\(\int \frac {\cos ^3(c+d x) (A+C \cos ^2(c+d x))}{a+a \cos (c+d x)} \, dx\) [39]

Optimal result

Integrand size = 33, antiderivative size = 156 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\frac {3 (4 A+5 C) x}{8 a}-\frac {(3 A+4 C) \sin (c+d x)}{a d}+\frac {3 (4 A+5 C) \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {(4 A+5 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {(3 A+4 C) \sin ^3(c+d x)}{3 a d} \]

[Out]

3/8*(4*A+5*C)*x/a-(3*A+4*C)*sin(d*x+c)/a/d+3/8*(4*A+5*C)*cos(d*x+c)*sin(d*x+c)/a/d+1/4*(4*A+5*C)*cos(d*x+c)^3*
sin(d*x+c)/a/d-(A+C)*cos(d*x+c)^4*sin(d*x+c)/d/(a+a*cos(d*x+c))+1/3*(3*A+4*C)*sin(d*x+c)^3/a/d

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3121, 2827, 2713, 2715, 8} \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\frac {(3 A+4 C) \sin ^3(c+d x)}{3 a d}-\frac {(3 A+4 C) \sin (c+d x)}{a d}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}+\frac {(4 A+5 C) \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac {3 (4 A+5 C) \sin (c+d x) \cos (c+d x)}{8 a d}+\frac {3 x (4 A+5 C)}{8 a} \]

[In]

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

[Out]

(3*(4*A + 5*C)*x)/(8*a) - ((3*A + 4*C)*Sin[c + d*x])/(a*d) + (3*(4*A + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(8*a*d)
+ ((4*A + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - ((A + C)*Cos[c + d*x]^4*Sin[c + d*x])/(d*(a + a*Cos[c +
d*x])) + ((3*A + 4*C)*Sin[c + d*x]^3)/(3*a*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2827

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 3121

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[a*(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)) - C*(a*c*m + b*d*(n + 1)) + (a*A*d*(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, 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*} \text {integral}& = -\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {\int \cos ^3(c+d x) (-a (3 A+4 C)+a (4 A+5 C) \cos (c+d x)) \, dx}{a^2} \\ & = -\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac {(3 A+4 C) \int \cos ^3(c+d x) \, dx}{a}+\frac {(4 A+5 C) \int \cos ^4(c+d x) \, dx}{a} \\ & = \frac {(4 A+5 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {(3 (4 A+5 C)) \int \cos ^2(c+d x) \, dx}{4 a}+\frac {(3 A+4 C) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a d} \\ & = -\frac {(3 A+4 C) \sin (c+d x)}{a d}+\frac {3 (4 A+5 C) \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {(4 A+5 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {(3 A+4 C) \sin ^3(c+d x)}{3 a d}+\frac {(3 (4 A+5 C)) \int 1 \, dx}{8 a} \\ & = \frac {3 (4 A+5 C) x}{8 a}-\frac {(3 A+4 C) \sin (c+d x)}{a d}+\frac {3 (4 A+5 C) \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {(4 A+5 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {(3 A+4 C) \sin ^3(c+d x)}{3 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.56 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.81 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (72 (4 A+5 C) d x \cos \left (\frac {d x}{2}\right )+72 (4 A+5 C) d x \cos \left (c+\frac {d x}{2}\right )-480 A \sin \left (\frac {d x}{2}\right )-552 C \sin \left (\frac {d x}{2}\right )-96 A \sin \left (c+\frac {d x}{2}\right )-168 C \sin \left (c+\frac {d x}{2}\right )-72 A \sin \left (c+\frac {3 d x}{2}\right )-120 C \sin \left (c+\frac {3 d x}{2}\right )-72 A \sin \left (2 c+\frac {3 d x}{2}\right )-120 C \sin \left (2 c+\frac {3 d x}{2}\right )+24 A \sin \left (2 c+\frac {5 d x}{2}\right )+40 C \sin \left (2 c+\frac {5 d x}{2}\right )+24 A \sin \left (3 c+\frac {5 d x}{2}\right )+40 C \sin \left (3 c+\frac {5 d x}{2}\right )-5 C \sin \left (3 c+\frac {7 d x}{2}\right )-5 C \sin \left (4 c+\frac {7 d x}{2}\right )+3 C \sin \left (4 c+\frac {9 d x}{2}\right )+3 C \sin \left (5 c+\frac {9 d x}{2}\right )\right )}{192 a d (1+\cos (c+d x))} \]

