\(\int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx\) [116]

Optimal result

Integrand size = 31, antiderivative size = 223 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {(21 A-8 B) x}{2 a^4}-\frac {8 (216 A-83 B) \sin (c+d x)}{105 a^4 d}+\frac {(21 A-8 B) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {(129 A-52 B) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {4 (216 A-83 B) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))}-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(2 A-B) \cos (c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^3} \]

[Out]

1/2*(21*A-8*B)*x/a^4-8/105*(216*A-83*B)*sin(d*x+c)/a^4/d+1/2*(21*A-8*B)*cos(d*x+c)*sin(d*x+c)/a^4/d-1/105*(129
*A-52*B)*cos(d*x+c)*sin(d*x+c)/a^4/d/(1+sec(d*x+c))^2-4/105*(216*A-83*B)*cos(d*x+c)*sin(d*x+c)/a^4/d/(1+sec(d*
x+c))-1/7*(A-B)*cos(d*x+c)*sin(d*x+c)/d/(a+a*sec(d*x+c))^4-1/5*(2*A-B)*cos(d*x+c)*sin(d*x+c)/a/d/(a+a*sec(d*x+
c))^3

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {4105, 3872, 2715, 8, 2717} \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {8 (216 A-83 B) \sin (c+d x)}{105 a^4 d}+\frac {(21 A-8 B) \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac {4 (216 A-83 B) \sin (c+d x) \cos (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac {(129 A-52 B) \sin (c+d x) \cos (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac {x (21 A-8 B)}{2 a^4}-\frac {(2 A-B) \sin (c+d x) \cos (c+d x)}{5 a d (a \sec (c+d x)+a)^3}-\frac {(A-B) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]

[In]

Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]

[Out]

((21*A - 8*B)*x)/(2*a^4) - (8*(216*A - 83*B)*Sin[c + d*x])/(105*a^4*d) + ((21*A - 8*B)*Cos[c + d*x]*Sin[c + d*
x])/(2*a^4*d) - ((129*A - 52*B)*Cos[c + d*x]*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (4*(216*A - 83*B
)*Cos[c + d*x]*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A - B)*Cos[c + d*x]*Sin[c + d*x])/(7*d*(a + a*
Sec[c + d*x])^4) - ((2*A - B)*Cos[c + d*x]*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)

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)}{7 d (a+a \sec (c+d x))^4}+\frac {\int \frac {\cos ^2(c+d x) (a (9 A-2 B)-5 a (A-B) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2} \\ & = -\frac {(A-B) \cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(2 A-B) \cos (c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\cos ^2(c+d x) \left (a^2 (73 A-24 B)-28 a^2 (2 A-B) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4} \\ & = -\frac {(129 A-52 B) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(2 A-B) \cos (c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\cos ^2(c+d x) \left (a^3 (477 A-176 B)-3 a^3 (129 A-52 B) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6} \\ & = -\frac {(129 A-52 B) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(2 A-B) \cos (c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {4 (216 A-83 B) \cos (c+d x) \sin (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {\int \cos ^2(c+d x) \left (105 a^4 (21 A-8 B)-8 a^4 (216 A-83 B) \sec (c+d x)\right ) \, dx}{105 a^8} \\ & = -\frac {(129 A-52 B) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(2 A-B) \cos (c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {4 (216 A-83 B) \cos (c+d x) \sin (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {(8 (216 A-83 B)) \int \cos (c+d x) \, dx}{105 a^4}+\frac {(21 A-8 B) \int \cos ^2(c+d x) \, dx}{a^4} \\ & = -\frac {8 (216 A-83 B) \sin (c+d x)}{105 a^4 d}+\frac {(21 A-8 B) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {(129 A-52 B) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(2 A-B) \cos (c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {4 (216 A-83 B) \cos (c+d x) \sin (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {(21 A-8 B) \int 1 \, dx}{2 a^4} \\ & = \frac {(21 A-8 B) x}{2 a^4}-\frac {8 (216 A-83 B) \sin (c+d x)}{105 a^4 d}+\frac {(21 A-8 B) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {(129 A-52 B) \cos (c+d x) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(2 A-B) \cos (c+d x) \sin (c+d x)}{5 a d (a+a \sec (c+d x))^3}-\frac {4 (216 A-83 B) \cos (c+d x) \sin (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(555\) vs. \(2(223)=446\).

