Integrand size = 12, antiderivative size = 98 \[ \int (c \sec (a+b x))^{7/2} \, dx=-\frac {6 c^4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b \sqrt {\cos (a+b x)} \sqrt {c \sec (a+b x)}}+\frac {6 c^3 \sqrt {c \sec (a+b x)} \sin (a+b x)}{5 b}+\frac {2 c (c \sec (a+b x))^{5/2} \sin (a+b x)}{5 b} \]
2/5*c*(c*sec(b*x+a))^(5/2)*sin(b*x+a)/b-6/5*c^4*(cos(1/2*a+1/2*b*x)^2)^(1/ 2)/cos(1/2*a+1/2*b*x)*EllipticE(sin(1/2*a+1/2*b*x),2^(1/2))/b/cos(b*x+a)^( 1/2)/(c*sec(b*x+a))^(1/2)+6/5*c^3*sin(b*x+a)*(c*sec(b*x+a))^(1/2)/b
Time = 0.24 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.63 \[ \int (c \sec (a+b x))^{7/2} \, dx=\frac {c (c \sec (a+b x))^{5/2} \left (-12 \cos ^{\frac {5}{2}}(a+b x) E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+7 \sin (a+b x)+3 \sin (3 (a+b x))\right )}{10 b} \]
(c*(c*Sec[a + b*x])^(5/2)*(-12*Cos[a + b*x]^(5/2)*EllipticE[(a + b*x)/2, 2 ] + 7*Sin[a + b*x] + 3*Sin[3*(a + b*x)]))/(10*b)
Time = 0.44 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {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.
\(\displaystyle \int (c \sec (a+b x))^{7/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (c \csc \left (a+b x+\frac {\pi }{2}\right )\right )^{7/2}dx\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {3}{5} c^2 \int (c \sec (a+b x))^{3/2}dx+\frac {2 c \sin (a+b x) (c \sec (a+b x))^{5/2}}{5 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{5} c^2 \int \left (c \csc \left (a+b x+\frac {\pi }{2}\right )\right )^{3/2}dx+\frac {2 c \sin (a+b x) (c \sec (a+b x))^{5/2}}{5 b}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {3}{5} c^2 \left (\frac {2 c \sin (a+b x) \sqrt {c \sec (a+b x)}}{b}-c^2 \int \frac {1}{\sqrt {c \sec (a+b x)}}dx\right )+\frac {2 c \sin (a+b x) (c \sec (a+b x))^{5/2}}{5 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{5} c^2 \left (\frac {2 c \sin (a+b x) \sqrt {c \sec (a+b x)}}{b}-c^2 \int \frac {1}{\sqrt {c \csc \left (a+b x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 c \sin (a+b x) (c \sec (a+b x))^{5/2}}{5 b}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {3}{5} c^2 \left (\frac {2 c \sin (a+b x) \sqrt {c \sec (a+b x)}}{b}-\frac {c^2 \int \sqrt {\cos (a+b x)}dx}{\sqrt {\cos (a+b x)} \sqrt {c \sec (a+b x)}}\right )+\frac {2 c \sin (a+b x) (c \sec (a+b x))^{5/2}}{5 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{5} c^2 \left (\frac {2 c \sin (a+b x) \sqrt {c \sec (a+b x)}}{b}-\frac {c^2 \int \sqrt {\sin \left (a+b x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (a+b x)} \sqrt {c \sec (a+b x)}}\right )+\frac {2 c \sin (a+b x) (c \sec (a+b x))^{5/2}}{5 b}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {3}{5} c^2 \left (\frac {2 c \sin (a+b x) \sqrt {c \sec (a+b x)}}{b}-\frac {2 c^2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b \sqrt {\cos (a+b x)} \sqrt {c \sec (a+b x)}}\right )+\frac {2 c \sin (a+b x) (c \sec (a+b x))^{5/2}}{5 b}\) |
(2*c*(c*Sec[a + b*x])^(5/2)*Sin[a + b*x])/(5*b) + (3*c^2*((-2*c^2*Elliptic E[(a + b*x)/2, 2])/(b*Sqrt[Cos[a + b*x]]*Sqrt[c*Sec[a + b*x]]) + (2*c*Sqrt [c*Sec[a + b*x]]*Sin[a + b*x])/b))/5
3.1.17.3.1 Defintions of rubi rules used
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 = 13.58 (sec) , antiderivative size = 415, normalized size of antiderivative = 4.23
method | result | size |
