\(\int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx\) [65]

Optimal result

Integrand size = 31, antiderivative size = 229 \[ \int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=-\frac {(8 A-21 B) x}{2 a^4}+\frac {8 (83 A-216 B) \sin (c+d x)}{105 a^4 d}-\frac {(8 A-21 B) \cos (c+d x) \sin (c+d x)}{2 a^4 d}+\frac {(52 A-129 B) \cos ^3(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac {4 (83 A-216 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))}+\frac {(A-B) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(A-2 B) \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {3056, 2813} \[ \int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=\frac {8 (83 A-216 B) \sin (c+d x)}{105 a^4 d}+\frac {(52 A-129 B) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac {4 (83 A-216 B) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac {(8 A-21 B) \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac {x (8 A-21 B)}{2 a^4}+\frac {(A-B) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac {(A-2 B) \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3} \]

[In]

Int[(Cos[c + d*x]^5*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^4,x]

[Out]

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

Rule 2813

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*a*c +
b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Cos[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 3056

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(555\) vs. \(2(229)=458\).

Time = 5.03 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.42 \[ \int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (-14700 (8 A-21 B) d x \cos \left (\frac {d x}{2}\right )-14700 (8 A-21 B) d x \cos \left (c+\frac {d x}{2}\right )-70560 A d x \cos \left (c+\frac {3 d x}{2}\right )+185220 B d x \cos \left (c+\frac {3 d x}{2}\right )-70560 A d x \cos \left (2 c+\frac {3 d x}{2}\right )+185220 B d x \cos \left (2 c+\frac {3 d x}{2}\right )-23520 A d x \cos \left (2 c+\frac {5 d x}{2}\right )+61740 B d x \cos \left (2 c+\frac {5 d x}{2}\right )-23520 A d x \cos \left (3 c+\frac {5 d x}{2}\right )+61740 B d x \cos \left (3 c+\frac {5 d x}{2}\right )-3360 A d x \cos \left (3 c+\frac {7 d x}{2}\right )+8820 B d x \cos \left (3 c+\frac {7 d x}{2}\right )-3360 A d x \cos \left (4 c+\frac {7 d x}{2}\right )+8820 B d x \cos \left (4 c+\frac {7 d x}{2}\right )+243320 A \sin \left (\frac {d x}{2}\right )-539490 B \sin \left (\frac {d x}{2}\right )-184520 A \sin \left (c+\frac {d x}{2}\right )+386190 B \sin \left (c+\frac {d x}{2}\right )+184464 A \sin \left (c+\frac {3 d x}{2}\right )-422478 B \sin \left (c+\frac {3 d x}{2}\right )-72240 A \sin \left (2 c+\frac {3 d x}{2}\right )+132930 B \sin \left (2 c+\frac {3 d x}{2}\right )+77168 A \sin \left (2 c+\frac {5 d x}{2}\right )-181461 B \sin \left (2 c+\frac {5 d x}{2}\right )-8400 A \sin \left (3 c+\frac {5 d x}{2}\right )+3675 B \sin \left (3 c+\frac {5 d x}{2}\right )+15164 A \sin \left (3 c+\frac {7 d x}{2}\right )-36003 B \sin \left (3 c+\frac {7 d x}{2}\right )+2940 A \sin \left (4 c+\frac {7 d x}{2}\right )-9555 B \sin \left (4 c+\frac {7 d x}{2}\right )+420 A \sin \left (4 c+\frac {9 d x}{2}\right )-945 B \sin \left (4 c+\frac {9 d x}{2}\right )+420 A \sin \left (5 c+\frac {9 d x}{2}\right )-945 B \sin \left (5 c+\frac {9 d x}{2}\right )+105 B \sin \left (5 c+\frac {11 d x}{2}\right )+105 B \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]^5*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^4,x]

[Out]

(Cos[(c + d*x)/2]*Sec[c/2]*(-14700*(8*A - 21*B)*d*x*Cos[(d*x)/2] - 14700*(8*A - 21*B)*d*x*Cos[c + (d*x)/2] - 7
0560*A*d*x*Cos[c + (3*d*x)/2] + 185220*B*d*x*Cos[c + (3*d*x)/2] - 70560*A*d*x*Cos[2*c + (3*d*x)/2] + 185220*B*
d*x*Cos[2*c + (3*d*x)/2] - 23520*A*d*x*Cos[2*c + (5*d*x)/2] + 61740*B*d*x*Cos[2*c + (5*d*x)/2] - 23520*A*d*x*C
os[3*c + (5*d*x)/2] + 61740*B*d*x*Cos[3*c + (5*d*x)/2] - 3360*A*d*x*Cos[3*c + (7*d*x)/2] + 8820*B*d*x*Cos[3*c
+ (7*d*x)/2] - 3360*A*d*x*Cos[4*c + (7*d*x)/2] + 8820*B*d*x*Cos[4*c + (7*d*x)/2] + 243320*A*Sin[(d*x)/2] - 539
490*B*Sin[(d*x)/2] - 184520*A*Sin[c + (d*x)/2] + 386190*B*Sin[c + (d*x)/2] + 184464*A*Sin[c + (3*d*x)/2] - 422
478*B*Sin[c + (3*d*x)/2] - 72240*A*Sin[2*c + (3*d*x)/2] + 132930*B*Sin[2*c + (3*d*x)/2] + 77168*A*Sin[2*c + (5
*d*x)/2] - 181461*B*Sin[2*c + (5*d*x)/2] - 8400*A*Sin[3*c + (5*d*x)/2] + 3675*B*Sin[3*c + (5*d*x)/2] + 15164*A
*Sin[3*c + (7*d*x)/2] - 36003*B*Sin[3*c + (7*d*x)/2] + 2940*A*Sin[4*c + (7*d*x)/2] - 9555*B*Sin[4*c + (7*d*x)/
2] + 420*A*Sin[4*c + (9*d*x)/2] - 945*B*Sin[4*c + (9*d*x)/2] + 420*A*Sin[5*c + (9*d*x)/2] - 945*B*Sin[5*c + (9
*d*x)/2] + 105*B*Sin[5*c + (11*d*x)/2] + 105*B*Sin[6*c + (11*d*x)/2]))/(6720*a^4*d*(1 + Cos[c + d*x])^4)

Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.54

method	result	size
parallelrisch	\(\frac {420 \left (\left (\frac {10964 A}{105}-\frac {9376 B}{35}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {2368 A}{105}-\frac {7873 B}{140}\right ) \cos \left (3 d x +3 c \right )+\left (A -2 B \right ) \cos \left (4 d x +4 c \right )+\frac {B \cos \left (5 d x +5 c \right )}{4}+\left (\frac {24992 A}{105}-\frac {42881 B}{70}\right ) \cos \left (d x +c \right )+\frac {16171 A}{105}-\frac {13914 B}{35}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-107520 \left (A -\frac {21 B}{8}\right ) x d}{26880 a^{4} d}\)	\(123\)
derivativedivides	\(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}+\frac {7 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {9 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}-\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}+13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B +49 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-111 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {16 \left (\left (-A +\frac {9 B}{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-A +\frac {7 B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-8 \left (8 A -21 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\)	\(191\)
default	\(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}+\frac {7 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {9 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}-\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}+13 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B +49 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-111 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {16 \left (\left (-A +\frac {9 B}{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-A +\frac {7 B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-8 \left (8 A -21 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\)	\(191\)
risch	\(-\frac {4 x A}{a^{4}}+\frac {21 B x}{2 a^{4}}-\frac {i B \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{4} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} A}{2 a^{4} d}+\frac {2 i {\mathrm e}^{i \left (d x +c \right )} B}{a^{4} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} A}{2 a^{4} d}-\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} B}{a^{4} d}+\frac {i B \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{4} d}+\frac {2 i \left (1050 A \,{\mathrm e}^{6 i \left (d x +c \right )}-2100 B \,{\mathrm e}^{6 i \left (d x +c \right )}+5250 A \,{\mathrm e}^{5 i \left (d x +c \right )}-11025 B \,{\mathrm e}^{5 i \left (d x +c \right )}+11900 A \,{\mathrm e}^{4 i \left (d x +c \right )}-25515 B \,{\mathrm e}^{4 i \left (d x +c \right )}+14840 A \,{\mathrm e}^{3 i \left (d x +c \right )}-32340 B \,{\mathrm e}^{3 i \left (d x +c \right )}+10794 A \,{\mathrm e}^{2 i \left (d x +c \right )}-23688 B \,{\mathrm e}^{2 i \left (d x +c \right )}+4298 A \,{\mathrm e}^{i \left (d x +c \right )}-9471 B \,{\mathrm e}^{i \left (d x +c \right )}+764 A -1653 B \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\)	\(303\)

[In]

int(cos(d*x+c)^5*(A+B*cos(d*x+c))/(a+cos(d*x+c)*a)^4,x,method=_RETURNVERBOSE)

[Out]

1/26880*(420*((10964/105*A-9376/35*B)*cos(2*d*x+2*c)+(2368/105*A-7873/140*B)*cos(3*d*x+3*c)+(A-2*B)*cos(4*d*x+
4*c)+1/4*B*cos(5*d*x+5*c)+(24992/105*A-42881/70*B)*cos(d*x+c)+16171/105*A-13914/35*B)*tan(1/2*d*x+1/2*c)*sec(1
/2*d*x+1/2*c)^6-107520*(A-21/8*B)*x*d)/a^4/d

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.04 \[ \int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=-\frac {105 \, {\left (8 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{4} + 420 \, {\left (8 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{3} + 630 \, {\left (8 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{2} + 420 \, {\left (8 \, A - 21 \, B\right )} d x \cos \left (d x + c\right ) + 105 \, {\left (8 \, A - 21 \, B\right )} d x - {\left (105 \, B \cos \left (d x + c\right )^{5} + 210 \, {\left (A - 2 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (592 \, A - 1509 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (1318 \, A - 3411 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (4472 \, A - 11619 \, B\right )} \cos \left (d x + c\right ) + 1328 \, A - 3456 \, B\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]