Time = 5.98 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.49 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (14700 (21 A-8 B) d x \cos \left (\frac {d x}{2}\right )+14700 (21 A-8 B) d x \cos \left (c+\frac {d x}{2}\right )+185220 A d x \cos \left (c+\frac {3 d x}{2}\right )-70560 B d x \cos \left (c+\frac {3 d x}{2}\right )+185220 A d x \cos \left (2 c+\frac {3 d x}{2}\right )-70560 B d x \cos \left (2 c+\frac {3 d x}{2}\right )+61740 A d x \cos \left (2 c+\frac {5 d x}{2}\right )-23520 B d x \cos \left (2 c+\frac {5 d x}{2}\right )+61740 A d x \cos \left (3 c+\frac {5 d x}{2}\right )-23520 B d x \cos \left (3 c+\frac {5 d x}{2}\right )+8820 A d x \cos \left (3 c+\frac {7 d x}{2}\right )-3360 B d x \cos \left (3 c+\frac {7 d x}{2}\right )+8820 A d x \cos \left (4 c+\frac {7 d x}{2}\right )-3360 B d x \cos \left (4 c+\frac {7 d x}{2}\right )-539490 A \sin \left (\frac {d x}{2}\right )+243320 B \sin \left (\frac {d x}{2}\right )+386190 A \sin \left (c+\frac {d x}{2}\right )-184520 B \sin \left (c+\frac {d x}{2}\right )-422478 A \sin \left (c+\frac {3 d x}{2}\right )+184464 B \sin \left (c+\frac {3 d x}{2}\right )+132930 A \sin \left (2 c+\frac {3 d x}{2}\right )-72240 B \sin \left (2 c+\frac {3 d x}{2}\right )-181461 A \sin \left (2 c+\frac {5 d x}{2}\right )+77168 B \sin \left (2 c+\frac {5 d x}{2}\right )+3675 A \sin \left (3 c+\frac {5 d x}{2}\right )-8400 B \sin \left (3 c+\frac {5 d x}{2}\right )-36003 A \sin \left (3 c+\frac {7 d x}{2}\right )+15164 B \sin \left (3 c+\frac {7 d x}{2}\right )-9555 A \sin \left (4 c+\frac {7 d x}{2}\right )+2940 B \sin \left (4 c+\frac {7 d x}{2}\right )-945 A \sin \left (4 c+\frac {9 d x}{2}\right )+420 B \sin \left (4 c+\frac {9 d x}{2}\right )-945 A \sin \left (5 c+\frac {9 d x}{2}\right )+420 B \sin \left (5 c+\frac {9 d x}{2}\right )+105 A \sin \left (5 c+\frac {11 d x}{2}\right )+105 A \sin \left (6 c+\frac {11 d x}{2}\right )\right )}{6720 a^4 d (1+\cos (c+d x))^4} \]

[In]

Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]

[Out]

