3.284 \(\int \frac {\sec ^4(x)}{(a-a \sin ^2(x))^2} \, dx\)

Optimal. Leaf size=37 \[ \frac {\tan ^7(x)}{7 a^2}+\frac {3 \tan ^5(x)}{5 a^2}+\frac {\tan ^3(x)}{a^2}+\frac {\tan (x)}{a^2} \]

[Out]

tan(x)/a^2+tan(x)^3/a^2+3/5*tan(x)^5/a^2+1/7*tan(x)^7/a^2

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3175, 3767} \[ \frac {\tan ^7(x)}{7 a^2}+\frac {3 \tan ^5(x)}{5 a^2}+\frac {\tan ^3(x)}{a^2}+\frac {\tan (x)}{a^2} \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]^4/(a - a*Sin[x]^2)^2,x]

[Out]

Tan[x]/a^2 + Tan[x]^3/a^2 + (3*Tan[x]^5)/(5*a^2) + Tan[x]^7/(7*a^2)

Rule 3175

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

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin {align*} \int \frac {\sec ^4(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx &=\frac {\int \sec ^8(x) \, dx}{a^2}\\ &=-\frac {\operatorname {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (x)\right )}{a^2}\\ &=\frac {\tan (x)}{a^2}+\frac {\tan ^3(x)}{a^2}+\frac {3 \tan ^5(x)}{5 a^2}+\frac {\tan ^7(x)}{7 a^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 41, normalized size = 1.11 \[ \frac {\frac {16 \tan (x)}{35}+\frac {1}{7} \tan (x) \sec ^6(x)+\frac {6}{35} \tan (x) \sec ^4(x)+\frac {8}{35} \tan (x) \sec ^2(x)}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^4/(a - a*Sin[x]^2)^2,x]

[Out]

((16*Tan[x])/35 + (8*Sec[x]^2*Tan[x])/35 + (6*Sec[x]^4*Tan[x])/35 + (Sec[x]^6*Tan[x])/7)/a^2

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 31, normalized size = 0.84 \[ \frac {{\left (16 \, \cos \relax (x)^{6} + 8 \, \cos \relax (x)^{4} + 6 \, \cos \relax (x)^{2} + 5\right )} \sin \relax (x)}{35 \, a^{2} \cos \relax (x)^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^4/(a-a*sin(x)^2)^2,x, algorithm="fricas")

[Out]

1/35*(16*cos(x)^6 + 8*cos(x)^4 + 6*cos(x)^2 + 5)*sin(x)/(a^2*cos(x)^7)

________________________________________________________________________________________

giac [A]  time = 0.13, size = 28, normalized size = 0.76 \[ \frac {5 \, \tan \relax (x)^{7} + 21 \, \tan \relax (x)^{5} + 35 \, \tan \relax (x)^{3} + 35 \, \tan \relax (x)}{35 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^4/(a-a*sin(x)^2)^2,x, algorithm="giac")

[Out]

1/35*(5*tan(x)^7 + 21*tan(x)^5 + 35*tan(x)^3 + 35*tan(x))/a^2

________________________________________________________________________________________

maple [A]  time = 0.24, size = 24, normalized size = 0.65 \[ \frac {\frac {\left (\tan ^{7}\relax (x )\right )}{7}+\frac {3 \left (\tan ^{5}\relax (x )\right )}{5}+\tan ^{3}\relax (x )+\tan \relax (x )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^4/(a-a*sin(x)^2)^2,x)

[Out]

1/a^2*(1/7*tan(x)^7+3/5*tan(x)^5+tan(x)^3+tan(x))

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 28, normalized size = 0.76 \[ \frac {5 \, \tan \relax (x)^{7} + 21 \, \tan \relax (x)^{5} + 35 \, \tan \relax (x)^{3} + 35 \, \tan \relax (x)}{35 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^4/(a-a*sin(x)^2)^2,x, algorithm="maxima")

[Out]

1/35*(5*tan(x)^7 + 21*tan(x)^5 + 35*tan(x)^3 + 35*tan(x))/a^2

________________________________________________________________________________________

mupad [B]  time = 13.84, size = 33, normalized size = 0.89 \[ \frac {\mathrm {tan}\relax (x)}{a^2}+\frac {{\mathrm {tan}\relax (x)}^3}{a^2}+\frac {3\,{\mathrm {tan}\relax (x)}^5}{5\,a^2}+\frac {{\mathrm {tan}\relax (x)}^7}{7\,a^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^4*(a - a*sin(x)^2)^2),x)

[Out]

tan(x)/a^2 + tan(x)^3/a^2 + (3*tan(x)^5)/(5*a^2) + tan(x)^7/(7*a^2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sec ^{4}{\relax (x )}}{\sin ^{4}{\relax (x )} - 2 \sin ^{2}{\relax (x )} + 1}\, dx}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**4/(a-a*sin(x)**2)**2,x)

[Out]

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

________________________________________________________________________________________