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

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

[Out]

14/15*a*(cos(1/2*x)^2)^(1/2)/cos(1/2*x)*EllipticE(sin(1/2*x),2^(1/2))*(a*cos(x)^3)^(1/2)/cos(x)^(3/2)+14/45*a*
sin(x)*(a*cos(x)^3)^(1/2)+2/9*a*cos(x)^2*sin(x)*(a*cos(x)^3)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 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]

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
]])

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*}

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Mathematica [A]  time = 0.07, 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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fricas [F]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \cos \relax (x)^{3}} a \cos \relax (x)^{3}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \cos \relax (x)^{3}\right )^{\frac {3}{2}}\,{d x} \]

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)

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maple [C]  time = 0.16, size = 198, normalized size = 2.96 \[ -\frac {2 \left (5 \left (\cos ^{6}\relax (x )\right )-21 i \cos \relax (x ) \sin \relax (x ) \EllipticF \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}+21 i \cos \relax (x ) \sin \relax (x ) \EllipticE \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}-21 i \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sin \relax (x )+21 i \sin \relax (x ) \EllipticE \left (\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, i\right ) \sqrt {\frac {1}{\cos \relax (x )+1}}\, \sqrt {\frac {\cos \relax (x )}{\cos \relax (x )+1}}+2 \left (\cos ^{4}\relax (x )\right )+14 \left (\cos ^{2}\relax (x )\right )-21 \cos \relax (x )\right ) \left (a \left (\cos ^{3}\relax (x )\right )\right )^{\frac {3}{2}}}{45 \cos \relax (x )^{5} \sin \relax (x )} \]

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)*EllipticF(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)*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)*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)*Ellipti
cE(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \cos \relax (x)^{3}\right )^{\frac {3}{2}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,{\cos \relax (x)}^3\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*cos(x)^3)^(3/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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