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 = 0.0224468, 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, Rules used = {2564, 14} \[ \frac{2}{3} \sin ^{\frac{3}{2}}(x)-\frac{2}{7} \sin ^{\frac{7}{2}}(x) \]

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

Rule 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \cos ^3(x) \sqrt{\sin (x)} \, dx &=\operatorname{Subst}\left (\int \sqrt{x} \left (1-x^2\right ) \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\sqrt{x}-x^{5/2}\right ) \, dx,x,\sin (x)\right )\\ &=\frac{2}{3} \sin ^{\frac{3}{2}}(x)-\frac{2}{7} \sin ^{\frac{7}{2}}(x)\\ \end{align*}

Mathematica [A]  time = 0.0111494, size = 18, normalized size = 0.86 \[ \frac{1}{21} \sin ^{\frac{3}{2}}(x) (3 \cos (2 x)+11) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((11 + 3*Cos[2*x])*Sin[x]^(3/2))/21

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Maple [A]  time = 0.02, size = 14, normalized size = 0.7 \begin{align*}{\frac{2}{3} \left ( \sin \left ( x \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{2}{7} \left ( \sin \left ( x \right ) \right ) ^{{\frac{7}{2}}}} \end{align*}

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 = 0.933042, size = 18, normalized size = 0.86 \begin{align*} -\frac{2}{7} \, \sin \left (x\right )^{\frac{7}{2}} + \frac{2}{3} \, \sin \left (x\right )^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x)^(1/2),x, algorithm="maxima")

[Out]

-2/7*sin(x)^(7/2) + 2/3*sin(x)^(3/2)

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Fricas [A]  time = 2.06587, size = 49, normalized size = 2.33 \begin{align*} \frac{2}{21} \,{\left (3 \, \cos \left (x\right )^{2} + 4\right )} \sin \left (x\right )^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x)^(1/2),x, algorithm="fricas")

[Out]

2/21*(3*cos(x)^2 + 4)*sin(x)^(3/2)

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Sympy [B]  time = 42.1372, size = 167, normalized size = 7.95 \begin{align*} \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} \end{align*}

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 [A]  time = 1.04997, size = 18, normalized size = 0.86 \begin{align*} -\frac{2}{7} \, \sin \left (x\right )^{\frac{7}{2}} + \frac{2}{3} \, \sin \left (x\right )^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^3*sin(x)^(1/2),x, algorithm="giac")

[Out]

-2/7*sin(x)^(7/2) + 2/3*sin(x)^(3/2)