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

Optimal. Leaf size=19 \[ 2 \sqrt{\sin (x)}-\frac{2}{5} \sin ^{\frac{5}{2}}(x) \]

[Out]

2*Sqrt[Sin[x]] - (2*Sin[x]^(5/2))/5

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Rubi [A]  time = 0.0232385, antiderivative size = 19, 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} \[ 2 \sqrt{\sin (x)}-\frac{2}{5} \sin ^{\frac{5}{2}}(x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^3/Sqrt[Sin[x]],x]

[Out]

2*Sqrt[Sin[x]] - (2*Sin[x]^(5/2))/5

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

Mathematica [A]  time = 0.0085595, size = 16, normalized size = 0.84 \[ \frac{1}{5} \sqrt{\sin (x)} (\cos (2 x)+9) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((9 + Cos[2*x])*Sqrt[Sin[x]])/5

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Maple [A]  time = 0.039, size = 14, normalized size = 0.7 \begin{align*} -{\frac{2}{5} \left ( \sin \left ( x \right ) \right ) ^{{\frac{5}{2}}}}+2\,\sqrt{\sin \left ( x \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^3/sin(x)^(1/2),x)

[Out]

-2/5*sin(x)^(5/2)+2*sin(x)^(1/2)

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Maxima [A]  time = 1.00441, size = 18, normalized size = 0.95 \begin{align*} -\frac{2}{5} \, \sin \left (x\right )^{\frac{5}{2}} + 2 \, \sqrt{\sin \left (x\right )} \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]

-2/5*sin(x)^(5/2) + 2*sqrt(sin(x))

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Fricas [A]  time = 2.17845, size = 45, normalized size = 2.37 \begin{align*} \frac{2}{5} \,{\left (\cos \left (x\right )^{2} + 4\right )} \sqrt{\sin \left (x\right )} \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]

2/5*(cos(x)^2 + 4)*sqrt(sin(x))

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Sympy [B]  time = 51.1132, size = 323, normalized size = 17. \begin{align*} \frac{10 \sqrt{2} \tan ^{5}{\left (\frac{x}{2} \right )}}{5 \sqrt{\frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1}} \tan ^{\frac{13}{2}}{\left (\frac{x}{2} \right )} + 15 \sqrt{\frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1}} \tan ^{\frac{9}{2}}{\left (\frac{x}{2} \right )} + 15 \sqrt{\frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1}} \tan ^{\frac{5}{2}}{\left (\frac{x}{2} \right )} + 5 \sqrt{\frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1}} \sqrt{\tan{\left (\frac{x}{2} \right )}}} + \frac{12 \sqrt{2} \tan ^{3}{\left (\frac{x}{2} \right )}}{5 \sqrt{\frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1}} \tan ^{\frac{13}{2}}{\left (\frac{x}{2} \right )} + 15 \sqrt{\frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1}} \tan ^{\frac{9}{2}}{\left (\frac{x}{2} \right )} + 15 \sqrt{\frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1}} \tan ^{\frac{5}{2}}{\left (\frac{x}{2} \right )} + 5 \sqrt{\frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1}} \sqrt{\tan{\left (\frac{x}{2} \right )}}} + \frac{10 \sqrt{2} \tan{\left (\frac{x}{2} \right )}}{5 \sqrt{\frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1}} \tan ^{\frac{13}{2}}{\left (\frac{x}{2} \right )} + 15 \sqrt{\frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1}} \tan ^{\frac{9}{2}}{\left (\frac{x}{2} \right )} + 15 \sqrt{\frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1}} \tan ^{\frac{5}{2}}{\left (\frac{x}{2} \right )} + 5 \sqrt{\frac{1}{\tan ^{2}{\left (\frac{x}{2} \right )} + 1}} \sqrt{\tan{\left (\frac{x}{2} \right )}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*sqrt(2)*tan(x/2)**5/(5*sqrt(1/(tan(x/2)**2 + 1))*tan(x/2)**(13/2) + 15*sqrt(1/(tan(x/2)**2 + 1))*tan(x/2)**
(9/2) + 15*sqrt(1/(tan(x/2)**2 + 1))*tan(x/2)**(5/2) + 5*sqrt(1/(tan(x/2)**2 + 1))*sqrt(tan(x/2))) + 12*sqrt(2
)*tan(x/2)**3/(5*sqrt(1/(tan(x/2)**2 + 1))*tan(x/2)**(13/2) + 15*sqrt(1/(tan(x/2)**2 + 1))*tan(x/2)**(9/2) + 1
5*sqrt(1/(tan(x/2)**2 + 1))*tan(x/2)**(5/2) + 5*sqrt(1/(tan(x/2)**2 + 1))*sqrt(tan(x/2))) + 10*sqrt(2)*tan(x/2
)/(5*sqrt(1/(tan(x/2)**2 + 1))*tan(x/2)**(13/2) + 15*sqrt(1/(tan(x/2)**2 + 1))*tan(x/2)**(9/2) + 15*sqrt(1/(ta
n(x/2)**2 + 1))*tan(x/2)**(5/2) + 5*sqrt(1/(tan(x/2)**2 + 1))*sqrt(tan(x/2)))

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Giac [A]  time = 1.11444, size = 18, normalized size = 0.95 \begin{align*} -\frac{2}{5} \, \sin \left (x\right )^{\frac{5}{2}} + 2 \, \sqrt{\sin \left (x\right )} \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]

-2/5*sin(x)^(5/2) + 2*sqrt(sin(x))