\(\int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx\) [66]

Optimal result

Integrand size = 31, antiderivative size = 185 \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=\frac {(A-4 B) x}{a^4}-\frac {(55 A-244 B) \sin (c+d x)}{105 a^4 d}+\frac {(25 A-88 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-4 B) \sin (c+d x)}{a^4 d (1+\cos (c+d x))}+\frac {(A-B) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(5 A-12 B) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3} \]

[Out]

(A-4*B)*x/a^4-1/105*(55*A-244*B)*sin(d*x+c)/a^4/d+1/105*(25*A-88*B)*cos(d*x+c)^2*sin(d*x+c)/a^4/d/(1+cos(d*x+c
))^2-(A-4*B)*sin(d*x+c)/a^4/d/(1+cos(d*x+c))+1/7*(A-B)*cos(d*x+c)^4*sin(d*x+c)/d/(a+a*cos(d*x+c))^4+1/35*(5*A-
12*B)*cos(d*x+c)^3*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^3

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3056, 3047, 3102, 12, 2814, 2727} \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=-\frac {(55 A-244 B) \sin (c+d x)}{105 a^4 d}+\frac {(25 A-88 B) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac {(A-4 B) \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}+\frac {x (A-4 B)}{a^4}+\frac {(A-B) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac {(5 A-12 B) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]

[In]

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

[Out]

((A - 4*B)*x)/a^4 - ((55*A - 244*B)*Sin[c + d*x])/(105*a^4*d) + ((25*A - 88*B)*Cos[c + d*x]^2*Sin[c + d*x])/(1
05*a^4*d*(1 + Cos[c + d*x])^2) - ((A - 4*B)*Sin[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) + ((A - B)*Cos[c + d*x]^4
*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((5*A - 12*B)*Cos[c + d*x]^3*Sin[c + d*x])/(35*a*d*(a + a*Cos[c
+ d*x])^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2727

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2814

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*(x/d)
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, 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])

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {(A-B) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {\int \frac {\cos ^3(c+d x) (4 a (A-B)-a (A-8 B) \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx}{7 a^2} \\ & = \frac {(A-B) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(5 A-12 B) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\cos ^2(c+d x) \left (3 a^2 (5 A-12 B)-2 a^2 (5 A-26 B) \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4} \\ & = \frac {(25 A-88 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac {(A-B) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(5 A-12 B) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\cos (c+d x) \left (2 a^3 (25 A-88 B)-a^3 (55 A-244 B) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{105 a^6} \\ & = \frac {(25 A-88 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac {(A-B) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(5 A-12 B) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {2 a^3 (25 A-88 B) \cos (c+d x)-a^3 (55 A-244 B) \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6} \\ & = -\frac {(55 A-244 B) \sin (c+d x)}{105 a^4 d}+\frac {(25 A-88 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac {(A-B) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(5 A-12 B) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {105 a^4 (A-4 B) \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^7} \\ & = -\frac {(55 A-244 B) \sin (c+d x)}{105 a^4 d}+\frac {(25 A-88 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac {(A-B) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(5 A-12 B) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {(A-4 B) \int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx}{a^3} \\ & = \frac {(A-4 B) x}{a^4}-\frac {(55 A-244 B) \sin (c+d x)}{105 a^4 d}+\frac {(25 A-88 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac {(A-B) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(5 A-12 B) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(A-4 B) \int \frac {1}{a+a \cos (c+d x)} \, dx}{a^3} \\ & = \frac {(A-4 B) x}{a^4}-\frac {(55 A-244 B) \sin (c+d x)}{105 a^4 d}+\frac {(25 A-88 B) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac {(A-B) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(5 A-12 B) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(A-4 B) \sin (c+d x)}{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. \(481\) vs. \(2(185)=370\).

