b*d^2*polylog(3,-a*exp(I*(f*x+
e))/(b+(-a^2+b^2)^(1/2)))/a^2/f^3/(-a^2+b^2)^(1/2)-2*b^3*d*(d*x+c)*polylog(2,-a*exp(I*(f*x+e))/(b-(-a^2+b^2)^(
1/2)))/a^2/(-a^2+b^2)^(3/2)/f^2+2*b^3*d*(d*x+c)*polylog(2,-a*exp(I*(f*x+e))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^
2)^(3/2)/f^2+2*I*b*(d*x+c)^2*ln(1+a*exp(I*(f*x+e))/(b-(-a^2+b^2)^(1/2)))/a^2/f/(-a^2+b^2)^(1/2)-2*I*b^2*d^2*po
lylog(2,-a*exp(I*(f*x+e))/(b-I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/f^3+b^2*(d*x+c)^2*sin(f*x+e)/a/(a^2-b^2)/f/(b+a
*cos(f*x+e))-2*I*b^2*d^2*polylog(2,-a*exp(I*(f*x+e))/(b+I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/f^3-I*b^3*(d*x+c)^2*
ln(1+a*exp(I*(f*x+e))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/f+4*b*d*(d*x+c)*polylog(2,-a*exp(I*(f*x+e))/(
b-(-a^2+b^2)^(1/2)))/a^2/f^2/(-a^2+b^2)^(1/2)-4*b*d*(d*x+c)*polylog(2,-a*exp(I*(f*x+e))/(b+(-a^2+b^2)^(1/2)))/
a^2/f^2/(-a^2+b^2)^(1/2)-I*b^2*(d*x+c)^2/a^2/(a^2-b^2)/f+2*I*b^3*d^2*polylog(3,-a*exp(I*(f*x+e))/(b+(-a^2+b^2)
^(1/2)))/a^2/(-a^2+b^2)^(3/2)/f^3

Rubi [A] (verified)

Time = 2.48 (sec) , antiderivative size = 1117, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {4276, 3405, 3402, 2296, 2221, 2611, 2320, 6724, 4618, 2317, 2438} \[ \int \frac {(c+d x)^2}{(a+b \sec (e+f x))^2} \, dx=-\frac {i (c+d x)^2 \log \left (\frac {e^{i (e+f x)} a}{b-\sqrt {b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} f}+\frac {i (c+d x)^2 \log \left (\frac {e^{i (e+f x)} a}{b+\sqrt {b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} f}-\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} f^2}+\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} f^2}-\frac {2 i d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} f^3}+\frac {2 i d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} f^3}-\frac {i (c+d x)^2 b^2}{a^2 \left (a^2-b^2\right ) f}+\frac {2 d (c+d x) \log \left (\frac {e^{i (e+f x)} a}{b-i \sqrt {a^2-b^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) f^2}+\frac {2 d (c+d x) \log \left (\frac {e^{i (e+f x)} a}{b+i \sqrt {a^2-b^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) f^2}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) f^3}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) f^3}+\frac {(c+d x)^2 \sin (e+f x) b^2}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {2 i (c+d x)^2 \log \left (\frac {e^{i (e+f x)} a}{b-\sqrt {b^2-a^2}}+1\right ) b}{a^2 \sqrt {b^2-a^2} f}-\frac {2 i (c+d x)^2 \log \left (\frac {e^{i (e+f x)} a}{b+\sqrt {b^2-a^2}}+1\right ) b}{a^2 \sqrt {b^2-a^2} f}+\frac {4 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} f^2}-\frac {4 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} f^2}+\frac {4 i d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} f^3}-\frac {4 i d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} f^3}+\frac {(c+d x)^3}{3 a^2 d} \]

[In]

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

[Out]

