\(\int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx\) [106]

Optimal result

Integrand size = 31, antiderivative size = 187 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\frac {(13 A-6 B) x}{2 a^3}-\frac {8 (19 A-9 B) \sin (c+d x)}{15 a^3 d}+\frac {(13 A-6 B) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(11 A-6 B) \cos (c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {4 (19 A-9 B) \cos (c+d x) \sin (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )} \]

[Out]

1/2*(13*A-6*B)*x/a^3-8/15*(19*A-9*B)*sin(d*x+c)/a^3/d+1/2*(13*A-6*B)*cos(d*x+c)*sin(d*x+c)/a^3/d-1/5*(A-B)*cos
(d*x+c)*sin(d*x+c)/d/(a+a*sec(d*x+c))^3-1/15*(11*A-6*B)*cos(d*x+c)*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^2-4/15*(19*
A-9*B)*cos(d*x+c)*sin(d*x+c)/d/(a^3+a^3*sec(d*x+c))

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {4105, 3872, 2715, 8, 2717} \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=-\frac {8 (19 A-9 B) \sin (c+d x)}{15 a^3 d}+\frac {(13 A-6 B) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac {4 (19 A-9 B) \sin (c+d x) \cos (c+d x)}{15 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac {x (13 A-6 B)}{2 a^3}-\frac {(11 A-6 B) \sin (c+d x) \cos (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac {(A-B) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]

[In]

Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]

[Out]

((13*A - 6*B)*x)/(2*a^3) - (8*(19*A - 9*B)*Sin[c + d*x])/(15*a^3*d) + ((13*A - 6*B)*Cos[c + d*x]*Sin[c + d*x])
/(2*a^3*d) - ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((11*A - 6*B)*Cos[c + d*x]*Sin
[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (4*(19*A - 9*B)*Cos[c + d*x]*Sin[c + d*x])/(15*d*(a^3 + a^3*Sec[c
+ d*x]))

Rule 8

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

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3872

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 4105

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[(-(A*b - a*B))*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(b*f*(
2*m + 1))), x] - Dist[1/(a^2*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[b*B*n - a*A*
(2*m + n + 1) + (A*b - a*B)*(m + n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[
A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)] &&  !GtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B) \cos (c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {\int \frac {\cos ^2(c+d x) (a (7 A-2 B)-4 a (A-B) \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2} \\ & = -\frac {(A-B) \cos (c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(11 A-6 B) \cos (c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac {\int \frac {\cos ^2(c+d x) \left (a^2 (43 A-18 B)-3 a^2 (11 A-6 B) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4} \\ & = -\frac {(A-B) \cos (c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(11 A-6 B) \cos (c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {4 (19 A-9 B) \cos (c+d x) \sin (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\int \cos ^2(c+d x) \left (15 a^3 (13 A-6 B)-8 a^3 (19 A-9 B) \sec (c+d x)\right ) \, dx}{15 a^6} \\ & = -\frac {(A-B) \cos (c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(11 A-6 B) \cos (c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {4 (19 A-9 B) \cos (c+d x) \sin (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac {(8 (19 A-9 B)) \int \cos (c+d x) \, dx}{15 a^3}+\frac {(13 A-6 B) \int \cos ^2(c+d x) \, dx}{a^3} \\ & = -\frac {8 (19 A-9 B) \sin (c+d x)}{15 a^3 d}+\frac {(13 A-6 B) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(11 A-6 B) \cos (c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {4 (19 A-9 B) \cos (c+d x) \sin (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {(13 A-6 B) \int 1 \, dx}{2 a^3} \\ & = \frac {(13 A-6 B) x}{2 a^3}-\frac {8 (19 A-9 B) \sin (c+d x)}{15 a^3 d}+\frac {(13 A-6 B) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(11 A-6 B) \cos (c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {4 (19 A-9 B) \cos (c+d x) \sin (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(435\) vs. \(2(187)=374\).