[In]

integrate(cos(d*x+c)^5*(A+B*cos(d*x+c))/(a+a*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/210*(105*(8*A - 21*B)*d*x*cos(d*x + c)^4 + 420*(8*A - 21*B)*d*x*cos(d*x + c)^3 + 630*(8*A - 21*B)*d*x*cos(d
*x + c)^2 + 420*(8*A - 21*B)*d*x*cos(d*x + c) + 105*(8*A - 21*B)*d*x - (105*B*cos(d*x + c)^5 + 210*(A - 2*B)*c
os(d*x + c)^4 + 4*(592*A - 1509*B)*cos(d*x + c)^3 + 4*(1318*A - 3411*B)*cos(d*x + c)^2 + (4472*A - 11619*B)*co
s(d*x + c) + 1328*A - 3456*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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1085 vs. \(2 (216) = 432\).

Time = 7.97 (sec) , antiderivative size = 1085, normalized size of antiderivative = 4.74 \[ \int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)**5*(A+B*cos(d*x+c))/(a+a*cos(d*x+c))**4,x)

[Out]

Piecewise((-3360*A*d*x*tan(c/2 + d*x/2)**4/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 +
840*a**4*d) - 6720*A*d*x*tan(c/2 + d*x/2)**2/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**
2 + 840*a**4*d) - 3360*A*d*x/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) -
15*A*tan(c/2 + d*x/2)**11/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 1
17*A*tan(c/2 + d*x/2)**9/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 526
*A*tan(c/2 + d*x/2)**7/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 3682*
A*tan(c/2 + d*x/2)**5/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 11165*
A*tan(c/2 + d*x/2)**3/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 6825*A
*tan(c/2 + d*x/2)/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 8820*B*d*x
*tan(c/2 + d*x/2)**4/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 17640*B
*d*x*tan(c/2 + d*x/2)**2/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 882
0*B*d*x/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 15*B*tan(c/2 + d*x/2
)**11/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 159*B*tan(c/2 + d*x/2)
**9/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 1002*B*tan(c/2 + d*x/2)*
*7/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 9114*B*tan(c/2 + d*x/2)**
5/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 29505*B*tan(c/2 + d*x/2)**
3/(840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 17535*B*tan(c/2 + d*x/2)/(
840*a**4*d*tan(c/2 + d*x/2)**4 + 1680*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d), Ne(d, 0)), (x*(A + B*cos(c))*c
os(c)**5/(a*cos(c) + a)**4, True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.59 \[ \int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=-\frac {3 \, B {\left (\frac {280 \, {\left (\frac {7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} + \frac {2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {5880 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - A {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} + \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {6720 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )}}{840 \, d} \]

[In]

integrate(cos(d*x+c)^5*(A+B*cos(d*x+c))/(a+a*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/840*(3*B*(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) - 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))/d

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.02 \[ \int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=-\frac {\frac {420 \, {\left (d x + c\right )} {\left (8 \, A - 21 \, B\right )}}{a^{4}} - \frac {840 \, {\left (2 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 147 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 189 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 805 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1365 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5145 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11655 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]

[In]

integrate(cos(d*x+c)^5*(A+B*cos(d*x+c))/(a+a*cos(d*x+c))^4,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.13 \[ \int \frac {\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,\left (A-B\right )}{4\,a^4}-\frac {5\,B}{2\,a^4}+\frac {3\,\left (4\,A-6\,B\right )}{4\,a^4}+\frac {3\,\left (5\,A-15\,B\right )}{8\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-B}{4\,a^4}+\frac {4\,A-6\,B}{8\,a^4}+\frac {5\,A-15\,B}{24\,a^4}\right )}{d}-\frac {x\,\left (8\,A-21\,B\right )}{2\,a^4}+\frac {\left (2\,A-9\,B\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A-7\,B\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,\left (A-B\right )}{40\,a^4}+\frac {4\,A-6\,B}{40\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B\right )}{56\,a^4\,d} \]

[In]

int((cos(c + d*x)^5*(A + B*cos(c + d*x)))/(a + a*cos(c + d*x))^4,x)

[Out]

(tan(c/2 + (d*x)/2)*((5*(A - B))/(4*a^4) - (5*B)/(2*a^4) + (3*(4*A - 6*B))/(4*a^4) + (3*(5*A - 15*B))/(8*a^4))
)/d - (tan(c/2 + (d*x)/2)^3*((A - B)/(4*a^4) + (4*A - 6*B)/(8*a^4) + (5*A - 15*B)/(24*a^4)))/d - (x*(8*A - 21*
B))/(2*a^4) + (tan(c/2 + (d*x)/2)^3*(2*A - 9*B) + tan(c/2 + (d*x)/2)*(2*A - 7*B))/(d*(2*a^4*tan(c/2 + (d*x)/2)
^2 + a^4*tan(c/2 + (d*x)/2)^4 + a^4)) + (tan(c/2 + (d*x)/2)^5*((3*(A - B))/(40*a^4) + (4*A - 6*B)/(40*a^4)))/d
- (tan(c/2 + (d*x)/2)^7*(A - B))/(56*a^4*d)