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

Optimal. Leaf size=90 \[ \frac{(A+4 C) \sin (c+d x)}{3 a^2 d}+\frac{2 C \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{2 C x}{a^2}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]

[Out]

(-2*C*x)/a^2 + ((A + 4*C)*Sin[c + d*x])/(3*a^2*d) + (2*C*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A + C)*C
os[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)

________________________________________________________________________________________

Rubi [A]  time = 0.241035, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {3042, 2968, 3023, 12, 2735, 2648} \[ \frac{(A+4 C) \sin (c+d x)}{3 a^2 d}+\frac{2 C \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{2 C x}{a^2}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*C*x)/a^2 + ((A + 4*C)*Sin[c + d*x])/(3*a^2*d) + (2*C*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A + C)*C
os[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)

Rule 3042

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)]

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 (c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx &=-\frac{(A+C) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \frac{\cos (c+d x) (a (A-2 C)+a (A+4 C) \cos (c+d x))}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac{(A+C) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int \frac{a (A-2 C) \cos (c+d x)+a (A+4 C) \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=\frac{(A+4 C) \sin (c+d x)}{3 a^2 d}-\frac{(A+C) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{\int -\frac{6 a^2 C \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{3 a^3}\\ &=\frac{(A+4 C) \sin (c+d x)}{3 a^2 d}-\frac{(A+C) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac{(2 C) \int \frac{\cos (c+d x)}{a+a \cos (c+d x)} \, dx}{a}\\ &=-\frac{2 C x}{a^2}+\frac{(A+4 C) \sin (c+d x)}{3 a^2 d}-\frac{(A+C) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{(2 C) \int \frac{1}{a+a \cos (c+d x)} \, dx}{a}\\ &=-\frac{2 C x}{a^2}+\frac{(A+4 C) \sin (c+d x)}{3 a^2 d}-\frac{(A+C) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac{2 C \sin (c+d x)}{d \left (a^2+a^2 \cos (c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 0.519144, size = 195, normalized size = 2.17 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec ^3\left (\frac{1}{2} (c+d x)\right ) \left (-12 A \sin \left (c+\frac{d x}{2}\right )+8 A \sin \left (c+\frac{3 d x}{2}\right )+12 A \sin \left (\frac{d x}{2}\right )-30 C \sin \left (c+\frac{d x}{2}\right )+41 C \sin \left (c+\frac{3 d x}{2}\right )+9 C \sin \left (2 c+\frac{3 d x}{2}\right )+3 C \sin \left (2 c+\frac{5 d x}{2}\right )+3 C \sin \left (3 c+\frac{5 d x}{2}\right )-36 C d x \cos \left (c+\frac{d x}{2}\right )-12 C d x \cos \left (c+\frac{3 d x}{2}\right )-12 C d x \cos \left (2 c+\frac{3 d x}{2}\right )+66 C \sin \left (\frac{d x}{2}\right )-36 C d x \cos \left (\frac{d x}{2}\right )\right )}{48 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]^3*(-36*C*d*x*Cos[(d*x)/2] - 36*C*d*x*Cos[c + (d*x)/2] - 12*C*d*x*Cos[c + (3*d*x)/2]
- 12*C*d*x*Cos[2*c + (3*d*x)/2] + 12*A*Sin[(d*x)/2] + 66*C*Sin[(d*x)/2] - 12*A*Sin[c + (d*x)/2] - 30*C*Sin[c
+ (d*x)/2] + 8*A*Sin[c + (3*d*x)/2] + 41*C*Sin[c + (3*d*x)/2] + 9*C*Sin[2*c + (3*d*x)/2] + 3*C*Sin[2*c + (5*d*
x)/2] + 3*C*Sin[3*c + (5*d*x)/2]))/(48*a^2*d)

