\(\int \frac {\cos (c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+a \sec (c+d x))^4} \, dx\) [480]

Optimal result

Integrand size = 39, antiderivative size = 176 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {(4 A-B) x}{a^4}+\frac {2 (332 A-80 B+3 C) \sin (c+d x)}{105 a^4 d}-\frac {(88 A-25 B-3 C) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(4 A-B) \sin (c+d x)}{a^4 d (1+\sec (c+d x))}-\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(12 A-5 B-2 C) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3} \]

[Out]

-(4*A-B)*x/a^4+2/105*(332*A-80*B+3*C)*sin(d*x+c)/a^4/d-1/105*(88*A-25*B-3*C)*sin(d*x+c)/a^4/d/(1+sec(d*x+c))^2
-(4*A-B)*sin(d*x+c)/a^4/d/(1+sec(d*x+c))-1/7*(A-B+C)*sin(d*x+c)/d/(a+a*sec(d*x+c))^4-1/35*(12*A-5*B-2*C)*sin(d
*x+c)/a/d/(a+a*sec(d*x+c))^3

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {4169, 4105, 3872, 2717, 8} \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {2 (332 A-80 B+3 C) \sin (c+d x)}{105 a^4 d}-\frac {(88 A-25 B-3 C) \sin (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac {(4 A-B) \sin (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac {x (4 A-B)}{a^4}-\frac {(12 A-5 B-2 C) \sin (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac {(A-B+C) \sin (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]

[In]

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

[Out]

-(((4*A - B)*x)/a^4) + (2*(332*A - 80*B + 3*C)*Sin[c + d*x])/(105*a^4*d) - ((88*A - 25*B - 3*C)*Sin[c + d*x])/
(105*a^4*d*(1 + Sec[c + d*x])^2) - ((4*A - B)*Sin[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sin[c +
d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((12*A - 5*B - 2*C)*Sin[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)

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)}{7 d (a+a \sec (c+d x))^4}+\frac {\int \frac {\cos (c+d x) (a (8 A-B+C)-a (4 A-4 B-3 C) \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2} \\ & = -\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(12 A-5 B-2 C) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\cos (c+d x) \left (a^2 (52 A-10 B+3 C)-3 a^2 (12 A-5 B-2 C) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4} \\ & = -\frac {(88 A-25 B-3 C) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(12 A-5 B-2 C) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac {\int \frac {\cos (c+d x) \left (a^3 (244 A-55 B+6 C)-2 a^3 (88 A-25 B-3 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6} \\ & = -\frac {(88 A-25 B-3 C) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(12 A-5 B-2 C) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(4 A-B) \sin (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}+\frac {\int \cos (c+d x) \left (2 a^4 (332 A-80 B+3 C)-105 a^4 (4 A-B) \sec (c+d x)\right ) \, dx}{105 a^8} \\ & = -\frac {(88 A-25 B-3 C) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(12 A-5 B-2 C) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(4 A-B) \sin (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}-\frac {(4 A-B) \int 1 \, dx}{a^4}+\frac {(2 (332 A-80 B+3 C)) \int \cos (c+d x) \, dx}{105 a^4} \\ & = -\frac {(4 A-B) x}{a^4}+\frac {2 (332 A-80 B+3 C) \sin (c+d x)}{105 a^4 d}-\frac {(88 A-25 B-3 C) \sin (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-B+C) \sin (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac {(12 A-5 B-2 C) \sin (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {(4 A-B) \sin (c+d x)}{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. \(567\) vs. \(2(176)=352\).