[In]

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

[Out]

(Cos[(c + d*x)/2]*Sec[c/2]*(72*(4*A + 5*C)*d*x*Cos[(d*x)/2] + 72*(4*A + 5*C)*d*x*Cos[c + (d*x)/2] - 480*A*Sin[
(d*x)/2] - 552*C*Sin[(d*x)/2] - 96*A*Sin[c + (d*x)/2] - 168*C*Sin[c + (d*x)/2] - 72*A*Sin[c + (3*d*x)/2] - 120
*C*Sin[c + (3*d*x)/2] - 72*A*Sin[2*c + (3*d*x)/2] - 120*C*Sin[2*c + (3*d*x)/2] + 24*A*Sin[2*c + (5*d*x)/2] + 4
0*C*Sin[2*c + (5*d*x)/2] + 24*A*Sin[3*c + (5*d*x)/2] + 40*C*Sin[3*c + (5*d*x)/2] - 5*C*Sin[3*c + (7*d*x)/2] -
5*C*Sin[4*c + (7*d*x)/2] + 3*C*Sin[4*c + (9*d*x)/2] + 3*C*Sin[5*c + (9*d*x)/2]))/(192*a*d*(1 + Cos[c + d*x]))

Maple [A] (verified)

Time = 2.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.58

method	result	size
parallelrisch	\(\frac {\left (\left (24 A +38 C \right ) \cos \left (2 d x +2 c \right )-2 C \cos \left (3 d x +3 c \right )+3 C \cos \left (4 d x +4 c \right )+\left (-48 A -82 C \right ) \cos \left (d x +c \right )-168 A -221 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+144 \left (A +\frac {5 C}{4}\right ) x d}{96 a d}\)	\(91\)
derivativedivides	\(\frac {-A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {2 \left (-\frac {3 A}{2}-\frac {25 C}{8}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-\frac {7 A}{2}-\frac {115 C}{24}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-\frac {5 A}{2}-\frac {109 C}{24}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-\frac {A}{2}-\frac {7 C}{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {3 \left (4 A +5 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\)	\(144\)
default	\(\frac {-A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C +\frac {2 \left (-\frac {3 A}{2}-\frac {25 C}{8}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-\frac {7 A}{2}-\frac {115 C}{24}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-\frac {5 A}{2}-\frac {109 C}{24}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-\frac {A}{2}-\frac {7 C}{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {3 \left (4 A +5 C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{d a}\)	\(144\)
risch	\(\frac {3 x A}{2 a}+\frac {15 C x}{8 a}+\frac {i {\mathrm e}^{i \left (d x +c \right )} A}{2 a d}+\frac {7 i {\mathrm e}^{i \left (d x +c \right )} C}{8 a d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} A}{2 a d}-\frac {7 i {\mathrm e}^{-i \left (d x +c \right )} C}{8 a d}-\frac {2 i A}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}-\frac {2 i C}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}+\frac {C \sin \left (4 d x +4 c \right )}{32 a d}-\frac {C \sin \left (3 d x +3 c \right )}{12 a d}+\frac {\sin \left (2 d x +2 c \right ) A}{4 a d}+\frac {\sin \left (2 d x +2 c \right ) C}{2 a d}\)	\(210\)
norman	\(\frac {\frac {3 \left (4 A +5 C \right ) x}{8 a}-\frac {\left (A +C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {15 \left (4 A +5 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {15 \left (4 A +5 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {15 \left (4 A +5 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {15 \left (4 A +5 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {3 \left (4 A +5 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {\left (8 A +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}-\frac {5 \left (24 A +31 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}-\frac {\left (32 A +45 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}-\frac {2 \left (33 A +43 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {\left (66 A +95 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\)	\(301\)