((-I)*b^2*(c + d*x)^2)/(a^2*(a^2 - b^2)*f) + (c + d*x)^3/(3*a^2*d) + (2*b^2*d*(c + d*x)*Log[1 + (a*E^(I*(e + f
*x)))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*f^2) + (2*b^2*d*(c + d*x)*Log[1 + (a*E^(I*(e + f*x)))/(b + I*
Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*f^2) - (I*b^3*(c + d*x)^2*Log[1 + (a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2
])])/(a^2*(-a^2 + b^2)^(3/2)*f) + ((2*I)*b*(c + d*x)^2*Log[1 + (a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2])])/(a
^2*Sqrt[-a^2 + b^2]*f) + (I*b^3*(c + d*x)^2*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 +
b^2)^(3/2)*f) - ((2*I)*b*(c + d*x)^2*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2
]*f) - ((2*I)*b^2*d^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*f^3) - ((2*
I)*b^2*d^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*f^3) - (2*b^3*d*(c + d
*x)*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^2) + (4*b*d*(c + d*x)
*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^2) + (2*b^3*d*(c + d*x)*Po
lyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^2) - (4*b*d*(c + d*x)*PolyL
og[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^2) - ((2*I)*b^3*d^2*PolyLog[3, -
((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^3) + ((4*I)*b*d^2*PolyLog[3, -((a*E^(
I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^3) + ((2*I)*b^3*d^2*PolyLog[3, -((a*E^(I*(e +
f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^3) - ((4*I)*b*d^2*PolyLog[3, -((a*E^(I*(e + f*x)))/
(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^3) + (b^2*(c + d*x)^2*Sin[e + f*x])/(a*(a^2 - b^2)*f*(b + a*
Cos[e + f*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 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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 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 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 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 3402

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

Rule 3405

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

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 4618

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)])/(Cos[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :>
Simp[I*((e + f*x)^(m + 1)/(b*f*(m + 1))), x] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a - Rt[-a^2 + b^2, 2] + I*
b*E^(I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a + Rt[-a^2 + b^2, 2] + I*b*E^(I*(c + d*x)))), x
]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]

