3.422 \(\int \frac{\sin (x)}{(5 \cos ^2(x)-2 \sin ^2(x))^{7/2}} \, dx\)

Optimal. Leaf size=55 \[ \frac{\cos (x)}{15 \sqrt{7 \cos ^2(x)-2}}-\frac{\cos (x)}{15 \left (7 \cos ^2(x)-2\right )^{3/2}}+\frac{\cos (x)}{10 \left (7 \cos ^2(x)-2\right )^{5/2}} \]

[Out]

Cos[x]/(10*(-2 + 7*Cos[x]^2)^(5/2)) - Cos[x]/(15*(-2 + 7*Cos[x]^2)^(3/2)) + Cos[x]/(15*Sqrt[-2 + 7*Cos[x]^2])

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Rubi [A]  time = 0.0535616, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4357, 192, 191} \[ \frac{\cos (x)}{15 \sqrt{7 \cos ^2(x)-2}}-\frac{\cos (x)}{15 \left (7 \cos ^2(x)-2\right )^{3/2}}+\frac{\cos (x)}{10 \left (7 \cos ^2(x)-2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]/(5*Cos[x]^2 - 2*Sin[x]^2)^(7/2),x]

[Out]

Cos[x]/(10*(-2 + 7*Cos[x]^2)^(5/2)) - Cos[x]/(15*(-2 + 7*Cos[x]^2)^(3/2)) + Cos[x]/(15*Sqrt[-2 + 7*Cos[x]^2])

Rule 4357

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[d/(
b*c), Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a
+ b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{\sin (x)}{\left (5 \cos ^2(x)-2 \sin ^2(x)\right )^{7/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{\left (-2+7 x^2\right )^{7/2}} \, dx,x,\cos (x)\right )\\ &=\frac{\cos (x)}{10 \left (-2+7 \cos ^2(x)\right )^{5/2}}+\frac{2}{5} \operatorname{Subst}\left (\int \frac{1}{\left (-2+7 x^2\right )^{5/2}} \, dx,x,\cos (x)\right )\\ &=\frac{\cos (x)}{10 \left (-2+7 \cos ^2(x)\right )^{5/2}}-\frac{\cos (x)}{15 \left (-2+7 \cos ^2(x)\right )^{3/2}}-\frac{2}{15} \operatorname{Subst}\left (\int \frac{1}{\left (-2+7 x^2\right )^{3/2}} \, dx,x,\cos (x)\right )\\ &=\frac{\cos (x)}{10 \left (-2+7 \cos ^2(x)\right )^{5/2}}-\frac{\cos (x)}{15 \left (-2+7 \cos ^2(x)\right )^{3/2}}+\frac{\cos (x)}{15 \sqrt{-2+7 \cos ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0923111, size = 37, normalized size = 0.67 \[ \frac{\cos (x) (56 \cos (2 x)+49 \cos (4 x)+67)}{15 \sqrt{2} (7 \cos (2 x)+3)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]/(5*Cos[x]^2 - 2*Sin[x]^2)^(7/2),x]

[Out]

(Cos[x]*(67 + 56*Cos[2*x] + 49*Cos[4*x]))/(15*Sqrt[2]*(3 + 7*Cos[2*x])^(5/2))

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Maple [A]  time = 0.03, size = 44, normalized size = 0.8 \begin{align*}{\frac{\cos \left ( x \right ) }{10} \left ( -2+7\, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{5}{2}}}}-{\frac{\cos \left ( x \right ) }{15} \left ( -2+7\, \left ( \cos \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{\cos \left ( x \right ) }{15}{\frac{1}{\sqrt{-2+7\, \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)/(5*cos(x)^2-2*sin(x)^2)^(7/2),x)

[Out]

1/10*cos(x)/(-2+7*cos(x)^2)^(5/2)-1/15*cos(x)/(-2+7*cos(x)^2)^(3/2)+1/15*cos(x)/(-2+7*cos(x)^2)^(1/2)

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Maxima [A]  time = 0.960126, size = 58, normalized size = 1.05 \begin{align*} \frac{\cos \left (x\right )}{15 \, \sqrt{7 \, \cos \left (x\right )^{2} - 2}} - \frac{\cos \left (x\right )}{15 \,{\left (7 \, \cos \left (x\right )^{2} - 2\right )}^{\frac{3}{2}}} + \frac{\cos \left (x\right )}{10 \,{\left (7 \, \cos \left (x\right )^{2} - 2\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(5*cos(x)^2-2*sin(x)^2)^(7/2),x, algorithm="maxima")

[Out]

1/15*cos(x)/sqrt(7*cos(x)^2 - 2) - 1/15*cos(x)/(7*cos(x)^2 - 2)^(3/2) + 1/10*cos(x)/(7*cos(x)^2 - 2)^(5/2)

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Fricas [A]  time = 3.54351, size = 155, normalized size = 2.82 \begin{align*} \frac{{\left (98 \, \cos \left (x\right )^{5} - 70 \, \cos \left (x\right )^{3} + 15 \, \cos \left (x\right )\right )} \sqrt{7 \, \cos \left (x\right )^{2} - 2}}{30 \,{\left (343 \, \cos \left (x\right )^{6} - 294 \, \cos \left (x\right )^{4} + 84 \, \cos \left (x\right )^{2} - 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(5*cos(x)^2-2*sin(x)^2)^(7/2),x, algorithm="fricas")

[Out]

1/30*(98*cos(x)^5 - 70*cos(x)^3 + 15*cos(x))*sqrt(7*cos(x)^2 - 2)/(343*cos(x)^6 - 294*cos(x)^4 + 84*cos(x)^2 -
8)

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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(sin(x)/(5*cos(x)**2-2*sin(x)**2)**(7/2),x)

[Out]

Timed out

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Giac [A]  time = 1.11863, size = 41, normalized size = 0.75 \begin{align*} \frac{{\left (14 \,{\left (7 \, \cos \left (x\right )^{2} - 5\right )} \cos \left (x\right )^{2} + 15\right )} \cos \left (x\right )}{30 \,{\left (7 \, \cos \left (x\right )^{2} - 2\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)/(5*cos(x)^2-2*sin(x)^2)^(7/2),x, algorithm="giac")

[Out]

1/30*(14*(7*cos(x)^2 - 5)*cos(x)^2 + 15)*cos(x)/(7*cos(x)^2 - 2)^(5/2)