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

Optimal. Leaf size=582 \[ -\frac{b^3 d \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a^2 f^2 \left (b^2-a^2\right )^{3/2}}+\frac{b^3 d \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )}{a^2 f^2 \left (b^2-a^2\right )^{3/2}}+\frac{2 b d \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a^2 f^2 \sqrt{b^2-a^2}}-\frac{2 b d \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )}{a^2 f^2 \sqrt{b^2-a^2}}-\frac{i b^3 (c+d x) \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a^2 f \left (b^2-a^2\right )^{3/2}}+\frac{i b^3 (c+d x) \log \left (1+\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )}{a^2 f \left (b^2-a^2\right )^{3/2}}+\frac{2 i b (c+d x) \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a^2 f \sqrt{b^2-a^2}}-\frac{2 i b (c+d x) \log \left (1+\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )}{a^2 f \sqrt{b^2-a^2}}+\frac{b^2 (c+d x) \sin (e+f x)}{a f \left (a^2-b^2\right ) (a \cos (e+f x)+b)}+\frac{b^2 d \log (a \cos (e+f x)+b)}{a^2 f^2 \left (a^2-b^2\right )}+\frac{(c+d x)^2}{2 a^2 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.04705, antiderivative size = 582, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4191, 3324, 3321, 2264, 2190, 2279, 2391, 2668, 31} \[ -\frac{b^3 d \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a^2 f^2 \left (b^2-a^2\right )^{3/2}}+\frac{b^3 d \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )}{a^2 f^2 \left (b^2-a^2\right )^{3/2}}+\frac{2 b d \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a^2 f^2 \sqrt{b^2-a^2}}-\frac{2 b d \text{PolyLog}\left (2,-\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )}{a^2 f^2 \sqrt{b^2-a^2}}-\frac{i b^3 (c+d x) \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a^2 f \left (b^2-a^2\right )^{3/2}}+\frac{i b^3 (c+d x) \log \left (1+\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )}{a^2 f \left (b^2-a^2\right )^{3/2}}+\frac{2 i b (c+d x) \log \left (1+\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right )}{a^2 f \sqrt{b^2-a^2}}-\frac{2 i b (c+d x) \log \left (1+\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right )}{a^2 f \sqrt{b^2-a^2}}+\frac{b^2 (c+d x) \sin (e+f x)}{a f \left (a^2-b^2\right ) (a \cos (e+f x)+b)}+\frac{b^2 d \log (a \cos (e+f x)+b)}{a^2 f^2 \left (a^2-b^2\right )}+\frac{(c+d x)^2}{2 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3324

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 3321

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 2264

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 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]

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{c+d x}{(a+b \sec (e+f x))^2} \, dx &=\int \left (\frac{c+d x}{a^2}+\frac{b^2 (c+d x)}{a^2 (b+a \cos (e+f x))^2}-\frac{2 b (c+d x)}{a^2 (b+a \cos (e+f x))}\right ) \, dx\\ &=\frac{(c+d x)^2}{2 a^2 d}-\frac{(2 b) \int \frac{c+d x}{b+a \cos (e+f x)} \, dx}{a^2}+\frac{b^2 \int \frac{c+d x}{(b+a \cos (e+f x))^2} \, dx}{a^2}\\ &=\frac{(c+d x)^2}{2 a^2 d}+\frac{b^2 (c+d x) \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)}{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}{b+a \cos (e+f x)} \, dx}{a^2 \left (a^2-b^2\right )}-\frac{\left (b^2 d\right ) \int \frac{\sin (e+f x)}{b+a \cos (e+f x)} \, dx}{a \left (a^2-b^2\right ) f}\\ &=\frac{(c+d x)^2}{2 a^2 d}+\frac{b^2 (c+d x) \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)}{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 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 b+2 \sqrt{-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \sqrt{-a^2+b^2}}+\frac{\left (b^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{b+x} \, dx,x,a \cos (e+f x)\right )}{a^2 \left (a^2-b^2\right ) f^2}\\ &=\frac{(c+d x)^2}{2 a^2 d}+\frac{2 i b (c+d x) \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) \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 d \log (b+a \cos (e+f x))}{a^2 \left (a^2-b^2\right ) f^2}+\frac{b^2 (c+d x) \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 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 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{(2 i b d) \int \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{(2 i b d) \int \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{(c+d x)^2}{2 a^2 d}-\frac{i b^3 (c+d x) \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) \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) \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) \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 d \log (b+a \cos (e+f x))}{a^2 \left (a^2-b^2\right ) f^2}+\frac{b^2 (c+d x) \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}-\frac{(2 b d) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 a x}{2 b-2 \sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \sqrt{-a^2+b^2} f^2}+\frac{(2 b d) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 a x}{2 b+2 \sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \sqrt{-a^2+b^2} f^2}+\frac{\left (i b^3 d\right ) \int \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 (i b^3 d\right ) \int \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{(c+d x)^2}{2 a^2 d}-\frac{i b^3 (c+d x) \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) \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) \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) \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 d \log (b+a \cos (e+f x))}{a^2 \left (a^2-b^2\right ) f^2}+\frac{2 b d \text{Li}_2\left (-\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 d \text{Li}_2\left (-\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) \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac{\left (b^3 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 a x}{2 b-2 \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^2}-\frac{\left (b^3 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 a x}{2 b+2 \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^2}\\ &=\frac{(c+d x)^2}{2 a^2 d}-\frac{i b^3 (c+d x) \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) \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) \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) \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 d \log (b+a \cos (e+f x))}{a^2 \left (a^2-b^2\right ) f^2}-\frac{b^3 d \text{Li}_2\left (-\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{2 b d \text{Li}_2\left (-\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^3 d \text{Li}_2\left (-\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{2 b d \text{Li}_2\left (-\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) \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}\\ \end{align*}