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*} \text {integral}& = \int \left (\frac {(c+d x)^2}{a^2}+\frac {b^2 (c+d x)^2}{a^2 (b+a \cos (e+f x))^2}-\frac {2 b (c+d x)^2}{a^2 (b+a \cos (e+f x))}\right ) \, dx \\ & = \frac {(c+d x)^3}{3 a^2 d}-\frac {(2 b) \int \frac {(c+d x)^2}{b+a \cos (e+f x)} \, dx}{a^2}+\frac {b^2 \int \frac {(c+d x)^2}{(b+a \cos (e+f x))^2} \, dx}{a^2} \\ & = \frac {(c+d x)^3}{3 a^2 d}+\frac {b^2 (c+d x)^2 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}-\frac {(4 b) \int \frac {e^{i (e+f x)} (c+d x)^2}{a+2 b e^{i (e+f x)}+a e^{2 i (e+f x)}} \, dx}{a^2}-\frac {b^3 \int \frac {(c+d x)^2}{b+a \cos (e+f x)} \, dx}{a^2 \left (a^2-b^2\right )}-\frac {\left (2 b^2 d\right ) \int \frac {(c+d x) \sin (e+f x)}{b+a \cos (e+f x)} \, dx}{a \left (a^2-b^2\right ) f} \\ & = -\frac {i b^2 (c+d x)^2}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^3}{3 a^2 d}+\frac {b^2 (c+d x)^2 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}-\frac {\left (2 b^3\right ) \int \frac {e^{i (e+f x)} (c+d x)^2}{a+2 b e^{i (e+f x)}+a e^{2 i (e+f x)}} \, dx}{a^2 \left (a^2-b^2\right )}-\frac {(4 b) \int \frac {e^{i (e+f x)} (c+d x)^2}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \sqrt {-a^2+b^2}}+\frac {(4 b) \int \frac {e^{i (e+f x)} (c+d x)^2}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \sqrt {-a^2+b^2}}-\frac {\left (2 b^2 d\right ) \int \frac {e^{i (e+f x)} (c+d x)}{i b-\sqrt {a^2-b^2}+i a e^{i (e+f x)}} \, dx}{a \left (a^2-b^2\right ) f}-\frac {\left (2 b^2 d\right ) \int \frac {e^{i (e+f x)} (c+d x)}{i b+\sqrt {a^2-b^2}+i a e^{i (e+f x)}} \, dx}{a \left (a^2-b^2\right ) f} \\ & = -\frac {i b^2 (c+d x)^2}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^3}{3 a^2 d}+\frac {2 b^2 d (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {2 b^2 d (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {2 i b (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}-\frac {2 i b (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {b^2 (c+d x)^2 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {\left (2 b^3\right ) \int \frac {e^{i (e+f x)} (c+d x)^2}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \left (-a^2+b^2\right )^{3/2}}-\frac {\left (2 b^3\right ) \int \frac {e^{i (e+f x)} (c+d x)^2}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \left (-a^2+b^2\right )^{3/2}}-\frac {\left (2 b^2 d^2\right ) \int \log \left (1+\frac {i a e^{i (e+f x)}}{i b-\sqrt {a^2-b^2}}\right ) \, dx}{a^2 \left (a^2-b^2\right ) f^2}-\frac {\left (2 b^2 d^2\right ) \int \log \left (1+\frac {i a e^{i (e+f x)}}{i b+\sqrt {a^2-b^2}}\right ) \, dx}{a^2 \left (a^2-b^2\right ) f^2}-\frac {(4 i b d) \int (c+d x) \log \left (1+\frac {2 a e^{i (e+f x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \sqrt {-a^2+b^2} f}+\frac {(4 i b d) \int (c+d x) \log \left (1+\frac {2 a e^{i (e+f x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \sqrt {-a^2+b^2} f} \\ & = -\frac {i b^2 (c+d x)^2}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^3}{3 a^2 d}+\frac {2 b^2 d (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {2 b^2 d (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac {i b^3 (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac {2 i b (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {i b^3 (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac {2 i b (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {4 b d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}-\frac {4 b d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {b^2 (c+d x)^2 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {\left (2 i b^2 d^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {i a x}{i b-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \left (a^2-b^2\right ) f^3}+\frac {\left (2 i b^2 d^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {i a x}{i b+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {\left (4 b d^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {2 a e^{i (e+f x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {\left (4 b d^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {2 a e^{i (e+f x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {\left (2 i b^3 d\right ) \int (c+d x) \log \left (1+\frac {2 a e^{i (e+f x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac {\left (2 i b^3 d\right ) \int (c+d x) \log \left (1+\frac {2 a e^{i (e+f x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \left (-a^2+b^2\right )^{3/2} f} \\ & = -\frac {i b^2 (c+d x)^2}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^3}{3 a^2 d}+\frac {2 b^2 d (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {2 b^2 d (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac {i b^3 (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac {2 i b (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {i b^3 (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac {2 i b (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}-\frac {2 i b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {2 i b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {2 b^3 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}+\frac {4 b d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {2 b^3 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}-\frac {4 b d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {b^2 (c+d x)^2 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {\left (4 i b d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {a x}{-b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \sqrt {-a^2+b^2} f^3}-\frac {\left (4 i b d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {\left (2 b^3 d^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {2 a e^{i (e+f x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}-\frac {\left (2 b^3 d^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {2 a e^{i (e+f x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \left (-a^2+b^2\right )^{3/2} f^2} \\ & = -\frac {i b^2 (c+d x)^2}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^3}{3 a^2 d}+\frac {2 b^2 d (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {2 b^2 d (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac {i b^3 (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac {2 i b (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {i b^3 (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac {2 i b (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}-\frac {2 i b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {2 i b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {2 b^3 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}+\frac {4 b d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {2 b^3 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}-\frac {4 b d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {4 i b d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}-\frac {4 i b d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {b^2 (c+d x)^2 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}-\frac {\left (2 i b^3 d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {a x}{-b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^3}+\frac {\left (2 i b^3 d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^3} \\ & = -\frac {i b^2 (c+d x)^2}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^3}{3 a^2 d}+\frac {2 b^2 d (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {2 b^2 d (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac {i b^3 (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac {2 i b (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {i b^3 (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac {2 i b (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}-\frac {2 i b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {2 i b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {2 b^3 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}+\frac {4 b d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {2 b^3 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}-\frac {4 b d (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}-\frac {2 i b^3 d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^3}+\frac {4 i b d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {2 i b^3 d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^3}-\frac {4 i b d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {b^2 (c+d x)^2 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(11147\) vs. \(2(1117)=2234\).

Time = 20.08 (sec) , antiderivative size = 11147, normalized size of antiderivative = 9.98 \[ \int \frac {(c+d x)^2}{(a+b \sec (e+f x))^2} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [F]

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

[In]

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

[Out]

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

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. 4274 vs. \(2 (991) = 1982\).

