\(\int \frac {1}{(a-a \sin ^2(x))^5} \, dx\) [63]

Optimal result

Integrand size = 11, antiderivative size = 51 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^5} \, dx=\frac {\tan (x)}{a^5}+\frac {4 \tan ^3(x)}{3 a^5}+\frac {6 \tan ^5(x)}{5 a^5}+\frac {4 \tan ^7(x)}{7 a^5}+\frac {\tan ^9(x)}{9 a^5} \]

[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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3254, 3852} \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^5} \, dx=\frac {\tan ^9(x)}{9 a^5}+\frac {4 \tan ^7(x)}{7 a^5}+\frac {6 \tan ^5(x)}{5 a^5}+\frac {4 \tan ^3(x)}{3 a^5}+\frac {\tan (x)}{a^5} \]

[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 3254

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 3852

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*} \text {integral}& = \frac {\int \sec ^{10}(x) \, dx}{a^5} \\ & = -\frac {\text {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*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^5} \, dx=\frac {\tan (x)+\frac {4 \tan ^3(x)}{3}+\frac {6 \tan ^5(x)}{5}+\frac {4 \tan ^7(x)}{7}+\frac {\tan ^9(x)}{9}}{a^5} \]

[In]

Integrate[(a - a*Sin[x]^2)^(-5),x]

[Out]

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

Maple [A] (verified)

Time = 1.46 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.63

method	result	size
default	\(\frac {\frac {\left (\tan ^{9}\left (x \right )\right )}{9}+\frac {4 \left (\tan ^{7}\left (x \right )\right )}{7}+\frac {6 \left (\tan ^{5}\left (x \right )\right )}{5}+\frac {4 \left (\tan ^{3}\left (x \right )\right )}{3}+\tan \left (x \right )}{a^{5}}\)	\(32\)
parallelrisch	\(\frac {\tan \left (x \right ) \left (\sec ^{8}\left (x \right )\right ) \left (128+\cos \left (8 x \right )+10 \cos \left (6 x \right )+46 \cos \left (4 x \right )+130 \cos \left (2 x \right )\right )}{315 a^{5}}\)	\(36\)
risch	\(\frac {256 i \left (126 \,{\mathrm e}^{8 i x}+84 \,{\mathrm e}^{6 i x}+36 \,{\mathrm e}^{4 i x}+9 \,{\mathrm e}^{2 i x}+1\right )}{315 \left ({\mathrm e}^{2 i x}+1\right )^{9} a^{5}}\)	\(46\)

[In]

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

[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))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^5} \, dx=\frac {{\left (128 \, \cos \left (x\right )^{8} + 64 \, \cos \left (x\right )^{6} + 48 \, \cos \left (x\right )^{4} + 40 \, \cos \left (x\right )^{2} + 35\right )} \sin \left (x\right )}{315 \, a^{5} \cos \left (x\right )^{9}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1083 vs. \(2 (51) = 102\).

Time = 14.84 (sec) , antiderivative size = 1083, normalized size of antiderivative = 21.24 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^5} \, dx=\text {Too large to display} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^5} \, dx=\frac {35 \, \tan \left (x\right )^{9} + 180 \, \tan \left (x\right )^{7} + 378 \, \tan \left (x\right )^{5} + 420 \, \tan \left (x\right )^{3} + 315 \, \tan \left (x\right )}{315 \, a^{5}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^5} \, dx=\frac {35 \, \tan \left (x\right )^{9} + 180 \, \tan \left (x\right )^{7} + 378 \, \tan \left (x\right )^{5} + 420 \, \tan \left (x\right )^{3} + 315 \, \tan \left (x\right )}{315 \, a^{5}} \]

[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

Mupad [B] (verification not implemented)

Time = 13.14 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a-a \sin ^2(x)\right )^5} \, dx=\frac {\mathrm {tan}\left (x\right )}{a^5}+\frac {4\,{\mathrm {tan}\left (x\right )}^3}{3\,a^5}+\frac {6\,{\mathrm {tan}\left (x\right )}^5}{5\,a^5}+\frac {4\,{\mathrm {tan}\left (x\right )}^7}{7\,a^5}+\frac {{\mathrm {tan}\left (x\right )}^9}{9\,a^5} \]

[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)