(Cos[(c + d*x)/2]*Sec[c/2]*(14700*(21*A - 8*B)*d*x*Cos[(d*x)/2] + 14700*(21*A - 8*B)*d*x*Cos[c + (d*x)/2] + 18
5220*A*d*x*Cos[c + (3*d*x)/2] - 70560*B*d*x*Cos[c + (3*d*x)/2] + 185220*A*d*x*Cos[2*c + (3*d*x)/2] - 70560*B*d
*x*Cos[2*c + (3*d*x)/2] + 61740*A*d*x*Cos[2*c + (5*d*x)/2] - 23520*B*d*x*Cos[2*c + (5*d*x)/2] + 61740*A*d*x*Co
s[3*c + (5*d*x)/2] - 23520*B*d*x*Cos[3*c + (5*d*x)/2] + 8820*A*d*x*Cos[3*c + (7*d*x)/2] - 3360*B*d*x*Cos[3*c +
(7*d*x)/2] + 8820*A*d*x*Cos[4*c + (7*d*x)/2] - 3360*B*d*x*Cos[4*c + (7*d*x)/2] - 539490*A*Sin[(d*x)/2] + 2433
20*B*Sin[(d*x)/2] + 386190*A*Sin[c + (d*x)/2] - 184520*B*Sin[c + (d*x)/2] - 422478*A*Sin[c + (3*d*x)/2] + 1844
64*B*Sin[c + (3*d*x)/2] + 132930*A*Sin[2*c + (3*d*x)/2] - 72240*B*Sin[2*c + (3*d*x)/2] - 181461*A*Sin[2*c + (5
*d*x)/2] + 77168*B*Sin[2*c + (5*d*x)/2] + 3675*A*Sin[3*c + (5*d*x)/2] - 8400*B*Sin[3*c + (5*d*x)/2] - 36003*A*
Sin[3*c + (7*d*x)/2] + 15164*B*Sin[3*c + (7*d*x)/2] - 9555*A*Sin[4*c + (7*d*x)/2] + 2940*B*Sin[4*c + (7*d*x)/2
] - 945*A*Sin[4*c + (9*d*x)/2] + 420*B*Sin[4*c + (9*d*x)/2] - 945*A*Sin[5*c + (9*d*x)/2] + 420*B*Sin[5*c + (9*
d*x)/2] + 105*A*Sin[5*c + (11*d*x)/2] + 105*A*Sin[6*c + (11*d*x)/2]))/(6720*a^4*d*(1 + Cos[c + d*x])^4)

Maple [A] (verified)

Time = 1.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.55

method	result	size
parallelrisch	\(\frac {-840 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (\frac {4688 A}{35}-\frac {5482 B}{105}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {7873 A}{280}-\frac {1184 B}{105}\right ) \cos \left (3 d x +3 c \right )+\left (A -\frac {B}{2}\right ) \cos \left (4 d x +4 c \right )-\frac {A \cos \left (5 d x +5 c \right )}{8}+\left (\frac {42881 A}{140}-\frac {12496 B}{105}\right ) \cos \left (d x +c \right )+\frac {6957 A}{35}-\frac {16171 B}{210}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+282240 d \left (A -\frac {8 B}{21}\right ) x}{26880 a^{4} d}\)	\(123\)
derivativedivides	\(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A -\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\frac {16 \left (-\frac {9 A}{2}+B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+16 \left (-\frac {7 A}{2}+B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+8 \left (21 A -8 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\)	\(187\)
default	\(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A -\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-111 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\frac {16 \left (-\frac {9 A}{2}+B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+16 \left (-\frac {7 A}{2}+B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+8 \left (21 A -8 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\)	\(187\)
norman	\(\frac {\frac {\left (21 A -8 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}+\frac {\left (21 A -8 B \right ) x}{2 a}+\frac {\left (A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{56 a d}+\frac {\left (21 A -8 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a}-\frac {\left (53 A -39 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{280 a d}-\frac {\left (167 A -65 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\left (501 A -263 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{420 a d}-\frac {\left (651 A -263 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{60 a d}-\frac {\left (843 A -319 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} a^{3}}\)	\(233\)
risch	\(\frac {21 A x}{2 a^{4}}-\frac {4 x B}{a^{4}}-\frac {i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{4} d}+\frac {2 i A \,{\mathrm e}^{i \left (d x +c \right )}}{a^{4} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} B}{2 a^{4} d}-\frac {2 i A \,{\mathrm e}^{-i \left (d x +c \right )}}{a^{4} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} B}{2 a^{4} d}+\frac {i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{4} d}-\frac {2 i \left (2100 A \,{\mathrm e}^{6 i \left (d x +c \right )}-1050 B \,{\mathrm e}^{6 i \left (d x +c \right )}+11025 A \,{\mathrm e}^{5 i \left (d x +c \right )}-5250 B \,{\mathrm e}^{5 i \left (d x +c \right )}+25515 A \,{\mathrm e}^{4 i \left (d x +c \right )}-11900 B \,{\mathrm e}^{4 i \left (d x +c \right )}+32340 A \,{\mathrm e}^{3 i \left (d x +c \right )}-14840 B \,{\mathrm e}^{3 i \left (d x +c \right )}+23688 A \,{\mathrm e}^{2 i \left (d x +c \right )}-10794 B \,{\mathrm e}^{2 i \left (d x +c \right )}+9471 \,{\mathrm e}^{i \left (d x +c \right )} A -4298 B \,{\mathrm e}^{i \left (d x +c \right )}+1653 A -764 B \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\)	\(303\)