Time = 0.64 (sec) , antiderivative size = 4274, normalized size of antiderivative = 3.83 \[ \int \frac {(c+d x)^2}{(a+b \sec (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/6*(2*(a^4*b - 2*a^2*b^3 + b^5)*d^2*f^3*x^3 + 6*(a^4*b - 2*a^2*b^3 + b^5)*c*d*f^3*x^2 + 6*(a^4*b - 2*a^2*b^3
+ b^5)*c^2*f^3*x - 6*(I*(2*a^4*b - a^2*b^3)*d^2*cos(f*x + e) + I*(2*a^3*b^2 - a*b^4)*d^2)*sqrt(-(a^2 - b^2)/a^
2)*polylog(3, -(b*cos(f*x + e) + I*b*sin(f*x + e) + (a*cos(f*x + e) + I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/a^2)
)/a) - 6*(-I*(2*a^4*b - a^2*b^3)*d^2*cos(f*x + e) - I*(2*a^3*b^2 - a*b^4)*d^2)*sqrt(-(a^2 - b^2)/a^2)*polylog(
3, -(b*cos(f*x + e) + I*b*sin(f*x + e) - (a*cos(f*x + e) + I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/a^2))/a) - 6*(-
I*(2*a^4*b - a^2*b^3)*d^2*cos(f*x + e) - I*(2*a^3*b^2 - a*b^4)*d^2)*sqrt(-(a^2 - b^2)/a^2)*polylog(3, -(b*cos(
f*x + e) - I*b*sin(f*x + e) + (a*cos(f*x + e) - I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/a^2))/a) - 6*(I*(2*a^4*b -
a^2*b^3)*d^2*cos(f*x + e) + I*(2*a^3*b^2 - a*b^4)*d^2)*sqrt(-(a^2 - b^2)/a^2)*polylog(3, -(b*cos(f*x + e) - I
*b*sin(f*x + e) - (a*cos(f*x + e) - I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/a^2))/a) + 2*((a^5 - 2*a^3*b^2 + a*b^4
)*d^2*f^3*x^3 + 3*(a^5 - 2*a^3*b^2 + a*b^4)*c*d*f^3*x^2 + 3*(a^5 - 2*a^3*b^2 + a*b^4)*c^2*f^3*x)*cos(f*x + e)
- 6*(I*(a^3*b^2 - a*b^4)*d^2*cos(f*x + e) + I*(a^2*b^3 - b^5)*d^2 + ((2*a^3*b^2 - a*b^4)*d^2*f*x + (2*a^3*b^2
- a*b^4)*c*d*f + ((2*a^4*b - a^2*b^3)*d^2*f*x + (2*a^4*b - a^2*b^3)*c*d*f)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2
))*dilog(-(b*cos(f*x + e) + I*b*sin(f*x + e) + (a*cos(f*x + e) + I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/a^2) + a)
/a + 1) - 6*(I*(a^3*b^2 - a*b^4)*d^2*cos(f*x + e) + I*(a^2*b^3 - b^5)*d^2 - ((2*a^3*b^2 - a*b^4)*d^2*f*x + (2*
a^3*b^2 - a*b^4)*c*d*f + ((2*a^4*b - a^2*b^3)*d^2*f*x + (2*a^4*b - a^2*b^3)*c*d*f)*cos(f*x + e))*sqrt(-(a^2 -
b^2)/a^2))*dilog(-(b*cos(f*x + e) + I*b*sin(f*x + e) - (a*cos(f*x + e) + I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/a
^2) + a)/a + 1) - 6*(-I*(a^3*b^2 - a*b^4)*d^2*cos(f*x + e) - I*(a^2*b^3 - b^5)*d^2 + ((2*a^3*b^2 - a*b^4)*d^2*
f*x + (2*a^3*b^2 - a*b^4)*c*d*f + ((2*a^4*b - a^2*b^3)*d^2*f*x + (2*a^4*b - a^2*b^3)*c*d*f)*cos(f*x + e))*sqrt
(-(a^2 - b^2)/a^2))*dilog(-(b*cos(f*x + e) - I*b*sin(f*x + e) + (a*cos(f*x + e) - I*a*sin(f*x + e))*sqrt(-(a^2