[In]

int(cos(d*x+c)^3*(A+C*cos(d*x+c)^2)/(a+cos(d*x+c)*a),x,method=_RETURNVERBOSE)

[Out]

1/96*(((24*A+38*C)*cos(2*d*x+2*c)-2*C*cos(3*d*x+3*c)+3*C*cos(4*d*x+4*c)+(-48*A-82*C)*cos(d*x+c)-168*A-221*C)*t
an(1/2*d*x+1/2*c)+144*(A+5/4*C)*x*d)/a/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.72 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\frac {9 \, {\left (4 \, A + 5 \, C\right )} d x \cos \left (d x + c\right ) + 9 \, {\left (4 \, A + 5 \, C\right )} d x + {\left (6 \, C \cos \left (d x + c\right )^{4} - 2 \, C \cos \left (d x + c\right )^{3} + {\left (12 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (12 \, A + 19 \, C\right )} \cos \left (d x + c\right ) - 48 \, A - 64 \, C\right )} \sin \left (d x + c\right )}{24 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]

[In]

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

[Out]

1/24*(9*(4*A + 5*C)*d*x*cos(d*x + c) + 9*(4*A + 5*C)*d*x + (6*C*cos(d*x + c)^4 - 2*C*cos(d*x + c)^3 + (12*A +
13*C)*cos(d*x + c)^2 - (12*A + 19*C)*cos(d*x + c) - 48*A - 64*C)*sin(d*x + c))/(a*d*cos(d*x + c) + a*d)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1795 vs. \(2 (131) = 262\).

Time = 1.82 (sec) , antiderivative size = 1795, normalized size of antiderivative = 11.51 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)**3*(A+C*cos(d*x+c)**2)/(a+a*cos(d*x+c)),x)

[Out]

Piecewise((36*A*d*x*tan(c/2 + d*x/2)**8/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan
(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 144*A*d*x*tan(c/2 + d*x/2)**6/(24*a*d*tan(c/2 + d*x/
2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 216*
A*d*x*tan(c/2 + d*x/2)**4/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)*
*4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 144*A*d*x*tan(c/2 + d*x/2)**2/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d
*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 36*A*d*x/(24*a*d*t
an(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 2
4*a*d) - 24*A*tan(c/2 + d*x/2)**9/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 +
d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 168*A*tan(c/2 + d*x/2)**7/(24*a*d*tan(c/2 + d*x/2)**8 + 96
*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 312*A*tan(c/2
+ d*x/2)**5/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*ta
n(c/2 + d*x/2)**2 + 24*a*d) - 216*A*tan(c/2 + d*x/2)**3/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)*
*6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 48*A*tan(c/2 + d*x/2)/(24*a*d*tan(c/
2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d
) + 45*C*d*x*tan(c/2 + d*x/2)**8/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 +
d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 180*C*d*x*tan(c/2 + d*x/2)**6/(24*a*d*tan(c/2 + d*x/2)**8 +
96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 270*C*d*x*t
an(c/2 + d*x/2)**4/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96
*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 180*C*d*x*tan(c/2 + d*x/2)**2/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/
2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) + 45*C*d*x/(24*a*d*tan(c/2
+ d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d)
- 24*C*tan(c/2 + d*x/2)**9/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)
**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 246*C*tan(c/2 + d*x/2)**7/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*ta
n(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 374*C*tan(c/2 + d*x/2
)**5/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 +
d*x/2)**2 + 24*a*d) - 314*C*tan(c/2 + d*x/2)**3/(24*a*d*tan(c/2 + d*x/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 14
4*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d) - 66*C*tan(c/2 + d*x/2)/(24*a*d*tan(c/2 + d*x
/2)**8 + 96*a*d*tan(c/2 + d*x/2)**6 + 144*a*d*tan(c/2 + d*x/2)**4 + 96*a*d*tan(c/2 + d*x/2)**2 + 24*a*d), Ne(d
, 0)), (x*(A + C*cos(c)**2)*cos(c)**3/(a*cos(c) + a), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (148) = 296\).