[In]

int(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/26880*(-840*tan(1/2*d*x+1/2*c)*((4688/35*A-5482/105*B)*cos(2*d*x+2*c)+(7873/280*A-1184/105*B)*cos(3*d*x+3*c)
+(A-1/2*B)*cos(4*d*x+4*c)-1/8*A*cos(5*d*x+5*c)+(42881/140*A-12496/105*B)*cos(d*x+c)+6957/35*A-16171/210*B)*sec
(1/2*d*x+1/2*c)^6+282240*d*(A-8/21*B)*x)/a^4/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.08 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {105 \, {\left (21 \, A - 8 \, B\right )} d x \cos \left (d x + c\right )^{4} + 420 \, {\left (21 \, A - 8 \, B\right )} d x \cos \left (d x + c\right )^{3} + 630 \, {\left (21 \, A - 8 \, B\right )} d x \cos \left (d x + c\right )^{2} + 420 \, {\left (21 \, A - 8 \, B\right )} d x \cos \left (d x + c\right ) + 105 \, {\left (21 \, A - 8 \, B\right )} d x + {\left (105 \, A \cos \left (d x + c\right )^{5} - 210 \, {\left (2 \, A - B\right )} \cos \left (d x + c\right )^{4} - 4 \, {\left (1509 \, A - 592 \, B\right )} \cos \left (d x + c\right )^{3} - 4 \, {\left (3411 \, A - 1318 \, B\right )} \cos \left (d x + c\right )^{2} - {\left (11619 \, A - 4472 \, B\right )} \cos \left (d x + c\right ) - 3456 \, A + 1328 \, B\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]

[In]

integrate(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x, algorithm="fricas")

[Out]

1/210*(105*(21*A - 8*B)*d*x*cos(d*x + c)^4 + 420*(21*A - 8*B)*d*x*cos(d*x + c)^3 + 630*(21*A - 8*B)*d*x*cos(d*
x + c)^2 + 420*(21*A - 8*B)*d*x*cos(d*x + c) + 105*(21*A - 8*B)*d*x + (105*A*cos(d*x + c)^5 - 210*(2*A - B)*co
s(d*x + c)^4 - 4*(1509*A - 592*B)*cos(d*x + c)^3 - 4*(3411*A - 1318*B)*cos(d*x + c)^2 - (11619*A - 4472*B)*cos
(d*x + c) - 3456*A + 1328*B)*sin(d*x + c))/(a^4*d*cos(d*x + c)^4 + 4*a^4*d*cos(d*x + c)^3 + 6*a^4*d*cos(d*x +
c)^2 + 4*a^4*d*cos(d*x + c) + a^4*d)

Sympy [F]

\[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {A \cos ^{2}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]

[In]

integrate(cos(d*x+c)**2*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))**4,x)

[Out]