Time = 7.04 (sec) , antiderivative size = 567, normalized size of antiderivative = 3.22 \[ \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))^4} \, dx=\frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (-7350 (4 A-B) d x \cos \left (\frac {d x}{2}\right )-7350 (4 A-B) d x \cos \left (c+\frac {d x}{2}\right )-17640 A d x \cos \left (c+\frac {3 d x}{2}\right )+4410 B d x \cos \left (c+\frac {3 d x}{2}\right )-17640 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+4410 B d x \cos \left (2 c+\frac {3 d x}{2}\right )-5880 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+1470 B d x \cos \left (2 c+\frac {5 d x}{2}\right )-5880 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+1470 B d x \cos \left (3 c+\frac {5 d x}{2}\right )-840 A d x \cos \left (3 c+\frac {7 d x}{2}\right )+210 B d x \cos \left (3 c+\frac {7 d x}{2}\right )-840 A d x \cos \left (4 c+\frac {7 d x}{2}\right )+210 B d x \cos \left (4 c+\frac {7 d x}{2}\right )+60830 A \sin \left (\frac {d x}{2}\right )-19880 B \sin \left (\frac {d x}{2}\right )+2520 C \sin \left (\frac {d x}{2}\right )-46130 A \sin \left (c+\frac {d x}{2}\right )+16520 B \sin \left (c+\frac {d x}{2}\right )-2520 C \sin \left (c+\frac {d x}{2}\right )+46116 A \sin \left (c+\frac {3 d x}{2}\right )-14280 B \sin \left (c+\frac {3 d x}{2}\right )+1764 C \sin \left (c+\frac {3 d x}{2}\right )-18060 A \sin \left (2 c+\frac {3 d x}{2}\right )+7560 B \sin \left (2 c+\frac {3 d x}{2}\right )-1260 C \sin \left (2 c+\frac {3 d x}{2}\right )+19292 A \sin \left (2 c+\frac {5 d x}{2}\right )-5600 B \sin \left (2 c+\frac {5 d x}{2}\right )+588 C \sin \left (2 c+\frac {5 d x}{2}\right )-2100 A \sin \left (3 c+\frac {5 d x}{2}\right )+1680 B \sin \left (3 c+\frac {5 d x}{2}\right )-420 C \sin \left (3 c+\frac {5 d x}{2}\right )+3791 A \sin \left (3 c+\frac {7 d x}{2}\right )-1040 B \sin \left (3 c+\frac {7 d x}{2}\right )+144 C \sin \left (3 c+\frac {7 d x}{2}\right )+735 A \sin \left (4 c+\frac {7 d x}{2}\right )+105 A \sin \left (4 c+\frac {9 d x}{2}\right )+105 A \sin \left (5 c+\frac {9 d x}{2}\right )\right )}{26880 a^4 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])^4,x]

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]^7*(-7350*(4*A - B)*d*x*Cos[(d*x)/2] - 7350*(4*A - B)*d*x*Cos[c + (d*x)/2] - 17640*A
*d*x*Cos[c + (3*d*x)/2] + 4410*B*d*x*Cos[c + (3*d*x)/2] - 17640*A*d*x*Cos[2*c + (3*d*x)/2] + 4410*B*d*x*Cos[2*
c + (3*d*x)/2] - 5880*A*d*x*Cos[2*c + (5*d*x)/2] + 1470*B*d*x*Cos[2*c + (5*d*x)/2] - 5880*A*d*x*Cos[3*c + (5*d
*x)/2] + 1470*B*d*x*Cos[3*c + (5*d*x)/2] - 840*A*d*x*Cos[3*c + (7*d*x)/2] + 210*B*d*x*Cos[3*c + (7*d*x)/2] - 8
40*A*d*x*Cos[4*c + (7*d*x)/2] + 210*B*d*x*Cos[4*c + (7*d*x)/2] + 60830*A*Sin[(d*x)/2] - 19880*B*Sin[(d*x)/2] +
2520*C*Sin[(d*x)/2] - 46130*A*Sin[c + (d*x)/2] + 16520*B*Sin[c + (d*x)/2] - 2520*C*Sin[c + (d*x)/2] + 46116*A
*Sin[c + (3*d*x)/2] - 14280*B*Sin[c + (3*d*x)/2] + 1764*C*Sin[c + (3*d*x)/2] - 18060*A*Sin[2*c + (3*d*x)/2] +
7560*B*Sin[2*c + (3*d*x)/2] - 1260*C*Sin[2*c + (3*d*x)/2] + 19292*A*Sin[2*c + (5*d*x)/2] - 5600*B*Sin[2*c + (5
*d*x)/2] + 588*C*Sin[2*c + (5*d*x)/2] - 2100*A*Sin[3*c + (5*d*x)/2] + 1680*B*Sin[3*c + (5*d*x)/2] - 420*C*Sin[
3*c + (5*d*x)/2] + 3791*A*Sin[3*c + (7*d*x)/2] - 1040*B*Sin[3*c + (7*d*x)/2] + 144*C*Sin[3*c + (7*d*x)/2] + 73
5*A*Sin[4*c + (7*d*x)/2] + 105*A*Sin[4*c + (9*d*x)/2] + 105*A*Sin[5*c + (9*d*x)/2]))/(26880*a^4*d)

