3.365 \(\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{(a+a \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=195 \[ \frac{(6 A-55 B+244 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A+25 B-88 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(B-4 C) \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}+\frac{x (B-4 C)}{a^4}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(2 A+5 B-12 C) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]

[Out]

((B - 4*C)*x)/a^4 + ((6*A - 55*B + 244*C)*Sin[c + d*x])/(105*a^4*d) + ((3*A + 25*B - 88*C)*Cos[c + d*x]^2*Sin[
c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - ((B - 4*C)*Sin[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - ((A - B + C
)*Cos[c + d*x]^4*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((2*A + 5*B - 12*C)*Cos[c + d*x]^3*Sin[c + d*x])
/(35*a*d*(a + a*Cos[c + d*x])^3)

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Rubi [A]  time = 0.648229, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {3041, 2977, 2968, 3023, 12, 2735, 2648} \[ \frac{(6 A-55 B+244 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A+25 B-88 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(B-4 C) \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}+\frac{x (B-4 C)}{a^4}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(2 A+5 B-12 C) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]

[Out]

((B - 4*C)*x)/a^4 + ((6*A - 55*B + 244*C)*Sin[c + d*x])/(105*a^4*d) + ((3*A + 25*B - 88*C)*Cos[c + d*x]^2*Sin[
c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])^2) - ((B - 4*C)*Sin[c + d*x])/(a^4*d*(1 + Cos[c + d*x])) - ((A - B + C
)*Cos[c + d*x]^4*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((2*A + 5*B - 12*C)*Cos[c + d*x]^3*Sin[c + d*x])
/(35*a*d*(a + a*Cos[c + d*x])^3)

Rule 3041

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

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 ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx &=-\frac{(A-B+C) \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) (a (3 A+4 B-4 C)+a (A-B+8 C) \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(2 A+5 B-12 C) \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 (2 A+5 B-12 C)+a^2 (3 A-10 B+52 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=\frac{(3 A+25 B-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(2 A+5 B-12 C) \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 (3 A+25 B-88 C)+a^3 (6 A-55 B+244 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=\frac{(3 A+25 B-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(2 A+5 B-12 C) \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 (3 A+25 B-88 C) \cos (c+d x)+a^3 (6 A-55 B+244 C) \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=\frac{(6 A-55 B+244 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A+25 B-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(2 A+5 B-12 C) \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 (B-4 C) \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^7}\\ &=\frac{(6 A-55 B+244 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A+25 B-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(2 A+5 B-12 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{(B-4 C) \int \frac{\cos (c+d x)}{a+a \cos (c+d x)} \, dx}{a^3}\\ &=\frac{(B-4 C) x}{a^4}+\frac{(6 A-55 B+244 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A+25 B-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(2 A+5 B-12 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{(B-4 C) \int \frac{1}{a+a \cos (c+d x)} \, dx}{a^3}\\ &=\frac{(B-4 C) x}{a^4}+\frac{(6 A-55 B+244 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A+25 B-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(2 A+5 B-12 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{(B-4 C) \sin (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 1.10958, size = 571, normalized size = 2.93 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (-2520 A \sin \left (c+\frac{d x}{2}\right )+1764 A \sin \left (c+\frac{3 d x}{2}\right )-1260 A \sin \left (2 c+\frac{3 d x}{2}\right )+588 A \sin \left (2 c+\frac{5 d x}{2}\right )-420 A \sin \left (3 c+\frac{5 d x}{2}\right )+144 A \sin \left (3 c+\frac{7 d x}{2}\right )+2520 A \sin \left (\frac{d x}{2}\right )+7350 d x (B-4 C) \cos \left (c+\frac{d x}{2}\right )+16520 B \sin \left (c+\frac{d x}{2}\right )-14280 B \sin \left (c+\frac{3 d x}{2}\right )+7560 B \sin \left (2 c+\frac{3 d x}{2}\right )-5600 B \sin \left (2 c+\frac{5 d x}{2}\right )+1680 B \sin \left (3 c+\frac{5 d x}{2}\right )-1040 B \sin \left (3 c+\frac{7 d x}{2}\right )+4410 B d x \cos \left (c+\frac{3 d x}{2}\right )+4410 B d x \cos \left (2 c+\frac{3 d x}{2}\right )+1470 B d x \cos \left (2 c+\frac{5 d x}{2}\right )+1470 B d x \cos \left (3 c+\frac{5 d x}{2}\right )+210 B d x \cos \left (3 c+\frac{7 d x}{2}\right )+210 B d x \cos \left (4 c+\frac{7 d x}{2}\right )+7350 d x (B-4 C) \cos \left (\frac{d x}{2}\right )-19880 B \sin \left (\frac{d x}{2}\right )-46130 C \sin \left (c+\frac{d x}{2}\right )+46116 C \sin \left (c+\frac{3 d x}{2}\right )-18060 C \sin \left (2 c+\frac{3 d x}{2}\right )+19292 C \sin \left (2 c+\frac{5 d x}{2}\right )-2100 C \sin \left (3 c+\frac{5 d x}{2}\right )+3791 C \sin \left (3 c+\frac{7 d x}{2}\right )+735 C \sin \left (4 c+\frac{7 d x}{2}\right )+105 C \sin \left (4 c+\frac{9 d x}{2}\right )+105 C \sin \left (5 c+\frac{9 d x}{2}\right )-17640 C d x \cos \left (c+\frac{3 d x}{2}\right )-17640 C d x \cos \left (2 c+\frac{3 d x}{2}\right )-5880 C d x \cos \left (2 c+\frac{5 d x}{2}\right )-5880 C d x \cos \left (3 c+\frac{5 d x}{2}\right )-840 C d x \cos \left (3 c+\frac{7 d x}{2}\right )-840 C d x \cos \left (4 c+\frac{7 d x}{2}\right )+60830 C \sin \left (\frac{d x}{2}\right )\right )}{1680 a^4 d (\cos (c+d x)+1)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]

