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

Optimal. Leaf size=245 \[ -\frac {8 (20 A-83 B+216 C) \sin (c+d x)}{105 a^4 d}-\frac {(10 A-52 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac {4 (20 A-83 B+216 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac {(2 A-8 B+21 C) \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac {x (2 A-8 B+21 C)}{2 a^4}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac {(B-2 C) \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3} \]

[Out]

1/2*(2*A-8*B+21*C)*x/a^4-8/105*(20*A-83*B+216*C)*sin(d*x+c)/a^4/d+1/2*(2*A-8*B+21*C)*cos(d*x+c)*sin(d*x+c)/a^4
/d-1/105*(10*A-52*B+129*C)*cos(d*x+c)^3*sin(d*x+c)/a^4/d/(1+cos(d*x+c))^2-4/105*(20*A-83*B+216*C)*cos(d*x+c)^2
*sin(d*x+c)/a^4/d/(1+cos(d*x+c))-1/7*(A-B+C)*cos(d*x+c)^5*sin(d*x+c)/d/(a+a*cos(d*x+c))^4+1/5*(B-2*C)*cos(d*x+
c)^4*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^3

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Rubi [A]  time = 0.69, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {3041, 2977, 2734} \[ -\frac {8 (20 A-83 B+216 C) \sin (c+d x)}{105 a^4 d}-\frac {(10 A-52 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac {4 (20 A-83 B+216 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac {(2 A-8 B+21 C) \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac {x (2 A-8 B+21 C)}{2 a^4}-\frac {(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac {(B-2 C) \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*A - 8*B + 21*C)*x)/(2*a^4) - (8*(20*A - 83*B + 216*C)*Sin[c + d*x])/(105*a^4*d) + ((2*A - 8*B + 21*C)*Cos[
c + d*x]*Sin[c + d*x])/(2*a^4*d) - ((10*A - 52*B + 129*C)*Cos[c + d*x]^3*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c +
d*x])^2) - (4*(20*A - 83*B + 216*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C
)*Cos[c + d*x]^5*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((B - 2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(5*a*d*(
a + a*Cos[c + d*x])^3)

Rule 2734

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 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 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)]

Rubi steps

\begin {align*} \int \frac {\cos ^4(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 ^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) (a (2 A+5 B-5 C)+a (2 A-2 B+9 C) \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(B-2 C) \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 (B-2 C)+a^2 (10 A-24 B+73 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac {(10 A-52 B+129 C) \cos ^3(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(B-2 C) \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 (10 A-52 B+129 C)+a^3 (50 A-176 B+477 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=-\frac {(10 A-52 B+129 C) \cos ^3(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(B-2 C) \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac {4 (20 A-83 B+216 C) \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 (20 A-83 B+216 C)+105 a^4 (2 A-8 B+21 C) \cos (c+d x)\right ) \, dx}{105 a^8}\\ &=\frac {(2 A-8 B+21 C) x}{2 a^4}-\frac {8 (20 A-83 B+216 C) \sin (c+d x)}{105 a^4 d}+\frac {(2 A-8 B+21 C) \cos (c+d x) \sin (c+d x)}{2 a^4 d}-\frac {(10 A-52 B+129 C) \cos ^3(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(B-2 C) \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}-\frac {4 (20 A-83 B+216 C) \cos ^2(c+d x) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end {align*}

