3.85 \(\int \sin ^3(x) \, dx\)

Optimal. Leaf size=13 \[ \frac{\cos ^3(x)}{3}-\cos (x) \]

[Out]

-Cos[x] + Cos[x]^3/3

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Rubi [A]  time = 0.0111025, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\cos ^3(x)}{3}-\cos (x) \]

Antiderivative was successfully verified.

[In]  Int[Sin[x]^3,x]

[Out]

-Cos[x] + Cos[x]^3/3

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Rubi in Sympy [A]  time = 0.642657, size = 8, normalized size = 0.62 \[ \frac{\cos ^{3}{\left (x \right )}}{3} - \cos{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(x)**3,x)

[Out]

cos(x)**3/3 - cos(x)

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Mathematica [A]  time = 0.00280369, size = 15, normalized size = 1.15 \[ \frac{1}{12} \cos (3 x)-\frac{3 \cos (x)}{4} \]

Antiderivative was successfully verified.

[In]  Integrate[Sin[x]^3,x]

[Out]

(-3*Cos[x])/4 + Cos[3*x]/12

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Maple [A]  time = 0.036, size = 11, normalized size = 0.9 \[ -{\frac{ \left ( 2+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \cos \left ( x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(x)^3,x)

[Out]

-1/3*(2+sin(x)^2)*cos(x)

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Maxima [A]  time = 1.39823, size = 15, normalized size = 1.15 \[ \frac{1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*cos(x)^3 - cos(x)

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Fricas [A]  time = 0.232486, size = 15, normalized size = 1.15 \[ \frac{1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*cos(x)^3 - cos(x)

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Sympy [A]  time = 0.040446, size = 8, normalized size = 0.62 \[ \frac{\cos ^{3}{\left (x \right )}}{3} - \cos{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)**3,x)

[Out]

cos(x)**3/3 - cos(x)

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GIAC/XCAS [A]  time = 0.208975, size = 15, normalized size = 1.15 \[ \frac{1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*cos(x)^3 - cos(x)