3.68 \(\int \cos ^5(x) \sin ^5(x) \, dx\)

Optimal. Leaf size=25 \[ \frac{\sin ^{10}(x)}{10}-\frac{\sin ^8(x)}{4}+\frac{\sin ^6(x)}{6} \]

[Out]

Sin[x]^6/6 - Sin[x]^8/4 + Sin[x]^10/10

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Rubi [A]  time = 0.0600106, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sin ^{10}(x)}{10}-\frac{\sin ^8(x)}{4}+\frac{\sin ^6(x)}{6} \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]^5*Sin[x]^5,x]

[Out]

Sin[x]^6/6 - Sin[x]^8/4 + Sin[x]^10/10

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Rubi in Sympy [A]  time = 3.44483, size = 19, normalized size = 0.76 \[ \frac{\sin ^{10}{\left (x \right )}}{10} - \frac{\sin ^{8}{\left (x \right )}}{4} + \frac{\sin ^{6}{\left (x \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)**5*sin(x)**5,x)

[Out]

sin(x)**10/10 - sin(x)**8/4 + sin(x)**6/6

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Mathematica [A]  time = 0.0156388, size = 25, normalized size = 1. \[ -\frac{5}{512} \cos (2 x)+\frac{5 \cos (6 x)}{3072}-\frac{\cos (10 x)}{5120} \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]^5*Sin[x]^5,x]

[Out]

(-5*Cos[2*x])/512 + (5*Cos[6*x])/3072 - Cos[10*x]/5120

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Maple [A]  time = 0.01, size = 28, normalized size = 1.1 \[ -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{6} \left ( \sin \left ( x \right ) \right ) ^{4}}{10}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{2} \left ( \cos \left ( x \right ) \right ) ^{6}}{20}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{6}}{60}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^5*sin(x)^5,x)

[Out]

-1/10*cos(x)^6*sin(x)^4-1/20*sin(x)^2*cos(x)^6-1/60*cos(x)^6

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Maxima [A]  time = 1.33287, size = 26, normalized size = 1.04 \[ \frac{1}{10} \, \sin \left (x\right )^{10} - \frac{1}{4} \, \sin \left (x\right )^{8} + \frac{1}{6} \, \sin \left (x\right )^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^5*sin(x)^5,x, algorithm="maxima")

[Out]

1/10*sin(x)^10 - 1/4*sin(x)^8 + 1/6*sin(x)^6

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Fricas [A]  time = 0.230845, size = 26, normalized size = 1.04 \[ -\frac{1}{10} \, \cos \left (x\right )^{10} + \frac{1}{4} \, \cos \left (x\right )^{8} - \frac{1}{6} \, \cos \left (x\right )^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^5*sin(x)^5,x, algorithm="fricas")

[Out]

-1/10*cos(x)^10 + 1/4*cos(x)^8 - 1/6*cos(x)^6

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Sympy [A]  time = 0.053033, size = 19, normalized size = 0.76 \[ \frac{\sin ^{10}{\left (x \right )}}{10} - \frac{\sin ^{8}{\left (x \right )}}{4} + \frac{\sin ^{6}{\left (x \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)**5*sin(x)**5,x)

[Out]

sin(x)**10/10 - sin(x)**8/4 + sin(x)**6/6

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GIAC/XCAS [A]  time = 0.212994, size = 26, normalized size = 1.04 \[ -\frac{1}{10} \, \cos \left (x\right )^{10} + \frac{1}{4} \, \cos \left (x\right )^{8} - \frac{1}{6} \, \cos \left (x\right )^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^5*sin(x)^5,x, algorithm="giac")

[Out]

-1/10*cos(x)^10 + 1/4*cos(x)^8 - 1/6*cos(x)^6