Mathematica [A]  time = 9.79327, size = 1037, normalized size = 1.78 \[ \frac{(b+a \cos (e+f x)) \left (d e \sin (e+f x) b^2-c f \sin (e+f x) b^2-d (e+f x) \sin (e+f x) b^2\right ) \sec ^2(e+f x)}{a (b-a) (a+b) f^2 (a+b \sec (e+f x))^2}+\frac{b \cos ^2\left (\frac{1}{2} (e+f x)\right ) (b+a \cos (e+f x)) \left (-\frac{2 \left (2 a^2-b^2\right ) (d e-c f) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{-a-b}}\right )}{\sqrt{-a-b} \sqrt{a-b}}-b d \log \left (\sec ^2\left (\frac{1}{2} (e+f x)\right )\right )+b d \log \left (-(b+a \cos (e+f x)) \sec ^2\left (\frac{1}{2} (e+f x)\right )\right )-\frac{i \left (2 a^2-b^2\right ) d \left (\log \left (i \tan \left (\frac{1}{2} (e+f x)\right )+1\right ) \log \left (\frac{i \left (\sqrt{a+b}-\sqrt{a-b} \tan \left (\frac{1}{2} (e+f x)\right )\right )}{\sqrt{a-b}+i \sqrt{a+b}}\right )-\log \left (1-i \tan \left (\frac{1}{2} (e+f x)\right )\right ) \log \left (\frac{\sqrt{a+b}-\sqrt{a-b} \tan \left (\frac{1}{2} (e+f x)\right )}{i \sqrt{a-b}+\sqrt{a+b}}\right )+\log \left (1-i \tan \left (\frac{1}{2} (e+f x)\right )\right ) \log \left (\frac{i \left (\sqrt{a-b} \tan \left (\frac{1}{2} (e+f x)\right )+\sqrt{a+b}\right )}{\sqrt{a-b}+i \sqrt{a+b}}\right )-\log \left (i \tan \left (\frac{1}{2} (e+f x)\right )+1\right ) \log \left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (e+f x)\right )+\sqrt{a+b}}{i \sqrt{a-b}+\sqrt{a+b}}\right )-\text{PolyLog}\left (2,\frac{\sqrt{a-b} \left (1-i \tan \left (\frac{1}{2} (e+f x)\right )\right )}{\sqrt{a-b}-i \sqrt{a+b}}\right )+\text{PolyLog}\left (2,\frac{\sqrt{a-b} \left (1-i \tan \left (\frac{1}{2} (e+f x)\right )\right )}{\sqrt{a-b}+i \sqrt{a+b}}\right )-\text{PolyLog}\left (2,\frac{\sqrt{a-b} \left (i \tan \left (\frac{1}{2} (e+f x)\right )+1\right )}{\sqrt{a-b}-i \sqrt{a+b}}\right )+\text{PolyLog}\left (2,\frac{\sqrt{a-b} \left (i \tan \left (\frac{1}{2} (e+f x)\right )+1\right )}{\sqrt{a-b}+i \sqrt{a+b}}\right )\right )}{\sqrt{a-b} \sqrt{a+b}}\right ) \left (\left (2 a^2-b^2\right ) (c f+d x f)+a b d \sin (e+f x)\right ) \left (\sqrt{a+b}-\sqrt{a-b} \tan \left (\frac{1}{2} (e+f x)\right )\right ) \left (\sqrt{a-b} \tan \left (\frac{1}{2} (e+f