3.74 \(\int \sqrt{\cos (x)} \sin ^3(x) \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{7} \cos ^{\frac{7}{2}}(x)-\frac{2}{3} \cos ^{\frac{3}{2}}(x) \]

[Out]

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

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Rubi [A]  time = 0.0356935, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2}{7} \cos ^{\frac{7}{2}}(x)-\frac{2}{3} \cos ^{\frac{3}{2}}(x) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[Cos[x]]*Sin[x]^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 2.2582, size = 19, normalized size = 0.9 \[ \frac{2 \cos ^{\frac{7}{2}}{\left (x \right )}}{7} - \frac{2 \cos ^{\frac{3}{2}}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(x)**3*cos(x)**(1/2),x)

[Out]

2*cos(x)**(7/2)/7 - 2*cos(x)**(3/2)/3

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Mathematica [A]  time = 0.0343083, size = 22, normalized size = 1.05 \[ \sqrt{\cos (x)} \left (\frac{1}{14} \cos (3 x)-\frac{19 \cos (x)}{42}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[Cos[x]]*Sin[x]^3,x]

[Out]

Sqrt[Cos[x]]*((-19*Cos[x])/42 + Cos[3*x]/14)

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Maple [B]  time = 0.091, size = 39, normalized size = 1.9 \[ -{\frac{8}{21}\sqrt{-2\, \left ( \sin \left ( x/2 \right ) \right ) ^{2}+1} \left ( 6\, \left ( \sin \left ( x/2 \right ) \right ) ^{6}-9\, \left ( \sin \left ( x/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(x)^3*cos(x)^(1/2),x)

[Out]

-8/21*(-2*sin(1/2*x)^2+1)^(1/2)*(6*sin(1/2*x)^6-9*sin(1/2*x)^4+sin(1/2*x)^2+1)

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Maxima [A]  time = 1.36288, size = 18, normalized size = 0.86 \[ \frac{2}{7} \, \cos \left (x\right )^{\frac{7}{2}} - \frac{2}{3} \, \cos \left (x\right )^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*cos(x)^(7/2) - 2/3*cos(x)^(3/2)

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Fricas [A]  time = 0.237176, size = 23, normalized size = 1.1 \[ \frac{2}{21} \,{\left (3 \, \cos \left (x\right )^{3} - 7 \, \cos \left (x\right )\right )} \sqrt{\cos \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*(3*cos(x)^3 - 7*cos(x))*sqrt(cos(x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)**3*cos(x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213291, size = 18, normalized size = 0.86 \[ \frac{2}{7} \, \cos \left (x\right )^{\frac{7}{2}} - \frac{2}{3} \, \cos \left (x\right )^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*cos(x)^(7/2) - 2/3*cos(x)^(3/2)