\(\int \frac {\cos (c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+a \sec (c+d x))^3} \, dx\) [471]

Optimal result

Integrand size = 39, antiderivative size = 141 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {(3 A-B) x}{a^3}+\frac {2 (36 A-11 B+C) \sin (c+d x)}{15 a^3 d}-\frac {(A-B+C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(9 A-4 B-C) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac {(3 A-B) \sin (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )} \]

[Out]

-(3*A-B)*x/a^3+2/15*(36*A-11*B+C)*sin(d*x+c)/a^3/d-1/5*(A-B+C)*sin(d*x+c)/d/(a+a*sec(d*x+c))^3-1/15*(9*A-4*B-C
)*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))

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {4169, 4105, 3872, 2717, 8} \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {2 (36 A-11 B+C) \sin (c+d x)}{15 a^3 d}-\frac {(3 A-B) \sin (c+d x)}{d \left (a^3 \sec (c+d x)+a^3\right )}-\frac {x (3 A-B)}{a^3}-\frac {(9 A-4 B-C) \sin (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac {(A-B+C) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]

[In]

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

[Out]

-(((3*A - B)*x)/a^3) + (2*(36*A - 11*B + C)*Sin[c + d*x])/(15*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*
Sec[c + d*x])^3) - ((9*A - 4*B - C)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((3*A - B)*Sin[c + d*x])/(
d*(a^3 + a^3*Sec[c + d*x]))

Rule 8

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

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]

Rule 4169

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

Rubi steps \begin{align*} \text {integral}& = -\frac {(A-B+C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {\int \frac {\cos (c+d x) (a (6 A-B+C)-a (3 A-3 B-2 C) \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2} \\ & = -\frac {(A-B+C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(9 A-4 B-C) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac {\int \frac {\cos (c+d x) \left (a^2 (27 A-7 B+2 C)-2 a^2 (9 A-4 B-C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4} \\ & = -\frac {(A-B+C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(9 A-4 B-C) \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 )}+\frac {\int \cos (c+d x) \left (2 a^3 (36 A-11 B+C)-15 a^3 (3 A-B) \sec (c+d x)\right ) \, dx}{15 a^6} \\ & = -\frac {(A-B+C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(9 A-4 B-C) \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 )}-\frac {(3 A-B) \int 1 \, dx}{a^3}+\frac {(2 (36 A-11 B+C)) \int \cos (c+d x) \, dx}{15 a^3} \\ & = -\frac {(3 A-B) x}{a^3}+\frac {2 (36 A-11 B+C) \sin (c+d x)}{15 a^3 d}-\frac {(A-B+C) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac {(9 A-4 B-C) \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 )} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(419\) vs. \(2(141)=282\).

Time = 3.85 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.97 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right ) \left (-300 (3 A-B) d x \cos \left (\frac {d x}{2}\right )-300 (3 A-B) d x \cos \left (c+\frac {d x}{2}\right )-450 A d x \cos \left (c+\frac {3 d x}{2}\right )+150 B d x \cos \left (c+\frac {3 d x}{2}\right )-450 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+150 B d x \cos \left (2 c+\frac {3 d x}{2}\right )-90 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+30 B d x \cos \left (2 c+\frac {5 d x}{2}\right )-90 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+30 B d x \cos \left (3 c+\frac {5 d x}{2}\right )+1755 A \sin \left (\frac {d x}{2}\right )-740 B \sin \left (\frac {d x}{2}\right )+160 C \sin \left (\frac {d x}{2}\right )-1125 A \sin \left (c+\frac {d x}{2}\right )+540 B \sin \left (c+\frac {d x}{2}\right )-120 C \sin \left (c+\frac {d x}{2}\right )+1215 A \sin \left (c+\frac {3 d x}{2}\right )-460 B \sin \left (c+\frac {3 d x}{2}\right )+80 C \sin \left (c+\frac {3 d x}{2}\right )-225 A \sin \left (2 c+\frac {3 d x}{2}\right )+180 B \sin \left (2 c+\frac {3 d x}{2}\right )-60 C \sin \left (2 c+\frac {3 d x}{2}\right )+363 A \sin \left (2 c+\frac {5 d x}{2}\right )-128 B \sin \left (2 c+\frac {5 d x}{2}\right )+28 C \sin \left (2 c+\frac {5 d x}{2}\right )+75 A \sin \left (3 c+\frac {5 d x}{2}\right )+15 A \sin \left (3 c+\frac {7 d x}{2}\right )+15 A \sin \left (4 c+\frac {7 d x}{2}\right )\right )}{960 a^3 d} \]

[In]

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

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]^5*(-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] + 160*C*Sin[(d*x)/2] - 1125*A*Sin[c + (d*x
)/2] + 540*B*Sin[c + (d*x)/2] - 120*C*Sin[c + (d*x)/2] + 1215*A*Sin[c + (3*d*x)/2] - 460*B*Sin[c + (3*d*x)/2]
+ 80*C*Sin[c + (3*d*x)/2] - 225*A*Sin[2*c + (3*d*x)/2] + 180*B*Sin[2*c + (3*d*x)/2] - 60*C*Sin[2*c + (3*d*x)/2
] + 363*A*Sin[2*c + (5*d*x)/2] - 128*B*Sin[2*c + (5*d*x)/2] + 28*C*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]))/(960*a^3*d)

