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

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

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Rubi [A]  time = 0.01, 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 = {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*}

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

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin (x)}{\cos ^{\frac {5}{2}}(2 x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [B]  time = 1.33, size = 39, normalized size = 2.44 \[ -\frac {{\left (4 \, \cos \relax (x)^{3} - 3 \, \cos \relax (x)\right )} \sqrt {2 \, \cos \relax (x)^{2} - 1}}{3 \, {\left (4 \, \cos \relax (x)^{4} - 4 \, \cos \relax (x)^{2} + 1\right )}} \]

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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giac [B]  time = 0.97, size = 46, normalized size = 2.88 \[ \frac {{\left ({\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 15\right )} \tan \left (\frac {1}{2} \, x\right )^{2} + 15\right )} \tan \left (\frac {1}{2} \, x\right )^{2} - 1}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 6 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{\frac {3}{2}}} \]

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)

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maple [B]  time = 0.18, size = 39, normalized size = 2.44

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

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

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maxima [B]  time = 1.02, size = 90, normalized size = 5.62 \[ -\frac {\sqrt {2} \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right )\right )\right ) + {\left (\sqrt {2} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right )\right )\right ) + \sqrt {2}\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )}{3 \, {\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac {3}{4}}} \]

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

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\relax (x )}}{\cos ^{\frac {5}{2}}{\left (2 x \right )}}\, dx \]

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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