3.262 \(\int \frac{\cos ^7(x)}{a-a \sin ^2(x)} \, dx\)

Optimal. Leaf size=29 \[ \frac{\sin ^5(x)}{5 a}-\frac{2 \sin ^3(x)}{3 a}+\frac{\sin (x)}{a} \]

[Out]

Sin[x]/a - (2*Sin[x]^3)/(3*a) + Sin[x]^5/(5*a)

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Rubi [A]  time = 0.0523097, antiderivative size = 29, normalized size of antiderivative = 1., 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, 2633} \[ \frac{\sin ^5(x)}{5 a}-\frac{2 \sin ^3(x)}{3 a}+\frac{\sin (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^7/(a - a*Sin[x]^2),x]

[Out]

Sin[x]/a - (2*Sin[x]^3)/(3*a) + Sin[x]^5/(5*a)

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 2633

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

Rubi steps

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

Mathematica [A]  time = 0.0030325, size = 27, normalized size = 0.93 \[ \frac{\frac{5 \sin (x)}{8}+\frac{5}{48} \sin (3 x)+\frac{1}{80} \sin (5 x)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^7/(a - a*Sin[x]^2),x]

[Out]

((5*Sin[x])/8 + (5*Sin[3*x])/48 + Sin[5*x]/80)/a

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Maple [A]  time = 0.035, size = 20, normalized size = 0.7 \begin{align*}{\frac{1}{a} \left ({\frac{ \left ( \sin \left ( x \right ) \right ) ^{5}}{5}}-{\frac{2\, \left ( \sin \left ( x \right ) \right ) ^{3}}{3}}+\sin \left ( x \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^7/(a-a*sin(x)^2),x)

[Out]

1/a*(1/5*sin(x)^5-2/3*sin(x)^3+sin(x))

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Maxima [A]  time = 1.0433, size = 30, normalized size = 1.03 \begin{align*} \frac{3 \, \sin \left (x\right )^{5} - 10 \, \sin \left (x\right )^{3} + 15 \, \sin \left (x\right )}{15 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^7/(a-a*sin(x)^2),x, algorithm="maxima")

[Out]

1/15*(3*sin(x)^5 - 10*sin(x)^3 + 15*sin(x))/a

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Fricas [A]  time = 2.53558, size = 61, normalized size = 2.1 \begin{align*} \frac{{\left (3 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 8\right )} \sin \left (x\right )}{15 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^7/(a-a*sin(x)^2),x, algorithm="fricas")

[Out]

1/15*(3*cos(x)^4 + 4*cos(x)^2 + 8)*sin(x)/a

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Sympy [B]  time = 35.1572, size = 311, normalized size = 10.72 \begin{align*} \frac{30 \tan ^{9}{\left (\frac{x}{2} \right )}}{15 a \tan ^{10}{\left (\frac{x}{2} \right )} + 75 a \tan ^{8}{\left (\frac{x}{2} \right )} + 150 a \tan ^{6}{\left (\frac{x}{2} \right )} + 150 a \tan ^{4}{\left (\frac{x}{2} \right )} + 75 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} + \frac{40 \tan ^{7}{\left (\frac{x}{2} \right )}}{15 a \tan ^{10}{\left (\frac{x}{2} \right )} + 75 a \tan ^{8}{\left (\frac{x}{2} \right )} + 150 a \tan ^{6}{\left (\frac{x}{2} \right )} + 150 a \tan ^{4}{\left (\frac{x}{2} \right )} + 75 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} + \frac{116 \tan ^{5}{\left (\frac{x}{2} \right )}}{15 a \tan ^{10}{\left (\frac{x}{2} \right )} + 75 a \tan ^{8}{\left (\frac{x}{2} \right )} + 150 a \tan ^{6}{\left (\frac{x}{2} \right )} + 150 a \tan ^{4}{\left (\frac{x}{2} \right )} + 75 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} + \frac{40 \tan ^{3}{\left (\frac{x}{2} \right )}}{15 a \tan ^{10}{\left (\frac{x}{2} \right )} + 75 a \tan ^{8}{\left (\frac{x}{2} \right )} + 150 a \tan ^{6}{\left (\frac{x}{2} \right )} + 150 a \tan ^{4}{\left (\frac{x}{2} \right )} + 75 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} + \frac{30 \tan{\left (\frac{x}{2} \right )}}{15 a \tan ^{10}{\left (\frac{x}{2} \right )} + 75 a \tan ^{8}{\left (\frac{x}{2} \right )} + 150 a \tan ^{6}{\left (\frac{x}{2} \right )} + 150 a \tan ^{4}{\left (\frac{x}{2} \right )} + 75 a \tan ^{2}{\left (\frac{x}{2} \right )} + 15 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**7/(a-a*sin(x)**2),x)

[Out]

30*tan(x/2)**9/(15*a*tan(x/2)**10 + 75*a*tan(x/2)**8 + 150*a*tan(x/2)**6 + 150*a*tan(x/2)**4 + 75*a*tan(x/2)**
2 + 15*a) + 40*tan(x/2)**7/(15*a*tan(x/2)**10 + 75*a*tan(x/2)**8 + 150*a*tan(x/2)**6 + 150*a*tan(x/2)**4 + 75*
a*tan(x/2)**2 + 15*a) + 116*tan(x/2)**5/(15*a*tan(x/2)**10 + 75*a*tan(x/2)**8 + 150*a*tan(x/2)**6 + 150*a*tan(
x/2)**4 + 75*a*tan(x/2)**2 + 15*a) + 40*tan(x/2)**3/(15*a*tan(x/2)**10 + 75*a*tan(x/2)**8 + 150*a*tan(x/2)**6
+ 150*a*tan(x/2)**4 + 75*a*tan(x/2)**2 + 15*a) + 30*tan(x/2)/(15*a*tan(x/2)**10 + 75*a*tan(x/2)**8 + 150*a*tan
(x/2)**6 + 150*a*tan(x/2)**4 + 75*a*tan(x/2)**2 + 15*a)

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Giac [A]  time = 1.14075, size = 30, normalized size = 1.03 \begin{align*} \frac{3 \, \sin \left (x\right )^{5} - 10 \, \sin \left (x\right )^{3} + 15 \, \sin \left (x\right )}{15 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^7/(a-a*sin(x)^2),x, algorithm="giac")

[Out]

1/15*(3*sin(x)^5 - 10*sin(x)^3 + 15*sin(x))/a