3.74 \(\int \cos ^5(x) \, dx\)

Optimal. Leaf size=19 \[ \frac{\sin ^5(x)}{5}-\frac{2 \sin ^3(x)}{3}+\sin (x) \]

[Out]

Sin[x] - (2*Sin[x]^3)/3 + Sin[x]^5/5

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Rubi [A]  time = 0.0170276, antiderivative size = 19, 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 \[ \frac{\sin ^5(x)}{5}-\frac{2 \sin ^3(x)}{3}+\sin (x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]^5,x]

[Out]

Sin[x] - (2*Sin[x]^3)/3 + Sin[x]^5/5

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Rubi in Sympy [A]  time = 0.71909, size = 17, normalized size = 0.89 \[ \frac{\sin ^{5}{\left (x \right )}}{5} - \frac{2 \sin ^{3}{\left (x \right )}}{3} + \sin{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)**5,x)

[Out]

sin(x)**5/5 - 2*sin(x)**3/3 + sin(x)

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Mathematica [A]  time = 0.00340846, size = 23, normalized size = 1.21 \[ \frac{5 \sin (x)}{8}+\frac{5}{48} \sin (3 x)+\frac{1}{80} \sin (5 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]^5,x]

[Out]

(5*Sin[x])/8 + (5*Sin[3*x])/48 + Sin[5*x]/80

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Maple [A]  time = 0., size = 17, normalized size = 0.9 \[{\frac{\sin \left ( x \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( x \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( x \right ) \right ) ^{2}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^5,x)

[Out]

1/5*(8/3+cos(x)^4+4/3*cos(x)^2)*sin(x)

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Maxima [A]  time = 1.35091, size = 20, normalized size = 1.05 \[ \frac{1}{5} \, \sin \left (x\right )^{5} - \frac{2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*sin(x)^5 - 2/3*sin(x)^3 + sin(x)

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Fricas [A]  time = 0.257404, size = 24, normalized size = 1.26 \[ \frac{1}{15} \,{\left (3 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 8\right )} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(3*cos(x)^4 + 4*cos(x)^2 + 8)*sin(x)

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Sympy [A]  time = 0.043716, size = 17, normalized size = 0.89 \[ \frac{\sin ^{5}{\left (x \right )}}{5} - \frac{2 \sin ^{3}{\left (x \right )}}{3} + \sin{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)**5,x)

[Out]

sin(x)**5/5 - 2*sin(x)**3/3 + sin(x)

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GIAC/XCAS [A]  time = 0.232312, size = 20, normalized size = 1.05 \[ \frac{1}{5} \, \sin \left (x\right )^{5} - \frac{2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*sin(x)^5 - 2/3*sin(x)^3 + sin(x)