default | \(\frac {2 \sqrt {c \sec \left (b x +a \right )}\, c^{3} \left (-3 i \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (b x +a \right )-\csc \left (b x +a \right )\right ), i\right ) \cos \left (b x +a \right )^{2}+3 i \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}\, \operatorname {EllipticE}\left (i \left (\cot \left (b x +a \right )-\csc \left (b x +a \right )\right ), i\right ) \cos \left (b x +a \right )^{2}-6 i \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (b x +a \right )-\csc \left (b x +a \right )\right ), i\right ) \cos \left (b x +a \right )+6 i \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}\, \operatorname {EllipticE}\left (i \left (\cot \left (b x +a \right )-\csc \left (b x +a \right )\right ), i\right ) \cos \left (b x +a \right )-3 i \operatorname {EllipticF}\left (i \left (\cot \left (b x +a \right )-\csc \left (b x +a \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}+3 i \operatorname {EllipticE}\left (i \left (\cot \left (b x +a \right )-\csc \left (b x +a \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}+3 \sin \left (b x +a \right )+\tan \left (b x +a \right )+\sec \left (b x +a \right ) \tan \left (b x +a \right )\right )}{5 b \left (\cos \left (b x +a \right )+1\right )}\) | \(415\) |
2/5/b*(c*sec(b*x+a))^(1/2)*c^3/(cos(b*x+a)+1)*(-3*I*(1/(cos(b*x+a)+1))^(1/ 2)*(cos(b*x+a)/(cos(b*x+a)+1))^(1/2)*EllipticF(I*(cot(b*x+a)-csc(b*x+a)),I )*cos(b*x+a)^2+3*I*(1/(cos(b*x+a)+1))^(1/2)*(cos(b*x+a)/(cos(b*x+a)+1))^(1 /2)*EllipticE(I*(cot(b*x+a)-csc(b*x+a)),I)*cos(b*x+a)^2-6*I*(1/(cos(b*x+a) +1))^(1/2)*(cos(b*x+a)/(cos(b*x+a)+1))^(1/2)*EllipticF(I*(cot(b*x+a)-csc(b *x+a)),I)*cos(b*x+a)+6*I*(1/(cos(b*x+a)+1))^(1/2)*(cos(b*x+a)/(cos(b*x+a)+ 1))^(1/2)*EllipticE(I*(cot(b*x+a)-csc(b*x+a)),I)*cos(b*x+a)-3*I*EllipticF( I*(cot(b*x+a)-csc(b*x+a)),I)*(1/(cos(b*x+a)+1))^(1/2)*(cos(b*x+a)/(cos(b*x +a)+1))^(1/2)+3*I*EllipticE(I*(cot(b*x+a)-csc(b*x+a)),I)*(1/(cos(b*x+a)+1) )^(1/2)*(cos(b*x+a)/(cos(b*x+a)+1))^(1/2)+3*sin(b*x+a)+tan(b*x+a)+sec(b*x+ a)*tan(b*x+a))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.28 \[ \int (c \sec (a+b x))^{7/2} \, dx=\frac {-3 i \, \sqrt {2} c^{\frac {7}{2}} \cos \left (b x + a\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 3 i \, \sqrt {2} c^{\frac {7}{2}} \cos \left (b x + a\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + 2 \, {\left (3 \, c^{3} \cos \left (b x + a\right )^{2} + c^{3}\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sin \left (b x + a\right )}{5 \, b \cos \left (b x + a\right )^{2}} \]
1/5*(-3*I*sqrt(2)*c^(7/2)*cos(b*x + a)^2*weierstrassZeta(-4, 0, weierstras sPInverse(-4, 0, cos(b*x + a) + I*sin(b*x + a))) + 3*I*sqrt(2)*c^(7/2)*cos (b*x + a)^2*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*x + a) - I*sin(b*x + a))) + 2*(3*c^3*cos(b*x + a)^2 + c^3)*sqrt(c/cos(b*x + a))* sin(b*x + a))/(b*cos(b*x + a)^2)
Timed out. \[ \int (c \sec (a+b x))^{7/2} \, dx=\text {Timed out} \]
\[ \int (c \sec (a+b x))^{7/2} \, dx=\int { \left (c \sec \left (b x + a\right )\right )^{\frac {7}{2}} \,d x } \]
\[ \int (c \sec (a+b x))^{7/2} \, dx=\int { \left (c \sec \left (b x + a\right )\right )^{\frac {7}{2}} \,d x } \]
Timed out. \[ \int (c \sec (a+b x))^{7/2} \, dx=\int {\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{7/2} \,d x \]