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

Optimal. Leaf size=122 \[ \frac{(3 A-4 B) \sin ^3(c+d x)}{3 a d}-\frac{(3 A-4 B) \sin (c+d x)}{a d}+\frac{(A-B) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}+\frac{3 (A-B) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{3 x (A-B)}{2 a} \]

[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)

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Rubi [A]  time = 0.171539, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {2977, 2748, 2635, 8, 2633} \[ \frac{(3 A-4 B) \sin ^3(c+d x)}{3 a d}-\frac{(3 A-4 B) \sin (c+d x)}{a d}+\frac{(A-B) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}+\frac{3 (A-B) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{3 x (A-B)}{2 a} \]

Antiderivative was successfully verified.

[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 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 2748

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 2635

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] && Integer
Q[2*n]

Rule 8

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

Rule 2633

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]

Rubi steps

\begin{align*} \int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx &=\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) \operatorname{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]  time = 0.534345, size = 249, normalized size = 2.04 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (36 d x (A-B) \cos \left (c+\frac{d x}{2}\right )+36 d x (A-B) \cos \left (\frac{d x}{2}\right )-12 A \sin \left (c+\frac{d x}{2}\right )-9 A \sin \left (c+\frac{3 d x}{2}\right )-9 A \sin \left (2 c+\frac{3 d x}{2}\right )+3 A \sin \left (2 c+\frac{5 d x}{2}\right )+3 A \sin \left (3 c+\frac{5 d x}{2}\right )-60 A \sin \left (\frac{d x}{2}\right )+21 B \sin \left (c+\frac{d x}{2}\right )+18 B \sin \left (c+\frac{3 d x}{2}\right )+18 B \sin \left (2 c+\frac{3 d x}{2}\right )-2 B \sin \left (2 c+\frac{5 d x}{2}\right )-2 B \sin \left (3 c+\frac{5 d x}{2}\right )+B \sin \left (3 c+\frac{7 d x}{2}\right )+B \sin \left (4 c+\frac{7 d x}{2}\right )+69 B \sin \left (\frac{d x}{2}\right )\right )}{24 a d (\cos (c+d x)+1)} \]

Antiderivative was successfully verified.

[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]))

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Maple [B]  time = 0.066, size = 281, normalized size = 2.3 \begin{align*} -{\frac{A}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{B}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-3\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}A}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}+5\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}B}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}+{\frac{16\,B}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-3}}-4\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}A}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}-{\frac{A}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-3}}+3\,{\frac{B\tan \left ( 1/2\,dx+c/2 \right ) }{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}+3\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) A}{da}}-3\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) B}{da}} \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))/(a+cos(d*x+c)*a),x)

[Out]

-1/d/a*A*tan(1/2*d*x+1/2*c)+1/d/a*B*tan(1/2*d*x+1/2*c)-3/d/a/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^5*A
+5/d/a/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^5*B+16/3/d/a/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c
)^3*B-4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^3*A-1/d/a/(1+tan(1/2*d*x+1/2*c)^2)^3*A*tan(1/2*d*x+1
/2*c)+3/d/a/(1+tan(1/2*d*x+1/2*c)^2)^3*B*tan(1/2*d*x+1/2*c)+3/d/a*arctan(tan(1/2*d*x+1/2*c))*A-3/d/a*arctan(ta
n(1/2*d*x+1/2*c))*B

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Maxima [B]  time = 1.52061, size = 419, normalized size = 3.43 \begin{align*} \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} \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))/(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

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Fricas [A]  time = 1.46638, size = 242, normalized size = 1.98 \begin{align*} \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 )}} \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))/(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)

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Sympy [A]  time = 7.63455, size = 1161, normalized size = 9.52 \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))/(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))

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Giac [A]  time = 1.24128, size = 204, normalized size = 1.67 \begin{align*} \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} \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))/(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