Time = 2.81 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.33 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (600 (13 A-6 B) d x \cos \left (\frac {d x}{2}\right )+600 (13 A-6 B) d x \cos \left (c+\frac {d x}{2}\right )+3900 A d x \cos \left (c+\frac {3 d x}{2}\right )-1800 B d x \cos \left (c+\frac {3 d x}{2}\right )+3900 A d x \cos \left (2 c+\frac {3 d x}{2}\right )-1800 B d x \cos \left (2 c+\frac {3 d x}{2}\right )+780 A d x \cos \left (2 c+\frac {5 d x}{2}\right )-360 B d x \cos \left (2 c+\frac {5 d x}{2}\right )+780 A d x \cos \left (3 c+\frac {5 d x}{2}\right )-360 B d x \cos \left (3 c+\frac {5 d x}{2}\right )-12760 A \sin \left (\frac {d x}{2}\right )+7020 B \sin \left (\frac {d x}{2}\right )+7560 A \sin \left (c+\frac {d x}{2}\right )-4500 B \sin \left (c+\frac {d x}{2}\right )-9230 A \sin \left (c+\frac {3 d x}{2}\right )+4860 B \sin \left (c+\frac {3 d x}{2}\right )+930 A \sin \left (2 c+\frac {3 d x}{2}\right )-900 B \sin \left (2 c+\frac {3 d x}{2}\right )-2782 A \sin \left (2 c+\frac {5 d x}{2}\right )+1452 B \sin \left (2 c+\frac {5 d x}{2}\right )-750 A \sin \left (3 c+\frac {5 d x}{2}\right )+300 B \sin \left (3 c+\frac {5 d x}{2}\right )-105 A \sin \left (3 c+\frac {7 d x}{2}\right )+60 B \sin \left (3 c+\frac {7 d x}{2}\right )-105 A \sin \left (4 c+\frac {7 d x}{2}\right )+60 B \sin \left (4 c+\frac {7 d x}{2}\right )+15 A \sin \left (4 c+\frac {9 d x}{2}\right )+15 A \sin \left (5 c+\frac {9 d x}{2}\right )\right )}{480 a^3 d (1+\cos (c+d x))^3} \]

[In]

Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]

[Out]

(Cos[(c + d*x)/2]*Sec[c/2]*(600*(13*A - 6*B)*d*x*Cos[(d*x)/2] + 600*(13*A - 6*B)*d*x*Cos[c + (d*x)/2] + 3900*A
*d*x*Cos[c + (3*d*x)/2] - 1800*B*d*x*Cos[c + (3*d*x)/2] + 3900*A*d*x*Cos[2*c + (3*d*x)/2] - 1800*B*d*x*Cos[2*c
+ (3*d*x)/2] + 780*A*d*x*Cos[2*c + (5*d*x)/2] - 360*B*d*x*Cos[2*c + (5*d*x)/2] + 780*A*d*x*Cos[3*c + (5*d*x)/
2] - 360*B*d*x*Cos[3*c + (5*d*x)/2] - 12760*A*Sin[(d*x)/2] + 7020*B*Sin[(d*x)/2] + 7560*A*Sin[c + (d*x)/2] - 4
500*B*Sin[c + (d*x)/2] - 9230*A*Sin[c + (3*d*x)/2] + 4860*B*Sin[c + (3*d*x)/2] + 930*A*Sin[2*c + (3*d*x)/2] -
900*B*Sin[2*c + (3*d*x)/2] - 2782*A*Sin[2*c + (5*d*x)/2] + 1452*B*Sin[2*c + (5*d*x)/2] - 750*A*Sin[3*c + (5*d*
x)/2] + 300*B*Sin[3*c + (5*d*x)/2] - 105*A*Sin[3*c + (7*d*x)/2] + 60*B*Sin[3*c + (7*d*x)/2] - 105*A*Sin[4*c +
(7*d*x)/2] + 60*B*Sin[4*c + (7*d*x)/2] + 15*A*Sin[4*c + (9*d*x)/2] + 15*A*Sin[5*c + (9*d*x)/2]))/(480*a^3*d*(1
+ Cos[c + d*x])^3)

