3.48 \(\int \frac{\cos (c+d x)}{a+a \sec (c+d x)} \, dx\)

Optimal. Leaf size=44 \[ \frac{2 \sin (c+d x)}{a d}-\frac{\sin (c+d x)}{d (a \sec (c+d x)+a)}-\frac{x}{a} \]

[Out]

-(x/a) + (2*Sin[c + d*x])/(a*d) - Sin[c + d*x]/(d*(a + a*Sec[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.0571631, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3819, 3787, 2637, 8} \[ \frac{2 \sin (c+d x)}{a d}-\frac{\sin (c+d x)}{d (a \sec (c+d x)+a)}-\frac{x}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x/a) + (2*Sin[c + d*x])/(a*d) - Sin[c + d*x]/(d*(a + a*Sec[c + d*x]))

Rule 3819

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(Cot[e + f*
x]*(d*Csc[e + f*x])^n)/(f*(a + b*Csc[e + f*x])), x] - Dist[1/a^2, Int[(d*Csc[e + f*x])^n*(a*(n - 1) - b*n*Csc[
e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 8

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

Rubi steps

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

Mathematica [B]  time = 0.222399, size = 89, normalized size = 2.02 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (c+\frac{d x}{2}\right )+\sin \left (c+\frac{3 d x}{2}\right )+\sin \left (2 c+\frac{3 d x}{2}\right )-2 d x \cos \left (c+\frac{d x}{2}\right )+5 \sin \left (\frac{d x}{2}\right )-2 d x \cos \left (\frac{d x}{2}\right )\right )}{4 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]/(a + a*Sec[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)

________________________________________________________________________________________

Maple [A]  time = 0.051, 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)/(a+a*sec(d*x+c)),x)

[Out]

1/a/d*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))

________________________________________________________________________________________

Maxima [B]  time = 1.65903, size = 124, normalized size = 2.82 \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)/(a+a*sec(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

________________________________________________________________________________________

Fricas [A]  time = 1.66372, size = 116, normalized size = 2.64 \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)/(a+a*sec(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)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cos{\left (c + d x \right )}}{\sec{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(c + d*x)/(sec(c + d*x) + 1), x)/a

________________________________________________________________________________________

Giac [A]  time = 1.22235, size = 78, normalized size = 1.77 \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)/(a+a*sec(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