Integrand size = 10, antiderivative size = 117 \[ \int \frac {1}{\left (a \cos ^3(x)\right )^{5/2}} \, dx=-\frac {154 \cos ^{\frac {3}{2}}(x) E\left (\left .\frac {x}{2}\right |2\right )}{195 a^2 \sqrt {a \cos ^3(x)}}+\frac {154 \cos (x) \sin (x)}{195 a^2 \sqrt {a \cos ^3(x)}}+\frac {154 \tan (x)}{585 a^2 \sqrt {a \cos ^3(x)}}+\frac {22 \sec ^2(x) \tan (x)}{117 a^2 \sqrt {a \cos ^3(x)}}+\frac {2 \sec ^4(x) \tan (x)}{13 a^2 \sqrt {a \cos ^3(x)}} \]
-154/195*cos(x)^(3/2)*(cos(1/2*x)^2)^(1/2)/cos(1/2*x)*EllipticE(sin(1/2*x) ,2^(1/2))/a^2/(a*cos(x)^3)^(1/2)+154/195*cos(x)*sin(x)/a^2/(a*cos(x)^3)^(1 /2)+154/585*tan(x)/a^2/(a*cos(x)^3)^(1/2)+22/117*sec(x)^2*tan(x)/a^2/(a*co s(x)^3)^(1/2)+2/13*sec(x)^4*tan(x)/a^2/(a*cos(x)^3)^(1/2)
Time = 0.14 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\left (a \cos ^3(x)\right )^{5/2}} \, dx=\frac {-462 \cos ^{\frac {3}{2}}(x) E\left (\left .\frac {x}{2}\right |2\right )+462 \cos (x) \sin (x)+2 \left (77+55 \sec ^2(x)+45 \sec ^4(x)\right ) \tan (x)}{585 a^2 \sqrt {a \cos ^3(x)}} \]
(-462*Cos[x]^(3/2)*EllipticE[x/2, 2] + 462*Cos[x]*Sin[x] + 2*(77 + 55*Sec[ x]^2 + 45*Sec[x]^4)*Tan[x])/(585*a^2*Sqrt[a*Cos[x]^3])
Time = 0.48 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.78, 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, 3116, 3042, 3116, 3042, 3116, 3042, 3116, 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.
| \(\displaystyle \int \frac {1}{\left (a \cos ^3(x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a \sin \left (x+\frac {\pi }{2}\right )^3\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3686 |
| \(\displaystyle \frac {\cos ^{\frac {3}{2}}(x) \int \frac {1}{\cos ^{\frac {15}{2}}(x)}dx}{a^2 \sqrt {a \cos ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cos ^{\frac {3}{2}}(x) \int \frac {1}{\sin \left (x+\frac {\pi }{2}\right )^{15/2}}dx}{a^2 \sqrt {a \cos ^3(x)}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\cos ^{\frac {3}{2}}(x) \left (\frac {11}{13} \int \frac {1}{\cos ^{\frac {11}{2}}(x)}dx+\frac {2 \sin (x)}{13 \cos ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \cos ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cos ^{\frac {3}{2}}(x) \left (\frac {11}{13} \int \frac {1}{\sin \left (x+\frac {\pi }{2}\right )^{11/2}}dx+\frac {2 \sin (x)}{13 \cos ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \cos ^3(x)}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\cos ^{\frac {3}{2}}(x) \left (\frac {11}{13} \left (\frac {7}{9} \int \frac {1}{\cos ^{\frac {7}{2}}(x)}dx+\frac {2 \sin (x)}{9 \cos ^{\frac {9}{2}}(x)}\right )+\frac {2 \sin (x)}{13 \cos ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \cos ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cos ^{\frac {3}{2}}(x) \left (\frac {11}{13} \left (\frac {7}{9} \int \frac {1}{\sin \left (x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 \sin (x)}{9 \cos ^{\frac {9}{2}}(x)}\right )+\frac {2 \sin (x)}{13 \cos ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \cos ^3(x)}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\cos ^{\frac {3}{2}}(x) \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \int \frac {1}{\cos ^{\frac {3}{2}}(x)}dx+\frac {2 \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}\right )+\frac {2 \sin (x)}{9 \cos ^{\frac {9}{2}}(x)}\right )+\frac {2 \sin (x)}{13 \cos ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \cos ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cos ^{\frac {3}{2}}(x) \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \int \frac {1}{\sin \left (x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}\right )+\frac {2 \sin (x)}{9 \cos ^{\frac {9}{2}}(x)}\right )+\frac {2 \sin (x)}{13 \cos ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \cos ^3(x)}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\cos ^{\frac {3}{2}}(x) \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \left (\frac {2 \sin (x)}{\sqrt {\cos (x)}}-\int \sqrt {\cos (x)}dx\right )+\frac {2 \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}\right )+\frac {2 \sin (x)}{9 \cos ^{\frac {9}{2}}(x)}\right )+\frac {2 \sin (x)}{13 \cos ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \cos ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cos ^{\frac {3}{2}}(x) \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \left (\frac {2 \sin (x)}{\sqrt {\cos (x)}}-\int \sqrt {\sin \left (x+\frac {\pi }{2}\right )}dx\right )+\frac {2 \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}\right )+\frac {2 \sin (x)}{9 \cos ^{\frac {9}{2}}(x)}\right )+\frac {2 \sin (x)}{13 \cos ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \cos ^3(x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\cos ^{\frac {3}{2}}(x) \left (\frac {2 \sin (x)}{13 \cos ^{\frac {13}{2}}(x)}+\frac {11}{13} \left (\frac {2 \sin (x)}{9 \cos ^{\frac {9}{2}}(x)}+\frac {7}{9} \left (\frac {2 \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}+\frac {3}{5} \left (\frac {2 \sin (x)}{\sqrt {\cos (x)}}-2 E\left (\left .\frac {x}{2}\right |2\right )\right )\right )\right )\right )}{a^2 \sqrt {a \cos ^3(x)}}\) |
