\(\int \frac {c+d x}{(a+b \sec (e+f x))^2} \, dx\) [41]

Optimal result

Integrand size = 18, antiderivative size = 582 \[ \int \frac {c+d x}{(a+b \sec (e+f x))^2} \, dx=\frac {(c+d x)^2}{2 a^2 d}-\frac {i b^3 (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac {2 i b (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {i b^3 (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac {2 i b (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {b^2 d \log (b+a \cos (e+f x))}{a^2 \left (a^2-b^2\right ) f^2}-\frac {b^3 d \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}+\frac {2 b d \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {b^3 d \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}-\frac {2 b d \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {b^2 (c+d x) \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))} \]

[Out]

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

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4276, 3405, 3402, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {c+d x}{(a+b \sec (e+f x))^2} \, dx=\frac {2 i b (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right )}{a^2 f \sqrt {b^2-a^2}}-\frac {2 i b (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{\sqrt {b^2-a^2}+b}\right )}{a^2 f \sqrt {b^2-a^2}}+\frac {b^2 (c+d x) \sin (e+f x)}{a f \left (a^2-b^2\right ) (a \cos (e+f x)+b)}+\frac {2 b d \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right )}{a^2 f^2 \sqrt {b^2-a^2}}-\frac {2 b d \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right )}{a^2 f^2 \sqrt {b^2-a^2}}+\frac {b^2 d \log (a \cos (e+f x)+b)}{a^2 f^2 \left (a^2-b^2\right )}-\frac {i b^3 (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right )}{a^2 f \left (b^2-a^2\right )^{3/2}}+\frac {i b^3 (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{\sqrt {b^2-a^2}+b}\right )}{a^2 f \left (b^2-a^2\right )^{3/2}}-\frac {b^3 d \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right )}{a^2 f^2 \left (b^2-a^2\right )^{3/2}}+\frac {b^3 d \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right )}{a^2 f^2 \left (b^2-a^2\right )^{3/2}}+\frac {(c+d x)^2}{2 a^2 d} \]

[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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, 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 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 2747

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 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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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) \text {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) \text {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 \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 d \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) \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {\left (b^3 d\right ) \text {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 ) \text {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 \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 {2 b d \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^3 d \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 {2 b d \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) \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 11.64 (sec) , antiderivative size = 1037, normalized size of antiderivative = 1.78 \[ \int \frac {c+d x}{(a+b \sec (e+f x))^2} \, dx=\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}+\frac {(b+a \cos (e+f x)) \sec ^2(e+f x) \left (b^2 d e \sin (e+f x)-b^2 c f \sin (e+f x)-b^2 d (e+f x) \sin (e+f x)\right )}{a (-a+b) (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) \arctan \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 (-\left ((b+a \cos (e+f x)) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )-\frac {i \left (2 a^2-b^2\right ) d \left (\log \left (1+i \tan \left (\frac {1}{2} (e+f x)\right )\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}+\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 )-\operatorname {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 )+\operatorname {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 )-\operatorname {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 )+\operatorname {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 )\right )}{\sqrt {a-b} \sqrt {a+b}}\right ) \sec ^2(e+f x) \left (\left (2 a^2-b^2\right ) (c f+d f x)+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}+\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )\right )}{a^2 \left (a^2-b^2\right ) f^2 (a+b \sec (e+f x))^2 \left (-\left (\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 (1+i \tan \left (\frac {1}{2} (e+f x)\right )\right )\right )\right )+a b d \sin (e+f x)\right )} \]

[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]))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1288 vs. \(2 (528 ) = 1056\).

Time = 0.64 (sec) , antiderivative size = 1289, normalized size of antiderivative = 2.21

method	result	size
risch	\(\text {Expression too large to display}\)	\(1289\)

[In]

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

[Out]

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

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. 2080 vs. \(2 (520) = 1040\).

Time = 0.49 (sec) , antiderivative size = 2080, normalized size of antiderivative = 3.57 \[ \int \frac {c+d x}{(a+b \sec (e+f x))^2} \, dx=\text {Too large to display} \]

[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(-(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) + ((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(-(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) - ((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(-(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) + ((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)*di
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 +
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((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) + (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((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) + (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((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) + (-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((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) + ((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)

Sympy [F]

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

[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)

Maxima [F(-2)]

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

[In]

integrate((d*x+c)/(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]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}