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Mathematica [A]  time = 2.87, size = 299, normalized size = 1.22 \[ \frac {2 \cos \left (\frac {1}{2} (c+d x)\right ) \left (210 \cos ^7\left (\frac {1}{2} (c+d x)\right ) (2 d x (2 A-8 B+21 C)+4 (B-4 C) \sin (c+d x)+C \sin (2 (c+d x)))+4 \tan \left (\frac {c}{2}\right ) (160 A-286 B+447 C) \cos ^5\left (\frac {1}{2} (c+d x)\right )-6 \tan \left (\frac {c}{2}\right ) (25 A-32 B+39 C) \cos ^3\left (\frac {1}{2} (c+d x)\right )+15 \tan \left (\frac {c}{2}\right ) (A-B+C) \cos \left (\frac {1}{2} (c+d x)\right )+15 \sec \left (\frac {c}{2}\right ) (A-B+C) \sin \left (\frac {d x}{2}\right )-8 \sec \left (\frac {c}{2}\right ) (260 A-764 B+1653 C) \sin \left (\frac {d x}{2}\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right )+4 \sec \left (\frac {c}{2}\right ) (160 A-286 B+447 C) \sin \left (\frac {d x}{2}\right ) \cos ^4\left (\frac {1}{2} (c+d x)\right )-6 \sec \left (\frac {c}{2}\right ) (25 A-32 B+39 C) \sin \left (\frac {d x}{2}\right ) \cos ^2\left (\frac {1}{2} (c+d x)\right )\right )}{105 a^4 d (\cos (c+d x)+1)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Cos[(c + d*x)/2]*(15*(A - B + C)*Sec[c/2]*Sin[(d*x)/2] - 6*(25*A - 32*B + 39*C)*Cos[(c + d*x)/2]^2*Sec[c/2]
*Sin[(d*x)/2] + 4*(160*A - 286*B + 447*C)*Cos[(c + d*x)/2]^4*Sec[c/2]*Sin[(d*x)/2] - 8*(260*A - 764*B + 1653*C
)*Cos[(c + d*x)/2]^6*Sec[c/2]*Sin[(d*x)/2] + 210*Cos[(c + d*x)/2]^7*(2*(2*A - 8*B + 21*C)*d*x + 4*(B - 4*C)*Si
n[c + d*x] + C*Sin[2*(c + d*x)]) + 15*(A - B + C)*Cos[(c + d*x)/2]*Tan[c/2] - 6*(25*A - 32*B + 39*C)*Cos[(c +
d*x)/2]^3*Tan[c/2] + 4*(160*A - 286*B + 447*C)*Cos[(c + d*x)/2]^5*Tan[c/2]))/(105*a^4*d*(1 + Cos[c + d*x])^4)

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fricas [A]  time = 0.45, size = 265, normalized size = 1.08 \[ \frac {105 \, {\left (2 \, A - 8 \, B + 21 \, C\right )} d x \cos \left (d x + c\right )^{4} + 420 \, {\left (2 \, A - 8 \, B + 21 \, C\right )} d x \cos \left (d x + c\right )^{3} + 630 \, {\left (2 \, A - 8 \, B + 21 \, C\right )} d x \cos \left (d x + c\right )^{2} + 420 \, {\left (2 \, A - 8 \, B + 21 \, C\right )} d x \cos \left (d x + c\right ) + 105 \, {\left (2 \, A - 8 \, B + 21 \, C\right )} d x + {\left (105 \, C \cos \left (d x + c\right )^{5} + 210 \, {\left (B - 2 \, C\right )} \cos \left (d x + c\right )^{4} - 4 \, {\left (130 \, A - 592 \, B + 1509 \, C\right )} \cos \left (d x + c\right )^{3} - 4 \, {\left (310 \, A - 1318 \, B + 3411 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (1070 \, A - 4472 \, B + 11619 \, C\right )} \cos \left (d x + c\right ) - 320 \, A + 1328 \, B - 3456 \, C\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/210*(105*(2*A - 8*B + 21*C)*d*x*cos(d*x + c)^4 + 420*(2*A - 8*B + 21*C)*d*x*cos(d*x + c)^3 + 630*(2*A - 8*B
+ 21*C)*d*x*cos(d*x + c)^2 + 420*(2*A - 8*B + 21*C)*d*x*cos(d*x + c) + 105*(2*A - 8*B + 21*C)*d*x + (105*C*cos
(d*x + c)^5 + 210*(B - 2*C)*cos(d*x + c)^4 - 4*(130*A - 592*B + 1509*C)*cos(d*x + c)^3 - 4*(310*A - 1318*B + 3
411*C)*cos(d*x + c)^2 - (1070*A - 4472*B + 11619*C)*cos(d*x + c) - 320*A + 1328*B - 3456*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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giac [A]  time = 0.67, size = 302, normalized size = 1.23 \[ \frac {\frac {420 \, {\left (d x + c\right )} {\left (2 \, A - 8 \, B + 21 \, C\right )}}{a^{4}} + \frac {840 \, {\left (2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 147 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 189 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 805 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1365 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1575 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5145 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 11655 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(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*(420*(d*x + c)*(2*A - 8*B + 21*C)/a^4 + 840*(2*B*tan(1/2*d*x + 1/2*c)^3 - 9*C*tan(1/2*d*x + 1/2*c)^3 + 2
*B*tan(1/2*d*x + 1/2*c) - 7*C*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 + 15*C*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 - 189*C*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 + 1365*C*a^24*tan(1/2*d*x + 1/2*c)^3 - 1575*A*a^24*tan(1/2*d*x +
1/2*c) + 5145*B*a^24*tan(1/2*d*x + 1/2*c) - 11655*C*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d

