3.371 \(\int \cos ^3(x) \sin ^3(x) \, dx\)

Optimal. Leaf size=17 \[ \frac{\sin ^4(x)}{4}-\frac{\sin ^6(x)}{6} \]

[Out]

Sin[x]^4/4 - Sin[x]^6/6

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Rubi [A]  time = 0.0351594, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\sin ^4(x)}{4}-\frac{\sin ^6(x)}{6} \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]^3*Sin[x]^3,x]

[Out]

Sin[x]^4/4 - Sin[x]^6/6

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Rubi in Sympy [A]  time = 2.42613, size = 12, normalized size = 0.71 \[ - \frac{\sin ^{6}{\left (x \right )}}{6} + \frac{\sin ^{4}{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)**3*sin(x)**3,x)

[Out]

-sin(x)**6/6 + sin(x)**4/4

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Mathematica [A]  time = 0.00985004, size = 17, normalized size = 1. \[ \frac{1}{192} \cos (6 x)-\frac{3}{64} \cos (2 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]^3*Sin[x]^3,x]

[Out]

(-3*Cos[2*x])/64 + Cos[6*x]/192

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Maple [A]  time = 0.01, size = 18, normalized size = 1.1 \[ -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4} \left ( \sin \left ( x \right ) \right ) ^{2}}{6}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^3*sin(x)^3,x)

[Out]

-1/6*cos(x)^4*sin(x)^2-1/12*cos(x)^4

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Maxima [A]  time = 1.3473, size = 18, normalized size = 1.06 \[ -\frac{1}{6} \, \sin \left (x\right )^{6} + \frac{1}{4} \, \sin \left (x\right )^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^3*sin(x)^3,x, algorithm="maxima")

[Out]

-1/6*sin(x)^6 + 1/4*sin(x)^4

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Fricas [A]  time = 0.217586, size = 18, normalized size = 1.06 \[ \frac{1}{6} \, \cos \left (x\right )^{6} - \frac{1}{4} \, \cos \left (x\right )^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^3*sin(x)^3,x, algorithm="fricas")

[Out]

1/6*cos(x)^6 - 1/4*cos(x)^4

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Sympy [A]  time = 0.044095, size = 12, normalized size = 0.71 \[ - \frac{\sin ^{6}{\left (x \right )}}{6} + \frac{\sin ^{4}{\left (x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)**3*sin(x)**3,x)

[Out]

-sin(x)**6/6 + sin(x)**4/4

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GIAC/XCAS [A]  time = 0.20138, size = 18, normalized size = 1.06 \[ \frac{1}{6} \, \cos \left (x\right )^{6} - \frac{1}{4} \, \cos \left (x\right )^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^3*sin(x)^3,x, algorithm="giac")

[Out]

1/6*cos(x)^6 - 1/4*cos(x)^4