Integrand size = 10, antiderivative size = 67 \[ \int \left (a \cos ^3(x)\right )^{3/2} \, dx=\frac {14 a \sqrt {a \cos ^3(x)} E\left (\left .\frac {x}{2}\right |2\right )}{15 \cos ^{\frac {3}{2}}(x)}+\frac {14}{45} a \sqrt {a \cos ^3(x)} \sin (x)+\frac {2}{9} a \cos ^2(x) \sqrt {a \cos ^3(x)} \sin (x) \]
14/15*a*(cos(1/2*x)^2)^(1/2)/cos(1/2*x)*EllipticE(sin(1/2*x),2^(1/2))*(a*c os(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)
Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.75 \[ \int \left (a \cos ^3(x)\right )^{3/2} \, dx=\frac {\left (a \cos ^3(x)\right )^{3/2} \left (168 E\left (\left .\frac {x}{2}\right |2\right )+\sqrt {\cos (x)} (38 \sin (2 x)+5 \sin (4 x))\right )}{180 \cos ^{\frac {9}{2}}(x)} \]
((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))
Time = 0.34 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.88, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 3686, 3042, 3115, 3042, 3115, 3042, 3119}
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.
\(\displaystyle \int \left (a \cos ^3(x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \sin \left (x+\frac {\pi }{2}\right )^3\right )^{3/2}dx\) |
\(\Big \downarrow \) 3686 |
\(\displaystyle \frac {a \sqrt {a \cos ^3(x)} \int \cos ^{\frac {9}{2}}(x)dx}{\cos ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \sqrt {a \cos ^3(x)} \int \sin \left (x+\frac {\pi }{2}\right )^{9/2}dx}{\cos ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {a \sqrt {a \cos ^3(x)} \left (\frac {7}{9} \int \cos ^{\frac {5}{2}}(x)dx+\frac {2}{9} \sin (x) \cos ^{\frac {7}{2}}(x)\right )}{\cos ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \sqrt {a \cos ^3(x)} \left (\frac {7}{9} \int \sin \left (x+\frac {\pi }{2}\right )^{5/2}dx+\frac {2}{9} \sin (x) \cos ^{\frac {7}{2}}(x)\right )}{\cos ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {a \sqrt {a \cos ^3(x)} \left (\frac {7}{9} \left (\frac {3}{5} \int \sqrt {\cos (x)}dx+\frac {2}{5} \sin (x) \cos ^{\frac {3}{2}}(x)\right )+\frac {2}{9} \sin (x) \cos ^{\frac {7}{2}}(x)\right )}{\cos ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \sqrt {a \cos ^3(x)} \left (\frac {7}{9} \left (\frac {3}{5} \int \sqrt {\sin \left (x+\frac {\pi }{2}\right )}dx+\frac {2}{5} \sin (x) \cos ^{\frac {3}{2}}(x)\right )+\frac {2}{9} \sin (x) \cos ^{\frac {7}{2}}(x)\right )}{\cos ^{\frac {3}{2}}(x)}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {a \sqrt {a \cos ^3(x)} \left (\frac {2}{9} \sin (x) \cos ^{\frac {7}{2}}(x)+\frac {7}{9} \left (\frac {6 E\left (\left .\frac {x}{2}\right |2\right )}{5}+\frac {2}{5} \sin (x) \cos ^{\frac {3}{2}}(x)\right )\right )}{\cos ^{\frac {3}{2}}(x)}\) |
(a*Sqrt[a*Cos[x]^3]*((2*Cos[x]^(7/2)*Sin[x])/9 + (7*((6*EllipticE[x/2, 2]) /5 + (2*Cos[x]^(3/2)*Sin[x])/5))/9))/Cos[x]^(3/2)
3.1.46.3.1 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[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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 = 6.04 (sec) , antiderivative size = 271, normalized size of antiderivative = 4.04
method | result | size |
default | \(\frac {2 \sqrt {a \left (\cos ^{3}\left (x \right )\right )}\, a \left (5 \left (\cos ^{3}\left (x \right )\right ) \sin \left (x \right )-21 i F\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}+21 i \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, E\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right )+5 \left (\cos ^{2}\left (x \right )\right ) \sin \left (x \right )-42 i \sec \left (x \right ) F\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}+42 i \sec \left (x \right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, E\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right )+7 \cos \left (x \right ) \sin \left (x \right )-21 i \left (\sec ^{2}\left (x \right )\right ) F\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}+21 i \left (\sec ^{2}\left (x \right )\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, E\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right )+7 \sin \left (x \right )+21 \tan \left (x \right )\right )}{45 \left (\cos \left (x \right )+1\right )}\) | \(271\) |
2/45*(a*cos(x)^3)^(1/2)*a/(cos(x)+1)*(5*cos(x)^3*sin(x)-21*I*EllipticF(I*( csc(x)-cot(x)),I)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)+21*I*(1/( cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticE(I*(csc(x)-cot(x)),I)+ 5*cos(x)^2*sin(x)-42*I*sec(x)*EllipticF(I*(csc(x)-cot(x)),I)*(1/(cos(x)+1) )^(1/2)*(cos(x)/(cos(x)+1))^(1/2)+42*I*sec(x)*(1/(cos(x)+1))^(1/2)*(cos(x) /(cos(x)+1))^(1/2)*EllipticE(I*(csc(x)-cot(x)),I)+7*cos(x)*sin(x)-21*I*sec (x)^2*EllipticF(I*(csc(x)-cot(x)),I)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+ 1))^(1/2)+21*I*sec(x)^2*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*Ell ipticE(I*(csc(x)-cot(x)),I)+7*sin(x)+21*tan(x))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.99 \[ \int \left (a \cos ^3(x)\right )^{3/2} \, dx=-\frac {7}{15} i \, \sqrt {2} a^{\frac {3}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) + \frac {7}{15} i \, \sqrt {2} a^{\frac {3}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) + \frac {2}{45} \, \sqrt {a \cos \left (x\right )^{3}} {\left (5 \, a \cos \left (x\right )^{2} + 7 \, a\right )} \sin \left (x\right ) \]
-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, weiers trassPInverse(-4, 0, cos(x) - I*sin(x))) + 2/45*sqrt(a*cos(x)^3)*(5*a*cos( x)^2 + 7*a)*sin(x)
Timed out. \[ \int \left (a \cos ^3(x)\right )^{3/2} \, dx=\text {Timed out} \]
\[ \int \left (a \cos ^3(x)\right )^{3/2} \, dx=\int { \left (a \cos \left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
\[ \int \left (a \cos ^3(x)\right )^{3/2} \, dx=\int { \left (a \cos \left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \left (a \cos ^3(x)\right )^{3/2} \, dx=\int {\left (a\,{\cos \left (x\right )}^3\right )}^{3/2} \,d x \]