Integrand size = 10, antiderivative size = 85 \[ \int \frac {1}{\sec ^{\frac {7}{2}}(a+b x)} \, dx=\frac {10 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right ) \sqrt {\sec (a+b x)}}{21 b}+\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}+\frac {10 \sin (a+b x)}{21 b \sqrt {\sec (a+b x)}} \]
2/7*sin(b*x+a)/b/sec(b*x+a)^(5/2)+10/21*sin(b*x+a)/b/sec(b*x+a)^(1/2)+10/2 1*(cos(1/2*a+1/2*b*x)^2)^(1/2)/cos(1/2*a+1/2*b*x)*EllipticF(sin(1/2*a+1/2* b*x),2^(1/2))*cos(b*x+a)^(1/2)*sec(b*x+a)^(1/2)/b
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\sec ^{\frac {7}{2}}(a+b x)} \, dx=\frac {\sqrt {\sec (a+b x)} \left (40 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )+26 \sin (2 (a+b x))+3 \sin (4 (a+b x))\right )}{84 b} \]
(Sqrt[Sec[a + b*x]]*(40*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2] + 26* Sin[2*(a + b*x)] + 3*Sin[4*(a + b*x)]))/(84*b)
Time = 0.40 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 4256, 3042, 4256, 3042, 4258, 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.
\(\displaystyle \int \frac {1}{\sec ^{\frac {7}{2}}(a+b x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\csc \left (a+b x+\frac {\pi }{2}\right )^{7/2}}dx\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle \frac {5}{7} \int \frac {1}{\sec ^{\frac {3}{2}}(a+b x)}dx+\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{7} \int \frac {1}{\csc \left (a+b x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle \frac {5}{7} \left (\frac {1}{3} \int \sqrt {\sec (a+b x)}dx+\frac {2 \sin (a+b x)}{3 b \sqrt {\sec (a+b x)}}\right )+\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{7} \left (\frac {1}{3} \int \sqrt {\csc \left (a+b x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (a+b x)}{3 b \sqrt {\sec (a+b x)}}\right )+\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {5}{7} \left (\frac {1}{3} \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} \int \frac {1}{\sqrt {\cos (a+b x)}}dx+\frac {2 \sin (a+b x)}{3 b \sqrt {\sec (a+b x)}}\right )+\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{7} \left (\frac {1}{3} \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} \int \frac {1}{\sqrt {\sin \left (a+b x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (a+b x)}{3 b \sqrt {\sec (a+b x)}}\right )+\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}+\frac {5}{7} \left (\frac {2 \sin (a+b x)}{3 b \sqrt {\sec (a+b x)}}+\frac {2 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{3 b}\right )\) |
(2*Sin[a + b*x])/(7*b*Sec[a + b*x]^(5/2)) + (5*((2*Sqrt[Cos[a + b*x]]*Elli pticF[(a + b*x)/2, 2]*Sqrt[Sec[a + b*x]])/(3*b) + (2*Sin[a + b*x])/(3*b*Sq rt[Sec[a + b*x]])))/7
3.1.16.3.1 Defintions of rubi rules used
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[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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]
Leaf count of result is larger than twice the leaf count of optimal. \(198\) vs. \(2(97)=194\).
Time = 6.20 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.34
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \left (48 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{9}-120 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{7}+128 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}-72 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}+5 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {-2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )+16 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{21 \sqrt {-2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+\sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, b}\) | \(199\) |
-2/21*((2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)*(48*cos(1/2* b*x+1/2*a)^9-120*cos(1/2*b*x+1/2*a)^7+128*cos(1/2*b*x+1/2*a)^5-72*cos(1/2* b*x+1/2*a)^3+5*(sin(1/2*b*x+1/2*a)^2)^(1/2)*(-2*cos(1/2*b*x+1/2*a)^2+1)^(1 /2)*EllipticF(cos(1/2*b*x+1/2*a),2^(1/2))+16*cos(1/2*b*x+1/2*a))/(-2*sin(1 /2*b*x+1/2*a)^4+sin(1/2*b*x+1/2*a)^2)^(1/2)/sin(1/2*b*x+1/2*a)/(2*cos(1/2* b*x+1/2*a)^2-1)^(1/2)/b
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\sec ^{\frac {7}{2}}(a+b x)} \, dx=\frac {\frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{3} + 5 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{\sqrt {\cos \left (b x + a\right )}} - 5 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 5 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{21 \, b} \]
1/21*(2*(3*cos(b*x + a)^3 + 5*cos(b*x + a))*sin(b*x + a)/sqrt(cos(b*x + a) ) - 5*I*sqrt(2)*weierstrassPInverse(-4, 0, cos(b*x + a) + I*sin(b*x + a)) + 5*I*sqrt(2)*weierstrassPInverse(-4, 0, cos(b*x + a) - I*sin(b*x + a)))/b
\[ \int \frac {1}{\sec ^{\frac {7}{2}}(a+b x)} \, dx=\int \frac {1}{\sec ^{\frac {7}{2}}{\left (a + b x \right )}}\, dx \]
\[ \int \frac {1}{\sec ^{\frac {7}{2}}(a+b x)} \, dx=\int { \frac {1}{\sec \left (b x + a\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {1}{\sec ^{\frac {7}{2}}(a+b x)} \, dx=\int { \frac {1}{\sec \left (b x + a\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sec ^{\frac {7}{2}}(a+b x)} \, dx=\int \frac {1}{{\left (\frac {1}{\cos \left (a+b\,x\right )}\right )}^{7/2}} \,d x \]