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

Optimal. Leaf size=72 \[ \frac{10 \sin (c+d x)}{3 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac{2 x}{a^2}-\frac{\sin (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]

[Out]

(-2*x)/a^2 + (10*Sin[c + d*x])/(3*a^2*d) - (2*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*(a
+ a*Sec[c + d*x])^2)

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Rubi [A]  time = 0.129373, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {3817, 4020, 3787, 2637, 8} \[ \frac{10 \sin (c+d x)}{3 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac{2 x}{a^2}-\frac{\sin (c+d x)}{3 d (a \sec (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/a^2 + (10*Sin[c + d*x])/(3*a^2*d) - (2*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - Sin[c + d*x]/(3*d*(a
+ a*Sec[c + d*x])^2)

Rule 3817

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

Rule 4020

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> -Simp[((A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(b*f*(2
*m + 1)), x] - Dist[1/(a^2*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[b*B*n - a*A*(2
*m + n + 1) + (A*b - a*B)*(m + n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*
b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)] &&  !GtQ[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))^2} \, dx &=-\frac{\sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac{\int \frac{\cos (c+d x) (-4 a+2 a \sec (c+d x))}{a+a \sec (c+d x)} \, dx}{3 a^2}\\ &=-\frac{2 \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac{\sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac{\int \cos (c+d x) \left (-10 a^2+6 a^2 \sec (c+d x)\right ) \, dx}{3 a^4}\\ &=-\frac{2 \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac{\sin (c+d x)}{3 d (a+a \sec (c+d x))^2}-\frac{2 \int 1 \, dx}{a^2}+\frac{10 \int \cos (c+d x) \, dx}{3 a^2}\\ &=-\frac{2 x}{a^2}+\frac{10 \sin (c+d x)}{3 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d (1+\sec (c+d x))}-\frac{\sin (c+d x)}{3 d (a+a \sec (c+d x))^2}\\ \end{align*}

Mathematica [B]  time = 0.477439, size = 151, normalized size = 2.1 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec ^3\left (\frac{1}{2} (c+d x)\right ) \left (-30 \sin \left (c+\frac{d x}{2}\right )+41 \sin \left (c+\frac{3 d x}{2}\right )+9 \sin \left (2 c+\frac{3 d x}{2}\right )+3 \sin \left (2 c+\frac{5 d x}{2}\right )+3 \sin \left (3 c+\frac{5 d x}{2}\right )-36 d x \cos \left (c+\frac{d x}{2}\right )-12 d x \cos \left (c+\frac{3 d x}{2}\right )-12 d x \cos \left (2 c+\frac{3 d x}{2}\right )+66 \sin \left (\frac{d x}{2}\right )-36 d x \cos \left (\frac{d x}{2}\right )\right )}{48 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.054, size = 88, normalized size = 1.2 \begin{align*} -{\frac{1}{6\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{5}{2\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}-4\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.67684, size = 159, normalized size = 2.21 \begin{align*} \frac{\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 )}}}{6 \, 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))^2,x, algorithm="maxima")

[Out]

1/6*((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)))
/d

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Fricas [A]  time = 1.62132, size = 230, normalized size = 3.19 \begin{align*} -\frac{6 \, d x \cos \left (d x + c\right )^{2} + 12 \, d x \cos \left (d x + c\right ) + 6 \, d x -{\left (3 \, \cos \left (d x + c\right )^{2} + 14 \, \cos \left (d x + c\right ) + 10\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+a*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/3*(6*d*x*cos(d*x + c)^2 + 12*d*x*cos(d*x + c) + 6*d*x - (3*cos(d*x + c)^2 + 14*cos(d*x + c) + 10)*sin(d*x +
c))/(a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cos{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.32418, size = 107, normalized size = 1.49 \begin{align*} -\frac{\frac{12 \,{\left (d x + c\right )}}{a^{2}} - \frac{12 \, \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^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, 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+a*sec(d*x+c))^2,x, algorithm="giac")

[Out]

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