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

Optimal result

Integrand size = 31, antiderivative size = 122 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx=\frac {3 (A-B) x}{2 a}-\frac {(3 A-4 B) \sin (c+d x)}{a d}+\frac {3 (A-B) \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {(A-B) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {(3 A-4 B) \sin ^3(c+d x)}{3 a d} \]

[Out]

3/2*(A-B)*x/a-(3*A-4*B)*sin(d*x+c)/a/d+3/2*(A-B)*cos(d*x+c)*sin(d*x+c)/a/d+(A-B)*cos(d*x+c)^3*sin(d*x+c)/d/(a+
a*cos(d*x+c))+1/3*(3*A-4*B)*sin(d*x+c)^3/a/d

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3056, 2827, 2715, 8, 2713} \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx=\frac {(3 A-4 B) \sin ^3(c+d x)}{3 a d}-\frac {(3 A-4 B) \sin (c+d x)}{a d}+\frac {(A-B) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}+\frac {3 (A-B) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac {3 x (A-B)}{2 a} \]

[In]

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

[Out]

(3*(A - B)*x)/(2*a) - ((3*A - 4*B)*Sin[c + d*x])/(a*d) + (3*(A - B)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + ((A -
B)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((3*A - 4*B)*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 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])

Rubi steps \begin{align*} \text {integral}& = \frac {(A-B) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {\int \cos ^2(c+d x) (3 a (A-B)-a (3 A-4 B) \cos (c+d x)) \, dx}{a^2} \\ & = \frac {(A-B) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac {(3 A-4 B) \int \cos ^3(c+d x) \, dx}{a}+\frac {(3 (A-B)) \int \cos ^2(c+d x) \, dx}{a} \\ & = \frac {3 (A-B) \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {(A-B) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {(3 (A-B)) \int 1 \, dx}{2 a}+\frac {(3 A-4 B) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a d} \\ & = \frac {3 (A-B) x}{2 a}-\frac {(3 A-4 B) \sin (c+d x)}{a d}+\frac {3 (A-B) \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {(A-B) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {(3 A-4 B) \sin ^3(c+d x)}{3 a d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(249\) vs. \(2(122)=244\).

Time = 1.23 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.04 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (36 (A-B) d x \cos \left (\frac {d x}{2}\right )+36 (A-B) d x \cos \left (c+\frac {d x}{2}\right )-60 A \sin \left (\frac {d x}{2}\right )+69 B \sin \left (\frac {d x}{2}\right )-12 A \sin \left (c+\frac {d x}{2}\right )+21 B \sin \left (c+\frac {d x}{2}\right )-9 A \sin \left (c+\frac {3 d x}{2}\right )+18 B \sin \left (c+\frac {3 d x}{2}\right )-9 A \sin \left (2 c+\frac {3 d x}{2}\right )+18 B \sin \left (2 c+\frac {3 d x}{2}\right )+3 A \sin \left (2 c+\frac {5 d x}{2}\right )-2 B \sin \left (2 c+\frac {5 d x}{2}\right )+3 A \sin \left (3 c+\frac {5 d x}{2}\right )-2 B \sin \left (3 c+\frac {5 d x}{2}\right )+B \sin \left (3 c+\frac {7 d x}{2}\right )+B \sin \left (4 c+\frac {7 d x}{2}\right )\right )}{24 a d (1+\cos (c+d x))} \]

[In]

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

[Out]

(Cos[(c + d*x)/2]*Sec[c/2]*(36*(A - B)*d*x*Cos[(d*x)/2] + 36*(A - B)*d*x*Cos[c + (d*x)/2] - 60*A*Sin[(d*x)/2]
+ 69*B*Sin[(d*x)/2] - 12*A*Sin[c + (d*x)/2] + 21*B*Sin[c + (d*x)/2] - 9*A*Sin[c + (3*d*x)/2] + 18*B*Sin[c + (3
*d*x)/2] - 9*A*Sin[2*c + (3*d*x)/2] + 18*B*Sin[2*c + (3*d*x)/2] + 3*A*Sin[2*c + (5*d*x)/2] - 2*B*Sin[2*c + (5*
d*x)/2] + 3*A*Sin[3*c + (5*d*x)/2] - 2*B*Sin[3*c + (5*d*x)/2] + B*Sin[3*c + (7*d*x)/2] + B*Sin[4*c + (7*d*x)/2
]))/(24*a*d*(1 + Cos[c + d*x]))

