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

Optimal. Leaf size=25 \[ -\frac{2 \sin (x)}{\sqrt{\sin ^3(x)}}-\frac{2}{3} \sqrt{\sin ^3(x)} \]

[Out]

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

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Rubi [A]  time = 0.0488918, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3207, 2564, 14} \[ -\frac{2 \sin (x)}{\sqrt{\sin ^3(x)}}-\frac{2}{3} \sqrt{\sin ^3(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3207

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

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

Mathematica [A]  time = 0.0226984, size = 20, normalized size = 0.8 \[ \frac{\sin (x) (\cos (2 x)-7)}{3 \sqrt{\sin ^3(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.577, size = 14, normalized size = 0.6 \begin{align*} -{\frac{2}{3} \left ( \sin \left ( x \right ) \right ) ^{{\frac{3}{2}}}}-2\,{\frac{1}{\sqrt{\sin \left ( x \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sin(x)^(3/2)-2/sin(x)^(1/2)

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Maxima [A]  time = 0.988047, size = 26, normalized size = 1.04 \begin{align*} -\frac{2}{3} \, \sqrt{\sin \left (x\right )^{3}} - \frac{2 \, \sin \left (x\right )}{\sqrt{\sin \left (x\right )^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.14133, size = 88, normalized size = 3.52 \begin{align*} -\frac{2 \,{\left (\cos \left (x\right )^{2} - 4\right )} \sqrt{-{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}}{3 \,{\left (\cos \left (x\right )^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(cos(x)^2 - 4)*sqrt(-(cos(x)^2 - 1)*sin(x))/(cos(x)^2 - 1)

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Sympy [A]  time = 3.24742, size = 36, normalized size = 1.44 \begin{align*} - \frac{8 \sin ^{3}{\left (x \right )}}{3 \sqrt{\sin ^{3}{\left (x \right )}}} - \frac{2 \sin{\left (x \right )} \cos ^{2}{\left (x \right )}}{\sqrt{\sin ^{3}{\left (x \right )}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*sin(x)**3/(3*sqrt(sin(x)**3)) - 2*sin(x)*cos(x)**2/sqrt(sin(x)**3)

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Giac [A]  time = 1.08893, size = 18, normalized size = 0.72 \begin{align*} -\frac{2}{3} \, \sin \left (x\right )^{\frac{3}{2}} - \frac{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)^3)^(1/2),x, algorithm="giac")

[Out]

-2/3*sin(x)^(3/2) - 2/sqrt(sin(x))