\(\int \frac {1}{(a-a \sec ^2(c+d x))^4} \, dx\) [151]

Optimal result

Integrand size = 15, antiderivative size = 73 \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^4} \, dx=\frac {x}{a^4}+\frac {\cot (c+d x)}{a^4 d}-\frac {\cot ^3(c+d x)}{3 a^4 d}+\frac {\cot ^5(c+d x)}{5 a^4 d}-\frac {\cot ^7(c+d x)}{7 a^4 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4205, 3554, 8} \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^4} \, dx=-\frac {\cot ^7(c+d x)}{7 a^4 d}+\frac {\cot ^5(c+d x)}{5 a^4 d}-\frac {\cot ^3(c+d x)}{3 a^4 d}+\frac {\cot (c+d x)}{a^4 d}+\frac {x}{a^4} \]

[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 8

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

Rule 3554

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 4205

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^4} \, dx=-\frac {\cot ^7(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},-\tan ^2(c+d x)\right )}{7 a^4 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(\frac {-\frac {1}{7 \tan \left (d x +c \right )^{7}}-\frac {1}{3 \tan \left (d x +c \right )^{3}}+\frac {1}{5 \tan \left (d x +c \right )^{5}}+\frac {1}{\tan \left (d x +c \right )}+\arctan \left (\tan \left (d x +c \right )\right )}{d \,a^{4}}\)	\(54\)
default	\(\frac {-\frac {1}{7 \tan \left (d x +c \right )^{7}}-\frac {1}{3 \tan \left (d x +c \right )^{3}}+\frac {1}{5 \tan \left (d x +c \right )^{5}}+\frac {1}{\tan \left (d x +c \right )}+\arctan \left (\tan \left (d x +c \right )\right )}{d \,a^{4}}\)	\(54\)
risch	\(\frac {x}{a^{4}}+\frac {8 i \left (105 \,{\mathrm e}^{12 i \left (d x +c \right )}-315 \,{\mathrm e}^{10 i \left (d x +c \right )}+770 \,{\mathrm e}^{8 i \left (d x +c \right )}-770 \,{\mathrm e}^{6 i \left (d x +c \right )}+609 \,{\mathrm e}^{4 i \left (d x +c \right )}-203 \,{\mathrm e}^{2 i \left (d x +c \right )}+44\right )}{105 d \,a^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}\)	\(97\)
parallelrisch	\(\frac {-15 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+189 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-189 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-1295 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+1295 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+13440 d x +9765 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-9765 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{13440 d \,a^{4}}\)	\(114\)
norman	\(\frac {\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{a}-\frac {1}{896 a d}+\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{640 a d}-\frac {37 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{384 a d}+\frac {93 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{128 a d}-\frac {93 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{128 a d}+\frac {37 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{384 a d}-\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{640 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{896 a d}}{a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}\)	\(174\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (67) = 134\).

Time = 0.24 (sec) , antiderivative size = 147, normalized size of antiderivative = 2.01 \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^4} \, dx=\frac {176 \, \cos \left (d x + c\right )^{7} - 406 \, \cos \left (d x + c\right )^{5} + 350 \, \cos \left (d x + c\right )^{3} + 105 \, {\left (d x \cos \left (d x + c\right )^{6} - 3 \, d x \cos \left (d x + c\right )^{4} + 3 \, d x \cos \left (d x + c\right )^{2} - d x\right )} \sin \left (d x + c\right ) - 105 \, \cos \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} - 3 \, a^{4} d \cos \left (d x + c\right )^{4} + 3 \, a^{4} d \cos \left (d x + c\right )^{2} - a^{4} d\right )} \sin \left (d x + c\right )} \]

[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))

Sympy [F]

\[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^4} \, dx=\frac {\int \frac {1}{\sec ^{8}{\left (c + d x \right )} - 4 \sec ^{6}{\left (c + d x \right )} + 6 \sec ^{4}{\left (c + d x \right )} - 4 \sec ^{2}{\left (c + d x \right )} + 1}\, dx}{a^{4}} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^4} \, dx=\frac {\frac {105 \, {\left (d x + c\right )}}{a^{4}} + \frac {105 \, \tan \left (d x + c\right )^{6} - 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} - 15}{a^{4} \tan \left (d x + c\right )^{7}}}{105 \, d} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (67) = 134\).

Time = 0.31 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.90 \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^4} \, dx=\frac {\frac {13440 \, {\left (d x + c\right )}}{a^{4}} + \frac {9765 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1295 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 189 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}} + \frac {15 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 189 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1295 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9765 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{13440 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 19.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^4} \, dx=\frac {x}{a^4}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^6-\frac {{\mathrm {tan}\left (c+d\,x\right )}^4}{3}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{5}-\frac {1}{7}}{a^4\,d\,{\mathrm {tan}\left (c+d\,x\right )}^7} \]

[In]

int(1/(a - a/cos(c + d*x)^2)^4,x)

[Out]

x/a^4 + (tan(c + d*x)^2/5 - tan(c + d*x)^4/3 + tan(c + d*x)^6 - 1/7)/(a^4*d*tan(c + d*x)^7)