Maple [A] (verified)

Time = 0.95 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.64

method	result	size
parallelrisch	\(\frac {\left (\left (3 A -B \right ) \cos \left (2 d x +2 c \right )+B \cos \left (3 d x +3 c \right )+\left (-6 A +17 B \right ) \cos \left (d x +c \right )-21 A +31 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+18 d x \left (A -B \right )}{12 a d}\)	\(78\)
derivativedivides	\(\frac {-A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2 \left (-\frac {3 A}{2}+\frac {5 B}{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\frac {8 B}{3}-2 A \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-\frac {A}{2}+\frac {3 B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+3 \left (A -B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\)	\(122\)
default	\(\frac {-A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {2 \left (-\frac {3 A}{2}+\frac {5 B}{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\frac {8 B}{3}-2 A \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (-\frac {A}{2}+\frac {3 B}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+3 \left (A -B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\)	\(122\)
risch	\(\frac {3 x A}{2 a}-\frac {3 B x}{2 a}+\frac {i {\mathrm e}^{i \left (d x +c \right )} A}{2 a d}-\frac {7 i {\mathrm e}^{i \left (d x +c \right )} B}{8 a d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} A}{2 a d}+\frac {7 i {\mathrm e}^{-i \left (d x +c \right )} B}{8 a d}-\frac {2 i A}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}+\frac {2 i B}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}+\frac {\sin \left (3 d x +3 c \right ) B}{12 a d}+\frac {\sin \left (2 d x +2 c \right ) A}{4 a d}-\frac {\sin \left (2 d x +2 c \right ) B}{4 a d}\)	\(192\)
norman	\(\frac {\frac {3 \left (A -B \right ) x}{2 a}-\frac {2 \left (A -2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {\left (A -B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {6 \left (A -B \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {9 \left (A -B \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {6 \left (A -B \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {3 \left (A -B \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {\left (7 A -9 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {\left (27 A -37 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {\left (39 A -49 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\)	\(241\)

[In]

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

[Out]

1/12*(((3*A-B)*cos(2*d*x+2*c)+B*cos(3*d*x+3*c)+(-6*A+17*B)*cos(d*x+c)-21*A+31*B)*tan(1/2*d*x+1/2*c)+18*d*x*(A-
B))/a/d

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx=\frac {9 \, {\left (A - B\right )} d x \cos \left (d x + c\right ) + 9 \, {\left (A - B\right )} d x + {\left (2 \, B \cos \left (d x + c\right )^{3} + {\left (3 \, A - B\right )} \cos \left (d x + c\right )^{2} - {\left (3 \, A - 7 \, B\right )} \cos \left (d x + c\right ) - 12 \, A + 16 \, B\right )} \sin \left (d x + c\right )}{6 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]

[In]

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

[Out]

1/6*(9*(A - B)*d*x*cos(d*x + c) + 9*(A - B)*d*x + (2*B*cos(d*x + c)^3 + (3*A - B)*cos(d*x + c)^2 - (3*A - 7*B)
*cos(d*x + c) - 12*A + 16*B)*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. 1161 vs. \(2 (105) = 210\).

