\(\int (a \cos ^3(x))^{5/2} \, dx\) [45]

Optimal result

Integrand size = 10, antiderivative size = 117 \[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\frac {26 a^2 \sqrt {a \cos ^3(x)} \operatorname {EllipticF}\left (\frac {x}{2},2\right )}{77 \cos ^{\frac {3}{2}}(x)}+\frac {78}{385} a^2 \cos (x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {26}{165} a^2 \cos ^3(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {2}{15} a^2 \cos ^5(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {26}{77} a^2 \sqrt {a \cos ^3(x)} \tan (x) \]

[Out]

26/77*a^2*(cos(1/2*x)^2)^(1/2)/cos(1/2*x)*EllipticF(sin(1/2*x),2^(1/2))*(a*cos(x)^3)^(1/2)/cos(x)^(3/2)+78/385
*a^2*cos(x)*sin(x)*(a*cos(x)^3)^(1/2)+26/165*a^2*cos(x)^3*sin(x)*(a*cos(x)^3)^(1/2)+2/15*a^2*cos(x)^5*sin(x)*(
a*cos(x)^3)^(1/2)+26/77*a^2*(a*cos(x)^3)^(1/2)*tan(x)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2715, 2720} \[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\frac {26}{165} a^2 \sin (x) \cos ^3(x) \sqrt {a \cos ^3(x)}+\frac {78}{385} a^2 \sin (x) \cos (x) \sqrt {a \cos ^3(x)}+\frac {26}{77} a^2 \tan (x) \sqrt {a \cos ^3(x)}+\frac {26 a^2 \operatorname {EllipticF}\left (\frac {x}{2},2\right ) \sqrt {a \cos ^3(x)}}{77 \cos ^{\frac {3}{2}}(x)}+\frac {2}{15} a^2 \sin (x) \cos ^5(x) \sqrt {a \cos ^3(x)} \]

[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 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 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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*} \text {integral}& = \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)} \operatorname {EllipticF}\left (\frac {x}{2},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] (verified)

Time = 0.15 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.52 \[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\frac {a \left (a \cos ^3(x)\right )^{3/2} \left (12480 \operatorname {EllipticF}\left (\frac {x}{2},2\right )+\sqrt {\cos (x)} (15465 \sin (x)+3657 \sin (3 x)+749 \sin (5 x)+77 \sin (7 x))\right )}{36960 \cos ^{\frac {9}{2}}(x)} \]

[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))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 12.33 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00

method	result	size
default	\(-\frac {2 \sqrt {a \left (\cos ^{3}\left (x \right )\right )}\, a^{2} \left (-77 \left (\cos ^{5}\left (x \right )\right ) \sin \left (x \right )-91 \left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )+195 i \sec \left (x \right ) F\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}+195 i \left (\sec ^{2}\left (x \right )\right ) F\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}-117 \cos \left (x \right ) \sin \left (x \right )-195 \tan \left (x \right )\right )}{1155}\)	\(117\)

[In]

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

[Out]

-2/1155*(a*cos(x)^3)^(1/2)*a^2*(-77*cos(x)^5*sin(x)-91*cos(x)^3*sin(x)+195*I*sec(x)*EllipticF(I*(csc(x)-cot(x)
),I)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)+195*I*sec(x)^2*EllipticF(I*(csc(x)-cot(x)),I)*(1/(cos(x)+1
))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)-117*cos(x)*sin(x)-195*tan(x))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.79 \[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\frac {195 i \, \sqrt {2} a^{\frac {5}{2}} \cos \left (x\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 195 i \, \sqrt {2} a^{\frac {5}{2}} \cos \left (x\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, {\left (77 \, a^{2} \cos \left (x\right )^{6} + 91 \, a^{2} \cos \left (x\right )^{4} + 117 \, a^{2} \cos \left (x\right )^{2} + 195 \, a^{2}\right )} \sqrt {a \cos \left (x\right )^{3}} \sin \left (x\right )}{1155 \, \cos \left (x\right )} \]

[In]

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

[Out]

1/1155*(195*I*sqrt(2)*a^(5/2)*cos(x)*weierstrassPInverse(-4, 0, cos(x) + I*sin(x)) - 195*I*sqrt(2)*a^(5/2)*cos
(x)*weierstrassPInverse(-4, 0, cos(x) - I*sin(x)) + 2*(77*a^2*cos(x)^6 + 91*a^2*cos(x)^4 + 117*a^2*cos(x)^2 +
195*a^2)*sqrt(a*cos(x)^3)*sin(x))/cos(x)

Sympy [F(-1)]

Timed out. \[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\text {Timed out} \]

[In]

integrate((a*cos(x)**3)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\int { \left (a \cos \left (x\right )^{3}\right )^{\frac {5}{2}} \,d x } \]

[In]

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

[Out]

integrate((a*cos(x)^3)^(5/2), x)

Giac [F]

\[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\int { \left (a \cos \left (x\right )^{3}\right )^{\frac {5}{2}} \,d x } \]

[In]

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

[Out]

integrate((a*cos(x)^3)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\int {\left (a\,{\cos \left (x\right )}^3\right )}^{5/2} \,d x \]

[In]

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

[Out]

int((a*cos(x)^3)^(5/2), x)