(Cos[x]^(3/2)*((2*Sin[x])/(13*Cos[x]^(13/2)) + (11*((2*Sin[x])/(9*Cos[x]^( 9/2)) + (7*((2*Sin[x])/(5*Cos[x]^(5/2)) + (3*(-2*EllipticE[x/2, 2] + (2*Si n[x])/Sqrt[Cos[x]]))/5))/9))/13))/(a^2*Sqrt[a*Cos[x]^3])
3.1.50.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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 = 8.31 (sec) , antiderivative size = 297, normalized size of antiderivative = 2.54
| method | result | size |
| default | \(-\frac {2 i \left (231 \left (\cos ^{3}\left (x \right )\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, E\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right )-231 \left (\cos ^{3}\left (x \right )\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, F\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right )+462 \left (\cos ^{2}\left (x \right )\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, E\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right )-462 \left (\cos ^{2}\left (x \right )\right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, F\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right )+231 \cos \left (x \right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, E\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right )-231 \cos \left (x \right ) \sqrt {\frac {1}{\cos \left (x \right )+1}}\, \sqrt {\frac {\cos \left (x \right )}{\cos \left (x \right )+1}}\, F\left (i \left (\csc \left (x \right )-\cot \left (x \right )\right ), i\right )+231 i \cos \left (x \right ) \sin \left (x \right )+77 i \sin \left (x \right )+77 i \tan \left (x \right )+55 i \sec \left (x \right ) \tan \left (x \right )+55 i \tan \left (x \right ) \left (\sec ^{2}\left (x \right )\right )+45 i \tan \left (x \right ) \left (\sec ^{3}\left (x \right )\right )+45 i \tan \left (x \right ) \left (\sec ^{4}\left (x \right )\right )\right )}{585 \left (\cos \left (x \right )+1\right ) a^{2} \sqrt {a \left (\cos ^{3}\left (x \right )\right )}}\) | \(297\) |
-2/585*I/(cos(x)+1)/a^2/(a*cos(x)^3)^(1/2)*(231*cos(x)^3*(1/(cos(x)+1))^(1 /2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticE(I*(csc(x)-cot(x)),I)-231*cos(x)^3* (1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticF(I*(csc(x)-cot(x)) ,I)+462*cos(x)^2*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticE( I*(csc(x)-cot(x)),I)-462*cos(x)^2*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1)) ^(1/2)*EllipticF(I*(csc(x)-cot(x)),I)+231*cos(x)*(1/(cos(x)+1))^(1/2)*(cos (x)/(cos(x)+1))^(1/2)*EllipticE(I*(csc(x)-cot(x)),I)-231*cos(x)*(1/(cos(x) +1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticF(I*(csc(x)-cot(x)),I)+231*I* cos(x)*sin(x)+77*I*sin(x)+77*I*tan(x)+55*I*sec(x)*tan(x)+55*I*tan(x)*sec(x )^2+45*I*tan(x)*sec(x)^3+45*I*tan(x)*sec(x)^4)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (a \cos ^3(x)\right )^{5/2}} \, dx=\frac {231 i \, \sqrt {2} \sqrt {a} \cos \left (x\right )^{8} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) - 231 i \, \sqrt {2} \sqrt {a} \cos \left (x\right )^{8} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) + 2 \, {\left (231 \, \cos \left (x\right )^{6} + 77 \, \cos \left (x\right )^{4} + 55 \, \cos \left (x\right )^{2} + 45\right )} \sqrt {a \cos \left (x\right )^{3}} \sin \left (x\right )}{585 \, a^{3} \cos \left (x\right )^{8}} \]
1/585*(231*I*sqrt(2)*sqrt(a)*cos(x)^8*weierstrassZeta(-4, 0, weierstrassPI nverse(-4, 0, cos(x) + I*sin(x))) - 231*I*sqrt(2)*sqrt(a)*cos(x)^8*weierst rassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(x) - I*sin(x))) + 2*(231*co s(x)^6 + 77*cos(x)^4 + 55*cos(x)^2 + 45)*sqrt(a*cos(x)^3)*sin(x))/(a^3*cos (x)^8)
Timed out. \[ \int \frac {1}{\left (a \cos ^3(x)\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (a \cos ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \cos \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{\left (a \cos ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \cos \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a \cos ^3(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (a\,{\cos \left (x\right )}^3\right )}^{5/2}} \,d x \]