3.431 \(\int \frac{\sin (x)}{\cos ^{\frac{5}{2}}(2 x)} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\cos (3 x)}{3 \cos ^{\frac{3}{2}}(2 x)} \]

[Out]

-Cos[3*x]/(3*Cos[2*x]^(3/2))

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Rubi [A]  time = 0.0134062, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4331} \[ -\frac{\cos (3 x)}{3 \cos ^{\frac{3}{2}}(2 x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[3*x]/(3*Cos[2*x]^(3/2))

Rule 4331

Int[(cos[(a_.) + (b_.)*(x_)]*(e_.))^(m_.)*sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[((m + 2)*(e*Cos[a + b*x]
)^(m + 1)*Cos[(m + 1)*(a + b*x)])/(d*e*(m + 1)), x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c - a*d, 0] && Eq
Q[d/b, Abs[m + 2]]

Rubi steps

\begin{align*} \int \frac{\sin (x)}{\cos ^{\frac{5}{2}}(2 x)} \, dx &=-\frac{\cos (3 x)}{3 \cos ^{\frac{3}{2}}(2 x)}\\ \end{align*}

Mathematica [A]  time = 0.0357061, size = 16, normalized size = 1. \[ -\frac{\cos (3 x)}{3 \cos ^{\frac{3}{2}}(2 x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cos[3*x]/(3*Cos[2*x]^(3/2))

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Maple [B]  time = 0.075, size = 39, normalized size = 2.4 \begin{align*}{\frac{\cos \left ( x \right ) \left ( 4\, \left ( \sin \left ( x \right ) \right ) ^{2}-1 \right ) }{12\, \left ( \sin \left ( x \right ) \right ) ^{4}-12\, \left ( \sin \left ( x \right ) \right ) ^{2}+3}\sqrt{-2\, \left ( \sin \left ( x \right ) \right ) ^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/(4*sin(x)^4-4*sin(x)^2+1)*(-2*sin(x)^2+1)^(1/2)*cos(x)*(4*sin(x)^2-1)

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Maxima [B]  time = 1.51037, size = 122, normalized size = 7.62 \begin{align*} -\frac{\sqrt{2} \sin \left (\frac{3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) \sin \left (\frac{3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right )\right )\right ) +{\left (\sqrt{2} \cos \left (\frac{3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right )\right )\right ) + \sqrt{2}\right )} \cos \left (\frac{3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )}{3 \,{\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac{3}{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(sqrt(2)*sin(3/2*arctan2(sin(4*x), cos(4*x) + 1))*sin(3/2*arctan2(sin(4*x), cos(4*x))) + (sqrt(2)*cos(3/2
*arctan2(sin(4*x), cos(4*x))) + sqrt(2))*cos(3/2*arctan2(sin(4*x), cos(4*x) + 1)))/(cos(4*x)^2 + sin(4*x)^2 +
2*cos(4*x) + 1)^(3/4)

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Fricas [B]  time = 3.198, size = 109, normalized size = 6.81 \begin{align*} -\frac{{\left (4 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )} \sqrt{2 \, \cos \left (x\right )^{2} - 1}}{3 \,{\left (4 \, \cos \left (x\right )^{4} - 4 \, \cos \left (x\right )^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(4*cos(x)^3 - 3*cos(x))*sqrt(2*cos(x)^2 - 1)/(4*cos(x)^4 - 4*cos(x)^2 + 1)

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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)/cos(2*x)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.27446, size = 62, normalized size = 3.88 \begin{align*} \frac{{\left ({\left (\tan \left (\frac{1}{2} \, x\right )^{2} - 15\right )} \tan \left (\frac{1}{2} \, x\right )^{2} + 15\right )} \tan \left (\frac{1}{2} \, x\right )^{2} - 1}{3 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{4} - 6 \, \tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(((tan(1/2*x)^2 - 15)*tan(1/2*x)^2 + 15)*tan(1/2*x)^2 - 1)/(tan(1/2*x)^4 - 6*tan(1/2*x)^2 + 1)^(3/2)