3.63 \(\int \frac {1}{(a-a \sin ^2(x))^5} \, dx\)

Optimal. Leaf size=51 \[ \frac {\tan ^9(x)}{9 a^5}+\frac {4 \tan ^7(x)}{7 a^5}+\frac {6 \tan ^5(x)}{5 a^5}+\frac {4 \tan ^3(x)}{3 a^5}+\frac {\tan (x)}{a^5} \]

[Out]

tan(x)/a^5+4/3*tan(x)^3/a^5+6/5*tan(x)^5/a^5+4/7*tan(x)^7/a^5+1/9*tan(x)^9/a^5

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Rubi [A]  time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3175, 3767} \[ \frac {\tan ^9(x)}{9 a^5}+\frac {4 \tan ^7(x)}{7 a^5}+\frac {6 \tan ^5(x)}{5 a^5}+\frac {4 \tan ^3(x)}{3 a^5}+\frac {\tan (x)}{a^5} \]

Antiderivative was successfully verified.

[In]

Int[(a - a*Sin[x]^2)^(-5),x]

[Out]

Tan[x]/a^5 + (4*Tan[x]^3)/(3*a^5) + (6*Tan[x]^5)/(5*a^5) + (4*Tan[x]^7)/(7*a^5) + Tan[x]^9/(9*a^5)

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 {1}{\left (a-a \sin ^2(x)\right )^5} \, dx &=\frac {\int \sec ^{10}(x) \, dx}{a^5}\\ &=-\frac {\operatorname {Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,-\tan (x)\right )}{a^5}\\ &=\frac {\tan (x)}{a^5}+\frac {4 \tan ^3(x)}{3 a^5}+\frac {6 \tan ^5(x)}{5 a^5}+\frac {4 \tan ^7(x)}{7 a^5}+\frac {\tan ^9(x)}{9 a^5}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 51, normalized size = 1.00 \[ \frac {\frac {128 \tan (x)}{315}+\frac {1}{9} \tan (x) \sec ^8(x)+\frac {8}{63} \tan (x) \sec ^6(x)+\frac {16}{105} \tan (x) \sec ^4(x)+\frac {64}{315} \tan (x) \sec ^2(x)}{a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - a*Sin[x]^2)^(-5),x]

[Out]

((128*Tan[x])/315 + (64*Sec[x]^2*Tan[x])/315 + (16*Sec[x]^4*Tan[x])/105 + (8*Sec[x]^6*Tan[x])/63 + (Sec[x]^8*T
an[x])/9)/a^5

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fricas [A]  time = 0.43, size = 37, normalized size = 0.73 \[ \frac {{\left (128 \, \cos \relax (x)^{8} + 64 \, \cos \relax (x)^{6} + 48 \, \cos \relax (x)^{4} + 40 \, \cos \relax (x)^{2} + 35\right )} \sin \relax (x)}{315 \, a^{5} \cos \relax (x)^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*sin(x)^2)^5,x, algorithm="fricas")

[Out]

1/315*(128*cos(x)^8 + 64*cos(x)^6 + 48*cos(x)^4 + 40*cos(x)^2 + 35)*sin(x)/(a^5*cos(x)^9)

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giac [A]  time = 0.12, size = 34, normalized size = 0.67 \[ \frac {35 \, \tan \relax (x)^{9} + 180 \, \tan \relax (x)^{7} + 378 \, \tan \relax (x)^{5} + 420 \, \tan \relax (x)^{3} + 315 \, \tan \relax (x)}{315 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*sin(x)^2)^5,x, algorithm="giac")

[Out]

1/315*(35*tan(x)^9 + 180*tan(x)^7 + 378*tan(x)^5 + 420*tan(x)^3 + 315*tan(x))/a^5

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maple [A]  time = 0.16, size = 32, normalized size = 0.63 \[ \frac {\frac {\left (\tan ^{9}\relax (x )\right )}{9}+\frac {4 \left (\tan ^{7}\relax (x )\right )}{7}+\frac {6 \left (\tan ^{5}\relax (x )\right )}{5}+\frac {4 \left (\tan ^{3}\relax (x )\right )}{3}+\tan \relax (x )}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-a*sin(x)^2)^5,x)

[Out]

1/a^5*(1/9*tan(x)^9+4/7*tan(x)^7+6/5*tan(x)^5+4/3*tan(x)^3+tan(x))

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maxima [A]  time = 0.34, size = 34, normalized size = 0.67 \[ \frac {35 \, \tan \relax (x)^{9} + 180 \, \tan \relax (x)^{7} + 378 \, \tan \relax (x)^{5} + 420 \, \tan \relax (x)^{3} + 315 \, \tan \relax (x)}{315 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*sin(x)^2)^5,x, algorithm="maxima")

[Out]

1/315*(35*tan(x)^9 + 180*tan(x)^7 + 378*tan(x)^5 + 420*tan(x)^3 + 315*tan(x))/a^5