Time = 0.29 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.25 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=-\frac {C {\left (\frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {109 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {115 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {75 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a + \frac {4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {45 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {12 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} + 12 \, A {\left (\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{12 \, d} \]

[In]

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

[Out]

-1/12*(C*((21*sin(d*x + c)/(cos(d*x + c) + 1) + 109*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 115*sin(d*x + c)^5/(
cos(d*x + c) + 1)^5 + 75*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/(a + 4*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6
*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + a*sin(d*x + c)^8/(cos(d*x +
c) + 1)^8) - 45*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + 12*sin(d*x + c)/(a*(cos(d*x + c) + 1))) + 12*A*((
sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a + 2*a*sin(d*x + c)^2/(cos(d*x + c)
+ 1)^2 + a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) - 3*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + sin(d*x + c)/
(a*(cos(d*x + c) + 1))))/d

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.15 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\frac {\frac {9 \, {\left (d x + c\right )} {\left (4 \, A + 5 \, C\right )}}{a} - \frac {24 \, {\left (A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a} - \frac {2 \, {\left (36 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 75 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 84 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 115 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 60 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 109 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a}}{24 \, d} \]

[In]

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

[Out]

1/24*(9*(d*x + c)*(4*A + 5*C)/a - 24*(A*tan(1/2*d*x + 1/2*c) + C*tan(1/2*d*x + 1/2*c))/a - 2*(36*A*tan(1/2*d*x
+ 1/2*c)^7 + 75*C*tan(1/2*d*x + 1/2*c)^7 + 84*A*tan(1/2*d*x + 1/2*c)^5 + 115*C*tan(1/2*d*x + 1/2*c)^5 + 60*A*
tan(1/2*d*x + 1/2*c)^3 + 109*C*tan(1/2*d*x + 1/2*c)^3 + 12*A*tan(1/2*d*x + 1/2*c) + 21*C*tan(1/2*d*x + 1/2*c))
/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*a))/d

Mupad [B] (verification not implemented)

Time = 1.14 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.98 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx=\frac {3\,A\,x}{2\,a}+\frac {15\,C\,x}{8\,a}-\frac {A\,\sin \left (c+d\,x\right )}{a\,d}-\frac {7\,C\,\sin \left (c+d\,x\right )}{4\,a\,d}+\frac {A\,\sin \left (2\,c+2\,d\,x\right )}{4\,a\,d}-\frac {A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d}+\frac {C\,\sin \left (2\,c+2\,d\,x\right )}{2\,a\,d}-\frac {C\,\sin \left (3\,c+3\,d\,x\right )}{12\,a\,d}+\frac {C\,\sin \left (4\,c+4\,d\,x\right )}{32\,a\,d}-\frac {C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d} \]

[In]

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

[Out]

(3*A*x)/(2*a) + (15*C*x)/(8*a) - (A*sin(c + d*x))/(a*d) - (7*C*sin(c + d*x))/(4*a*d) + (A*sin(2*c + 2*d*x))/(4
*a*d) - (A*tan(c/2 + (d*x)/2))/(a*d) + (C*sin(2*c + 2*d*x))/(2*a*d) - (C*sin(3*c + 3*d*x))/(12*a*d) + (C*sin(4
*c + 4*d*x))/(32*a*d) - (C*tan(c/2 + (d*x)/2))/(a*d)