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]

3/2*x/a-2*sin(d*x+c)/a/d+3/2*cos(d*x+c)*sin(d*x+c)/a/d-cos(d*x+c)*sin(d*x+c)/d/(a+a*sec(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 74, normalized size of antiderivative = 1.00, 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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 2637

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

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 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]

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 = 0.25, 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)

________________________________________________________________________________________

fricas [A]  time = 0.82, size = 57, normalized size = 0.77 \[ \frac {3 \, d x \cos \left (d x + c\right ) + 3 \, d x + {\left (\cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 4\right )} \sin \left (d x + c\right )}{2 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.33, size = 73, normalized size = 0.99 \[ \frac {\frac {3 \, {\left (d x + c\right )}}{a} - \frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} - \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a}}{2 \, d} \]

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

________________________________________________________________________________________

maple [A]  time = 0.51, size = 103, normalized size = 1.39 \[ -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a} \]

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

________________________________________________________________________________________

maxima [A]  time = 0.69, size = 133, normalized size = 1.80 \[ -\frac {\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \]

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

________________________________________________________________________________________

mupad [B]  time = 0.72, size = 89, normalized size = 1.20 \[ -\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (c+d\,x\right )}{2}+3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sin(c/2 + (d*x)/2) - (3*cos(c/2 + (d*x)/2)*(c + d*x))/2 + 3*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2) - 2*cos(
c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2))/(a*d*cos(c/2 + (d*x)/2))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\cos ^{2}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]

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

________________________________________________________________________________________