3.47 \(\int \sqrt{a \cos ^3(x)} \, dx\)

Optimal. Leaf size=44 \[ \frac{2}{3} \tan (x) \sqrt{a \cos ^3(x)}+\frac{2 F\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \cos ^3(x)}}{3 \cos ^{\frac{3}{2}}(x)} \]

[Out]

(2*Sqrt[a*Cos[x]^3]*EllipticF[x/2, 2])/(3*Cos[x]^(3/2)) + (2*Sqrt[a*Cos[x]^3]*Tan[x])/3

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Rubi [A]  time = 0.0283456, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2635, 2641} \[ \frac{2}{3} \tan (x) \sqrt{a \cos ^3(x)}+\frac{2 F\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \cos ^3(x)}}{3 \cos ^{\frac{3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Cos[x]^3],x]

[Out]

(2*Sqrt[a*Cos[x]^3]*EllipticF[x/2, 2])/(3*Cos[x]^(3/2)) + (2*Sqrt[a*Cos[x]^3]*Tan[x])/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 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \sqrt{a \cos ^3(x)} \, dx &=\frac{\sqrt{a \cos ^3(x)} \int \cos ^{\frac{3}{2}}(x) \, dx}{\cos ^{\frac{3}{2}}(x)}\\ &=\frac{2}{3} \sqrt{a \cos ^3(x)} \tan (x)+\frac{\sqrt{a \cos ^3(x)} \int \frac{1}{\sqrt{\cos (x)}} \, dx}{3 \cos ^{\frac{3}{2}}(x)}\\ &=\frac{2 \sqrt{a \cos ^3(x)} F\left (\left .\frac{x}{2}\right |2\right )}{3 \cos ^{\frac{3}{2}}(x)}+\frac{2}{3} \sqrt{a \cos ^3(x)} \tan (x)\\ \end{align*}

Mathematica [A]  time = 0.0254141, size = 37, normalized size = 0.84 \[ \frac{2 \sqrt{a \cos ^3(x)} \left (F\left (\left .\frac{x}{2}\right |2\right )+\sin (x) \sqrt{\cos (x)}\right )}{3 \cos ^{\frac{3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a*Cos[x]^3]*(EllipticF[x/2, 2] + Sqrt[Cos[x]]*Sin[x]))/(3*Cos[x]^(3/2))

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Maple [C]  time = 0.262, size = 76, normalized size = 1.7 \begin{align*}{\frac{ \left ( -2+2\,\cos \left ( x \right ) \right ) \left ( \cos \left ( x \right ) +1 \right ) ^{2}}{3\, \left ( \cos \left ( x \right ) \right ) ^{2} \left ( \sin \left ( x \right ) \right ) ^{3}} \left ( -i\sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) \sin \left ( x \right ) + \left ( \cos \left ( x \right ) \right ) ^{2}-\cos \left ( x \right ) \right ) \sqrt{a \left ( \cos \left ( x \right ) \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*cos(x)^3)^(1/2),x)

[Out]

2/3*(-1+cos(x))*(-I*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticF(I*(-1+cos(x))/sin(x),I)*sin(x)+co
s(x)^2-cos(x))*(cos(x)+1)^2*(a*cos(x)^3)^(1/2)/cos(x)^2/sin(x)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \cos \left (x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*cos(x)^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{a \cos \left (x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*cos(x)^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \cos \left (x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*cos(x)^3), x)