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*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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Rubi [A]  time = 0.0263289, antiderivative size = 51, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0059237, size = 51, normalized size = 1. \[ \frac{\frac{128 \tan (x)}{315}+\frac{1}{9} \tan (x) \sec ^8(x)+\frac{8}{63} \tan (x) \sec ^6(x)+\frac{16}{105} \tan (x) \sec ^4(x)+\frac{64}{315} \tan (x) \sec ^2(x)}{a^5} \]

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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Maple [A]  time = 0.031, size = 32, normalized size = 0.6 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{ \left ( \tan \left ( x \right ) \right ) ^{9}}{9}}+{\frac{4\, \left ( \tan \left ( x \right ) \right ) ^{7}}{7}}+{\frac{6\, \left ( \tan \left ( x \right ) \right ) ^{5}}{5}}+{\frac{4\, \left ( \tan \left ( x \right ) \right ) ^{3}}{3}}+\tan \left ( x \right ) \right ) } \end{align*}

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 = 0.958202, size = 46, normalized size = 0.9 \begin{align*} \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}} \end{align*}

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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Fricas [A]  time = 1.86624, size = 123, normalized size = 2.41 \begin{align*} \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}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.15653, size = 46, normalized size = 0.9 \begin{align*} \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}} \end{align*}

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