3.352 \(\int \cos ^{2 m}(x) \sin ^{2 m}(x) \, dx\)

Optimal. Leaf size=68 \[ \frac{\sin ^{2 m+1}(x) \cos ^{2 m-1}(x) \cos ^2(x)^{\frac{1}{2}-m} \, _2F_1\left (\frac{1}{2} (1-2 m),\frac{1}{2} (2 m+1);\frac{1}{2} (2 m+3);\sin ^2(x)\right )}{2 m+1} \]

[Out]

(Cos[x]^(-1 + 2*m)*(Cos[x]^2)^(1/2 - m)*Hypergeometric2F1[(1 - 2*m)/2, (1 + 2*m)
/2, (3 + 2*m)/2, Sin[x]^2]*Sin[x]^(1 + 2*m))/(1 + 2*m)

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Rubi [A]  time = 0.0647825, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\sin ^{2 m+1}(x) \cos ^{2 m-1}(x) \cos ^2(x)^{\frac{1}{2}-m} \, _2F_1\left (\frac{1}{2} (1-2 m),\frac{1}{2} (2 m+1);\frac{1}{2} (2 m+3);\sin ^2(x)\right )}{2 m+1} \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]^(2*m)*Sin[x]^(2*m),x]

[Out]

(Cos[x]^(-1 + 2*m)*(Cos[x]^2)^(1/2 - m)*Hypergeometric2F1[(1 - 2*m)/2, (1 + 2*m)
/2, (3 + 2*m)/2, Sin[x]^2]*Sin[x]^(1 + 2*m))/(1 + 2*m)

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Rubi in Sympy [A]  time = 2.33874, size = 49, normalized size = 0.72 \[ \frac{\left (\cos ^{2}{\left (x \right )}\right )^{- m - \frac{1}{2}} \sin ^{2 m + 1}{\left (x \right )} \cos ^{2 m + 1}{\left (x \right )}{{}_{2}F_{1}\left (\begin{matrix} m + \frac{1}{2}, - m + \frac{1}{2} \\ m + \frac{3}{2} \end{matrix}\middle |{\sin ^{2}{\left (x \right )}} \right )}}{2 m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)**(2*m)*sin(x)**(2*m),x)

[Out]

(cos(x)**2)**(-m - 1/2)*sin(x)**(2*m + 1)*cos(x)**(2*m + 1)*hyper((m + 1/2, -m +
1/2), (m + 3/2,), sin(x)**2)/(2*m + 1)

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Mathematica [A]  time = 0.145133, size = 59, normalized size = 0.87 \[ -\frac{\sin ^{2 m+1}(x) \sin ^2(x)^{-m-\frac{1}{2}} \cos ^{2 m+1}(x) \, _2F_1\left (\frac{1}{2}-m,m+\frac{1}{2};m+\frac{3}{2};\cos ^2(x)\right )}{2 m+1} \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]^(2*m)*Sin[x]^(2*m),x]

[Out]

-((Cos[x]^(1 + 2*m)*Hypergeometric2F1[1/2 - m, 1/2 + m, 3/2 + m, Cos[x]^2]*Sin[x
]^(1 + 2*m)*(Sin[x]^2)^(-1/2 - m))/(1 + 2*m))

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Maple [F]  time = 0.397, size = 0, normalized size = 0. \[ \int \left ( \cos \left ( x \right ) \right ) ^{2\,m} \left ( \sin \left ( x \right ) \right ) ^{2\,m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^(2*m)*sin(x)^(2*m),x)

[Out]

int(cos(x)^(2*m)*sin(x)^(2*m),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \cos \left (x\right )^{2 \, m} \sin \left (x\right )^{2 \, m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^(2*m)*sin(x)^(2*m),x, algorithm="maxima")

[Out]

integrate(cos(x)^(2*m)*sin(x)^(2*m), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\cos \left (x\right )^{2 \, m} \sin \left (x\right )^{2 \, m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^(2*m)*sin(x)^(2*m),x, algorithm="fricas")

[Out]

integral(cos(x)^(2*m)*sin(x)^(2*m), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sin ^{2 m}{\left (x \right )} \cos ^{2 m}{\left (x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)**(2*m)*sin(x)**(2*m),x)

[Out]

Integral(sin(x)**(2*m)*cos(x)**(2*m), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \cos \left (x\right )^{2 \, m} \sin \left (x\right )^{2 \, m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^(2*m)*sin(x)^(2*m),x, algorithm="giac")

[Out]

integrate(cos(x)^(2*m)*sin(x)^(2*m), x)