Time = 4.77 (sec) , antiderivative size = 481, normalized size of antiderivative = 2.60 \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (7350 (A-4 B) d x \cos \left (\frac {d x}{2}\right )+7350 (A-4 B) d x \cos \left (c+\frac {d x}{2}\right )+4410 A d x \cos \left (c+\frac {3 d x}{2}\right )-17640 B d x \cos \left (c+\frac {3 d x}{2}\right )+4410 A d x \cos \left (2 c+\frac {3 d x}{2}\right )-17640 B d x \cos \left (2 c+\frac {3 d x}{2}\right )+1470 A d x \cos \left (2 c+\frac {5 d x}{2}\right )-5880 B d x \cos \left (2 c+\frac {5 d x}{2}\right )+1470 A d x \cos \left (3 c+\frac {5 d x}{2}\right )-5880 B d x \cos \left (3 c+\frac {5 d x}{2}\right )+210 A d x \cos \left (3 c+\frac {7 d x}{2}\right )-840 B d x \cos \left (3 c+\frac {7 d x}{2}\right )+210 A d x \cos \left (4 c+\frac {7 d x}{2}\right )-840 B d x \cos \left (4 c+\frac {7 d x}{2}\right )-19880 A \sin \left (\frac {d x}{2}\right )+60830 B \sin \left (\frac {d x}{2}\right )+16520 A \sin \left (c+\frac {d x}{2}\right )-46130 B \sin \left (c+\frac {d x}{2}\right )-14280 A \sin \left (c+\frac {3 d x}{2}\right )+46116 B \sin \left (c+\frac {3 d x}{2}\right )+7560 A \sin \left (2 c+\frac {3 d x}{2}\right )-18060 B \sin \left (2 c+\frac {3 d x}{2}\right )-5600 A \sin \left (2 c+\frac {5 d x}{2}\right )+19292 B \sin \left (2 c+\frac {5 d x}{2}\right )+1680 A \sin \left (3 c+\frac {5 d x}{2}\right )-2100 B \sin \left (3 c+\frac {5 d x}{2}\right )-1040 A \sin \left (3 c+\frac {7 d x}{2}\right )+3791 B \sin \left (3 c+\frac {7 d x}{2}\right )+735 B \sin \left (4 c+\frac {7 d x}{2}\right )+105 B \sin \left (4 c+\frac {9 d x}{2}\right )+105 B \sin \left (5 c+\frac {9 d x}{2}\right )\right )}{1680 a^4 d (1+\cos (c+d x))^4} \]

[In]

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

[Out]

(Cos[(c + d*x)/2]*Sec[c/2]*(7350*(A - 4*B)*d*x*Cos[(d*x)/2] + 7350*(A - 4*B)*d*x*Cos[c + (d*x)/2] + 4410*A*d*x
*Cos[c + (3*d*x)/2] - 17640*B*d*x*Cos[c + (3*d*x)/2] + 4410*A*d*x*Cos[2*c + (3*d*x)/2] - 17640*B*d*x*Cos[2*c +
(3*d*x)/2] + 1470*A*d*x*Cos[2*c + (5*d*x)/2] - 5880*B*d*x*Cos[2*c + (5*d*x)/2] + 1470*A*d*x*Cos[3*c + (5*d*x)
/2] - 5880*B*d*x*Cos[3*c + (5*d*x)/2] + 210*A*d*x*Cos[3*c + (7*d*x)/2] - 840*B*d*x*Cos[3*c + (7*d*x)/2] + 210*
A*d*x*Cos[4*c + (7*d*x)/2] - 840*B*d*x*Cos[4*c + (7*d*x)/2] - 19880*A*Sin[(d*x)/2] + 60830*B*Sin[(d*x)/2] + 16
520*A*Sin[c + (d*x)/2] - 46130*B*Sin[c + (d*x)/2] - 14280*A*Sin[c + (3*d*x)/2] + 46116*B*Sin[c + (3*d*x)/2] +
7560*A*Sin[2*c + (3*d*x)/2] - 18060*B*Sin[2*c + (3*d*x)/2] - 5600*A*Sin[2*c + (5*d*x)/2] + 19292*B*Sin[2*c + (
5*d*x)/2] + 1680*A*Sin[3*c + (5*d*x)/2] - 2100*B*Sin[3*c + (5*d*x)/2] - 1040*A*Sin[3*c + (7*d*x)/2] + 3791*B*S
in[3*c + (7*d*x)/2] + 735*B*Sin[4*c + (7*d*x)/2] + 105*B*Sin[4*c + (9*d*x)/2] + 105*B*Sin[5*c + (9*d*x)/2]))/(
1680*a^4*d*(1 + Cos[c + d*x])^4)