- b^2)/a^2) + a)/a + 1) - 6*(-I*(a^3*b^2 - a*b^4)*d^2*cos(f*x + e) - I*(a^2*b^3 - b^5)*d^2 - ((2*a^3*b^2 - a*
b^4)*d^2*f*x + (2*a^3*b^2 - a*b^4)*c*d*f + ((2*a^4*b - a^2*b^3)*d^2*f*x + (2*a^4*b - a^2*b^3)*c*d*f)*cos(f*x +
e))*sqrt(-(a^2 - b^2)/a^2))*dilog(-(b*cos(f*x + e) - I*b*sin(f*x + e) - (a*cos(f*x + e) - I*a*sin(f*x + e))*s
qrt(-(a^2 - b^2)/a^2) + a)/a + 1) - 3*(2*(a^2*b^3 - b^5)*d^2*e - 2*(a^2*b^3 - b^5)*c*d*f + 2*((a^3*b^2 - a*b^4
)*d^2*e - (a^3*b^2 - a*b^4)*c*d*f)*cos(f*x + e) + (-I*(2*a^3*b^2 - a*b^4)*d^2*e^2 + 2*I*(2*a^3*b^2 - a*b^4)*c*
d*e*f - I*(2*a^3*b^2 - a*b^4)*c^2*f^2 + (-I*(2*a^4*b - a^2*b^3)*d^2*e^2 + 2*I*(2*a^4*b - a^2*b^3)*c*d*e*f - I*
(2*a^4*b - a^2*b^3)*c^2*f^2)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2))*log(2*a*cos(f*x + e) + 2*I*a*sin(f*x + e) +
2*a*sqrt(-(a^2 - b^2)/a^2) + 2*b) - 3*(2*(a^2*b^3 - b^5)*d^2*e - 2*(a^2*b^3 - b^5)*c*d*f + 2*((a^3*b^2 - a*b^
4)*d^2*e - (a^3*b^2 - a*b^4)*c*d*f)*cos(f*x + e) + (I*(2*a^3*b^2 - a*b^4)*d^2*e^2 - 2*I*(2*a^3*b^2 - a*b^4)*c*
d*e*f + I*(2*a^3*b^2 - a*b^4)*c^2*f^2 + (I*(2*a^4*b - a^2*b^3)*d^2*e^2 - 2*I*(2*a^4*b - a^2*b^3)*c*d*e*f + I*(
2*a^4*b - a^2*b^3)*c^2*f^2)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2))*log(2*a*cos(f*x + e) - 2*I*a*sin(f*x + e) +
2*a*sqrt(-(a^2 - b^2)/a^2) + 2*b) - 3*(2*(a^2*b^3 - b^5)*d^2*e - 2*(a^2*b^3 - b^5)*c*d*f + 2*((a^3*b^2 - a*b^4
)*d^2*e - (a^3*b^2 - a*b^4)*c*d*f)*cos(f*x + e) + (-I*(2*a^3*b^2 - a*b^4)*d^2*e^2 + 2*I*(2*a^3*b^2 - a*b^4)*c*
d*e*f - I*(2*a^3*b^2 - a*b^4)*c^2*f^2 + (-I*(2*a^4*b - a^2*b^3)*d^2*e^2 + 2*I*(2*a^4*b - a^2*b^3)*c*d*e*f - I*
(2*a^4*b - a^2*b^3)*c^2*f^2)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2))*log(-2*a*cos(f*x + e) + 2*I*a*sin(f*x + e)
+ 2*a*sqrt(-(a^2 - b^2)/a^2) - 2*b) - 3*(2*(a^2*b^3 - b^5)*d^2*e - 2*(a^2*b^3 - b^5)*c*d*f + 2*((a^3*b^2 - a*b
^4)*d^2*e - (a^3*b^2 - a*b^4)*c*d*f)*cos(f*x + e) + (I*(2*a^3*b^2 - a*b^4)*d^2*e^2 - 2*I*(2*a^3*b^2 - a*b^4)*c
*d*e*f + I*(2*a^3*b^2 - a*b^4)*c^2*f^2 + (I*(2*a^4*b - a^2*b^3)*d^2*e^2 - 2*I*(2*a^4*b - a^2*b^3)*c*d*e*f + I*
(2*a^4*b - a^2*b^3)*c^2*f^2)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2))*log(-2*a*cos(f*x + e) - 2*I*a*sin(f*x + e)
+ 2*a*sqrt(-(a^2 - b^2)/a^2) - 2*b) + 3*(2*(a^2*b^3 - b^5)*d^2*f*x + 2*(a^2*b^3 - b^5)*d^2*e + 2*((a^3*b^2 - a
*b^4)*d^2*f*x + (a^3*b^2 - a*b^4)*d^2*e)*cos(f*x + e) - (I*(2*a^3*b^2 - a*b^4)*d^2*f^2*x^2 + 2*I*(2*a^3*b^2 -
a*b^4)*c*d*f^2*x - I*(2*a^3*b^2 - a*b^4)*d^2*e^2 + 2*I*(2*a^3*b^2 - a*b^4)*c*d*e*f + (I*(2*a^4*b - a^2*b^3)*d^