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.71

method	result	size
parallelrisch	\(\frac {\left (\frac {2 \left (39 A -\frac {32 B}{3}+\frac {7 C}{3}\right ) \cos \left (2 d x +2 c \right )}{5}+A \cos \left (3 d x +3 c \right )+\frac {\left (243 A -68 B +8 C \right ) \cos \left (d x +c \right )}{5}+\frac {174 A}{5}-\frac {152 B}{15}+\frac {22 C}{15}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-48 x \left (A -\frac {B}{3}\right ) d}{16 a^{3} d}\)	\(100\)
derivativedivides	\(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A +\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {8 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-8 \left (3 A -B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\)	\(175\)
default	\(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A +\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {8 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-8 \left (3 A -B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}\)	\(175\)
norman	\(\frac {\frac {\left (3 A -B \right ) x}{a}-\frac {\left (3 A -B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a}+\frac {\left (A -B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{20 a d}-\frac {\left (3 A -2 B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 a d}+\frac {\left (15 A -2 B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 a d}-\frac {\left (25 A -7 B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {\left (42 A -17 B +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{10 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) a^{2}}\)	\(205\)
risch	\(-\frac {3 A x}{a^{3}}+\frac {B x}{a^{3}}-\frac {i A \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{3} d}+\frac {i A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 a^{3} d}+\frac {2 i \left (90 A \,{\mathrm e}^{4 i \left (d x +c \right )}-45 B \,{\mathrm e}^{4 i \left (d x +c \right )}+15 C \,{\mathrm e}^{4 i \left (d x +c \right )}+300 A \,{\mathrm e}^{3 i \left (d x +c \right )}-135 B \,{\mathrm e}^{3 i \left (d x +c \right )}+30 C \,{\mathrm e}^{3 i \left (d x +c \right )}+420 A \,{\mathrm e}^{2 i \left (d x +c \right )}-185 B \,{\mathrm e}^{2 i \left (d x +c \right )}+40 C \,{\mathrm e}^{2 i \left (d x +c \right )}+270 A \,{\mathrm e}^{i \left (d x +c \right )}-115 B \,{\mathrm e}^{i \left (d x +c \right )}+20 C \,{\mathrm e}^{i \left (d x +c \right )}+72 A -32 B +7 C \right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\)	\(229\)

[In]

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

[Out]

1/16*((2/5*(39*A-32/3*B+7/3*C)*cos(2*d*x+2*c)+A*cos(3*d*x+3*c)+1/5*(243*A-68*B+8*C)*cos(d*x+c)+174/5*A-152/15*
B+22/15*C)*tan(1/2*d*x+1/2*c)*sec(1/2*d*x+1/2*c)^4-48*x*(A-1/3*B)*d)/a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.29 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {15 \, {\left (3 \, A - B\right )} d x \cos \left (d x + c\right )^{3} + 45 \, {\left (3 \, A - B\right )} d x \cos \left (d x + c\right )^{2} + 45 \, {\left (3 \, A - B\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (3 \, A - B\right )} d x - {\left (15 \, A \cos \left (d x + c\right )^{3} + {\left (117 \, A - 32 \, B + 7 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (57 \, A - 17 \, B + 2 \, C\right )} \cos \left (d x + c\right ) + 72 \, A - 22 \, B + 2 \, C\right )} \sin \left (d x + c\right )}{15 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]

[In]

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

[Out]

-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 + 7*C)*cos(d*x + c)^2 + 3*(57*A - 17*B + 2*C)*cos(d*x + c
) + 72*A - 22*B + 2*C)*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]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (135) = 270\).

Time = 0.31 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.09 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {3 \, A {\left (\frac {40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} - B {\left (\frac {\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac {120 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} + \frac {C {\left (\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3}}}{60 \, d} \]

[In]

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

[Out]

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 -
120*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) + C*(15*sin(d*x + c)/(cos(d*x + c) + 1) - 10*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*sin(d*x + c)^5/(c
os(d*x + c) + 1)^5)/a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.46 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {60 \, {\left (d x + c\right )} {\left (3 \, A - B\right )}}{a^{3}} - \frac {120 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{3}} - \frac {3 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 30 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 20 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 10 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 255 \, A a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, B a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 15 \, C a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{60 \, d} \]

[In]

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

[Out]

-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 + 3*C*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 - 10*C*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) + 15*C*a^12*tan(1/2*d*x + 1/2*c))/a^15)/d

Mupad [B] (verification not implemented)

Time = 16.09 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.16 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (A-B+C\right )}{4\,a^3}+\frac {4\,A-2\,B}{2\,a^3}+\frac {6\,A-2\,C}{4\,a^3}\right )}{d}-\frac {x\,\left (3\,A-B\right )}{a^3}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-B+C}{6\,a^3}+\frac {4\,A-2\,B}{12\,a^3}\right )}{d}+\frac {2\,A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (A-B+C\right )}{20\,a^3\,d} \]

[In]

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

[Out]

(tan(c/2 + (d*x)/2)*((3*(A - B + C))/(4*a^3) + (4*A - 2*B)/(2*a^3) + (6*A - 2*C)/(4*a^3)))/d - (x*(3*A - B))/a
^3 - (tan(c/2 + (d*x)/2)^3*((A - B + C)/(6*a^3) + (4*A - 2*B)/(12*a^3)))/d + (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 + C))/(20*a^3*d)