Integrand size = 29, antiderivative size = 136 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=-\frac {(3 A-B) x}{a^3}+\frac {2 (36 A-11 B) \sin (c+d x)}{15 a^3 d}-\frac {(A-B) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(9 A-4 B) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(3 A-B) \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )} \] Output:
-(3*A-B)*x/a^3+2/15*(36*A-11*B)*sin(d*x+c)/a^3/d-1/5*(A-B)*sin(d*x+c)/d/(a +a*sec(d*x+c))^3-1/15*(9*A-4*B)*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^2-(3*A-B)* sin(d*x+c)/d/(a^3+a^3*sec(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(365\) vs. \(2(136)=272\).
Time = 2.31 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.68 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (-300 (3 A-B) d x \cos \left (\frac {d x}{2}\right )-300 (3 A-B) d x \cos \left (c+\frac {d x}{2}\right )-450 A d x \cos \left (c+\frac {3 d x}{2}\right )+150 B d x \cos \left (c+\frac {3 d x}{2}\right )-450 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+150 B d x \cos \left (2 c+\frac {3 d x}{2}\right )-90 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+30 B d x \cos \left (2 c+\frac {5 d x}{2}\right )-90 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+30 B d x \cos \left (3 c+\frac {5 d x}{2}\right )+1755 A \sin \left (\frac {d x}{2}\right )-740 B \sin \left (\frac {d x}{2}\right )-1125 A \sin \left (c+\frac {d x}{2}\right )+540 B \sin \left (c+\frac {d x}{2}\right )+1215 A \sin \left (c+\frac {3 d x}{2}\right )-460 B \sin \left (c+\frac {3 d x}{2}\right )-225 A \sin \left (2 c+\frac {3 d x}{2}\right )+180 B \sin \left (2 c+\frac {3 d x}{2}\right )+363 A \sin \left (2 c+\frac {5 d x}{2}\right )-128 B \sin \left (2 c+\frac {5 d x}{2}\right )+75 A \sin \left (3 c+\frac {5 d x}{2}\right )+15 A \sin \left (3 c+\frac {7 d x}{2}\right )+15 A \sin \left (4 c+\frac {7 d x}{2}\right )\right )}{120 a^3 d (1+\cos (c+d x))^3} \] Input:
Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]
Output:
(Cos[(c + d*x)/2]*Sec[c/2]*(-300*(3*A - B)*d*x*Cos[(d*x)/2] - 300*(3*A - B )*d*x*Cos[c + (d*x)/2] - 450*A*d*x*Cos[c + (3*d*x)/2] + 150*B*d*x*Cos[c + (3*d*x)/2] - 450*A*d*x*Cos[2*c + (3*d*x)/2] + 150*B*d*x*Cos[2*c + (3*d*x)/ 2] - 90*A*d*x*Cos[2*c + (5*d*x)/2] + 30*B*d*x*Cos[2*c + (5*d*x)/2] - 90*A* d*x*Cos[3*c + (5*d*x)/2] + 30*B*d*x*Cos[3*c + (5*d*x)/2] + 1755*A*Sin[(d*x )/2] - 740*B*Sin[(d*x)/2] - 1125*A*Sin[c + (d*x)/2] + 540*B*Sin[c + (d*x)/ 2] + 1215*A*Sin[c + (3*d*x)/2] - 460*B*Sin[c + (3*d*x)/2] - 225*A*Sin[2*c + (3*d*x)/2] + 180*B*Sin[2*c + (3*d*x)/2] + 363*A*Sin[2*c + (5*d*x)/2] - 1 28*B*Sin[2*c + (5*d*x)/2] + 75*A*Sin[3*c + (5*d*x)/2] + 15*A*Sin[3*c + (7* d*x)/2] + 15*A*Sin[4*c + (7*d*x)/2]))/(120*a^3*d*(1 + Cos[c + d*x])^3)
Time = 0.98 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {3042, 4508, 3042, 4508, 3042, 4508, 3042, 4274, 24, 3042, 3117}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a \sec (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x) (a (6 A-B)-3 a (A-B) \sec (c+d x))}{(\sec (c+d x) a+a)^2}dx}{5 a^2}-\frac {(A-B) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (6 A-B)-3 a (A-B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {(A-B) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\int \frac {\cos (c+d x) \left (a^2 (27 A-7 B)-2 a^2 (9 A-4 B) \sec (c+d x)\right )}{\sec (c+d x) a+a}dx}{3 a^2}-\frac {a (9 A-4 B) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {a^2 (27 A-7 B)-2 a^2 (9 A-4 B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}dx}{3 a^2}-\frac {a (9 A-4 B) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4508 |
| \(\displaystyle \frac {\frac {\frac {\int \cos (c+d x) \left (2 a^3 (36 A-11 B)-15 a^3 (3 A-B) \sec (c+d x)\right )dx}{a^2}-\frac {15 a^2 (3 A-B) \sin (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (9 A-4 B) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {2 a^3 (36 A-11 B)-15 a^3 (3 A-B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}-\frac {15 a^2 (3 A-B) \sin (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (9 A-4 B) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {\frac {2 a^3 (36 A-11 B) \int \cos (c+d x)dx-15 a^3 (3 A-B) \int 1dx}{a^2}-\frac {15 a^2 (3 A-B) \sin (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (9 A-4 B) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {\frac {2 a^3 (36 A-11 B) \int \cos (c+d x)dx-15 a^3 x (3 A-B)}{a^2}-\frac {15 a^2 (3 A-B) \sin (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (9 A-4 B) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 a^3 (36 A-11 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx-15 a^3 x (3 A-B)}{a^2}-\frac {15 a^2 (3 A-B) \sin (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (9 A-4 B) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 a^3 (36 A-11 B) \sin (c+d x)}{d}-15 a^3 x (3 A-B)}{a^2}-\frac {15 a^2 (3 A-B) \sin (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}-\frac {a (9 A-4 B) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2}}{5 a^2}-\frac {(A-B) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