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.57

method	result	size
parallelrisch	\(\frac {-5840 \left (\left (\frac {31 A}{73}-\frac {2741 B}{1460}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {13 A}{146}-\frac {148 B}{365}\right ) \cos \left (3 d x +3 c \right )-\frac {21 B \cos \left (4 d x +4 c \right )}{1168}+\left (A -\frac {1562 B}{365}\right ) \cos \left (d x +c \right )+\frac {47 A}{73}-\frac {16171 B}{5840}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sec ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6720 d x \left (A -4 B \right )}{6720 a^{4} d}\)	\(106\)
derivativedivides	\(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}-A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}-\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}-15 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {16 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+16 \left (A -4 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\)	\(162\)
default	\(\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{7}-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{7}-A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{5}+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{3}-\frac {23 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3}-15 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+49 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {16 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+16 \left (A -4 B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}\)	\(162\)
risch	\(\frac {x A}{a^{4}}-\frac {4 B x}{a^{4}}-\frac {i {\mathrm e}^{i \left (d x +c \right )} B}{2 a^{4} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} B}{2 a^{4} d}-\frac {4 i \left (210 A \,{\mathrm e}^{6 i \left (d x +c \right )}-525 B \,{\mathrm e}^{6 i \left (d x +c \right )}+945 A \,{\mathrm e}^{5 i \left (d x +c \right )}-2625 B \,{\mathrm e}^{5 i \left (d x +c \right )}+2065 A \,{\mathrm e}^{4 i \left (d x +c \right )}-5950 B \,{\mathrm e}^{4 i \left (d x +c \right )}+2485 A \,{\mathrm e}^{3 i \left (d x +c \right )}-7420 B \,{\mathrm e}^{3 i \left (d x +c \right )}+1785 A \,{\mathrm e}^{2 i \left (d x +c \right )}-5397 B \,{\mathrm e}^{2 i \left (d x +c \right )}+700 A \,{\mathrm e}^{i \left (d x +c \right )}-2149 B \,{\mathrm e}^{i \left (d x +c \right )}+130 A -382 B \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\)	\(226\)
norman	\(\frac {\frac {\left (A -4 B \right ) x}{a}+\frac {\left (A -4 B \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {\left (A -22 B \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{84 a d}+\frac {5 \left (A -4 B \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {10 \left (A -4 B \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {10 \left (A -4 B \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {5 \left (A -4 B \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {\left (A -B \right ) \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 a d}-\frac {5 \left (3 A -13 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {\left (5 A -12 B \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{140 a d}-\frac {25 \left (20 A -83 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{84 a d}-\frac {\left (55 A -244 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{84 a d}-\frac {\left (107 A -452 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a d}-\frac {\left (995 A -4118 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 a d}-\frac {\left (1241 A -5084 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{84 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} a^{3}}\)	\(368\)

[In]

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

[Out]

1/6720*(-5840*((31/73*A-2741/1460*B)*cos(2*d*x+2*c)+(13/146*A-148/365*B)*cos(3*d*x+3*c)-21/1168*B*cos(4*d*x+4*
c)+(A-1562/365*B)*cos(d*x+c)+47/73*A-16171/5840*B)*tan(1/2*d*x+1/2*c)*sec(1/2*d*x+1/2*c)^6+6720*d*x*(A-4*B))/a
^4/d

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.15 \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=\frac {105 \, {\left (A - 4 \, B\right )} d x \cos \left (d x + c\right )^{4} + 420 \, {\left (A - 4 \, B\right )} d x \cos \left (d x + c\right )^{3} + 630 \, {\left (A - 4 \, B\right )} d x \cos \left (d x + c\right )^{2} + 420 \, {\left (A - 4 \, B\right )} d x \cos \left (d x + c\right ) + 105 \, {\left (A - 4 \, B\right )} d x + {\left (105 \, B \cos \left (d x + c\right )^{4} - 4 \, {\left (65 \, A - 296 \, B\right )} \cos \left (d x + c\right )^{3} - 4 \, {\left (155 \, A - 659 \, B\right )} \cos \left (d x + c\right )^{2} - {\left (535 \, A - 2236 \, B\right )} \cos \left (d x + c\right ) - 160 \, A + 664 \, B\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]

