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

Optimal. Leaf size=185 \[ -\frac{(55 A-244 B) \sin (c+d x)}{105 a^4 d}+\frac{(25 A-88 B) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(A-4 B) \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}+\frac{x (A-4 B)}{a^4}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(5 A-12 B) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.679183, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {2977, 2968, 3023, 12, 2735, 2648} \[ -\frac{(55 A-244 B) \sin (c+d x)}{105 a^4 d}+\frac{(25 A-88 B) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(A-4 B) \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}+\frac{x (A-4 B)}{a^4}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(5 A-12 B) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[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 2977

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 2968

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 3023

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]

Rule 12

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

Rule 2735

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 2648

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]

Rubi steps

\begin{align*} \int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx &=\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]  time = 0.854941, size = 481, normalized size = 2.6 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (7350 d x (A-4 B) \cos \left (c+\frac{d x}{2}\right )+7350 d x (A-4 B) \cos \left (\frac{d x}{2}\right )+16520 A \sin \left (c+\frac{d x}{2}\right )-14280 A \sin \left (c+\frac{3 d x}{2}\right )+7560 A \sin \left (2 c+\frac{3 d x}{2}\right )-5600 A \sin \left (2 c+\frac{5 d x}{2}\right )+1680 A \sin \left (3 c+\frac{5 d x}{2}\right )-1040 A \sin \left (3 c+\frac{7 d x}{2}\right )+4410 A d x \cos \left (c+\frac{3 d x}{2}\right )+4410 A d x \cos \left (2 c+\frac{3 d x}{2}\right )+1470 A d x \cos \left (2 c+\frac{5 d x}{2}\right )+1470 A d x \cos \left (3 c+\frac{5 d x}{2}\right )+210 A d x \cos \left (3 c+\frac{7 d x}{2}\right )+210 A d x \cos \left (4 c+\frac{7 d x}{2}\right )-19880 A \sin \left (\frac{d x}{2}\right )-46130 B \sin \left (c+\frac{d x}{2}\right )+46116 B \sin \left (c+\frac{3 d x}{2}\right )-18060 B \sin \left (2 c+\frac{3 d x}{2}\right )+19292 B \sin \left (2 c+\frac{5 d x}{2}\right )-2100 B \sin \left (3 c+\frac{5 d x}{2}\right )+3791 B \sin \left (3 c+\frac{7 d x}{2}\right )+735 B \sin \left (4 c+\frac{7 d x}{2}\right )+105 B \sin \left (4 c+\frac{9 d x}{2}\right )+105 B \sin \left (5 c+\frac{9 d x}{2}\right )-17640 B d x \cos \left (c+\frac{3 d x}{2}\right )-17640 B d x \cos \left (2 c+\frac{3 d x}{2}\right )-5880 B d x \cos \left (2 c+\frac{5 d x}{2}\right )-5880 B d x \cos \left (3 c+\frac{5 d x}{2}\right )-840 B d x \cos \left (3 c+\frac{7 d x}{2}\right )-840 B d x \cos \left (4 c+\frac{7 d x}{2}\right )+60830 B \sin \left (\frac{d x}{2}\right )\right )}{1680 a^4 d (\cos (c+d x)+1)^4} \]

Antiderivative was successfully verified.

[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]  time = 0.058, size = 229, normalized size = 1.2 \begin{align*}{\frac{A}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{B}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{A}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{7\,B}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{11\,A}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{23\,B}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{15\,A}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{49\,B}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{B\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{4} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) A}{d{a}^{4}}}-8\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) B}{d{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/56/d/a^4*tan(1/2*d*x+1/2*c)^7*A-1/56/d/a^4*B*tan(1/2*d*x+1/2*c)^7-1/8/d/a^4*A*tan(1/2*d*x+1/2*c)^5+7/40/d/a^
4*B*tan(1/2*d*x+1/2*c)^5+11/24/d/a^4*tan(1/2*d*x+1/2*c)^3*A-23/24/d/a^4*B*tan(1/2*d*x+1/2*c)^3-15/8/d/a^4*A*ta
n(1/2*d*x+1/2*c)+49/8/d/a^4*B*tan(1/2*d*x+1/2*c)+2/d/a^4*B*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)+2/d/a^4
*arctan(tan(1/2*d*x+1/2*c))*A-8/d/a^4*arctan(tan(1/2*d*x+1/2*c))*B

________________________________________________________________________________________

Maxima [A]  time = 1.47777, size = 366, normalized size = 1.98 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [A]  time = 1.31615, size = 572, normalized size = 3.09 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [A]  time = 42.4924, size = 578, normalized size = 3.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Giac [A]  time = 1.28351, size = 254, normalized size = 1.37 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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