3.342 \(\int \cos ^2(x) \sin ^5(x) \, dx\)

Optimal. Leaf size=25 \[ -\frac{1}{7} \cos ^7(x)+\frac{2 \cos ^5(x)}{5}-\frac{\cos ^3(x)}{3} \]

[Out]

-Cos[x]^3/3 + (2*Cos[x]^5)/5 - Cos[x]^7/7

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Rubi [A]  time = 0.0467748, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{1}{7} \cos ^7(x)+\frac{2 \cos ^5(x)}{5}-\frac{\cos ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]^2*Sin[x]^5,x]

[Out]

-Cos[x]^3/3 + (2*Cos[x]^5)/5 - Cos[x]^7/7

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Rubi in Sympy [A]  time = 3.10453, size = 20, normalized size = 0.8 \[ - \frac{\cos ^{7}{\left (x \right )}}{7} + \frac{2 \cos ^{5}{\left (x \right )}}{5} - \frac{\cos ^{3}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)**2*sin(x)**5,x)

[Out]

-cos(x)**7/7 + 2*cos(x)**5/5 - cos(x)**3/3

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Mathematica [A]  time = 0.0166215, size = 31, normalized size = 1.24 \[ -\frac{5 \cos (x)}{64}-\frac{1}{192} \cos (3 x)+\frac{3}{320} \cos (5 x)-\frac{1}{448} \cos (7 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]^2*Sin[x]^5,x]

[Out]

(-5*Cos[x])/64 - Cos[3*x]/192 + (3*Cos[5*x])/320 - Cos[7*x]/448

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Maple [A]  time = 0.016, size = 28, normalized size = 1.1 \[ -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3} \left ( \sin \left ( x \right ) \right ) ^{4}}{7}}-{\frac{4\, \left ( \sin \left ( x \right ) \right ) ^{2} \left ( \cos \left ( x \right ) \right ) ^{3}}{35}}-{\frac{8\, \left ( \cos \left ( x \right ) \right ) ^{3}}{105}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^2*sin(x)^5,x)

[Out]

-1/7*cos(x)^3*sin(x)^4-4/35*sin(x)^2*cos(x)^3-8/105*cos(x)^3

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Maxima [A]  time = 1.3685, size = 26, normalized size = 1.04 \[ -\frac{1}{7} \, \cos \left (x\right )^{7} + \frac{2}{5} \, \cos \left (x\right )^{5} - \frac{1}{3} \, \cos \left (x\right )^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^2*sin(x)^5,x, algorithm="maxima")

[Out]

-1/7*cos(x)^7 + 2/5*cos(x)^5 - 1/3*cos(x)^3

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Fricas [A]  time = 0.226975, size = 26, normalized size = 1.04 \[ -\frac{1}{7} \, \cos \left (x\right )^{7} + \frac{2}{5} \, \cos \left (x\right )^{5} - \frac{1}{3} \, \cos \left (x\right )^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^2*sin(x)^5,x, algorithm="fricas")

[Out]

-1/7*cos(x)^7 + 2/5*cos(x)^5 - 1/3*cos(x)^3

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Sympy [A]  time = 0.050655, size = 20, normalized size = 0.8 \[ - \frac{\cos ^{7}{\left (x \right )}}{7} + \frac{2 \cos ^{5}{\left (x \right )}}{5} - \frac{\cos ^{3}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)**2*sin(x)**5,x)

[Out]

-cos(x)**7/7 + 2*cos(x)**5/5 - cos(x)**3/3

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GIAC/XCAS [A]  time = 0.205304, size = 26, normalized size = 1.04 \[ -\frac{1}{7} \, \cos \left (x\right )^{7} + \frac{2}{5} \, \cos \left (x\right )^{5} - \frac{1}{3} \, \cos \left (x\right )^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^2*sin(x)^5,x, algorithm="giac")

[Out]

-1/7*cos(x)^7 + 2/5*cos(x)^5 - 1/3*cos(x)^3