x)\right )+\sqrt{a+b}\right ) \sec ^2(e+f x)}{a^2 \left (a^2-b^2\right ) f^2 (a+b \sec (e+f x))^2 \left (a b d \sin (e+f x)-\left (2 a^2-b^2\right ) \left (d e-c f-i d \log \left (1-i \tan \left (\frac{1}{2} (e+f x)\right )\right )+i d \log \left (i \tan \left (\frac{1}{2} (e+f x)\right )+1\right )\right )\right )}+\frac{(e+f x) (-2 d e+2 c f+d (e+f x)) (b+a \cos (e+f x))^2 \sec ^2(e+f x)}{2 a^2 f^2 (a+b \sec (e+f x))^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((e + f*x)*(-2*d*e + 2*c*f + d*(e + f*x))*(b + a*Cos[e + f*x])^2*Sec[e + f*x]^2)/(2*a^2*f^2*(a + b*Sec[e + f*x
])^2) + ((b + a*Cos[e + f*x])*Sec[e + f*x]^2*(b^2*d*e*Sin[e + f*x] - b^2*c*f*Sin[e + f*x] - b^2*d*(e + f*x)*Si
n[e + f*x]))/(a*(-a + b)*(a + b)*f^2*(a + b*Sec[e + f*x])^2) + (b*Cos[(e + f*x)/2]^2*(b + a*Cos[e + f*x])*((-2
*(2*a^2 - b^2)*(d*e - c*f)*ArcTan[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[-a - b]])/(Sqrt[-a - b]*Sqrt[a - b]) - b
*d*Log[Sec[(e + f*x)/2]^2] + b*d*Log[-((b + a*Cos[e + f*x])*Sec[(e + f*x)/2]^2)] - (I*(2*a^2 - b^2)*d*(Log[1 +
I*Tan[(e + f*x)/2]]*Log[(I*(Sqrt[a + b] - Sqrt[a - b]*Tan[(e + f*x)/2]))/(Sqrt[a - b] + I*Sqrt[a + b])] - Log
[1 - I*Tan[(e + f*x)/2]]*Log[(Sqrt[a + b] - Sqrt[a - b]*Tan[(e + f*x)/2])/(I*Sqrt[a - b] + Sqrt[a + b])] + Log
[1 - I*Tan[(e + f*x)/2]]*Log[(I*(Sqrt[a + b] + Sqrt[a - b]*Tan[(e + f*x)/2]))/(Sqrt[a - b] + I*Sqrt[a + b])] -
Log[1 + I*Tan[(e + f*x)/2]]*Log[(Sqrt[a + b] + Sqrt[a - b]*Tan[(e + f*x)/2])/(I*Sqrt[a - b] + Sqrt[a + b])] -
PolyLog[2, (Sqrt[a - b]*(1 - I*Tan[(e + f*x)/2]))/(Sqrt[a - b] - I*Sqrt[a + b])] + PolyLog[2, (Sqrt[a - b]*(1
- I*Tan[(e + f*x)/2]))/(Sqrt[a - b] + I*Sqrt[a + b])] - PolyLog[2, (Sqrt[a - b]*(1 + I*Tan[(e + f*x)/2]))/(Sq
rt[a - b] - I*Sqrt[a + b])] + PolyLog[2, (Sqrt[a - b]*(1 + I*Tan[(e + f*x)/2]))/(Sqrt[a - b] + I*Sqrt[a + b])]
))/(Sqrt[a - b]*Sqrt[a + b]))*Sec[e + f*x]^2*((2*a^2 - b^2)*(c*f + d*f*x) + a*b*d*Sin[e + f*x])*(Sqrt[a + b] -
Sqrt[a - b]*Tan[(e + f*x)/2])*(Sqrt[a + b] + Sqrt[a - b]*Tan[(e + f*x)/2]))/(a^2*(a^2 - b^2)*f^2*(a + b*Sec[e
+ f*x])^2*(-((2*a^2 - b^2)*(d*e - c*f - I*d*Log[1 - I*Tan[(e + f*x)/2]] + I*d*Log[1 + I*Tan[(e + f*x)/2]])) +
a*b*d*Sin[e + f*x]))

