Integrand size = 10, antiderivative size = 40 \[ \int \frac {1}{\sqrt {\text {sech}(a+b x)}} \, dx=-\frac {2 i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{b} \] Output:
-2*I*cosh(b*x+a)^(1/2)*EllipticE(I*sinh(1/2*a+1/2*b*x),2^(1/2))*sech(b*x+a )^(1/2)/b
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {\text {sech}(a+b x)}} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)}} \] Input:
Integrate[1/Sqrt[Sech[a + b*x]],x]
Output:
((-2*I)*EllipticE[(I/2)*(a + b*x), 2])/(b*Sqrt[Cosh[a + b*x]]*Sqrt[Sech[a + b*x]])
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {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.
\(\displaystyle \int \frac {1}{\sqrt {\text {sech}(a+b x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {\csc \left (i a+i b x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \int \sqrt {\cosh (a+b x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \int \sqrt {\sin \left (i a+i b x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle -\frac {2 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b}\) |
Input:
Int[1/Sqrt[Sech[a + b*x]],x]
Output:
((-2*I)*Sqrt[Cosh[a + b*x]]*EllipticE[(I/2)*(a + b*x), 2]*Sqrt[Sech[a + b* x]])/b
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*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. \(134\) vs. \(2(38)=76\).
Time = 0.83 (sec) , antiderivative size = 135, normalized size of antiderivative = 3.38
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )}{\sqrt {2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, b}\) | \(135\) |
risch | \(\frac {\sqrt {2}}{b \sqrt {\frac {{\mathrm e}^{b x +a}}{1+{\mathrm e}^{2 b x +2 a}}}}+\frac {\left (-\frac {2 \left (1+{\mathrm e}^{2 b x +2 a}\right )}{\sqrt {{\mathrm e}^{b x +a} \left (1+{\mathrm e}^{2 b x +2 a}\right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{b x +a}-i\right )}\, \sqrt {i {\mathrm e}^{b x +a}}\, \left (-2 i \operatorname {EllipticE}\left (\sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \operatorname {EllipticF}\left (\sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 b x +3 a}+{\mathrm e}^{b x +a}}}\right ) \sqrt {2}\, \sqrt {{\mathrm e}^{b x +a} \left (1+{\mathrm e}^{2 b x +2 a}\right )}}{b \sqrt {\frac {{\mathrm e}^{b x +a}}{1+{\mathrm e}^{2 b x +2 a}}}\, \left (1+{\mathrm e}^{2 b x +2 a}\right )}\) | \(230\) |
Input:
int(1/sech(b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2*((2*cosh(1/2*b*x+1/2*a)^2-1)*sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-sinh(1/2*b* x+1/2*a)^2)^(1/2)*(-2*cosh(1/2*b*x+1/2*a)^2+1)^(1/2)*EllipticE(cosh(1/2*b* x+1/2*a),2^(1/2))/(2*sinh(1/2*b*x+1/2*a)^4+sinh(1/2*b*x+1/2*a)^2)^(1/2)/si nh(1/2*b*x+1/2*a)/(2*cosh(1/2*b*x+1/2*a)^2-1)^(1/2)/b
Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (36) = 72\).
Time = 0.08 (sec) , antiderivative size = 150, normalized size of antiderivative = 3.75 \[ \int \frac {1}{\sqrt {\text {sech}(a+b x)}} \, dx=-\frac {\sqrt {2} {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )} \sqrt {\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1}} + 2 \, {\left (\sqrt {2} \cosh \left (b x + a\right ) + \sqrt {2} \sinh \left (b x + a\right )\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )\right )}{b \cosh \left (b x + a\right ) + b \sinh \left (b x + a\right )} \] Input:
integrate(1/sech(b*x+a)^(1/2),x, algorithm="fricas")
Output:
-(sqrt(2)*(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a) ^2 + 1)*sqrt((cosh(b*x + a) + sinh(b*x + a))/(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 + 1)) + 2*(sqrt(2)*cosh(b*x + a) + s qrt(2)*sinh(b*x + a))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, co sh(b*x + a) + sinh(b*x + a))))/(b*cosh(b*x + a) + b*sinh(b*x + a))
\[ \int \frac {1}{\sqrt {\text {sech}(a+b x)}} \, dx=\int \frac {1}{\sqrt {\operatorname {sech}{\left (a + b x \right )}}}\, dx \] Input:
integrate(1/sech(b*x+a)**(1/2),x)
Output:
Integral(1/sqrt(sech(a + b*x)), x)
\[ \int \frac {1}{\sqrt {\text {sech}(a+b x)}} \, dx=\int { \frac {1}{\sqrt {\operatorname {sech}\left (b x + a\right )}} \,d x } \] Input:
integrate(1/sech(b*x+a)^(1/2),x, algorithm="maxima")
Output:
integrate(1/sqrt(sech(b*x + a)), x)
\[ \int \frac {1}{\sqrt {\text {sech}(a+b x)}} \, dx=\int { \frac {1}{\sqrt {\operatorname {sech}\left (b x + a\right )}} \,d x } \] Input:
integrate(1/sech(b*x+a)^(1/2),x, algorithm="giac")
Output:
integrate(1/sqrt(sech(b*x + a)), x)
Timed out. \[ \int \frac {1}{\sqrt {\text {sech}(a+b x)}} \, dx=\int \frac {1}{\sqrt {\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}}} \,d x \] Input:
int(1/(1/cosh(a + b*x))^(1/2),x)
Output:
int(1/(1/cosh(a + b*x))^(1/2), x)
\[ \int \frac {1}{\sqrt {\text {sech}(a+b x)}} \, dx=\int \frac {\sqrt {\mathrm {sech}\left (b x +a \right )}}{\mathrm {sech}\left (b x +a \right )}d x \] Input:
int(1/sech(b*x+a)^(1/2),x)
Output:
int(sqrt(sech(a + b*x))/sech(a + b*x),x)