3.1.46 \(\int (a \cos ^3(x))^{3/2} \, dx\) [46]

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

[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 = 0.02, 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 = {3286, 2715, 2719} \begin {gather*} \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)} \end {gather*}

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 2715

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] && Integ
erQ[2*n]

Rule 2719

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

Rule 3286

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 = 0.05, size = 50, normalized size = 0.75 \begin {gather*} \frac {\left (a \cos ^3(x)\right )^{3/2} \left (168 E\left (\left .\frac {x}{2}\right |2\right )+\sqrt {\cos (x)} (38 \sin (2 x)+5 \sin (4 x))\right )}{180 \cos ^{\frac {9}{2}}(x)} \end {gather*}

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] Result contains complex when optimal does not.
time = 0.08, size = 198, normalized size = 2.96

method	result	size
default	\(-\frac {2 \left (5 \left (\cos ^{6}\left (x \right )\right )+21 i \cos \left (x \right ) \sin \left (x \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}, i\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}-21 i \cos \left (x \right ) \sin \left (x \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}, i\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}+21 i \sin \left (x \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}, i\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}-21 i \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (x \right )\right )}{\sin \left (x \right )}, i\right ) \sin \left (x \right )+2 \left (\cos ^{4}\left (x \right )\right )+14 \left (\cos ^{2}\left (x \right )\right )-21 \cos \left (x \right )\right ) \left (a \left (\cos ^{3}\left (x \right )\right )\right )^{\frac {3}{2}}}{45 \cos \left (x \right )^{5} \sin \left (x \right )}\)	\(198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.12, size = 66, normalized size = 0.99 \begin {gather*} -\frac {7}{15} i \, \sqrt {2} a^{\frac {3}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) + \frac {7}{15} i \, \sqrt {2} a^{\frac {3}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) + \frac {2}{45} \, \sqrt {a \cos \left (x\right )^{3}} {\left (5 \, a \cos \left (x\right )^{2} + 7 \, a\right )} \sin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7/15*I*sqrt(2)*a^(3/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(x) + I*sin(x))) + 7/15*I*sqrt(2)
*a^(3/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(x) - I*sin(x))) + 2/45*sqrt(a*cos(x)^3)*(5*a*co
s(x)^2 + 7*a)*sin(x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3063 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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