Input:
Int[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]
Output:
-1/5*((A - B)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])^3) + (-1/3*(a*(9*A - 4 *B)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])^2) + ((-15*a^2*(3*A - B)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) + (-15*a^3*(3*A - B)*x + (2*a^3*(36*A - 11 *B)*Sin[c + d*x])/d)/a^2)/(3*a^2))/(5*a^2)
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
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] - Simp[1/(a^2*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Cs c[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]
Time = 0.36 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.65
| method | result | size |
| parallelrisch | \(\frac {729 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (\frac {26 A}{81}-\frac {64 B}{729}\right ) \cos \left (2 d x +2 c \right )+\frac {5 A \cos \left (3 d x +3 c \right )}{243}+\left (A -\frac {68 B}{243}\right ) \cos \left (d x +c \right )+\frac {58 A}{81}-\frac {152 B}{729}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-720 \left (A -\frac {B}{3}\right ) d x}{240 a^{3} d}\) | \(89\) |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A +\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}+17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\frac {8 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-8 \left (3 A -B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(136\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A +\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}+17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\frac {8 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-8 \left (3 A -B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\) | \(136\) |
| norman | \(\frac {-\frac {\left (3 A -B \right ) x}{a}+\frac {\left (A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 a d}-\frac {\left (3 A -B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}+\frac {\left (25 A -7 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}-\frac {\left (27 A -17 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{60 a d}+\frac {\left (45 A -17 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) a^{2}}\) | \(158\) |
| risch | \(-\frac {3 A x}{a^{3}}+\frac {x B}{a^{3}}-\frac {i A \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{3} d}+\frac {i A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 a^{3} d}+\frac {2 i \left (90 A \,{\mathrm e}^{4 i \left (d x +c \right )}-45 B \,{\mathrm e}^{4 i \left (d x +c \right )}+300 A \,{\mathrm e}^{3 i \left (d x +c \right )}-135 B \,{\mathrm e}^{3 i \left (d x +c \right )}+420 A \,{\mathrm e}^{2 i \left (d x +c \right )}-185 B \,{\mathrm e}^{2 i \left (d x +c \right )}+270 \,{\mathrm e}^{i \left (d x +c \right )} A -115 B \,{\mathrm e}^{i \left (d x +c \right )}+72 A -32 B \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\) | \(178\) |
Input:
int(cos(d*x+c)*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE )
Output:
1/240*(729*tan(1/2*d*x+1/2*c)*((26/81*A-64/729*B)*cos(2*d*x+2*c)+5/243*A*c os(3*d*x+3*c)+(A-68/243*B)*cos(d*x+c)+58/81*A-152/729*B)*sec(1/2*d*x+1/2*c )^4-720*(A-1/3*B)*d*x)/a^3/d
Time = 0.09 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.27 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=-\frac {15 \, {\left (3 \, A - B\right )} d x \cos \left (d x + c\right )^{3} + 45 \, {\left (3 \, A - B\right )} d x \cos \left (d x + c\right )^{2} + 45 \, {\left (3 \, A - B\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (3 \, A - B\right )} d x - {\left (15 \, A \cos \left (d x + c\right )^{3} + {\left (117 \, A - 32 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (57 \, A - 17 \, B\right )} \cos \left (d x + c\right ) + 72 \, A - 22 \, B\right )} \sin \left (d x + c\right )}{15 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \] Input:
