3.5.31 \(\int \frac {\sin (x)}{\cos ^{\frac {5}{2}}(2 x)} \, dx\) [431]

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

[Out]

-1/3*cos(3*x)/cos(2*x)^(3/2)

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Rubi [A]
time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 = {4416} \begin {gather*} -\frac {\cos (3 x)}{3 \cos ^{\frac {3}{2}}(2 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*Cos[3*x]/Cos[2*x]^(3/2)

Rule 4416

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] &&
EqQ[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*}

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Mathematica [A]
time = 0.03, size = 16, normalized size = 1.00 \begin {gather*} -\frac {\cos (3 x)}{3 \cos ^{\frac {3}{2}}(2 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*Cos[3*x]/Cos[2*x]^(3/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs. \(2(12)=24\).
time = 0.15, size = 39, normalized size = 2.44

method	result	size
default	\(\frac {\sqrt {1-2 \left (\sin ^{2}\left (x \right )\right )}\, \cos \left (x \right ) \left (4 \left (\sin ^{2}\left (x \right )\right )-1\right )}{12 \left (\sin ^{4}\left (x \right )\right )-12 \left (\sin ^{2}\left (x \right )\right )+3}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (12) = 24\).
time = 3.83, size = 90, normalized size = 5.62 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (12) = 24\).
time = 1.27, size = 39, normalized size = 2.44 \begin {gather*} -\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 {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\left (x \right )}}{\cos ^{\frac {5}{2}}{\left (2 x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(x)/cos(2*x)**(5/2), x)

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Giac [A]
time = 0.99, size = 22, normalized size = 1.38 \begin {gather*} -\frac {{\left (4 \, \cos \left (x\right )^{2} - 3\right )} \cos \left (x\right )}{3 \, {\left (2 \, \cos \left (x\right )^{2} - 1\right )}^{\frac {3}{2}}} \end {gather*}

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

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Mupad [B]
time = 0.35, size = 12, normalized size = 0.75 \begin {gather*} -\frac {\cos \left (3\,x\right )}{3\,{\cos \left (2\,x\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(3*x)/(3*cos(2*x)^(3/2))

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