Maple [A] (verified)

Time = 0.99 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.57

method	result	size
parallelrisch	\(\frac {15 \left (\left (-\frac {1856 A}{15}+\frac {312 B}{5}\right ) \cos \left (2 d x +2 c \right )+\left (-6 A +4 B \right ) \cos \left (3 d x +3 c \right )+A \cos \left (4 d x +4 c \right )+\left (-\frac {2002 A}{5}+\frac {972 B}{5}\right ) \cos \left (d x +c \right )-\frac {4303 A}{15}+\frac {696 B}{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+6240 d \left (A -\frac {6 B}{13}\right ) x}{960 a^{3} d}\)	\(107\)
derivativedivides	\(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B -31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\frac {16 \left (-\frac {7 A}{4}+\frac {B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+16 \left (-\frac {5 A}{4}+\frac {B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+4 \left (13 A -6 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\)	\(163\)
default	\(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B -31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\frac {16 \left (-\frac {7 A}{4}+\frac {B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+16 \left (-\frac {5 A}{4}+\frac {B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+4 \left (13 A -6 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\)	\(163\)
norman	\(\frac {\frac {\left (13 A -6 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}+\frac {\left (13 A -6 B \right ) x}{2 a}-\frac {\left (A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{20 a d}+\frac {\left (13 A -6 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a}+\frac {\left (17 A -12 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{30 a d}-\frac {\left (51 A -25 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}-\frac {\left (131 A -60 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 a d}-\frac {\left (194 A -99 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{30 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} a^{2}}\)	\(207\)
risch	\(\frac {13 A x}{2 a^{3}}-\frac {3 x B}{a^{3}}-\frac {i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {3 i A \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{3} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} B}{2 a^{3} d}-\frac {3 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 a^{3} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} B}{2 a^{3} d}+\frac {i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{3} d}-\frac {2 i \left (150 A \,{\mathrm e}^{4 i \left (d x +c \right )}-90 B \,{\mathrm e}^{4 i \left (d x +c \right )}+525 A \,{\mathrm e}^{3 i \left (d x +c \right )}-300 B \,{\mathrm e}^{3 i \left (d x +c \right )}+745 A \,{\mathrm e}^{2 i \left (d x +c \right )}-420 B \,{\mathrm e}^{2 i \left (d x +c \right )}+485 \,{\mathrm e}^{i \left (d x +c \right )} A -270 B \,{\mathrm e}^{i \left (d x +c \right )}+127 A -72 B \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\)	\(255\)

[In]

int(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/960*(15*((-1856/15*A+312/5*B)*cos(2*d*x+2*c)+(-6*A+4*B)*cos(3*d*x+3*c)+A*cos(4*d*x+4*c)+(-2002/5*A+972/5*B)*
cos(d*x+c)-4303/15*A+696/5*B)*tan(1/2*d*x+1/2*c)*sec(1/2*d*x+1/2*c)^4+6240*d*(A-6/13*B)*x)/a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.02 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\frac {15 \, {\left (13 \, A - 6 \, B\right )} d x \cos \left (d x + c\right )^{3} + 45 \, {\left (13 \, A - 6 \, B\right )} d x \cos \left (d x + c\right )^{2} + 45 \, {\left (13 \, A - 6 \, B\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (13 \, A - 6 \, B\right )} d x + {\left (15 \, A \cos \left (d x + c\right )^{4} - 15 \, {\left (3 \, A - 2 \, B\right )} \cos \left (d x + c\right )^{3} - {\left (479 \, A - 234 \, B\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (239 \, A - 114 \, B\right )} \cos \left (d x + c\right ) - 304 \, A + 144 \, B\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]

