Integrand size = 10, antiderivative size = 117 \[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\frac {26 a^2 \sqrt {a \cos ^3(x)} \operatorname {EllipticF}\left (\frac {x}{2},2\right )}{77 \cos ^{\frac {3}{2}}(x)}+\frac {78}{385} a^2 \cos (x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {26}{165} a^2 \cos ^3(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {2}{15} a^2 \cos ^5(x) \sqrt {a \cos ^3(x)} \sin (x)+\frac {26}{77} a^2 \sqrt {a \cos ^3(x)} \tan (x) \]
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)
Time = 0.15 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.52 \[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\frac {a \left (a \cos ^3(x)\right )^{3/2} \left (12480 \operatorname {EllipticF}\left (\frac {x}{2},2\right )+\sqrt {\cos (x)} (15465 \sin (x)+3657 \sin (3 x)+749 \sin (5 x)+77 \sin (7 x))\right )}{36960 \cos ^{\frac {9}{2}}(x)} \]
(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] + 77*Sin[7*x])))/(36960*Cos[x]^(9/2))
Time = 0.48 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.81, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {3042, 3686, 3042, 3115, 3042, 3115, 3042, 3115, 3042, 3115, 3042, 3120}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (a \cos ^3(x)\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sin \left (x+\frac {\pi }{2}\right )^3\right )^{5/2}dx\) |
| \(\Big \downarrow \) 3686 |
| \(\displaystyle \frac {a^2 \sqrt {a \cos ^3(x)} \int \cos ^{\frac {15}{2}}(x)dx}{\cos ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^2 \sqrt {a \cos ^3(x)} \int \sin \left (x+\frac {\pi }{2}\right )^{15/2}dx}{\cos ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a^2 \sqrt {a \cos ^3(x)} \left (\frac {13}{15} \int \cos ^{\frac {11}{2}}(x)dx+\frac {2}{15} \sin (x) \cos ^{\frac {13}{2}}(x)\right )}{\cos ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^2 \sqrt {a \cos ^3(x)} \left (\frac {13}{15} \int \sin \left (x+\frac {\pi }{2}\right )^{11/2}dx+\frac {2}{15} \sin (x) \cos ^{\frac {13}{2}}(x)\right )}{\cos ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a^2 \sqrt {a \cos ^3(x)} \left (\frac {13}{15} \left (\frac {9}{11} \int \cos ^{\frac {7}{2}}(x)dx+\frac {2}{11} \sin (x) \cos ^{\frac {9}{2}}(x)\right )+\frac {2}{15} \sin (x) \cos ^{\frac {13}{2}}(x)\right )}{\cos ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^2 \sqrt {a \cos ^3(x)} \left (\frac {13}{15} \left (\frac {9}{11} \int \sin \left (x+\frac {\pi }{2}\right )^{7/2}dx+\frac {2}{11} \sin (x) \cos ^{\frac {9}{2}}(x)\right )+\frac {2}{15} \sin (x) \cos ^{\frac {13}{2}}(x)\right )}{\cos ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a^2 \sqrt {a \cos ^3(x)} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \int \cos ^{\frac {3}{2}}(x)dx+\frac {2}{7} \sin (x) \cos ^{\frac {5}{2}}(x)\right )+\frac {2}{11} \sin (x) \cos ^{\frac {9}{2}}(x)\right )+\frac {2}{15} \sin (x) \cos ^{\frac {13}{2}}(x)\right )}{\cos ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^2 \sqrt {a \cos ^3(x)} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \int \sin \left (x+\frac {\pi }{2}\right )^{3/2}dx+\frac {2}{7} \sin (x) \cos ^{\frac {5}{2}}(x)\right )+\frac {2}{11} \sin (x) \cos ^{\frac {9}{2}}(x)\right )+\frac {2}{15} \sin (x) \cos ^{\frac {13}{2}}(x)\right )}{\cos ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a^2 \sqrt {a \cos ^3(x)} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (x)}}dx+\frac {2}{3} \sin (x) \sqrt {\cos (x)}\right )+\frac {2}{7} \sin (x) \cos ^{\frac {5}{2}}(x)\right )+\frac {2}{11} \sin (x) \cos ^{\frac {9}{2}}(x)\right )+\frac {2}{15} \sin (x) \cos ^{\frac {13}{2}}(x)\right )}{\cos ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^2 \sqrt {a \cos ^3(x)} \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (x+\frac {\pi }{2}\right )}}dx+\frac {2}{3} \sin (x) \sqrt {\cos (x)}\right )+\frac {2}{7} \sin (x) \cos ^{\frac {5}{2}}(x)\right )+\frac {2}{11} \sin (x) \cos ^{\frac {9}{2}}(x)\right )+\frac {2}{15} \sin (x) \cos ^{\frac {13}{2}}(x)\right )}{\cos ^{\frac {3}{2}}(x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {a^2 \sqrt {a \cos ^3(x)} \left (\frac {2}{15} \sin (x) \cos ^{\frac {13}{2}}(x)+\frac {13}{15} \left (\frac {2}{11} \sin (x) \cos ^{\frac {9}{2}}(x)+\frac {9}{11} \left (\frac {2}{7} \sin (x) \cos ^{\frac {5}{2}}(x)+\frac {5}{7} \left (\frac {2 \operatorname {EllipticF}\left (\frac {x}{2},2\right )}{3}+\frac {2}{3} \sin (x) \sqrt {\cos (x)}\right )\right )\right )\right )}{\cos ^{\frac {3}{2}}(x)}\) |
(a^2*Sqrt[a*Cos[x]^3]*((2*Cos[x]^(13/2)*Sin[x])/15 + (13*((2*Cos[x]^(9/2)* Sin[x])/11 + (9*((2*Cos[x]^(5/2)*Sin[x])/7 + (5*((2*EllipticF[x/2, 2])/3 + (2*Sqrt[Cos[x]]*Sin[x])/3))/7))/11))/15))/Cos[x]^(3/2)
3.1.45.3.1 Defintions of rubi rules used
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] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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]
Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^ n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Si n[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]])
Result contains complex when optimal does not.
Time = 12.33 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00
| method | result | size |
| default | \(-\frac {2 \sqrt {a \left (\cos ^{3}\left (x \right )\right )}\, a^{2} \left (-77 \left (\cos ^{5}\left (x \right )\right ) \sin \left (x \right )-91 \left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )+195 i \sec \left (x \right ) F\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}+195 i \left (\sec ^{2}\left (x \right )\right ) F\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}-117 \cos \left (x \right ) \sin \left (x \right )-195 \tan \left (x \right )\right )}{1155}\) | \(117\) |
-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))
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 \[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\frac {195 i \, \sqrt {2} a^{\frac {5}{2}} \cos \left (x\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - 195 i \, \sqrt {2} a^{\frac {5}{2}} \cos \left (x\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, {\left (77 \, a^{2} \cos \left (x\right )^{6} + 91 \, a^{2} \cos \left (x\right )^{4} + 117 \, a^{2} \cos \left (x\right )^{2} + 195 \, a^{2}\right )} \sqrt {a \cos \left (x\right )^{3}} \sin \left (x\right )}{1155 \, \cos \left (x\right )} \]
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 + 19 5*a^2)*sqrt(a*cos(x)^3)*sin(x))/cos(x)
Timed out. \[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\text {Timed out} \]
\[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\int { \left (a \cos \left (x\right )^{3}\right )^{\frac {5}{2}} \,d x } \]
\[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\int { \left (a \cos \left (x\right )^{3}\right )^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int \left (a \cos ^3(x)\right )^{5/2} \, dx=\int {\left (a\,{\cos \left (x\right )}^3\right )}^{5/2} \,d x \]