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.67

method	result	size
parallelrisch	\(\frac {2368 \left (\left (\frac {2741 A}{592}-\frac {155 B}{148}+\frac {39 C}{592}\right ) \cos \left (2 d x +2 c \right )+\left (A -\frac {65 B}{296}+\frac {9 C}{296}\right ) \cos \left (3 d x +3 c \right )+\frac {105 A \cos \left (4 d x +4 c \right )}{2368}+\left (\frac {781 A}{74}-\frac {365 B}{148}+\frac {51 C}{296}\right ) \cos \left (d x +c \right )+\frac {16171 A}{2368}-\frac {235 B}{148}+\frac {51 C}{592}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-26880 \left (A -\frac {B}{4}\right ) x d}{6720 a^{4} d}\)	\(118\)
derivativedivides	\(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B +\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {16 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-16 \left (4 A -B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\)	\(217\)
default	\(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} C}{7}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A}{5}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B +\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} C}{5}-\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {16 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-16 \left (4 A -B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\)	\(217\)
norman	\(\frac {\frac {\left (4 A -B \right ) x}{a}-\frac {\left (4 A -B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a}-\frac {\left (A -B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{56 a d}+\frac {\left (7 A -5 B +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{40 a d}-\frac {\left (65 A -15 B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\left (71 A -11 B +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 a d}-\frac {\left (79 A -37 B +9 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{84 a d}+\frac {\left (119 A -35 B +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) a^{3}}\)	\(236\)
risch	\(-\frac {4 A x}{a^{4}}+\frac {x B}{a^{4}}-\frac {i A \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{4} d}+\frac {i A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 a^{4} d}+\frac {2 i \left (1050 A \,{\mathrm e}^{6 i \left (d x +c \right )}-420 B \,{\mathrm e}^{6 i \left (d x +c \right )}+105 C \,{\mathrm e}^{6 i \left (d x +c \right )}+5250 A \,{\mathrm e}^{5 i \left (d x +c \right )}-1890 B \,{\mathrm e}^{5 i \left (d x +c \right )}+315 C \,{\mathrm e}^{5 i \left (d x +c \right )}+11900 A \,{\mathrm e}^{4 i \left (d x +c \right )}-4130 B \,{\mathrm e}^{4 i \left (d x +c \right )}+630 C \,{\mathrm e}^{4 i \left (d x +c \right )}+14840 A \,{\mathrm e}^{3 i \left (d x +c \right )}-4970 B \,{\mathrm e}^{3 i \left (d x +c \right )}+630 C \,{\mathrm e}^{3 i \left (d x +c \right )}+10794 A \,{\mathrm e}^{2 i \left (d x +c \right )}-3570 B \,{\mathrm e}^{2 i \left (d x +c \right )}+441 C \,{\mathrm e}^{2 i \left (d x +c \right )}+4298 A \,{\mathrm e}^{i \left (d x +c \right )}-1400 B \,{\mathrm e}^{i \left (d x +c \right )}+147 C \,{\mathrm e}^{i \left (d x +c \right )}+764 A -260 B +36 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\)	\(301\)

[In]

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

[Out]

1/6720*(2368*((2741/592*A-155/148*B+39/592*C)*cos(2*d*x+2*c)+(A-65/296*B+9/296*C)*cos(3*d*x+3*c)+105/2368*A*co
s(4*d*x+4*c)+(781/74*A-365/148*B+51/296*C)*cos(d*x+c)+16171/2368*A-235/148*B+51/592*C)*tan(1/2*d*x+1/2*c)*sec(
1/2*d*x+1/2*c)^6-26880*(A-1/4*B)*x*d)/a^4/d

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.33 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {105 \, {\left (4 \, A - B\right )} d x \cos \left (d x + c\right )^{4} + 420 \, {\left (4 \, A - B\right )} d x \cos \left (d x + c\right )^{3} + 630 \, {\left (4 \, A - B\right )} d x \cos \left (d x + c\right )^{2} + 420 \, {\left (4 \, A - B\right )} d x \cos \left (d x + c\right ) + 105 \, {\left (4 \, A - B\right )} d x - {\left (105 \, A \cos \left (d x + c\right )^{4} + 4 \, {\left (296 \, A - 65 \, B + 9 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (2636 \, A - 620 \, B + 39 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (2236 \, A - 535 \, B + 24 \, C\right )} \cos \left (d x + c\right ) + 664 \, A - 160 \, B + 6 \, C\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]

[In]

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

[Out]

-1/105*(105*(4*A - B)*d*x*cos(d*x + c)^4 + 420*(4*A - B)*d*x*cos(d*x + c)^3 + 630*(4*A - B)*d*x*cos(d*x + c)^2
+ 420*(4*A - B)*d*x*cos(d*x + c) + 105*(4*A - B)*d*x - (105*A*cos(d*x + c)^4 + 4*(296*A - 65*B + 9*C)*cos(d*x
+ c)^3 + (2636*A - 620*B + 39*C)*cos(d*x + c)^2 + (2236*A - 535*B + 24*C)*cos(d*x + c) + 664*A - 160*B + 6*C)
*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]

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

[In]

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

[Out]

(Integral(A*cos(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) +
Integral(B*cos(c + d*x)*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) + Integral(C*cos(c + d*x)*sec(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))/a**4

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (168) = 336\).