[In]

integrate(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/30*(15*(13*A - 6*B)*d*x*cos(d*x + c)^3 + 45*(13*A - 6*B)*d*x*cos(d*x + c)^2 + 45*(13*A - 6*B)*d*x*cos(d*x +
c) + 15*(13*A - 6*B)*d*x + (15*A*cos(d*x + c)^4 - 15*(3*A - 2*B)*cos(d*x + c)^3 - (479*A - 234*B)*cos(d*x + c)
^2 - 3*(239*A - 114*B)*cos(d*x + c) - 304*A + 144*B)*sin(d*x + c))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c
)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)

Sympy [F]

\[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {A \cos ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]

[In]

integrate(cos(d*x+c)**2*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))**3,x)

[Out]

(Integral(A*cos(c + d*x)**2/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x) + Integral(B*cos(c
+ d*x)**2*sec(c + d*x)/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x))/a**3

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.72 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=-\frac {A {\left (\frac {60 \, {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{3} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {465 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {40 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {780 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} - 3 \, B {\left (\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{60 \, d} \]

[In]

integrate(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/60*(A*(60*(5*sin(d*x + c)/(cos(d*x + c) + 1) + 7*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^3 + 2*a^3*sin(d*x
+ c)^2/(cos(d*x + c) + 1)^2 + a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (465*sin(d*x + c)/(cos(d*x + c) + 1)
- 40*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 - 780*arctan(sin(d*x + c
)/(cos(d*x + c) + 1))/a^3) - 3*B*(40*sin(d*x + c)/((a^3 + a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x +
c) + 1)) + (85*sin(d*x + c)/(cos(d*x + c) + 1) - 10*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + sin(d*x + c)^5/(cos(
d*x + c) + 1)^5)/a^3 - 120*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3))/d

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.07 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {30 \, {\left (d x + c\right )} {\left (13 \, A - 6 \, B\right )}}{a^{3}} - \frac {60 \, {\left (7 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 465 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 255 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]

[In]

integrate(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/60*(30*(d*x + c)*(13*A - 6*B)/a^3 - 60*(7*A*tan(1/2*d*x + 1/2*c)^3 - 2*B*tan(1/2*d*x + 1/2*c)^3 + 5*A*tan(1/
2*d*x + 1/2*c) - 2*B*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^3) - (3*A*a^12*tan(1/2*d*x + 1/2*
c)^5 - 3*B*a^12*tan(1/2*d*x + 1/2*c)^5 - 40*A*a^12*tan(1/2*d*x + 1/2*c)^3 + 30*B*a^12*tan(1/2*d*x + 1/2*c)^3 +
465*A*a^12*tan(1/2*d*x + 1/2*c) - 255*B*a^12*tan(1/2*d*x + 1/2*c))/a^15)/d

Mupad [B] (verification not implemented)

Time = 14.10 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\frac {x\,\left (13\,A-6\,B\right )}{2\,a^3}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (A-B\right )}{2\,a^3}+\frac {3\,\left (5\,A-3\,B\right )}{4\,a^3}+\frac {10\,A-2\,B}{4\,a^3}\right )}{d}-\frac {\left (7\,A-2\,B\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (5\,A-2\,B\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-B}{4\,a^3}+\frac {5\,A-3\,B}{12\,a^3}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A-B\right )}{20\,a^3\,d} \]

[In]

int((cos(c + d*x)^2*(A + B/cos(c + d*x)))/(a + a/cos(c + d*x))^3,x)

[Out]

(x*(13*A - 6*B))/(2*a^3) - (tan(c/2 + (d*x)/2)*((3*(A - B))/(2*a^3) + (3*(5*A - 3*B))/(4*a^3) + (10*A - 2*B)/(
4*a^3)))/d - (tan(c/2 + (d*x)/2)^3*(7*A - 2*B) + tan(c/2 + (d*x)/2)*(5*A - 2*B))/(d*(2*a^3*tan(c/2 + (d*x)/2)^
2 + a^3*tan(c/2 + (d*x)/2)^4 + a^3)) + (tan(c/2 + (d*x)/2)^3*((A - B)/(4*a^3) + (5*A - 3*B)/(12*a^3)))/d - (ta
n(c/2 + (d*x)/2)^5*(A - B))/(20*a^3*d)