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mupad [B]  time = 13.34, size = 43, normalized size = 0.84 \[ \frac {\mathrm {tan}\relax (x)}{a^5}+\frac {4\,{\mathrm {tan}\relax (x)}^3}{3\,a^5}+\frac {6\,{\mathrm {tan}\relax (x)}^5}{5\,a^5}+\frac {4\,{\mathrm {tan}\relax (x)}^7}{7\,a^5}+\frac {{\mathrm {tan}\relax (x)}^9}{9\,a^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - a*sin(x)^2)^5,x)

[Out]

tan(x)/a^5 + (4*tan(x)^3)/(3*a^5) + (6*tan(x)^5)/(5*a^5) + (4*tan(x)^7)/(7*a^5) + tan(x)^9/(9*a^5)

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sympy [B]  time = 69.51, size = 1083, normalized size = 21.24 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-a*sin(x)**2)**5,x)

[Out]

-630*tan(x/2)**17/(315*a**5*tan(x/2)**18 - 2835*a**5*tan(x/2)**16 + 11340*a**5*tan(x/2)**14 - 26460*a**5*tan(x
/2)**12 + 39690*a**5*tan(x/2)**10 - 39690*a**5*tan(x/2)**8 + 26460*a**5*tan(x/2)**6 - 11340*a**5*tan(x/2)**4 +
2835*a**5*tan(x/2)**2 - 315*a**5) + 1680*tan(x/2)**15/(315*a**5*tan(x/2)**18 - 2835*a**5*tan(x/2)**16 + 11340
*a**5*tan(x/2)**14 - 26460*a**5*tan(x/2)**12 + 39690*a**5*tan(x/2)**10 - 39690*a**5*tan(x/2)**8 + 26460*a**5*t
an(x/2)**6 - 11340*a**5*tan(x/2)**4 + 2835*a**5*tan(x/2)**2 - 315*a**5) - 9576*tan(x/2)**13/(315*a**5*tan(x/2)
**18 - 2835*a**5*tan(x/2)**16 + 11340*a**5*tan(x/2)**14 - 26460*a**5*tan(x/2)**12 + 39690*a**5*tan(x/2)**10 -
39690*a**5*tan(x/2)**8 + 26460*a**5*tan(x/2)**6 - 11340*a**5*tan(x/2)**4 + 2835*a**5*tan(x/2)**2 - 315*a**5) +
10224*tan(x/2)**11/(315*a**5*tan(x/2)**18 - 2835*a**5*tan(x/2)**16 + 11340*a**5*tan(x/2)**14 - 26460*a**5*tan
(x/2)**12 + 39690*a**5*tan(x/2)**10 - 39690*a**5*tan(x/2)**8 + 26460*a**5*tan(x/2)**6 - 11340*a**5*tan(x/2)**4
+ 2835*a**5*tan(x/2)**2 - 315*a**5) - 21316*tan(x/2)**9/(315*a**5*tan(x/2)**18 - 2835*a**5*tan(x/2)**16 + 113
40*a**5*tan(x/2)**14 - 26460*a**5*tan(x/2)**12 + 39690*a**5*tan(x/2)**10 - 39690*a**5*tan(x/2)**8 + 26460*a**5
*tan(x/2)**6 - 11340*a**5*tan(x/2)**4 + 2835*a**5*tan(x/2)**2 - 315*a**5) + 10224*tan(x/2)**7/(315*a**5*tan(x/
2)**18 - 2835*a**5*tan(x/2)**16 + 11340*a**5*tan(x/2)**14 - 26460*a**5*tan(x/2)**12 + 39690*a**5*tan(x/2)**10
- 39690*a**5*tan(x/2)**8 + 26460*a**5*tan(x/2)**6 - 11340*a**5*tan(x/2)**4 + 2835*a**5*tan(x/2)**2 - 315*a**5)
- 9576*tan(x/2)**5/(315*a**5*tan(x/2)**18 - 2835*a**5*tan(x/2)**16 + 11340*a**5*tan(x/2)**14 - 26460*a**5*tan
(x/2)**12 + 39690*a**5*tan(x/2)**10 - 39690*a**5*tan(x/2)**8 + 26460*a**5*tan(x/2)**6 - 11340*a**5*tan(x/2)**4
+ 2835*a**5*tan(x/2)**2 - 315*a**5) + 1680*tan(x/2)**3/(315*a**5*tan(x/2)**18 - 2835*a**5*tan(x/2)**16 + 1134
0*a**5*tan(x/2)**14 - 26460*a**5*tan(x/2)**12 + 39690*a**5*tan(x/2)**10 - 39690*a**5*tan(x/2)**8 + 26460*a**5*
tan(x/2)**6 - 11340*a**5*tan(x/2)**4 + 2835*a**5*tan(x/2)**2 - 315*a**5) - 630*tan(x/2)/(315*a**5*tan(x/2)**18
- 2835*a**5*tan(x/2)**16 + 11340*a**5*tan(x/2)**14 - 26460*a**5*tan(x/2)**12 + 39690*a**5*tan(x/2)**10 - 3969
0*a**5*tan(x/2)**8 + 26460*a**5*tan(x/2)**6 - 11340*a**5*tan(x/2)**4 + 2835*a**5*tan(x/2)**2 - 315*a**5)

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