3.146 \(\int (a-a \sec ^2(c+d x))^2 \, dx\)

Optimal. Leaf size=38 \[ \frac{a^2 \tan ^3(c+d x)}{3 d}-\frac{a^2 \tan (c+d x)}{d}+a^2 x \]

[Out]

a^2*x - (a^2*Tan[c + d*x])/d + (a^2*Tan[c + d*x]^3)/(3*d)

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Rubi [A]  time = 0.0300588, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4120, 3473, 8} \[ \frac{a^2 \tan ^3(c+d x)}{3 d}-\frac{a^2 \tan (c+d x)}{d}+a^2 x \]

Antiderivative was successfully verified.

[In]

Int[(a - a*Sec[c + d*x]^2)^2,x]

[Out]

a^2*x - (a^2*Tan[c + d*x])/d + (a^2*Tan[c + d*x]^3)/(3*d)

Rule 4120

Int[(u_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[b^p, Int[ActivateTrig[u*tan[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

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

Rubi steps

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

Mathematica [A]  time = 0.0203196, size = 42, normalized size = 1.11 \[ a^2 \left (\frac{\tan ^3(c+d x)}{3 d}+\frac{\tan ^{-1}(\tan (c+d x))}{d}-\frac{\tan (c+d x)}{d}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a - a*Sec[c + d*x]^2)^2,x]

[Out]

a^2*(ArcTan[Tan[c + d*x]]/d - Tan[c + d*x]/d + Tan[c + d*x]^3/(3*d))

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Maple [A]  time = 0.024, size = 49, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( dx+c \right ) -2\,{a}^{2}\tan \left ( dx+c \right ) -{a}^{2} \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \tan \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0022, size = 61, normalized size = 1.61 \begin{align*} a^{2} x + \frac{{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{2}}{3 \, d} - \frac{2 \, a^{2} \tan \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.475513, size = 128, normalized size = 3.37 \begin{align*} \frac{3 \, a^{2} d x \cos \left (d x + c\right )^{3} -{\left (4 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*sec(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

1/3*(3*a^2*d*x*cos(d*x + c)^3 - (4*a^2*cos(d*x + c)^2 - a^2)*sin(d*x + c))/(d*cos(d*x + c)^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} \left (\int 1\, dx + \int - 2 \sec ^{2}{\left (c + d x \right )}\, dx + \int \sec ^{4}{\left (c + d x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*(Integral(1, x) + Integral(-2*sec(c + d*x)**2, x) + Integral(sec(c + d*x)**4, x))

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Giac [A]  time = 1.27645, size = 53, normalized size = 1.39 \begin{align*} \frac{a^{2} \tan \left (d x + c\right )^{3} + 3 \,{\left (d x + c\right )} a^{2} - 3 \, a^{2} \tan \left (d x + c\right )}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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