3.183 \(\int \frac{1}{(a+a \tan ^2(c+d x))^2} \, dx\)

Optimal. Leaf size=55 \[ \frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}+\frac{3 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{3 x}{8 a^2} \]

[Out]

(3*x)/(8*a^2) + (3*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a^2*d)

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Rubi [A]  time = 0.0336857, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3657, 12, 2635, 8} \[ \frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}+\frac{3 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{3 x}{8 a^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Tan[c + d*x]^2)^(-2),x]

[Out]

(3*x)/(8*a^2) + (3*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a^2*d)

Rule 3657

Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+a \tan ^2(c+d x)\right )^2} \, dx &=\int \frac{\cos ^4(c+d x)}{a^2} \, dx\\ &=\frac{\int \cos ^4(c+d x) \, dx}{a^2}\\ &=\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a^2 d}+\frac{3 \int \cos ^2(c+d x) \, dx}{4 a^2}\\ &=\frac{3 \cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a^2 d}+\frac{3 \int 1 \, dx}{8 a^2}\\ &=\frac{3 x}{8 a^2}+\frac{3 \cos (c+d x) \sin (c+d x)}{8 a^2 d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a^2 d}\\ \end{align*}

Mathematica [A]  time = 0.0443636, size = 36, normalized size = 0.65 \[ \frac{12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x))}{32 a^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Tan[c + d*x]^2)^(-2),x]

[Out]

(12*(c + d*x) + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)])/(32*a^2*d)

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Maple [A]  time = 0.017, size = 69, normalized size = 1.3 \begin{align*}{\frac{\tan \left ( dx+c \right ) }{4\,d{a}^{2} \left ( \left ( \tan \left ( dx+c \right ) \right ) ^{2}+1 \right ) ^{2}}}+{\frac{3\,\tan \left ( dx+c \right ) }{8\,d{a}^{2} \left ( \left ( \tan \left ( dx+c \right ) \right ) ^{2}+1 \right ) }}+{\frac{3\,\arctan \left ( \tan \left ( dx+c \right ) \right ) }{8\,d{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/d/a^2*tan(d*x+c)/(tan(d*x+c)^2+1)^2+3/8/d/a^2*tan(d*x+c)/(tan(d*x+c)^2+1)+3/8/d/a^2*arctan(tan(d*x+c))

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Maxima [A]  time = 1.49575, size = 90, normalized size = 1.64 \begin{align*} \frac{\frac{3 \, \tan \left (d x + c\right )^{3} + 5 \, \tan \left (d x + c\right )}{a^{2} \tan \left (d x + c\right )^{4} + 2 \, a^{2} \tan \left (d x + c\right )^{2} + a^{2}} + \frac{3 \,{\left (d x + c\right )}}{a^{2}}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*tan(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/8*((3*tan(d*x + c)^3 + 5*tan(d*x + c))/(a^2*tan(d*x + c)^4 + 2*a^2*tan(d*x + c)^2 + a^2) + 3*(d*x + c)/a^2)/
d

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Fricas [A]  time = 1.04385, size = 204, normalized size = 3.71 \begin{align*} \frac{3 \, d x \tan \left (d x + c\right )^{4} + 6 \, d x \tan \left (d x + c\right )^{2} + 3 \, \tan \left (d x + c\right )^{3} + 3 \, d x + 5 \, \tan \left (d x + c\right )}{8 \,{\left (a^{2} d \tan \left (d x + c\right )^{4} + 2 \, a^{2} d \tan \left (d x + c\right )^{2} + a^{2} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*tan(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

1/8*(3*d*x*tan(d*x + c)^4 + 6*d*x*tan(d*x + c)^2 + 3*tan(d*x + c)^3 + 3*d*x + 5*tan(d*x + c))/(a^2*d*tan(d*x +
c)^4 + 2*a^2*d*tan(d*x + c)^2 + a^2*d)

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Sympy [A]  time = 1.05706, size = 248, normalized size = 4.51 \begin{align*} \begin{cases} \frac{3 d x \tan ^{4}{\left (c + d x \right )}}{8 a^{2} d \tan ^{4}{\left (c + d x \right )} + 16 a^{2} d \tan ^{2}{\left (c + d x \right )} + 8 a^{2} d} + \frac{6 d x \tan ^{2}{\left (c + d x \right )}}{8 a^{2} d \tan ^{4}{\left (c + d x \right )} + 16 a^{2} d \tan ^{2}{\left (c + d x \right )} + 8 a^{2} d} + \frac{3 d x}{8 a^{2} d \tan ^{4}{\left (c + d x \right )} + 16 a^{2} d \tan ^{2}{\left (c + d x \right )} + 8 a^{2} d} + \frac{3 \tan ^{3}{\left (c + d x \right )}}{8 a^{2} d \tan ^{4}{\left (c + d x \right )} + 16 a^{2} d \tan ^{2}{\left (c + d x \right )} + 8 a^{2} d} + \frac{5 \tan{\left (c + d x \right )}}{8 a^{2} d \tan ^{4}{\left (c + d x \right )} + 16 a^{2} d \tan ^{2}{\left (c + d x \right )} + 8 a^{2} d} & \text{for}\: d \neq 0 \\\frac{x}{\left (a \tan ^{2}{\left (c \right )} + a\right )^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*tan(d*x+c)**2)**2,x)

[Out]

Piecewise((3*d*x*tan(c + d*x)**4/(8*a**2*d*tan(c + d*x)**4 + 16*a**2*d*tan(c + d*x)**2 + 8*a**2*d) + 6*d*x*tan
(c + d*x)**2/(8*a**2*d*tan(c + d*x)**4 + 16*a**2*d*tan(c + d*x)**2 + 8*a**2*d) + 3*d*x/(8*a**2*d*tan(c + d*x)*
*4 + 16*a**2*d*tan(c + d*x)**2 + 8*a**2*d) + 3*tan(c + d*x)**3/(8*a**2*d*tan(c + d*x)**4 + 16*a**2*d*tan(c + d
*x)**2 + 8*a**2*d) + 5*tan(c + d*x)/(8*a**2*d*tan(c + d*x)**4 + 16*a**2*d*tan(c + d*x)**2 + 8*a**2*d), Ne(d, 0
)), (x/(a*tan(c)**2 + a)**2, True))

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Giac [A]  time = 1.29306, size = 69, normalized size = 1.25 \begin{align*} \frac{\frac{3 \,{\left (d x + c\right )}}{a^{2}} + \frac{3 \, \tan \left (d x + c\right )^{3} + 5 \, \tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2} a^{2}}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*tan(d*x+c)^2)^2,x, algorithm="giac")

[Out]

1/8*(3*(d*x + c)/a^2 + (3*tan(d*x + c)^3 + 5*tan(d*x + c))/((tan(d*x + c)^2 + 1)^2*a^2))/d