[Out]

(Cos[(c + d*x)/2]*Sec[c/2]*(7350*(B - 4*C)*d*x*Cos[(d*x)/2] + 7350*(B - 4*C)*d*x*Cos[c + (d*x)/2] + 4410*B*d*x
*Cos[c + (3*d*x)/2] - 17640*C*d*x*Cos[c + (3*d*x)/2] + 4410*B*d*x*Cos[2*c + (3*d*x)/2] - 17640*C*d*x*Cos[2*c +
(3*d*x)/2] + 1470*B*d*x*Cos[2*c + (5*d*x)/2] - 5880*C*d*x*Cos[2*c + (5*d*x)/2] + 1470*B*d*x*Cos[3*c + (5*d*x)
/2] - 5880*C*d*x*Cos[3*c + (5*d*x)/2] + 210*B*d*x*Cos[3*c + (7*d*x)/2] - 840*C*d*x*Cos[3*c + (7*d*x)/2] + 210*
B*d*x*Cos[4*c + (7*d*x)/2] - 840*C*d*x*Cos[4*c + (7*d*x)/2] + 2520*A*Sin[(d*x)/2] - 19880*B*Sin[(d*x)/2] + 608
30*C*Sin[(d*x)/2] - 2520*A*Sin[c + (d*x)/2] + 16520*B*Sin[c + (d*x)/2] - 46130*C*Sin[c + (d*x)/2] + 1764*A*Sin
[c + (3*d*x)/2] - 14280*B*Sin[c + (3*d*x)/2] + 46116*C*Sin[c + (3*d*x)/2] - 1260*A*Sin[2*c + (3*d*x)/2] + 7560
*B*Sin[2*c + (3*d*x)/2] - 18060*C*Sin[2*c + (3*d*x)/2] + 588*A*Sin[2*c + (5*d*x)/2] - 5600*B*Sin[2*c + (5*d*x)
/2] + 19292*C*Sin[2*c + (5*d*x)/2] - 420*A*Sin[3*c + (5*d*x)/2] + 1680*B*Sin[3*c + (5*d*x)/2] - 2100*C*Sin[3*c
+ (5*d*x)/2] + 144*A*Sin[3*c + (7*d*x)/2] - 1040*B*Sin[3*c + (7*d*x)/2] + 3791*C*Sin[3*c + (7*d*x)/2] + 735*C
*Sin[4*c + (7*d*x)/2] + 105*C*Sin[4*c + (9*d*x)/2] + 105*C*Sin[5*c + (9*d*x)/2]))/(1680*a^4*d*(1 + Cos[c + d*x
])^4)