(Integral(A*cos(c + d*x)**2/(sec(c + d*x)**4 + 4*sec(c + d*x)**3 + 6*sec(c + d*x)**2 + 4*sec(c + d*x) + 1), x)
+ Integral(B*cos(c + d*x)**2*sec(c + d*x)/(sec(c + d*x)**4 + 4*sec(c + d*x)**3 + 6*sec(c + d*x)**2 + 4*sec(c
+ d*x) + 1), x))/a**4

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.63 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=-\frac {3 \, A {\left (\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} + \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {5880 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - B {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} + \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {6720 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )}}{840 \, d} \]

[In]

integrate(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/840*(3*A*(280*(7*sin(d*x + c)/(cos(d*x + c) + 1) + 9*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^4 + 2*a^4*sin(
d*x + c)^2/(cos(d*x + c) + 1)^2 + a^4*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (3885*sin(d*x + c)/(cos(d*x + c)
+ 1) - 455*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 63*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5*sin(d*x + c)^7/(co
s(d*x + c) + 1)^7)/a^4 - 5880*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^4) - B*(1680*sin(d*x + c)/((a^4 + a^4*
sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)) + (5145*sin(d*x + c)/(cos(d*x + c) + 1) - 805*sin(d*x
+ c)^3/(cos(d*x + c) + 1)^3 + 147*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^
7)/a^4 - 6720*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^4))/d

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.04 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\frac {420 \, {\left (d x + c\right )} {\left (21 \, A - 8 \, B\right )}}{a^{4}} - \frac {840 \, {\left (9 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 189 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 147 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1365 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 805 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 11655 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5145 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]

[In]

integrate(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x, algorithm="giac")

[Out]

1/840*(420*(d*x + c)*(21*A - 8*B)/a^4 - 840*(9*A*tan(1/2*d*x + 1/2*c)^3 - 2*B*tan(1/2*d*x + 1/2*c)^3 + 7*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^4) + (15*A*a^24*tan(1/2*d*x +
1/2*c)^7 - 15*B*a^24*tan(1/2*d*x + 1/2*c)^7 - 189*A*a^24*tan(1/2*d*x + 1/2*c)^5 + 147*B*a^24*tan(1/2*d*x + 1/2
*c)^5 + 1365*A*a^24*tan(1/2*d*x + 1/2*c)^3 - 805*B*a^24*tan(1/2*d*x + 1/2*c)^3 - 11655*A*a^24*tan(1/2*d*x + 1/
2*c) + 5145*B*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d

Mupad [B] (verification not implemented)

Time = 14.00 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\frac {21\,A\,d\,x}{2}-4\,B\,d\,x}{a^4\,d}-\frac {\left (9\,A-2\,B\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (7\,A-2\,B\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^4\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {111\,A}{8}-\frac {49\,B}{8}\right )}{a^4\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {13\,A}{8}-\frac {23\,B}{24}\right )}{a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {9\,A}{40}-\frac {7\,B}{40}\right )}{a^4\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {A}{56}-\frac {B}{56}\right )}{a^4\,d} \]

[In]

int((cos(c + d*x)^2*(A + B/cos(c + d*x)))/(a + a/cos(c + d*x))^4,x)

[Out]

((21*A*d*x)/2 - 4*B*d*x)/(a^4*d) - (tan(c/2 + (d*x)/2)^3*(9*A - 2*B) + tan(c/2 + (d*x)/2)*(7*A - 2*B))/(a^4*d*
(tan(c/2 + (d*x)/2)^2 + 1)^2) - (tan(c/2 + (d*x)/2)*((111*A)/8 - (49*B)/8))/(a^4*d) + (tan(c/2 + (d*x)/2)^3*((
13*A)/8 - (23*B)/24))/(a^4*d) - (tan(c/2 + (d*x)/2)^5*((9*A)/40 - (7*B)/40))/(a^4*d) + (tan(c/2 + (d*x)/2)^7*(
A/56 - B/56))/(a^4*d)