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

Optimal. Leaf size=43 \[ \frac{\sin (c+d x)}{a d}+\frac{\sin (c+d x)}{a d (\cos (c+d x)+1)}-\frac{x}{a} \]

[Out]

-(x/a) + Sin[c + d*x]/(a*d) + Sin[c + d*x]/(a*d*(1 + Cos[c + d*x]))

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Rubi [A]  time = 0.0798836, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2746, 12, 2735, 2648} \[ \frac{\sin (c+d x)}{a d}+\frac{\sin (c+d x)}{a d (\cos (c+d x)+1)}-\frac{x}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x/a) + Sin[c + d*x]/(a*d) + Sin[c + d*x]/(a*d*(1 + Cos[c + d*x]))

Rule 2746

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b^2
*Cos[e + f*x])/(d*f), x] + Dist[1/d, Int[Simp[a^2*d - b*(b*c - 2*a*d)*Sin[e + f*x], x]/(c + d*Sin[e + f*x]), x
], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

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 ^2(c+d x)}{a+a \cos (c+d x)} \, dx &=\frac{\sin (c+d x)}{a d}-\frac{\int \frac{a \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{a}\\ &=\frac{\sin (c+d x)}{a d}-\int \frac{\cos (c+d x)}{a+a \cos (c+d x)} \, dx\\ &=-\frac{x}{a}+\frac{\sin (c+d x)}{a d}+\int \frac{1}{a+a \cos (c+d x)} \, dx\\ &=-\frac{x}{a}+\frac{\sin (c+d x)}{a d}+\frac{\sin (c+d x)}{d (a+a \cos (c+d x))}\\ \end{align*}

Mathematica [B]  time = 0.200899, size = 89, normalized size = 2.07 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (c+\frac{d x}{2}\right )+\sin \left (c+\frac{3 d x}{2}\right )+\sin \left (2 c+\frac{3 d x}{2}\right )-2 d x \cos \left (c+\frac{d x}{2}\right )+5 \sin \left (\frac{d x}{2}\right )-2 d x \cos \left (\frac{d x}{2}\right )\right )}{4 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]*(-2*d*x*Cos[(d*x)/2] - 2*d*x*Cos[c + (d*x)/2] + 5*Sin[(d*x)/2] + Sin[c + (d*x)/2] +
Sin[c + (3*d*x)/2] + Sin[2*c + (3*d*x)/2]))/(4*a*d)

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Maple [A]  time = 0.044, size = 68, normalized size = 1.6 \begin{align*}{\frac{1}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) }{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}-2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d/a*tan(1/2*d*x+1/2*c)+2/d/a*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)-2/a/d*arctan(tan(1/2*d*x+1/2*c))

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Maxima [B]  time = 1.66696, size = 124, normalized size = 2.88 \begin{align*} -\frac{\frac{2 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{2 \, \sin \left (d x + c\right )}{{\left (a + \frac{a \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{\sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a - 2*sin(d*x + c)/((a + a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(c
os(d*x + c) + 1)) - sin(d*x + c)/(a*(cos(d*x + c) + 1)))/d

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Fricas [A]  time = 1.58301, size = 116, normalized size = 2.7 \begin{align*} -\frac{d x \cos \left (d x + c\right ) + d x -{\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*x*cos(d*x + c) + d*x - (cos(d*x + c) + 2)*sin(d*x + c))/(a*d*cos(d*x + c) + a*d)

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Sympy [A]  time = 1.97009, size = 129, normalized size = 3. \begin{align*} \begin{cases} - \frac{d x \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} - \frac{d x}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} + \frac{\tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} + \frac{3 \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} & \text{for}\: d \neq 0 \\\frac{x \cos ^{2}{\left (c \right )}}{a \cos{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-d*x*tan(c/2 + d*x/2)**2/(a*d*tan(c/2 + d*x/2)**2 + a*d) - d*x/(a*d*tan(c/2 + d*x/2)**2 + a*d) + ta
n(c/2 + d*x/2)**3/(a*d*tan(c/2 + d*x/2)**2 + a*d) + 3*tan(c/2 + d*x/2)/(a*d*tan(c/2 + d*x/2)**2 + a*d), Ne(d,
0)), (x*cos(c)**2/(a*cos(c) + a), True))

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Giac [A]  time = 1.36168, size = 78, normalized size = 1.81 \begin{align*} -\frac{\frac{d x + c}{a} - \frac{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a} - \frac{2 \, \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}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((d*x + c)/a - tan(1/2*d*x + 1/2*c)/a - 2*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*a))/d