Time = 1.27 (sec) , antiderivative size = 1161, normalized size of antiderivative = 9.52 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((9*A*d*x*tan(c/2 + d*x/2)**6/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 18*a*d*tan(c/
2 + d*x/2)**2 + 6*a*d) + 27*A*d*x*tan(c/2 + d*x/2)**4/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4
+ 18*a*d*tan(c/2 + d*x/2)**2 + 6*a*d) + 27*A*d*x*tan(c/2 + d*x/2)**2/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c
/2 + d*x/2)**4 + 18*a*d*tan(c/2 + d*x/2)**2 + 6*a*d) + 9*A*d*x/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d
*x/2)**4 + 18*a*d*tan(c/2 + d*x/2)**2 + 6*a*d) - 6*A*tan(c/2 + d*x/2)**7/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*t
an(c/2 + d*x/2)**4 + 18*a*d*tan(c/2 + d*x/2)**2 + 6*a*d) - 36*A*tan(c/2 + d*x/2)**5/(6*a*d*tan(c/2 + d*x/2)**6
+ 18*a*d*tan(c/2 + d*x/2)**4 + 18*a*d*tan(c/2 + d*x/2)**2 + 6*a*d) - 42*A*tan(c/2 + d*x/2)**3/(6*a*d*tan(c/2
+ d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 18*a*d*tan(c/2 + d*x/2)**2 + 6*a*d) - 12*A*tan(c/2 + d*x/2)/(6*a*d*
tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 18*a*d*tan(c/2 + d*x/2)**2 + 6*a*d) - 9*B*d*x*tan(c/2 + d*x
/2)**6/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 18*a*d*tan(c/2 + d*x/2)**2 + 6*a*d) - 27*B*d*
x*tan(c/2 + d*x/2)**4/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 18*a*d*tan(c/2 + d*x/2)**2 + 6
*a*d) - 27*B*d*x*tan(c/2 + d*x/2)**2/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 18*a*d*tan(c/2
+ d*x/2)**2 + 6*a*d) - 9*B*d*x/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 18*a*d*tan(c/2 + d*x/
2)**2 + 6*a*d) + 6*B*tan(c/2 + d*x/2)**7/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 + 18*a*d*tan(
c/2 + d*x/2)**2 + 6*a*d) + 48*B*tan(c/2 + d*x/2)**5/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d*x/2)**4 +
18*a*d*tan(c/2 + d*x/2)**2 + 6*a*d) + 50*B*tan(c/2 + d*x/2)**3/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan(c/2 + d
*x/2)**4 + 18*a*d*tan(c/2 + d*x/2)**2 + 6*a*d) + 24*B*tan(c/2 + d*x/2)/(6*a*d*tan(c/2 + d*x/2)**6 + 18*a*d*tan
(c/2 + d*x/2)**4 + 18*a*d*tan(c/2 + d*x/2)**2 + 6*a*d), Ne(d, 0)), (x*(A + B*cos(c))*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. 310 vs. \(2 (116) = 232\).

Time = 0.30 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.54 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx=\frac {B {\left (\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a + \frac {3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {9 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {3 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - 3 \, A {\left (\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{3 \, d} \]

[In]

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

[Out]

1/3*(B*((9*sin(d*x + c)/(cos(d*x + c) + 1) + 16*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 15*sin(d*x + c)^5/(cos(d
*x + c) + 1)^5)/(a + 3*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + a*sin
(d*x + c)^6/(cos(d*x + c) + 1)^6) - 9*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + 3*sin(d*x + c)/(a*(cos(d*x +
c) + 1))) - 3*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 = 0.31 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.24 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx=\frac {\frac {9 \, {\left (d x + c\right )} {\left (A - B\right )}}{a} - \frac {6 \, {\left (A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a} - \frac {2 \, {\left (9 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a}}{6 \, d} \]

[In]

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

[Out]

1/6*(9*(d*x + c)*(A - B)/a - 6*(A*tan(1/2*d*x + 1/2*c) - B*tan(1/2*d*x + 1/2*c))/a - 2*(9*A*tan(1/2*d*x + 1/2*
c)^5 - 15*B*tan(1/2*d*x + 1/2*c)^5 + 12*A*tan(1/2*d*x + 1/2*c)^3 - 16*B*tan(1/2*d*x + 1/2*c)^3 + 3*A*tan(1/2*d
*x + 1/2*c) - 9*B*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*a))/d

Mupad [B] (verification not implemented)

Time = 1.50 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.13 \[ \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx=\frac {3\,x\,\left (A-B\right )}{2\,a}-\frac {\left (3\,A-5\,B\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (4\,A-\frac {16\,B}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (A-3\,B\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A-B\right )}{a\,d} \]

[In]

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

[Out]

(3*x*(A - B))/(2*a) - (tan(c/2 + (d*x)/2)^5*(3*A - 5*B) + tan(c/2 + (d*x)/2)^3*(4*A - (16*B)/3) + tan(c/2 + (d
*x)/2)*(A - 3*B))/(d*(a + 3*a*tan(c/2 + (d*x)/2)^2 + 3*a*tan(c/2 + (d*x)/2)^4 + a*tan(c/2 + (d*x)/2)^6)) - (ta
n(c/2 + (d*x)/2)*(A - B))/(a*d)