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

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

[Out]

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

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Rubi [A]  time = 0.0357328, 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 \[ \frac{2}{3} \sin ^{\frac{3}{2}}(x)-\frac{2}{7} \sin ^{\frac{7}{2}}(x) \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]^3*Sqrt[Sin[x]],x]

[Out]

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

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Rubi in Sympy [A]  time = 2.25573, size = 19, normalized size = 0.9 \[ - \frac{2 \sin ^{\frac{7}{2}}{\left (x \right )}}{7} + \frac{2 \sin ^{\frac{3}{2}}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sin(x)**(7/2)/7 + 2*sin(x)**(3/2)/3

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Mathematica [A]  time = 0.0276347, size = 22, normalized size = 1.05 \[ \sqrt{\sin (x)} \left (\frac{19 \sin (x)}{42}+\frac{1}{14} \sin (3 x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.037, size = 14, normalized size = 0.7 \[{\frac{2}{3} \left ( \sin \left ( x \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{2}{7} \left ( \sin \left ( x \right ) \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^3*sin(x)^(1/2),x)

[Out]

2/3*sin(x)^(3/2)-2/7*sin(x)^(7/2)

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Maxima [A]  time = 1.49179, size = 18, normalized size = 0.86 \[ -\frac{2}{7} \, \sin \left (x\right )^{\frac{7}{2}} + \frac{2}{3} \, \sin \left (x\right )^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/7*sin(x)^(7/2) + 2/3*sin(x)^(3/2)

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Fricas [A]  time = 0.226532, size = 19, normalized size = 0.9 \[ \frac{2}{21} \,{\left (3 \, \cos \left (x\right )^{2} + 4\right )} \sin \left (x\right )^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*(3*cos(x)^2 + 4)*sin(x)^(3/2)

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Sympy [A]  time = 103.477, size = 167, normalized size = 7.95 \[ \frac{28 \sqrt{2} \sqrt{\frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1}} \tan ^{\frac{11}{2}}{\left (\frac{x}{2} \right )}}{21 \tan ^{6}{\left (\frac{x}{2} \right )} + 63 \tan ^{4}{\left (\frac{x}{2} \right )} + 63 \tan ^{2}{\left (\frac{x}{2} \right )} + 21} + \frac{8 \sqrt{2} \sqrt{\frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1}} \tan ^{\frac{7}{2}}{\left (\frac{x}{2} \right )}}{21 \tan ^{6}{\left (\frac{x}{2} \right )} + 63 \tan ^{4}{\left (\frac{x}{2} \right )} + 63 \tan ^{2}{\left (\frac{x}{2} \right )} + 21} + \frac{28 \sqrt{2} \sqrt{\frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1}} \tan ^{\frac{3}{2}}{\left (\frac{x}{2} \right )}}{21 \tan ^{6}{\left (\frac{x}{2} \right )} + 63 \tan ^{4}{\left (\frac{x}{2} \right )} + 63 \tan ^{2}{\left (\frac{x}{2} \right )} + 21} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

28*sqrt(2)*sqrt(1/(tan(x/2)**2 + 1))*tan(x/2)**(11/2)/(21*tan(x/2)**6 + 63*tan(x
/2)**4 + 63*tan(x/2)**2 + 21) + 8*sqrt(2)*sqrt(1/(tan(x/2)**2 + 1))*tan(x/2)**(7
/2)/(21*tan(x/2)**6 + 63*tan(x/2)**4 + 63*tan(x/2)**2 + 21) + 28*sqrt(2)*sqrt(1/
(tan(x/2)**2 + 1))*tan(x/2)**(3/2)/(21*tan(x/2)**6 + 63*tan(x/2)**4 + 63*tan(x/2
)**2 + 21)

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GIAC/XCAS [A]  time = 0.201983, size = 18, normalized size = 0.86 \[ -\frac{2}{7} \, \sin \left (x\right )^{\frac{7}{2}} + \frac{2}{3} \, \sin \left (x\right )^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/7*sin(x)^(7/2) + 2/3*sin(x)^(3/2)