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maple [A]  time = 0.13, size = 429, normalized size = 1.75 \[ \frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{56 d \,a^{4}}-\frac {B \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{4}}+\frac {C \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{4}}-\frac {A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}+\frac {7 B \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}-\frac {9 C \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}+\frac {11 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{24 d \,a^{4}}-\frac {23 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{4}}+\frac {13 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}-\frac {15 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {49 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {111 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {2 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {9 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {7 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{d \,a^{4}}-\frac {8 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{d \,a^{4}}+\frac {21 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{d \,a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*(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*B*tan(1/2*d*x+1/2*c)^7+1/56/d/a^4*C*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)^5-9/40/d/a^4*C*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+13/8/d/a^4*C*tan(1/2*d*x+1/2*c)^3-15/8/d/a^4*A*tan(1/2*d*
x+1/2*c)+49/8/d/a^4*B*tan(1/2*d*x+1/2*c)-111/8/d/a^4*C*tan(1/2*d*x+1/2*c)+2/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^2*B
*tan(1/2*d*x+1/2*c)^3-9/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^2*C*tan(1/2*d*x+1/2*c)^3+2/d/a^4/(1+tan(1/2*d*x+1/2*c)^
2)^2*B*tan(1/2*d*x+1/2*c)-7/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^2*C*tan(1/2*d*x+1/2*c)+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+21/d/a^4*arctan(tan(1/2*d*x+1/2*c))*C

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maxima [B]  time = 0.49, size = 474, normalized size = 1.93 \[ -\frac {3 \, C {\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 )} - 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} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(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*(3*C*(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) - 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*si
n(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

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mupad [B]  time = 1.21, size = 283, normalized size = 1.16 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (A+5\,B-15\,C\right )}{8\,a^4}-\frac {3\,\left (2\,A-4\,B+6\,C\right )}{4\,a^4}-\frac {5\,\left (A-B+C\right )}{4\,a^4}+\frac {4\,A-20\,C}{8\,a^4}\right )}{d}+\frac {\left (2\,B-9\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,B-7\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^4\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {2\,A-4\,B+6\,C}{40\,a^4}+\frac {3\,\left (A-B+C\right )}{40\,a^4}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {2\,A-4\,B+6\,C}{8\,a^4}-\frac {A+5\,B-15\,C}{24\,a^4}+\frac {A-B+C}{4\,a^4}\right )}{d}+\frac {x\,\left (2\,A-8\,B+21\,C\right )}{2\,a^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B+C\right )}{56\,a^4\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(tan(c/2 + (d*x)/2)*((3*(A + 5*B - 15*C))/(8*a^4) - (3*(2*A - 4*B + 6*C))/(4*a^4) - (5*(A - B + C))/(4*a^4) +
(4*A - 20*C)/(8*a^4)))/d + (tan(c/2 + (d*x)/2)^3*(2*B - 9*C) + tan(c/2 + (d*x)/2)*(2*B - 7*C))/(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*((2*A - 4*B + 6*C)/(40*a^4) + (3*(A
- B + C))/(40*a^4)))/d + (tan(c/2 + (d*x)/2)^3*((2*A - 4*B + 6*C)/(8*a^4) - (A + 5*B - 15*C)/(24*a^4) + (A -
B + C)/(4*a^4)))/d + (x*(2*A - 8*B + 21*C))/(2*a^4) + (tan(c/2 + (d*x)/2)^7*(A - B + C))/(56*a^4*d)

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sympy [A]  time = 52.69, size = 1624, normalized size = 6.63 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((840*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 + 8
40*a**4*d) + 1680*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) + 840*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) - 75*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) + 190*A*t
an(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) - 910*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) - 2765*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) - 1575*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) - 3360*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) - 6720*B*d*x*ta
n(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*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) + 117*B*tan(c/2 + d*x/2)**9/(84
0*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*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) + 3682*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) + 11165*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) + 6825*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) + 8820*C*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*C*d*x*tan(c/2 + d*x/2)**2/(84
0*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*C*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*C*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*C*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*C*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*C*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*C*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*C*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*cos(c)**2)*cos(c)**4/(a*co
s(c) + a)**4, True))

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