Integrand size = 10, antiderivative size = 62 \[ \int \sec ^{\frac {5}{2}}(a+b x) \, dx=\frac {2 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right ) \sqrt {\sec (a+b x)}}{3 b}+\frac {2 \sec ^{\frac {3}{2}}(a+b x) \sin (a+b x)}{3 b} \] Output:
2/3*cos(b*x+a)^(1/2)*InverseJacobiAM(1/2*a+1/2*b*x,2^(1/2))*sec(b*x+a)^(1/ 2)/b+2/3*sec(b*x+a)^(3/2)*sin(b*x+a)/b
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.74 \[ \int \sec ^{\frac {5}{2}}(a+b x) \, dx=\frac {2 \sec ^{\frac {3}{2}}(a+b x) \left (\cos ^{\frac {3}{2}}(a+b x) \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )+\sin (a+b x)\right )}{3 b} \] Input:
Integrate[Sec[a + b*x]^(5/2),x]
Output:
(2*Sec[a + b*x]^(3/2)*(Cos[a + b*x]^(3/2)*EllipticF[(a + b*x)/2, 2] + Sin[ a + b*x]))/(3*b)
Time = 0.32 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4255, 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.
\(\displaystyle \int \sec ^{\frac {5}{2}}(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc \left (a+b x+\frac {\pi }{2}\right )^{5/2}dx\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {1}{3} \int \sqrt {\sec (a+b x)}dx+\frac {2 \sin (a+b x) \sec ^{\frac {3}{2}}(a+b x)}{3 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \int \sqrt {\csc \left (a+b x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (a+b x) \sec ^{\frac {3}{2}}(a+b x)}{3 b}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {1}{3} \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} \int \frac {1}{\sqrt {\cos (a+b x)}}dx+\frac {2 \sin (a+b x) \sec ^{\frac {3}{2}}(a+b x)}{3 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} \int \frac {1}{\sqrt {\sin \left (a+b x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (a+b x) \sec ^{\frac {3}{2}}(a+b x)}{3 b}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 \sin (a+b x) \sec ^{\frac {3}{2}}(a+b x)}{3 b}+\frac {2 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{3 b}\) |
Input:
Int[Sec[a + b*x]^(5/2),x]
Output:
(2*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2]*Sqrt[Sec[a + b*x]])/(3*b) + (2*Sec[a + b*x]^(3/2)*Sin[a + b*x])/(3*b)
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[(-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. \(212\) vs. \(2(53)=106\).
Time = 0.87 (sec) , antiderivative size = 213, normalized size of antiderivative = 3.44
method | result | size |
default | \(-\frac {2 \left (-2 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right ) \sqrt {\left (2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}}{3 \sqrt {-2 \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+\sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \left (2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )^{\frac {3}{2}} \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) b}\) | \(213\) |
Input:
int(sec(b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3*(-2*(sin(1/2*b*x+1/2*a)^2)^(1/2)*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*Ell ipticF(cos(1/2*b*x+1/2*a),2^(1/2))*sin(1/2*b*x+1/2*a)^2-2*sin(1/2*b*x+1/2* a)^2*cos(1/2*b*x+1/2*a)+(sin(1/2*b*x+1/2*a)^2)^(1/2)*(2*sin(1/2*b*x+1/2*a) ^2-1)^(1/2)*EllipticF(cos(1/2*b*x+1/2*a),2^(1/2)))*((2*cos(1/2*b*x+1/2*a)^ 2-1)*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)^(3/2)/sin(1/2*b*x+1/2*a)/b
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.42 \[ \int \sec ^{\frac {5}{2}}(a+b x) \, dx=\frac {-i \, \sqrt {2} \cos \left (b x + a\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, \sqrt {2} \cos \left (b x + a\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + \frac {2 \, \sin \left (b x + a\right )}{\sqrt {\cos \left (b x + a\right )}}}{3 \, b \cos \left (b x + a\right )} \] Input:
integrate(sec(b*x+a)^(5/2),x, algorithm="fricas")
Output:
1/3*(-I*sqrt(2)*cos(b*x + a)*weierstrassPInverse(-4, 0, cos(b*x + a) + I*s in(b*x + a)) + I*sqrt(2)*cos(b*x + a)*weierstrassPInverse(-4, 0, cos(b*x + a) - I*sin(b*x + a)) + 2*sin(b*x + a)/sqrt(cos(b*x + a)))/(b*cos(b*x + a) )
\[ \int \sec ^{\frac {5}{2}}(a+b x) \, dx=\int \sec ^{\frac {5}{2}}{\left (a + b x \right )}\, dx \] Input:
integrate(sec(b*x+a)**(5/2),x)
Output:
Integral(sec(a + b*x)**(5/2), x)
\[ \int \sec ^{\frac {5}{2}}(a+b x) \, dx=\int { \sec \left (b x + a\right )^{\frac {5}{2}} \,d x } \] Input:
integrate(sec(b*x+a)^(5/2),x, algorithm="maxima")
Output:
integrate(sec(b*x + a)^(5/2), x)
\[ \int \sec ^{\frac {5}{2}}(a+b x) \, dx=\int { \sec \left (b x + a\right )^{\frac {5}{2}} \,d x } \] Input:
integrate(sec(b*x+a)^(5/2),x, algorithm="giac")
Output:
integrate(sec(b*x + a)^(5/2), x)
Timed out. \[ \int \sec ^{\frac {5}{2}}(a+b x) \, dx=\int {\left (\frac {1}{\cos \left (a+b\,x\right )}\right )}^{5/2} \,d x \] Input:
int((1/cos(a + b*x))^(5/2),x)
Output:
int((1/cos(a + b*x))^(5/2), x)
\[ \int \sec ^{\frac {5}{2}}(a+b x) \, dx=\int \sqrt {\sec \left (b x +a \right )}\, \sec \left (b x +a \right )^{2}d x \] Input:
int(sec(b*x+a)^(5/2),x)
Output:
int(sqrt(sec(a + b*x))*sec(a + b*x)**2,x)