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Maple [B]  time = 0.194, size = 1289, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/a^2*d*x^2+1/a^2*c*x+2*I*b/(a^2-b^2)/f^2*d/(-a^2+b^2)^(1/2)*ln((-a*exp(I*(f*x+e))+(-a^2+b^2)^(1/2)-b)/(-b+(
-a^2+b^2)^(1/2)))*e-2*I*b^3/a^2/(a^2-b^2)^(3/2)/f*c*arctan(1/2*(2*a*exp(I*(f*x+e))+2*b)/(a^2-b^2)^(1/2))+b^2/a
^2/(a^2-b^2)/f^2*d*ln(exp(2*I*(f*x+e))*a+2*b*exp(I*(f*x+e))+a)-2*b^2/a^2/(a^2-b^2)/f^2*d*ln(exp(I*(f*x+e)))-I*
b^3/a^2/(a^2-b^2)/f*d/(-a^2+b^2)^(1/2)*ln((-a*exp(I*(f*x+e))+(-a^2+b^2)^(1/2)-b)/(-b+(-a^2+b^2)^(1/2)))*x+2*I*
b^3/a^2/(a^2-b^2)^(3/2)/f^2*d*e*arctan(1/2*(2*a*exp(I*(f*x+e))+2*b)/(a^2-b^2)^(1/2))+2*I*b^2*(d*x+c)*(b*exp(I*
(f*x+e))+a)/a^2/(a^2-b^2)/f/(exp(2*I*(f*x+e))*a+2*b*exp(I*(f*x+e))+a)+I*b^3/a^2/(a^2-b^2)/f^2*d/(-a^2+b^2)^(1/
2)*ln((a*exp(I*(f*x+e))+(-a^2+b^2)^(1/2)+b)/(b+(-a^2+b^2)^(1/2)))*e-2*I*b/(a^2-b^2)/f^2*d/(-a^2+b^2)^(1/2)*ln(
(a*exp(I*(f*x+e))+(-a^2+b^2)^(1/2)+b)/(b+(-a^2+b^2)^(1/2)))*e+I*b^3/a^2/(a^2-b^2)/f*d/(-a^2+b^2)^(1/2)*ln((a*e
xp(I*(f*x+e))+(-a^2+b^2)^(1/2)+b)/(b+(-a^2+b^2)^(1/2)))*x+2*I*b/(a^2-b^2)/f*d/(-a^2+b^2)^(1/2)*ln((-a*exp(I*(f
*x+e))+(-a^2+b^2)^(1/2)-b)/(-b+(-a^2+b^2)^(1/2)))*x+2*b/(a^2-b^2)/f^2*d/(-a^2+b^2)^(1/2)*dilog((-a*exp(I*(f*x+
e))+(-a^2+b^2)^(1/2)-b)/(-b+(-a^2+b^2)^(1/2)))-2*b/(a^2-b^2)/f^2*d/(-a^2+b^2)^(1/2)*dilog((a*exp(I*(f*x+e))+(-
a^2+b^2)^(1/2)+b)/(b+(-a^2+b^2)^(1/2)))-2*I*b/(a^2-b^2)/f*d/(-a^2+b^2)^(1/2)*ln((a*exp(I*(f*x+e))+(-a^2+b^2)^(
1/2)+b)/(b+(-a^2+b^2)^(1/2)))*x-4*I*b/(a^2-b^2)^(3/2)/f^2*d*e*arctan(1/2*(2*a*exp(I*(f*x+e))+2*b)/(a^2-b^2)^(1
/2))+4*I*b/(a^2-b^2)^(3/2)/f*c*arctan(1/2*(2*a*exp(I*(f*x+e))+2*b)/(a^2-b^2)^(1/2))-I*b^3/a^2/(a^2-b^2)/f^2*d/
(-a^2+b^2)^(1/2)*ln((-a*exp(I*(f*x+e))+(-a^2+b^2)^(1/2)-b)/(-b+(-a^2+b^2)^(1/2)))*e-b^3/a^2/(a^2-b^2)/f^2*d/(-
a^2+b^2)^(1/2)*dilog((-a*exp(I*(f*x+e))+(-a^2+b^2)^(1/2)-b)/(-b+(-a^2+b^2)^(1/2)))+b^3/a^2/(a^2-b^2)/f^2*d/(-a
^2+b^2)^(1/2)*dilog((a*exp(I*(f*x+e))+(-a^2+b^2)^(1/2)+b)/(b+(-a^2+b^2)^(1/2)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.35607, size = 4639, normalized size = 7.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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