2*f^2*x^2 + 2*I*(2*a^4*b - a^2*b^3)*c*d*f^2*x - I*(2*a^4*b - a^2*b^3)*d^2*e^2 + 2*I*(2*a^4*b - a^2*b^3)*c*d*e*
f)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2))*log((b*cos(f*x + e) + I*b*sin(f*x + e) + (a*cos(f*x + e) + I*a*sin(f*
x + e))*sqrt(-(a^2 - b^2)/a^2) + a)/a) + 3*(2*(a^2*b^3 - b^5)*d^2*f*x + 2*(a^2*b^3 - b^5)*d^2*e + 2*((a^3*b^2
- a*b^4)*d^2*f*x + (a^3*b^2 - a*b^4)*d^2*e)*cos(f*x + e) - (-I*(2*a^3*b^2 - a*b^4)*d^2*f^2*x^2 - 2*I*(2*a^3*b^
2 - a*b^4)*c*d*f^2*x + I*(2*a^3*b^2 - a*b^4)*d^2*e^2 - 2*I*(2*a^3*b^2 - a*b^4)*c*d*e*f + (-I*(2*a^4*b - a^2*b^
3)*d^2*f^2*x^2 - 2*I*(2*a^4*b - a^2*b^3)*c*d*f^2*x + I*(2*a^4*b - a^2*b^3)*d^2*e^2 - 2*I*(2*a^4*b - a^2*b^3)*c
*d*e*f)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2))*log((b*cos(f*x + e) + I*b*sin(f*x + e) - (a*cos(f*x + e) + I*a*s
in(f*x + e))*sqrt(-(a^2 - b^2)/a^2) + a)/a) + 3*(2*(a^2*b^3 - b^5)*d^2*f*x + 2*(a^2*b^3 - b^5)*d^2*e + 2*((a^3
*b^2 - a*b^4)*d^2*f*x + (a^3*b^2 - a*b^4)*d^2*e)*cos(f*x + e) - (-I*(2*a^3*b^2 - a*b^4)*d^2*f^2*x^2 - 2*I*(2*a
^3*b^2 - a*b^4)*c*d*f^2*x + I*(2*a^3*b^2 - a*b^4)*d^2*e^2 - 2*I*(2*a^3*b^2 - a*b^4)*c*d*e*f + (-I*(2*a^4*b - a
^2*b^3)*d^2*f^2*x^2 - 2*I*(2*a^4*b - a^2*b^3)*c*d*f^2*x + I*(2*a^4*b - a^2*b^3)*d^2*e^2 - 2*I*(2*a^4*b - a^2*b
^3)*c*d*e*f)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2))*log((b*cos(f*x + e) - I*b*sin(f*x + e) + (a*cos(f*x + e) -
I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/a^2) + a)/a) + 3*(2*(a^2*b^3 - b^5)*d^2*f*x + 2*(a^2*b^3 - b^5)*d^2*e + 2*
((a^3*b^2 - a*b^4)*d^2*f*x + (a^3*b^2 - a*b^4)*d^2*e)*cos(f*x + e) - (I*(2*a^3*b^2 - a*b^4)*d^2*f^2*x^2 + 2*I*
(2*a^3*b^2 - a*b^4)*c*d*f^2*x - I*(2*a^3*b^2 - a*b^4)*d^2*e^2 + 2*I*(2*a^3*b^2 - a*b^4)*c*d*e*f + (I*(2*a^4*b
- a^2*b^3)*d^2*f^2*x^2 + 2*I*(2*a^4*b - a^2*b^3)*c*d*f^2*x - I*(2*a^4*b - a^2*b^3)*d^2*e^2 + 2*I*(2*a^4*b - a^
2*b^3)*c*d*e*f)*cos(f*x + e))*sqrt(-(a^2 - b^2)/a^2))*log((b*cos(f*x + e) - I*b*sin(f*x + e) - (a*cos(f*x + e)
- I*a*sin(f*x + e))*sqrt(-(a^2 - b^2)/a^2) + a)/a) + 6*((a^3*b^2 - a*b^4)*d^2*f^2*x^2 + 2*(a^3*b^2 - a*b^4)*c
*d*f^2*x + (a^3*b^2 - a*b^4)*c^2*f^2)*sin(f*x + e))/((a^7 - 2*a^5*b^2 + a^3*b^4)*f^3*cos(f*x + e) + (a^6*b - 2
*a^4*b^3 + a^2*b^5)*f^3)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^2}{(a+b \sec (e+f x))^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more de

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}