3.32 \(\int \sin ^2(x) \, dx\)

Optimal. Leaf size=14 \[ \frac{x}{2}-\frac{1}{2} \sin (x) \cos (x) \]

[Out]

x/2 - (Cos[x]*Sin[x])/2

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Rubi [A]  time = 0.0110247, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{x}{2}-\frac{1}{2} \sin (x) \cos (x) \]

Antiderivative was successfully verified.

[In]  Int[Sin[x]^2,x]

[Out]

x/2 - (Cos[x]*Sin[x])/2

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Rubi in Sympy [A]  time = 0.502193, size = 10, normalized size = 0.71 \[ \frac{x}{2} - \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(x)**2,x)

[Out]

x/2 - sin(x)*cos(x)/2

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Mathematica [A]  time = 0.00279825, size = 14, normalized size = 1. \[ \frac{x}{2}-\frac{1}{4} \sin (2 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Sin[x]^2,x]

[Out]

x/2 - Sin[2*x]/4

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Maple [A]  time = 0.002, size = 11, normalized size = 0.8 \[{\frac{x}{2}}-{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(x)^2,x)

[Out]

1/2*x-1/2*cos(x)*sin(x)

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Maxima [A]  time = 1.38067, size = 14, normalized size = 1. \[ \frac{1}{2} \, x - \frac{1}{4} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x - 1/4*sin(2*x)

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Fricas [A]  time = 0.217297, size = 14, normalized size = 1. \[ -\frac{1}{2} \, \cos \left (x\right ) \sin \left (x\right ) + \frac{1}{2} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*cos(x)*sin(x) + 1/2*x

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Sympy [A]  time = 0.037585, size = 10, normalized size = 0.71 \[ \frac{x}{2} - \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)**2,x)

[Out]

x/2 - sin(x)*cos(x)/2

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GIAC/XCAS [A]  time = 0.202506, size = 14, normalized size = 1. \[ \frac{1}{2} \, x - \frac{1}{4} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x - 1/4*sin(2*x)