Integrand size = 21, antiderivative size = 100 \[ \int \frac {\sec ^6(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx=-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^2 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {6 \sqrt {b \sec (c+d x)} \sin (c+d x)}{5 b^3 d}+\frac {2 (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 b^5 d} \]
2/5*(b*sec(d*x+c))^(5/2)*sin(d*x+c)/b^5/d-6/5*(cos(1/2*d*x+1/2*c)^2)^(1/2) /cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/b^2/d/cos(d*x+c) ^(1/2)/(b*sec(d*x+c))^(1/2)+6/5*sin(d*x+c)*(b*sec(d*x+c))^(1/2)/b^3/d
Time = 0.35 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.64 \[ \int \frac {\sec ^6(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx=\frac {-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)}}+2 \left (3+\sec ^2(c+d x)\right ) \tan (c+d x)}{5 b^2 d \sqrt {b \sec (c+d x)}} \]
((-6*EllipticE[(c + d*x)/2, 2])/Sqrt[Cos[c + d*x]] + 2*(3 + Sec[c + d*x]^2 )*Tan[c + d*x])/(5*b^2*d*Sqrt[b*Sec[c + d*x]])
Time = 0.45 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2030, 3042, 4255, 3042, 4255, 3042, 4258, 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.
| \(\displaystyle \int \frac {\sec ^6(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2030 |
| \(\displaystyle \frac {\int (b \sec (c+d x))^{7/2}dx}{b^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}dx}{b^6}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {3}{5} b^2 \int (b \sec (c+d x))^{3/2}dx+\frac {2 b \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 d}}{b^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{5} b^2 \int \left (b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx+\frac {2 b \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 d}}{b^6}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {3}{5} b^2 \left (\frac {2 b \sin (c+d x) \sqrt {b \sec (c+d x)}}{d}-b^2 \int \frac {1}{\sqrt {b \sec (c+d x)}}dx\right )+\frac {2 b \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 d}}{b^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{5} b^2 \left (\frac {2 b \sin (c+d x) \sqrt {b \sec (c+d x)}}{d}-b^2 \int \frac {1}{\sqrt {b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 d}}{b^6}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {\frac {3}{5} b^2 \left (\frac {2 b \sin (c+d x) \sqrt {b \sec (c+d x)}}{d}-\frac {b^2 \int \sqrt {\cos (c+d x)}dx}{\sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}\right )+\frac {2 b \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 d}}{b^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{5} b^2 \left (\frac {2 b \sin (c+d x) \sqrt {b \sec (c+d x)}}{d}-\frac {b^2 \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}\right )+\frac {2 b \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 d}}{b^6}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {3}{5} b^2 \left (\frac {2 b \sin (c+d x) \sqrt {b \sec (c+d x)}}{d}-\frac {2 b^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}\right )+\frac {2 b \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 d}}{b^6}\) |
((2*b*(b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d) + (3*b^2*((-2*b^2*Ellipti cE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]]) + (2*b*Sqr t[b*Sec[c + d*x]]*Sin[c + d*x])/d))/5)/b^6
3.2.22.3.1 Defintions of rubi rules used
Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m Int[(b*v) ^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
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[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Result contains complex when optimal does not.
Time = 0.62 (sec) , antiderivative size = 420, normalized size of antiderivative = 4.20
| method | result | size |
| default | \(\frac {\frac {6 i \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )}{5}-\frac {6 i \operatorname {EllipticE}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )}{5}+\frac {12 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right )}{5}-\frac {12 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right )}{5}+\frac {6 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right ) \sec \left (d x +c \right )}{5}-\frac {6 i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticE}\left (i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), i\right ) \sec \left (d x +c \right )}{5}+\frac {6 \tan \left (d x +c \right )}{5}+\frac {2 \sec \left (d x +c \right ) \tan \left (d x +c \right )}{5}+\frac {2 \tan \left (d x +c \right ) \sec \left (d x +c \right )^{2}}{5}}{d \left (\cos \left (d x +c \right )+1\right ) \sqrt {b \sec \left (d x +c \right )}\, b^{2}}\) | \(420\) |
2/5/d/(cos(d*x+c)+1)/(b*sec(d*x+c))^(1/2)/b^2*(3*I*(1/(cos(d*x+c)+1))^(1/2 )*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(I*(-cot(d*x+c)+csc(d*x+c)),I )*cos(d*x+c)-3*I*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2 )*EllipticE(I*(-cot(d*x+c)+csc(d*x+c)),I)*cos(d*x+c)+6*I*(1/(cos(d*x+c)+1) )^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*EllipticF(I*(-cot(d*x+c)+csc(d*x +c)),I)-6*I*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*Ell ipticE(I*(-cot(d*x+c)+csc(d*x+c)),I)+3*I*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x +c)/(cos(d*x+c)+1))^(1/2)*EllipticF(I*(-cot(d*x+c)+csc(d*x+c)),I)*sec(d*x+ c)-3*I*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*Elliptic E(I*(-cot(d*x+c)+csc(d*x+c)),I)*sec(d*x+c)+3*tan(d*x+c)+sec(d*x+c)*tan(d*x +c)+tan(d*x+c)*sec(d*x+c)^2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.23 \[ \int \frac {\sec ^6(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx=\frac {-3 i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{5 \, b^{3} d \cos \left (d x + c\right )^{2}} \]
1/5*(-3*I*sqrt(2)*sqrt(b)*cos(d*x + c)^2*weierstrassZeta(-4, 0, weierstras sPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 3*I*sqrt(2)*sqrt(b)*cos (d*x + c)^2*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 2*(3*cos(d*x + c)^2 + 1)*sqrt(b/cos(d*x + c))*sin(d* x + c))/(b^3*d*cos(d*x + c)^2)
\[ \int \frac {\sec ^6(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx=\int \frac {\sec ^{6}{\left (c + d x \right )}}{\left (b \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\sec ^6(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{6}}{\left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sec ^6(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{6}}{\left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sec ^6(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^6\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]