Integrand size = 10, antiderivative size = 85 \[ \int \sec ^{\frac {7}{2}}(a+b x) \, dx=-\frac {6 \sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {\sec (a+b x)}}{5 b}+\frac {6 \sqrt {\sec (a+b x)} \sin (a+b x)}{5 b}+\frac {2 \sec ^{\frac {5}{2}}(a+b x) \sin (a+b x)}{5 b} \]
2/5*sec(b*x+a)^(5/2)*sin(b*x+a)/b+6/5*sin(b*x+a)*sec(b*x+a)^(1/2)/b-6/5*(c os(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))*cos(b*x+a)^(1/2)*sec(b*x+a)^(1/2)/b
Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.69 \[ \int \sec ^{\frac {7}{2}}(a+b x) \, dx=\frac {\sec ^{\frac {5}{2}}(a+b x) \left (-12 \cos ^{\frac {5}{2}}(a+b x) E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+7 \sin (a+b x)+3 \sin (3 (a+b x))\right )}{10 b} \]
(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 = 0.41 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, 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.
\(\displaystyle \int \sec ^{\frac {7}{2}}(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc \left (a+b x+\frac {\pi }{2}\right )^{7/2}dx\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {3}{5} \int \sec ^{\frac {3}{2}}(a+b x)dx+\frac {2 \sin (a+b x) \sec ^{\frac {5}{2}}(a+b x)}{5 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{5} \int \csc \left (a+b x+\frac {\pi }{2}\right )^{3/2}dx+\frac {2 \sin (a+b x) \sec ^{\frac {5}{2}}(a+b x)}{5 b}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {3}{5} \left (\frac {2 \sin (a+b x) \sqrt {\sec (a+b x)}}{b}-\int \frac {1}{\sqrt {\sec (a+b x)}}dx\right )+\frac {2 \sin (a+b x) \sec ^{\frac {5}{2}}(a+b x)}{5 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{5} \left (\frac {2 \sin (a+b x) \sqrt {\sec (a+b x)}}{b}-\int \frac {1}{\sqrt {\csc \left (a+b x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 \sin (a+b x) \sec ^{\frac {5}{2}}(a+b x)}{5 b}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {3}{5} \left (\frac {2 \sin (a+b x) \sqrt {\sec (a+b x)}}{b}-\sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} \int \sqrt {\cos (a+b x)}dx\right )+\frac {2 \sin (a+b x) \sec ^{\frac {5}{2}}(a+b x)}{5 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{5} \left (\frac {2 \sin (a+b x) \sqrt {\sec (a+b x)}}{b}-\sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} \int \sqrt {\sin \left (a+b x+\frac {\pi }{2}\right )}dx\right )+\frac {2 \sin (a+b x) \sec ^{\frac {5}{2}}(a+b x)}{5 b}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {2 \sin (a+b x) \sec ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {3}{5} \left (\frac {2 \sin (a+b x) \sqrt {\sec (a+b x)}}{b}-\frac {2 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b}\right )\) |
(2*Sec[a + b*x]^(5/2)*Sin[a + b*x])/(5*b) + (3*((-2*Sqrt[Cos[a + b*x]]*Ell ipticE[(a + b*x)/2, 2]*Sqrt[Sec[a + b*x]])/b + (2*Sqrt[Sec[a + b*x]]*Sin[a + b*x])/b))/5
3.1.9.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(357\) vs. \(2(97)=194\).
Time = 8.78 (sec) , antiderivative size = 358, normalized size of antiderivative = 4.21
method | result | size |
default | \(-\frac {2 \sqrt {-\left (-2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \left (24 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{6} \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-12 \operatorname {EllipticE}\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}-24 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4} \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+12 \operatorname {EllipticE}\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+8 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-3 \operatorname {EllipticE}\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\right ) \sqrt {-2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+\sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}}{5 \left (8 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}-12 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+6 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{3} \sqrt {2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, b}\) | \(358\) |
-2/5*(-(-2*cos(1/2*b*x+1/2*a)^2+1)*sin(1/2*b*x+1/2*a)^2)^(1/2)/(8*sin(1/2* b*x+1/2*a)^6-12*sin(1/2*b*x+1/2*a)^4+6*sin(1/2*b*x+1/2*a)^2-1)/sin(1/2*b*x +1/2*a)^3*(24*sin(1/2*b*x+1/2*a)^6*cos(1/2*b*x+1/2*a)-12*EllipticE(cos(1/2 *b*x+1/2*a),2^(1/2))*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*(sin(1/2*b*x+1/2*a)^ 2)^(1/2)*sin(1/2*b*x+1/2*a)^4-24*sin(1/2*b*x+1/2*a)^4*cos(1/2*b*x+1/2*a)+1 2*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*( sin(1/2*b*x+1/2*a)^2)^(1/2)*sin(1/2*b*x+1/2*a)^2+8*sin(1/2*b*x+1/2*a)^2*co s(1/2*b*x+1/2*a)-3*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))*(2*sin(1/2*b*x+1/ 2*a)^2-1)^(1/2)*(sin(1/2*b*x+1/2*a)^2)^(1/2))*(-2*sin(1/2*b*x+1/2*a)^4+sin (1/2*b*x+1/2*a)^2)^(1/2)/(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 = 110, normalized size of antiderivative = 1.29 \[ \int \sec ^{\frac {7}{2}}(a+b x) \, dx=\frac {-3 i \, \sqrt {2} \cos \left (b x + a\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 3 i \, \sqrt {2} \cos \left (b x + a\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + \frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right )}{\sqrt {\cos \left (b x + a\right )}}}{5 \, b \cos \left (b x + a\right )^{2}} \]
1/5*(-3*I*sqrt(2)*cos(b*x + a)^2*weierstrassZeta(-4, 0, weierstrassPInvers e(-4, 0, cos(b*x + a) + I*sin(b*x + a))) + 3*I*sqrt(2)*cos(b*x + a)^2*weie rstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*x + a) - I*sin(b*x + a ))) + 2*(3*cos(b*x + a)^2 + 1)*sin(b*x + a)/sqrt(cos(b*x + a)))/(b*cos(b*x + a)^2)
Timed out. \[ \int \sec ^{\frac {7}{2}}(a+b x) \, dx=\text {Timed out} \]
\[ \int \sec ^{\frac {7}{2}}(a+b x) \, dx=\int { \sec \left (b x + a\right )^{\frac {7}{2}} \,d x } \]
\[ \int \sec ^{\frac {7}{2}}(a+b x) \, dx=\int { \sec \left (b x + a\right )^{\frac {7}{2}} \,d x } \]
Timed out. \[ \int \sec ^{\frac {7}{2}}(a+b x) \, dx=\int {\left (\frac {1}{\cos \left (a+b\,x\right )}\right )}^{7/2} \,d x \]