3.45 \(\int (a \cos ^3(x))^{5/2} \, dx\)

Optimal. Leaf size=117 \[ \frac{2}{15} a^2 \sin (x) \cos ^5(x) \sqrt{a \cos ^3(x)}+\frac{26}{165} a^2 \sin (x) \cos ^3(x) \sqrt{a \cos ^3(x)}+\frac{78}{385} a^2 \sin (x) \cos (x) \sqrt{a \cos ^3(x)}+\frac{26}{77} a^2 \tan (x) \sqrt{a \cos ^3(x)}+\frac{26 a^2 F\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \cos ^3(x)}}{77 \cos ^{\frac{3}{2}}(x)} \]

[Out]

(26*a^2*Sqrt[a*Cos[x]^3]*EllipticF[x/2, 2])/(77*Cos[x]^(3/2)) + (78*a^2*Cos[x]*Sqrt[a*Cos[x]^3]*Sin[x])/385 +
(26*a^2*Cos[x]^3*Sqrt[a*Cos[x]^3]*Sin[x])/165 + (2*a^2*Cos[x]^5*Sqrt[a*Cos[x]^3]*Sin[x])/15 + (26*a^2*Sqrt[a*C
os[x]^3]*Tan[x])/77

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Rubi [A]  time = 0.065901, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2635, 2641} \[ \frac{2}{15} a^2 \sin (x) \cos ^5(x) \sqrt{a \cos ^3(x)}+\frac{26}{165} a^2 \sin (x) \cos ^3(x) \sqrt{a \cos ^3(x)}+\frac{78}{385} a^2 \sin (x) \cos (x) \sqrt{a \cos ^3(x)}+\frac{26}{77} a^2 \tan (x) \sqrt{a \cos ^3(x)}+\frac{26 a^2 F\left (\left .\frac{x}{2}\right |2\right ) \sqrt{a \cos ^3(x)}}{77 \cos ^{\frac{3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cos[x]^3)^(5/2),x]

[Out]

(26*a^2*Sqrt[a*Cos[x]^3]*EllipticF[x/2, 2])/(77*Cos[x]^(3/2)) + (78*a^2*Cos[x]*Sqrt[a*Cos[x]^3]*Sin[x])/385 +
(26*a^2*Cos[x]^3*Sqrt[a*Cos[x]^3]*Sin[x])/165 + (2*a^2*Cos[x]^5*Sqrt[a*Cos[x]^3]*Sin[x])/15 + (26*a^2*Sqrt[a*C
os[x]^3]*Tan[x])/77

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 \left (a \cos ^3(x)\right )^{5/2} \, dx &=\frac{\left (a^2 \sqrt{a \cos ^3(x)}\right ) \int \cos ^{\frac{15}{2}}(x) \, dx}{\cos ^{\frac{3}{2}}(x)}\\ &=\frac{2}{15} a^2 \cos ^5(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{\left (13 a^2 \sqrt{a \cos ^3(x)}\right ) \int \cos ^{\frac{11}{2}}(x) \, dx}{15 \cos ^{\frac{3}{2}}(x)}\\ &=\frac{26}{165} a^2 \cos ^3(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{2}{15} a^2 \cos ^5(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{\left (39 a^2 \sqrt{a \cos ^3(x)}\right ) \int \cos ^{\frac{7}{2}}(x) \, dx}{55 \cos ^{\frac{3}{2}}(x)}\\ &=\frac{78}{385} a^2 \cos (x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{26}{165} a^2 \cos ^3(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{2}{15} a^2 \cos ^5(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{\left (39 a^2 \sqrt{a \cos ^3(x)}\right ) \int \cos ^{\frac{3}{2}}(x) \, dx}{77 \cos ^{\frac{3}{2}}(x)}\\ &=\frac{78}{385} a^2 \cos (x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{26}{165} a^2 \cos ^3(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{2}{15} a^2 \cos ^5(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{26}{77} a^2 \sqrt{a \cos ^3(x)} \tan (x)+\frac{\left (13 a^2 \sqrt{a \cos ^3(x)}\right ) \int \frac{1}{\sqrt{\cos (x)}} \, dx}{77 \cos ^{\frac{3}{2}}(x)}\\ &=\frac{26 a^2 \sqrt{a \cos ^3(x)} F\left (\left .\frac{x}{2}\right |2\right )}{77 \cos ^{\frac{3}{2}}(x)}+\frac{78}{385} a^2 \cos (x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{26}{165} a^2 \cos ^3(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{2}{15} a^2 \cos ^5(x) \sqrt{a \cos ^3(x)} \sin (x)+\frac{26}{77} a^2 \sqrt{a \cos ^3(x)} \tan (x)\\ \end{align*}

Mathematica [A]  time = 0.1198, size = 61, normalized size = 0.52 \[ \frac{a \left (a \cos ^3(x)\right )^{3/2} \left (12480 F\left (\left .\frac{x}{2}\right |2\right )+(15465 \sin (x)+3657 \sin (3 x)+749 \sin (5 x)+77 \sin (7 x)) \sqrt{\cos (x)}\right )}{36960 \cos ^{\frac{9}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cos[x]^3)^(5/2),x]

[Out]

(a*(a*Cos[x]^3)^(3/2)*(12480*EllipticF[x/2, 2] + Sqrt[Cos[x]]*(15465*Sin[x] + 3657*Sin[3*x] + 749*Sin[5*x] + 7
7*Sin[7*x])))/(36960*Cos[x]^(9/2))

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Maple [C]  time = 0.301, size = 114, normalized size = 1. \begin{align*} -{\frac{ \left ( -2+2\,\cos \left ( x \right ) \right ) \left ( \cos \left ( x \right ) +1 \right ) ^{2}}{1155\, \left ( \sin \left ( x \right ) \right ) ^{3} \left ( \cos \left ( x \right ) \right ) ^{8}} \left ( -77\, \left ( \cos \left ( x \right ) \right ) ^{8}+77\, \left ( \cos \left ( x \right ) \right ) ^{7}-91\, \left ( \cos \left ( x \right ) \right ) ^{6}+195\,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 ) +91\, \left ( \cos \left ( x \right ) \right ) ^{5}-117\, \left ( \cos \left ( x \right ) \right ) ^{4}+117\, \left ( \cos \left ( x \right ) \right ) ^{3}-195\, \left ( \cos \left ( x \right ) \right ) ^{2}+195\,\cos \left ( x \right ) \right ) \left ( a \left ( \cos \left ( x \right ) \right ) ^{3} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*cos(x)^3)^(5/2),x)

[Out]

-2/1155*(-1+cos(x))*(-77*cos(x)^8+77*cos(x)^7-91*cos(x)^6+195*I*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)
*EllipticF(I*(-1+cos(x))/sin(x),I)*sin(x)+91*cos(x)^5-117*cos(x)^4+117*cos(x)^3-195*cos(x)^2+195*cos(x))*(cos(
x)+1)^2*(a*cos(x)^3)^(5/2)/sin(x)^3/cos(x)^8

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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{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(x)^3)^(5/2),x, algorithm="maxima")

[Out]

integrate((a*cos(x)^3)^(5/2), 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^{2} \cos \left (x\right )^{6}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(x)^3)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*cos(x)^3)*a^2*cos(x)^6, 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)**(5/2),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{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(x)^3)^(5/2),x, algorithm="giac")

[Out]

integrate((a*cos(x)^3)^(5/2), x)