3.26 \(\int \cos (x) \sin (x) \, dx\)

Optimal. Leaf size=8 \[ \frac{\sin ^2(x)}{2} \]

[Out]

Sin[x]^2/2

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Rubi [A]  time = 0.0114538, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{\sin ^2(x)}{2} \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]*Sin[x],x]

[Out]

Sin[x]^2/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)*sin(x),x)

[Out]

Integral(x, (x, sin(x)))

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Mathematica [A]  time = 0.00207701, size = 8, normalized size = 1. \[ -\frac{1}{2} \cos ^2(x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]*Sin[x],x]

[Out]

-Cos[x]^2/2

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Maple [A]  time = 0.003, size = 7, normalized size = 0.9 \[{\frac{ \left ( \sin \left ( x \right ) \right ) ^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)*sin(x),x)

[Out]

1/2*sin(x)^2

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Maxima [A]  time = 1.34008, size = 8, normalized size = 1. \[ -\frac{1}{2} \, \cos \left (x\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*cos(x)^2

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Fricas [A]  time = 0.235346, size = 8, normalized size = 1. \[ -\frac{1}{2} \, \cos \left (x\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*cos(x)^2

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Sympy [A]  time = 0.03586, size = 5, normalized size = 0.62 \[ \frac{\sin ^{2}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)*sin(x),x)

[Out]

sin(x)**2/2

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GIAC/XCAS [A]  time = 0.218294, size = 8, normalized size = 1. \[ -\frac{1}{2} \, \cos \left (x\right )^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*cos(x)^2