3.31 \(\int \sin ^5(x) \, dx\)

Optimal. Leaf size=21 \[ -\frac{1}{5} \cos ^5(x)+\frac{2 \cos ^3(x)}{3}-\cos (x) \]

[Out]

-Cos[x] + (2*Cos[x]^3)/3 - Cos[x]^5/5

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Rubi [A]  time = 0.0126419, antiderivative size = 21, 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 \[ -\frac{1}{5} \cos ^5(x)+\frac{2 \cos ^3(x)}{3}-\cos (x) \]

Antiderivative was successfully verified.

[In]  Int[Sin[x]^5,x]

[Out]

-Cos[x] + (2*Cos[x]^3)/3 - Cos[x]^5/5

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Rubi in Sympy [A]  time = 0.744482, size = 17, normalized size = 0.81 \[ - \frac{\cos ^{5}{\left (x \right )}}{5} + \frac{2 \cos ^{3}{\left (x \right )}}{3} - \cos{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(x)**5,x)

[Out]

-cos(x)**5/5 + 2*cos(x)**3/3 - cos(x)

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Mathematica [A]  time = 0.00320367, size = 23, normalized size = 1.1 \[ -\frac{5 \cos (x)}{8}+\frac{5}{48} \cos (3 x)-\frac{1}{80} \cos (5 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Sin[x]^5,x]

[Out]

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

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Maple [A]  time = 0.053, size = 17, normalized size = 0.8 \[ -{\frac{\cos \left ( x \right ) }{5} \left ({\frac{8}{3}}+ \left ( \sin \left ( x \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( x \right ) \right ) ^{2}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(x)^5,x)

[Out]

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

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Maxima [A]  time = 1.48142, size = 23, normalized size = 1.1 \[ -\frac{1}{5} \, \cos \left (x\right )^{5} + \frac{2}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*cos(x)^5 + 2/3*cos(x)^3 - cos(x)

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Fricas [A]  time = 0.216444, size = 23, normalized size = 1.1 \[ -\frac{1}{5} \, \cos \left (x\right )^{5} + \frac{2}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*cos(x)^5 + 2/3*cos(x)^3 - cos(x)

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Sympy [A]  time = 0.046102, size = 17, normalized size = 0.81 \[ - \frac{\cos ^{5}{\left (x \right )}}{5} + \frac{2 \cos ^{3}{\left (x \right )}}{3} - \cos{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)**5,x)

[Out]

-cos(x)**5/5 + 2*cos(x)**3/3 - cos(x)

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GIAC/XCAS [A]  time = 0.223361, size = 23, normalized size = 1.1 \[ -\frac{1}{5} \, \cos \left (x\right )^{5} + \frac{2}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*cos(x)^5 + 2/3*cos(x)^3 - cos(x)