________________________________________________________________________________________

Maple [A]  time = 0.03, size = 130, normalized size = 1.4 \begin{align*} -{\frac{A}{6\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{C}{6\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{A}{2\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{5\,C}{2\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{C\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{2} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) }}-4\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) C}{d{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/d/a^2*tan(1/2*d*x+1/2*c)^3*A-1/6/d/a^2*C*tan(1/2*d*x+1/2*c)^3+1/2/d/a^2*A*tan(1/2*d*x+1/2*c)+5/2/d/a^2*C*
tan(1/2*d*x+1/2*c)+2/d/a^2*C*tan(1/2*d*x+1/2*c)/(tan(1/2*d*x+1/2*c)^2+1)-4/d/a^2*arctan(tan(1/2*d*x+1/2*c))*C

________________________________________________________________________________________

Maxima [A]  time = 1.52127, size = 223, normalized size = 2.48 \begin{align*} \frac{C{\left (\frac{\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac{24 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac{12 \, \sin \left (d x + c\right )}{{\left (a^{2} + \frac{a^{2} \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 )}}\right )} + \frac{A{\left (\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(C*((15*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/a^2 - 24*arctan(sin(d*x + c
)/(cos(d*x + c) + 1))/a^2 + 12*sin(d*x + c)/((a^2 + a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1
))) + A*(3*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/a^2)/d

________________________________________________________________________________________

Fricas [A]  time = 1.32886, size = 261, normalized size = 2.9 \begin{align*} -\frac{6 \, C d x \cos \left (d x + c\right )^{2} + 12 \, C d x \cos \left (d x + c\right ) + 6 \, C d x -{\left (3 \, C \cos \left (d x + c\right )^{2} + 2 \,{\left (A + 7 \, C\right )} \cos \left (d x + c\right ) + A + 10 \, C\right )} \sin \left (d x + c\right )}{3 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(6*C*d*x*cos(d*x + c)^2 + 12*C*d*x*cos(d*x + c) + 6*C*d*x - (3*C*cos(d*x + c)^2 + 2*(A + 7*C)*cos(d*x + c
) + A + 10*C)*sin(d*x + c))/(a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)

________________________________________________________________________________________

Sympy [A]  time = 6.21401, size = 335, normalized size = 3.72 \begin{align*} \begin{cases} - \frac{A \tan ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} + \frac{2 A \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} + \frac{3 A \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} - \frac{12 C d x \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} - \frac{12 C d x}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} - \frac{C \tan ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} + \frac{14 C \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} + \frac{27 C \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 6 a^{2} d} & \text{for}\: d \neq 0 \\\frac{x \left (A + C \cos ^{2}{\left (c \right )}\right ) \cos{\left (c \right )}}{\left (a \cos{\left (c \right )} + a\right )^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-A*tan(c/2 + d*x/2)**5/(6*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) + 2*A*tan(c/2 + d*x/2)**3/(6*a**2*
d*tan(c/2 + d*x/2)**2 + 6*a**2*d) + 3*A*tan(c/2 + d*x/2)/(6*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) - 12*C*d*x*
tan(c/2 + d*x/2)**2/(6*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) - 12*C*d*x/(6*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**
2*d) - C*tan(c/2 + d*x/2)**5/(6*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d) + 14*C*tan(c/2 + d*x/2)**3/(6*a**2*d*ta
n(c/2 + d*x/2)**2 + 6*a**2*d) + 27*C*tan(c/2 + d*x/2)/(6*a**2*d*tan(c/2 + d*x/2)**2 + 6*a**2*d), Ne(d, 0)), (x
*(A + C*cos(c)**2)*cos(c)/(a*cos(c) + a)**2, True))

________________________________________________________________________________________

Giac [A]  time = 1.25654, size = 154, normalized size = 1.71 \begin{align*} -\frac{\frac{12 \,{\left (d x + c\right )} C}{a^{2}} - \frac{12 \, 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^{2}} + \frac{A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 15 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{6}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(12*(d*x + c)*C/a^2 - 12*C*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*a^2) + (A*a^4*tan(1/2*d*x +
1/2*c)^3 + C*a^4*tan(1/2*d*x + 1/2*c)^3 - 3*A*a^4*tan(1/2*d*x + 1/2*c) - 15*C*a^4*tan(1/2*d*x + 1/2*c))/a^6)/
d