3.91 \(\int \cos (x) \sin ^2(x) \, dx\)

Optimal. Leaf size=8 \[ \frac{\sin ^3(x)}{3} \]

[Out]

Sin[x]^3/3

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Rubi [A]  time = 0.0186326, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{\sin ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]*Sin[x]^2,x]

[Out]

Sin[x]^3/3

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Rubi in Sympy [A]  time = 1.09323, size = 5, normalized size = 0.62 \[ \frac{\sin ^{3}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sin(x)**3/3

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Mathematica [A]  time = 0.00176855, size = 8, normalized size = 1. \[ \frac{\sin ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]*Sin[x]^2,x]

[Out]

Sin[x]^3/3

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Maple [A]  time = 0.003, size = 7, normalized size = 0.9 \[{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)*sin(x)^2,x)

[Out]

1/3*sin(x)^3

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Maxima [A]  time = 1.33929, size = 8, normalized size = 1. \[ \frac{1}{3} \, \sin \left (x\right )^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sin(x)^3

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Fricas [A]  time = 0.272542, size = 14, normalized size = 1.75 \[ -\frac{1}{3} \,{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(cos(x)^2 - 1)*sin(x)

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Sympy [A]  time = 0.039305, size = 5, normalized size = 0.62 \[ \frac{\sin ^{3}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)*sin(x)**2,x)

[Out]

sin(x)**3/3

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GIAC/XCAS [A]  time = 0.200659, size = 8, normalized size = 1. \[ \frac{1}{3} \, \sin \left (x\right )^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sin(x)^3