[In]

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

[Out]

1/105*(105*(A - 4*B)*d*x*cos(d*x + c)^4 + 420*(A - 4*B)*d*x*cos(d*x + c)^3 + 630*(A - 4*B)*d*x*cos(d*x + c)^2
+ 420*(A - 4*B)*d*x*cos(d*x + c) + 105*(A - 4*B)*d*x + (105*B*cos(d*x + c)^4 - 4*(65*A - 296*B)*cos(d*x + c)^3
- 4*(155*A - 659*B)*cos(d*x + c)^2 - (535*A - 2236*B)*cos(d*x + c) - 160*A + 664*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. 578 vs. \(2 (172) = 344\).

Time = 5.07 (sec) , antiderivative size = 578, normalized size of antiderivative = 3.12 \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=\begin {cases} \frac {840 A d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {840 A d x}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {15 A \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {90 A \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {280 A \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {1190 A \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {1575 A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {3360 B d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {3360 B d x}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {15 B \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {132 B \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {658 B \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {4340 B \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {6825 B \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \left (A + B \cos {\left (c \right )}\right ) \cos ^{4}{\left (c \right )}}{\left (a \cos {\left (c \right )} + a\right )^{4}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((840*A*d*x*tan(c/2 + d*x/2)**2/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 840*A*d*x/(840*a**4*d
*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 15*A*tan(c/2 + d*x/2)**9/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) -
90*A*tan(c/2 + d*x/2)**7/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 280*A*tan(c/2 + d*x/2)**5/(840*a**4*d
*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 1190*A*tan(c/2 + d*x/2)**3/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d)
- 1575*A*tan(c/2 + d*x/2)/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 3360*B*d*x*tan(c/2 + d*x/2)**2/(840*
a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 3360*B*d*x/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 15*B*tan
(c/2 + d*x/2)**9/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 132*B*tan(c/2 + d*x/2)**7/(840*a**4*d*tan(c/2
+ d*x/2)**2 + 840*a**4*d) - 658*B*tan(c/2 + d*x/2)**5/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 4340*B*
tan(c/2 + d*x/2)**3/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 6825*B*tan(c/2 + d*x/2)/(840*a**4*d*tan(c/
2 + d*x/2)**2 + 840*a**4*d), Ne(d, 0)), (x*(A + B*cos(c))*cos(c)**4/(a*cos(c) + a)**4, True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.46 \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=\frac {B {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} + \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {6720 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - 5 \, A {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {336 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )}}{840 \, d} \]

[In]

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

[Out]

1/840*(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) - 5*A*((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))/d

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.02 \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {840 \, {\left (d x + c\right )} {\left (A - 4 \, B\right )}}{a^{4}} + \frac {1680 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 147 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 805 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1575 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5145 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]

[In]

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

[Out]

1/840*(840*(d*x + c)*(A - 4*B)/a^4 + 1680*B*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 - 105*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 + 385*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 - 1575*A*a^24*t
an(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 = 0.43 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.09 \[ \int \frac {\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx=\frac {A\,d\,x-4\,B\,d\,x}{a^4\,d}-\frac {\left (\frac {52\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}-\frac {764\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {143\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}-\frac {16\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (\frac {5\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{28}-\frac {8\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}+\frac {B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}}{a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}+\frac {2\,B\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^4\,d} \]

[In]

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

[Out]

(A*d*x - 4*B*d*x)/(a^4*d) - ((B*sin(c/2 + (d*x)/2))/56 - (A*sin(c/2 + (d*x)/2))/56 + cos(c/2 + (d*x)/2)^2*((5*
A*sin(c/2 + (d*x)/2))/28 - (8*B*sin(c/2 + (d*x)/2))/35) - cos(c/2 + (d*x)/2)^4*((16*A*sin(c/2 + (d*x)/2))/21 -
(143*B*sin(c/2 + (d*x)/2))/105) + cos(c/2 + (d*x)/2)^6*((52*A*sin(c/2 + (d*x)/2))/21 - (764*B*sin(c/2 + (d*x)
/2))/105))/(a^4*d*cos(c/2 + (d*x)/2)^7) + (2*B*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2))/(a^4*d)