3.46 \(\int (a \cos ^3(x))^{3/2} \, dx\)

Optimal. Leaf size=67 \[ \frac{2}{9} a \sin (x) \cos ^2(x) \sqrt{a \cos ^3(x)}+\frac{14}{45} a \sin (x) \sqrt{a \cos ^3(x)}+\frac{14 a E\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \cos ^3(x)}}{15 \cos ^{\frac{3}{2}}(x)} \]

[Out]

(14*a*Sqrt[a*Cos[x]^3]*EllipticE[x/2, 2])/(15*Cos[x]^(3/2)) + (14*a*Sqrt[a*Cos[x]^3]*Sin[x])/45 + (2*a*Cos[x]^
2*Sqrt[a*Cos[x]^3]*Sin[x])/9

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Rubi [A]  time = 0.0408263, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2635, 2639} \[ \frac{2}{9} a \sin (x) \cos ^2(x) \sqrt{a \cos ^3(x)}+\frac{14}{45} a \sin (x) \sqrt{a \cos ^3(x)}+\frac{14 a E\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \cos ^3(x)}}{15 \cos ^{\frac{3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cos[x]^3)^(3/2),x]

[Out]

(14*a*Sqrt[a*Cos[x]^3]*EllipticE[x/2, 2])/(15*Cos[x]^(3/2)) + (14*a*Sqrt[a*Cos[x]^3]*Sin[x])/45 + (2*a*Cos[x]^
2*Sqrt[a*Cos[x]^3]*Sin[x])/9

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 2639

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

Rubi steps

\begin{align*} \int \left (a \cos ^3(x)\right )^{3/2} \, dx &=\frac{\left (a \sqrt{a \cos ^3(x)}\right ) \int \cos ^{\frac{9}{2}}(x) \, dx}{\cos ^{\frac{3}{2}}(x)}\\ &=\frac{2}{9} a \cos ^2(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{\left (7 a \sqrt{a \cos ^3(x)}\right ) \int \cos ^{\frac{5}{2}}(x) \, dx}{9 \cos ^{\frac{3}{2}}(x)}\\ &=\frac{14}{45} a \sqrt{a \cos ^3(x)} \sin (x)+\frac{2}{9} a \cos ^2(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{\left (7 a \sqrt{a \cos ^3(x)}\right ) \int \sqrt{\cos (x)} \, dx}{15 \cos ^{\frac{3}{2}}(x)}\\ &=\frac{14 a \sqrt{a \cos ^3(x)} E\left (\left .\frac{x}{2}\right |2\right )}{15 \cos ^{\frac{3}{2}}(x)}+\frac{14}{45} a \sqrt{a \cos ^3(x)} \sin (x)+\frac{2}{9} a \cos ^2(x) \sqrt{a \cos ^3(x)} \sin (x)\\ \end{align*}

Mathematica [A]  time = 0.0645819, size = 50, normalized size = 0.75 \[ \frac{\left (a \cos ^3(x)\right )^{3/2} \left (168 E\left (\left .\frac{x}{2}\right |2\right )+(38 \sin (2 x)+5 \sin (4 x)) \sqrt{\cos (x)}\right )}{180 \cos ^{\frac{9}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cos[x]^3)^(3/2),x]

[Out]

((a*Cos[x]^3)^(3/2)*(168*EllipticE[x/2, 2] + Sqrt[Cos[x]]*(38*Sin[2*x] + 5*Sin[4*x])))/(180*Cos[x]^(9/2))

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Maple [C]  time = 0.254, size = 198, normalized size = 3. \begin{align*} -{\frac{2}{45\, \left ( \cos \left ( x \right ) \right ) ^{5}\sin \left ( x \right ) } \left ( 5\, \left ( \cos \left ( x \right ) \right ) ^{6}+21\,i\cos \left ( x \right ) \sin \left ( x \right ){\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}-21\,i\cos \left ( x \right ) \sin \left ( x \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}+21\,i\sin \left ( x \right ){\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}-21\,i\sin \left ( x \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}+2\, \left ( \cos \left ( x \right ) \right ) ^{4}+14\, \left ( \cos \left ( x \right ) \right ) ^{2}-21\,\cos \left ( x \right ) \right ) \left ( a \left ( \cos \left ( x \right ) \right ) ^{3} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45*(5*cos(x)^6+21*I*cos(x)*sin(x)*EllipticE(I*(-1+cos(x))/sin(x),I)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1)
)^(1/2)-21*I*cos(x)*sin(x)*EllipticF(I*(-1+cos(x))/sin(x),I)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)+21
*I*sin(x)*EllipticE(I*(-1+cos(x))/sin(x),I)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)-21*I*sin(x)*Ellipti
cF(I*(-1+cos(x))/sin(x),I)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)+2*cos(x)^4+14*cos(x)^2-21*cos(x))*(a
*cos(x)^3)^(3/2)/cos(x)^5/sin(x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \cos \left (x\right )^{3}\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integral(sqrt(a*cos(x)^3)*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)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \cos \left (x\right )^{3}\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*cos(x)^3)^(3/2), x)