3.151 \(\int \frac{1}{(a-a \sec ^2(c+d x))^4} \, dx\)

Optimal. Leaf size=73 \[ -\frac{\cot ^7(c+d x)}{7 a^4 d}+\frac{\cot ^5(c+d x)}{5 a^4 d}-\frac{\cot ^3(c+d x)}{3 a^4 d}+\frac{\cot (c+d x)}{a^4 d}+\frac{x}{a^4} \]

[Out]

x/a^4 + Cot[c + d*x]/(a^4*d) - Cot[c + d*x]^3/(3*a^4*d) + Cot[c + d*x]^5/(5*a^4*d) - Cot[c + d*x]^7/(7*a^4*d)

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Rubi [A]  time = 0.0453561, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4120, 3473, 8} \[ -\frac{\cot ^7(c+d x)}{7 a^4 d}+\frac{\cot ^5(c+d x)}{5 a^4 d}-\frac{\cot ^3(c+d x)}{3 a^4 d}+\frac{\cot (c+d x)}{a^4 d}+\frac{x}{a^4} \]

Antiderivative was successfully verified.

[In]

Int[(a - a*Sec[c + d*x]^2)^(-4),x]

[Out]

x/a^4 + Cot[c + d*x]/(a^4*d) - Cot[c + d*x]^3/(3*a^4*d) + Cot[c + d*x]^5/(5*a^4*d) - Cot[c + d*x]^7/(7*a^4*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 \frac{1}{\left (a-a \sec ^2(c+d x)\right )^4} \, dx &=\frac{\int \cot ^8(c+d x) \, dx}{a^4}\\ &=-\frac{\cot ^7(c+d x)}{7 a^4 d}-\frac{\int \cot ^6(c+d x) \, dx}{a^4}\\ &=\frac{\cot ^5(c+d x)}{5 a^4 d}-\frac{\cot ^7(c+d x)}{7 a^4 d}+\frac{\int \cot ^4(c+d x) \, dx}{a^4}\\ &=-\frac{\cot ^3(c+d x)}{3 a^4 d}+\frac{\cot ^5(c+d x)}{5 a^4 d}-\frac{\cot ^7(c+d x)}{7 a^4 d}-\frac{\int \cot ^2(c+d x) \, dx}{a^4}\\ &=\frac{\cot (c+d x)}{a^4 d}-\frac{\cot ^3(c+d x)}{3 a^4 d}+\frac{\cot ^5(c+d x)}{5 a^4 d}-\frac{\cot ^7(c+d x)}{7 a^4 d}+\frac{\int 1 \, dx}{a^4}\\ &=\frac{x}{a^4}+\frac{\cot (c+d x)}{a^4 d}-\frac{\cot ^3(c+d x)}{3 a^4 d}+\frac{\cot ^5(c+d x)}{5 a^4 d}-\frac{\cot ^7(c+d x)}{7 a^4 d}\\ \end{align*}

Mathematica [C]  time = 0.0175616, size = 36, normalized size = 0.49 \[ -\frac{\cot ^7(c+d x) \text{Hypergeometric2F1}\left (-\frac{7}{2},1,-\frac{5}{2},-\tan ^2(c+d x)\right )}{7 a^4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - a*Sec[c + d*x]^2)^(-4),x]

[Out]

-(Cot[c + d*x]^7*Hypergeometric2F1[-7/2, 1, -5/2, -Tan[c + d*x]^2])/(7*a^4*d)

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Maple [A]  time = 0.05, size = 79, normalized size = 1.1 \begin{align*}{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{4}}}-{\frac{1}{7\,d{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{7}}}-{\frac{1}{3\,d{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}}+{\frac{1}{5\,d{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{5}}}+{\frac{1}{d{a}^{4}\tan \left ( dx+c \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-a*sec(d*x+c)^2)^4,x)

[Out]

1/d/a^4*arctan(tan(d*x+c))-1/7/d/a^4/tan(d*x+c)^7-1/3/d/a^4/tan(d*x+c)^3+1/5/d/a^4/tan(d*x+c)^5+1/d/a^4/tan(d*
x+c)

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Maxima [A]  time = 1.5127, size = 81, normalized size = 1.11 \begin{align*} \frac{\frac{105 \,{\left (d x + c\right )}}{a^{4}} + \frac{105 \, \tan \left (d x + c\right )^{6} - 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} - 15}{a^{4} \tan \left (d x + c\right )^{7}}}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*sec(d*x+c)^2)^4,x, algorithm="maxima")

[Out]

1/105*(105*(d*x + c)/a^4 + (105*tan(d*x + c)^6 - 35*tan(d*x + c)^4 + 21*tan(d*x + c)^2 - 15)/(a^4*tan(d*x + c)
^7))/d

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Fricas [B]  time = 0.494052, size = 374, normalized size = 5.12 \begin{align*} \frac{176 \, \cos \left (d x + c\right )^{7} - 406 \, \cos \left (d x + c\right )^{5} + 350 \, \cos \left (d x + c\right )^{3} + 105 \,{\left (d x \cos \left (d x + c\right )^{6} - 3 \, d x \cos \left (d x + c\right )^{4} + 3 \, d x \cos \left (d x + c\right )^{2} - d x\right )} \sin \left (d x + c\right ) - 105 \, \cos \left (d x + c\right )}{105 \,{\left (a^{4} d \cos \left (d x + c\right )^{6} - 3 \, a^{4} d \cos \left (d x + c\right )^{4} + 3 \, a^{4} d \cos \left (d x + c\right )^{2} - a^{4} d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*sec(d*x+c)^2)^4,x, algorithm="fricas")

[Out]

1/105*(176*cos(d*x + c)^7 - 406*cos(d*x + c)^5 + 350*cos(d*x + c)^3 + 105*(d*x*cos(d*x + c)^6 - 3*d*x*cos(d*x
+ c)^4 + 3*d*x*cos(d*x + c)^2 - d*x)*sin(d*x + c) - 105*cos(d*x + c))/((a^4*d*cos(d*x + c)^6 - 3*a^4*d*cos(d*x
+ c)^4 + 3*a^4*d*cos(d*x + c)^2 - a^4*d)*sin(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{\sec ^{8}{\left (c + d x \right )} - 4 \sec ^{6}{\left (c + d x \right )} + 6 \sec ^{4}{\left (c + d x \right )} - 4 \sec ^{2}{\left (c + d x \right )} + 1}\, dx}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sec(c + d*x)**8 - 4*sec(c + d*x)**6 + 6*sec(c + d*x)**4 - 4*sec(c + d*x)**2 + 1), x)/a**4

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Giac [B]  time = 1.1497, size = 188, normalized size = 2.58 \begin{align*} \frac{\frac{13440 \,{\left (d x + c\right )}}{a^{4}} + \frac{9765 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 1295 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 189 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15}{a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}} + \frac{15 \, a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 189 \, a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1295 \, a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 9765 \, a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{28}}}{13440 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*sec(d*x+c)^2)^4,x, algorithm="giac")

[Out]

1/13440*(13440*(d*x + c)/a^4 + (9765*tan(1/2*d*x + 1/2*c)^6 - 1295*tan(1/2*d*x + 1/2*c)^4 + 189*tan(1/2*d*x +
1/2*c)^2 - 15)/(a^4*tan(1/2*d*x + 1/2*c)^7) + (15*a^24*tan(1/2*d*x + 1/2*c)^7 - 189*a^24*tan(1/2*d*x + 1/2*c)^
5 + 1295*a^24*tan(1/2*d*x + 1/2*c)^3 - 9765*a^24*tan(1/2*d*x + 1/2*c))/a^28)/d