Time = 0.32 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.02 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {A {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} + \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {6720 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - 5 \, B {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {336 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} + \frac {3 \, C {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]

[In]

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

[Out]

1/840*(A*(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) - 5*B*((315
*sin(d*x + c)/(cos(d*x + c) + 1) - 77*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 21*sin(d*x + c)^5/(cos(d*x + c) +
1)^5 - 3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 336*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^4) + 3*C*(35
*sin(d*x + c)/(cos(d*x + c) + 1) - 35*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 21*sin(d*x + c)^5/(cos(d*x + c) +
1)^5 - 5*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4)/d

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.45 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {\frac {840 \, {\left (d x + c\right )} {\left (4 \, A - B\right )}}{a^{4}} - \frac {1680 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 147 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 63 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 805 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 385 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5145 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1575 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 105 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]

[In]

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

[Out]

-1/840*(840*(d*x + c)*(4*A - B)/a^4 - 1680*A*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*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 + 15*C*a^24*tan(1/2*d*x + 1/2*c)^7 - 147*A*a^24*
tan(1/2*d*x + 1/2*c)^5 + 105*B*a^24*tan(1/2*d*x + 1/2*c)^5 - 63*C*a^24*tan(1/2*d*x + 1/2*c)^5 + 805*A*a^24*tan
(1/2*d*x + 1/2*c)^3 - 385*B*a^24*tan(1/2*d*x + 1/2*c)^3 + 105*C*a^24*tan(1/2*d*x + 1/2*c)^3 - 5145*A*a^24*tan(
1/2*d*x + 1/2*c) + 1575*B*a^24*tan(1/2*d*x + 1/2*c) - 105*C*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d

Mupad [B] (verification not implemented)

Time = 17.56 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.66 \[ \int \frac {\cos (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {\frac {35\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {167\,A\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{160}+\frac {307\,A\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{480}+\frac {2263\,A\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{13440}+\frac {A\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{128}-\frac {B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}-\frac {B\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{4}-\frac {7\,B\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{48}-\frac {13\,B\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{336}+\frac {3\,C\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{160}+\frac {C\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{160}+\frac {3\,C\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{560}-\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {35\,A\,\left (c+d\,x\right )}{16}-\frac {35\,B\,\left (c+d\,x\right )}{64}\right )-\frac {21\,A\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,\left (c+d\,x\right )}{16}-\frac {7\,A\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,\left (c+d\,x\right )}{16}-\frac {A\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,\left (c+d\,x\right )}{16}+\frac {21\,B\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,\left (c+d\,x\right )}{64}+\frac {7\,B\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,\left (c+d\,x\right )}{64}+\frac {B\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,\left (c+d\,x\right )}{64}}{a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]

[In]

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

[Out]

((35*A*sin(c/2 + (d*x)/2))/64 + (167*A*sin((3*c)/2 + (3*d*x)/2))/160 + (307*A*sin((5*c)/2 + (5*d*x)/2))/480 +
(2263*A*sin((7*c)/2 + (7*d*x)/2))/13440 + (A*sin((9*c)/2 + (9*d*x)/2))/128 - (B*sin(c/2 + (d*x)/2))/8 - (B*sin
((3*c)/2 + (3*d*x)/2))/4 - (7*B*sin((5*c)/2 + (5*d*x)/2))/48 - (13*B*sin((7*c)/2 + (7*d*x)/2))/336 + (3*C*sin(
(3*c)/2 + (3*d*x)/2))/160 + (C*sin((5*c)/2 + (5*d*x)/2))/160 + (3*C*sin((7*c)/2 + (7*d*x)/2))/560 - cos(c/2 +
(d*x)/2)*((35*A*(c + d*x))/16 - (35*B*(c + d*x))/64) - (21*A*cos((3*c)/2 + (3*d*x)/2)*(c + d*x))/16 - (7*A*cos
((5*c)/2 + (5*d*x)/2)*(c + d*x))/16 - (A*cos((7*c)/2 + (7*d*x)/2)*(c + d*x))/16 + (21*B*cos((3*c)/2 + (3*d*x)/
2)*(c + d*x))/64 + (7*B*cos((5*c)/2 + (5*d*x)/2)*(c + d*x))/64 + (B*cos((7*c)/2 + (7*d*x)/2)*(c + d*x))/64)/(a
^4*d*cos(c/2 + (d*x)/2)^7)