3.2.49 \(\int \frac {1}{(a-a \sec ^2(c+d x))^2} \, dx\) [149]

Optimal. Leaf size=37 \[ \frac {x}{a^2}+\frac {\cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4205, 3554, 8} \begin {gather*} -\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {\cot (c+d x)}{a^2 d}+\frac {x}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a - a*Sec[c + d*x]^2)^(-2),x]

[Out]

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.03, size = 36, normalized size = 0.97 \begin {gather*} -\frac {\cot ^3(c+d x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(c+d x)\right )}{3 a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(Cot[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(a^2*d)

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Maple [A]
time = 0.05, size = 34, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {\arctan \left (\tan \left (d x +c \right )\right )-\frac {1}{3 \tan \left (d x +c \right )^{3}}+\frac {1}{\tan \left (d x +c \right )}}{d \,a^{2}}\)	\(34\)
default	\(\frac {\arctan \left (\tan \left (d x +c \right )\right )-\frac {1}{3 \tan \left (d x +c \right )^{3}}+\frac {1}{\tan \left (d x +c \right )}}{d \,a^{2}}\)	\(34\)
risch	\(\frac {x}{a^{2}}+\frac {4 i \left (3 \,{\mathrm e}^{4 i \left (d x +c \right )}-3 \,{\mathrm e}^{2 i \left (d x +c \right )}+2\right )}{3 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}\)	\(53\)
norman	\(\frac {\frac {x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {1}{24 a d}+\frac {5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d}-\frac {5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d}+\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 a d}}{a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}\)	\(98\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 40, normalized size = 1.08 \begin {gather*} \frac {\frac {3 \, {\left (d x + c\right )}}{a^{2}} + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{a^{2} \tan \left (d x + c\right )^{3}}}{3 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (35) = 70\).
time = 3.52, size = 75, normalized size = 2.03 \begin {gather*} \frac {4 \, \cos \left (d x + c\right )^{3} + 3 \, {\left (d x \cos \left (d x + c\right )^{2} - d x\right )} \sin \left (d x + c\right ) - 3 \, \cos \left (d x + c\right )}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{\sec ^{4}{\left (c + d x \right )} - 2 \sec ^{2}{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (35) = 70\).
time = 0.44, size = 80, normalized size = 2.16 \begin {gather*} \frac {\frac {24 \, {\left (d x + c\right )}}{a^{2}} + \frac {15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} + \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}}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(24*(d*x + c)/a^2 + (15*tan(1/2*d*x + 1/2*c)^2 - 1)/(a^2*tan(1/2*d*x + 1/2*c)^3) + (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

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Mupad [B]
time = 4.41, size = 31, normalized size = 0.84 \begin {gather*} \frac {x}{a^2}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2-\frac {1}{3}}{a^2\,d\,{\mathrm {tan}\left (c+d\,x\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - a/cos(c + d*x)^2)^2,x)

[Out]

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

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