integrate(cos(d*x+c)*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^3,x, algorithm="fri cas")
Output:
-1/15*(15*(3*A - B)*d*x*cos(d*x + c)^3 + 45*(3*A - B)*d*x*cos(d*x + c)^2 + 45*(3*A - B)*d*x*cos(d*x + c) + 15*(3*A - B)*d*x - (15*A*cos(d*x + c)^3 + (117*A - 32*B)*cos(d*x + c)^2 + 3*(57*A - 17*B)*cos(d*x + c) + 72*A - 22* 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)
\[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {A \cos {\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(cos(d*x+c)*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))**3,x)
Output:
(Integral(A*cos(c + d*x)/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x) + Integral(B*cos(c + d*x)*sec(c + d*x)/(sec(c + d*x)**3 + 3* sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x))/a**3
Time = 0.11 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.70 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\frac {3 \, A {\left (\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} - B {\left (\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{60 \, d} \] Input:
integrate(cos(d*x+c)*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^3,x, algorithm="max ima")
Output:
1/60*(3*A*(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 - 12 0*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3) - B*((105*sin(d*x + c)/(cos (d*x + c) + 1) - 20*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 - 120*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a ^3))/d
Time = 0.15 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.15 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {60 \, {\left (d x + c\right )} {\left (3 \, A - B\right )}}{a^{3}} - \frac {120 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 30 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 20 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 255 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \] Input:
integrate(cos(d*x+c)*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^3,x, algorithm="gia c")
Output:
-1/60*(60*(d*x + c)*(3*A - B)/a^3 - 120*A*tan(1/2*d*x + 1/2*c)/((tan(1/2*d *x + 1/2*c)^2 + 1)*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 - 30*A*a^12*tan(1/2*d*x + 1/2*c)^3 + 20*B*a^12*tan(1/2* d*x + 1/2*c)^3 + 255*A*a^12*tan(1/2*d*x + 1/2*c) - 105*B*a^12*tan(1/2*d*x + 1/2*c))/a^15)/d
Time = 11.78 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.14 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,A}{2\,a^3}+\frac {3\,\left (A-B\right )}{4\,a^3}+\frac {4\,A-2\,B}{2\,a^3}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-B}{6\,a^3}+\frac {4\,A-2\,B}{12\,a^3}\right )}{d}-\frac {x\,\left (3\,A-B\right )}{a^3}+\frac {2\,A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A-B\right )}{20\,a^3\,d} \] Input:
int((cos(c + d*x)*(A + B/cos(c + d*x)))/(a + a/cos(c + d*x))^3,x)
Output:
(tan(c/2 + (d*x)/2)*((3*A)/(2*a^3) + (3*(A - B))/(4*a^3) + (4*A - 2*B)/(2* a^3)))/d - (tan(c/2 + (d*x)/2)^3*((A - B)/(6*a^3) + (4*A - 2*B)/(12*a^3))) /d - (x*(3*A - B))/a^3 + (2*A*tan(c/2 + (d*x)/2))/(d*(a^3*tan(c/2 + (d*x)/ 2)^2 + a^3)) + (tan(c/2 + (d*x)/2)^5*(A - B))/(20*a^3*d)
Time = 0.18 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.28 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a -3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} b -27 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a +17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} b +225 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a -85 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b -180 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a d x +60 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b d x +375 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a -105 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b -180 a d x +60 b d x}{60 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )} \] Input:
int(cos(d*x+c)*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^3,x)
Output:
(3*tan((c + d*x)/2)**7*a - 3*tan((c + d*x)/2)**7*b - 27*tan((c + d*x)/2)** 5*a + 17*tan((c + d*x)/2)**5*b + 225*tan((c + d*x)/2)**3*a - 85*tan((c + d *x)/2)**3*b - 180*tan((c + d*x)/2)**2*a*d*x + 60*tan((c + d*x)/2)**2*b*d*x + 375*tan((c + d*x)/2)*a - 105*tan((c + d*x)/2)*b - 180*a*d*x + 60*b*d*x) /(60*a**3*d*(tan((c + d*x)/2)**2 + 1))