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

Optimal. Leaf size=74 \[ -\frac{2 \sin (c+d x)}{a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{\sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 x}{2 a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0816637, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3819, 3787, 2635, 8, 2637} \[ -\frac{2 \sin (c+d x)}{a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{\sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 x}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x)/(2*a) - (2*Sin[c + d*x])/(a*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - (Cos[c + d*x]*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 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 2637

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

Rubi steps

\begin{align*} \int \frac{\cos ^2(c+d x)}{a+a \sec (c+d x)} \, dx &=-\frac{\cos (c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac{\int \cos ^2(c+d x) (-3 a+2 a \sec (c+d x)) \, dx}{a^2}\\ &=-\frac{\cos (c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac{2 \int \cos (c+d x) \, dx}{a}+\frac{3 \int \cos ^2(c+d x) \, dx}{a}\\ &=-\frac{2 \sin (c+d x)}{a d}+\frac{3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac{\cos (c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac{3 \int 1 \, dx}{2 a}\\ &=\frac{3 x}{2 a}-\frac{2 \sin (c+d x)}{a d}+\frac{3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac{\cos (c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.232865, size = 117, normalized size = 1.58 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \left (-4 \sin \left (c+\frac{d x}{2}\right )-3 \sin \left (c+\frac{3 d x}{2}\right )-3 \sin \left (2 c+\frac{3 d x}{2}\right )+\sin \left (2 c+\frac{5 d x}{2}\right )+\sin \left (3 c+\frac{5 d x}{2}\right )+12 d x \cos \left (c+\frac{d x}{2}\right )-20 \sin \left (\frac{d x}{2}\right )+12 d x \cos \left (\frac{d x}{2}\right )\right )}{16 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[c/2]*Sec[(c + d*x)/2]*(12*d*x*Cos[(d*x)/2] + 12*d*x*Cos[c + (d*x)/2] - 20*Sin[(d*x)/2] - 4*Sin[c + (d*x)/
2] - 3*Sin[c + (3*d*x)/2] - 3*Sin[2*c + (3*d*x)/2] + Sin[2*c + (5*d*x)/2] + Sin[3*c + (5*d*x)/2]))/(16*a*d)

________________________________________________________________________________________

Maple [A]  time = 0.053, size = 103, normalized size = 1.4 \begin{align*} -{\frac{1}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-3\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{1}{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 ) ^{-2}}+3\,{\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+a*sec(d*x+c)),x)

[Out]

-1/a/d*tan(1/2*d*x+1/2*c)-3/a/d/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3-1/a/d/(1+tan(1/2*d*x+1/2*c)^2)
^2*tan(1/2*d*x+1/2*c)+3/a/d*arctan(tan(1/2*d*x+1/2*c))

________________________________________________________________________________________

Maxima [A]  time = 1.75684, size = 180, normalized size = 2.43 \begin{align*} -\frac{\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 )}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((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

________________________________________________________________________________________

Fricas [A]  time = 1.63877, size = 149, normalized size = 2.01 \begin{align*} \frac{3 \, d x \cos \left (d x + c\right ) + 3 \, d x +{\left (\cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 4\right )} \sin \left (d x + c\right )}{2 \,{\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)^2/(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

1/2*(3*d*x*cos(d*x + c) + 3*d*x + (cos(d*x + c)^2 - cos(d*x + c) - 4)*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 ^{2}{\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)**2/(a+a*sec(d*x+c)),x)

[Out]

Integral(cos(c + d*x)**2/(sec(c + d*x) + 1), x)/a

________________________________________________________________________________________

Giac [A]  time = 1.28448, size = 99, normalized size = 1.34 \begin{align*} \frac{\frac{3 \,{\left (d x + c\right )}}{a} - \frac{2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a} - \frac{2 \,{\left (3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + \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 )}^{2} a}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(3*(d*x + c)/a - 2*tan(1/2*d*x + 1/2*c)/a - 2*(3*tan(1/2*d*x + 1/2*c)^3 + tan(1/2*d*x + 1/2*c))/((tan(1/2*
d*x + 1/2*c)^2 + 1)^2*a))/d