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Maple [A]  time = 0.034, size = 307, normalized size = 1.6 \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{C}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{3\,A}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{B}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{7\,C}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{A}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{11\,B}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{23\,C}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{A}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{15\,B}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{49\,C}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{C\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{4} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) }}+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) B}{d{a}^{4}}}-8\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) C}{d{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^4,x)

[Out]

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

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Maxima [A]  time = 1.57601, size = 481, normalized size = 2.47 \begin{align*} \frac{C{\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 \, A{\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

1/840*(C*(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*A*(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

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Fricas [A]  time = 1.99902, size = 606, normalized size = 3.11 \begin{align*} \frac{105 \,{\left (B - 4 \, C\right )} d x \cos \left (d x + c\right )^{4} + 420 \,{\left (B - 4 \, C\right )} d x \cos \left (d x + c\right )^{3} + 630 \,{\left (B - 4 \, C\right )} d x \cos \left (d x + c\right )^{2} + 420 \,{\left (B - 4 \, C\right )} d x \cos \left (d x + c\right ) + 105 \,{\left (B - 4 \, C\right )} d x +{\left (105 \, C \cos \left (d x + c\right )^{4} + 4 \,{\left (9 \, A - 65 \, B + 296 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (39 \, A - 620 \, B + 2636 \, C\right )} \cos \left (d x + c\right )^{2} +{\left (24 \, A - 535 \, B + 2236 \, C\right )} \cos \left (d x + c\right ) + 6 \, A - 160 \, B + 664 \, 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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 58.6102, size = 746, normalized size = 3.83 \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)**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+a*cos(d*x+c))**4,x)

[Out]

Piecewise((-15*A*tan(c/2 + d*x/2)**9/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 48*A*tan(c/2 + d*x/2)**7/
(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 42*A*tan(c/2 + d*x/2)**5/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840
*a**4*d) + 105*A*tan(c/2 + d*x/2)/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 840*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) + 840*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) - 90*B*tan(c/2 + d*x/2)**7/(840*a**4*d*ta
n(c/2 + d*x/2)**2 + 840*a**4*d) + 280*B*tan(c/2 + d*x/2)**5/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 11
90*B*tan(c/2 + d*x/2)**3/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 1575*B*tan(c/2 + d*x/2)/(840*a**4*d*t
an(c/2 + d*x/2)**2 + 840*a**4*d) - 3360*C*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*C*d*x/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 15*C*tan(c/2 + d*x/2)**9/(840*a**4*d*tan(c/2 +
d*x/2)**2 + 840*a**4*d) + 132*C*tan(c/2 + d*x/2)**7/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) - 658*C*tan(
c/2 + d*x/2)**5/(840*a**4*d*tan(c/2 + d*x/2)**2 + 840*a**4*d) + 4340*C*tan(c/2 + d*x/2)**3/(840*a**4*d*tan(c/2
+ d*x/2)**2 + 840*a**4*d) + 6825*C*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) + C*cos(c)**2)*cos(c)**3/(a*cos(c) + a)**4, True))

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Giac [A]  time = 1.24765, size = 344, normalized size = 1.76 \begin{align*} \frac{\frac{840 \,{\left (d x + c\right )}{\left (B - 4 \, C\right )}}{a^{4}} + \frac{1680 \, C \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} - 63 \, 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} - 147 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 105 \, 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} + 805 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 105 \, 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 ) - 5145 \, C 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)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+a*cos(d*x+c))^4,x, algorithm="giac")

[Out]

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