\(\int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx\) [140]

Optimal result

Integrand size = 14, antiderivative size = 106 \[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\frac {2 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a d} \] Output:

2*(a+b)^(1/2)*coth(d*x+c)*EllipticPi((a+b*sech(d*x+c))^(1/2)/(a+b)^(1/2),( 
a+b)/a,((a+b)/(a-b))^(1/2))*(b*(1-sech(d*x+c))/(a+b))^(1/2)*(-b*(1+sech(d* 
x+c))/(a-b))^(1/2)/a/d
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\frac {2 b \sqrt {b+a \cosh (c+d x)} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a+b} \sqrt {a \cosh (c+d x)}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1+\text {sech}(c+d x))}{-a+b}} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a} \sqrt {a+b} d \sqrt {a \cosh (c+d x)} \sqrt {-\frac {b (-1+\text {sech}(c+d x))}{a+b}} \sqrt {a+b \text {sech}(c+d x)}} \] Input:

Integrate[1/Sqrt[a + b*Sech[c + d*x]],x]
 

Output:

(2*b*Sqrt[b + a*Cosh[c + d*x]]*EllipticPi[(a + b)/a, ArcSin[(Sqrt[a]*Sqrt[ 
b + a*Cosh[c + d*x]])/(Sqrt[a + b]*Sqrt[a*Cosh[c + d*x]])], (a + b)/(a - b 
)]*Sqrt[(b*(1 + Sech[c + d*x]))/(-a + b)]*Tanh[(c + d*x)/2])/(Sqrt[a]*Sqrt 
[a + b]*d*Sqrt[a*Cosh[c + d*x]]*Sqrt[-((b*(-1 + Sech[c + d*x]))/(a + b))]* 
Sqrt[a + b*Sech[c + d*x]])
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 4271}

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.

\(\displaystyle \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 4271

\(\displaystyle \frac {2 \sqrt {a+b} \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}\)

Input:

Int[1/Sqrt[a + b*Sech[c + d*x]],x]
 

Output:

(2*Sqrt[a + b]*Coth[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sech[ 
c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + 
 b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a*d)
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4271

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a 
 + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b) 
*((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[ 
c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && 
NeQ[a^2 - b^2, 0]
 

Maple [F]

\[\int \frac {1}{\sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}}d x\]

Input:

int(1/(a+b*sech(d*x+c))^(1/2),x)
 

Output:

int(1/(a+b*sech(d*x+c))^(1/2),x)
 

Fricas [F]

\[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \] Input:

integrate(1/(a+b*sech(d*x+c))^(1/2),x, algorithm="fricas")
 

Output:

integral(1/sqrt(b*sech(d*x + c) + a), x)
 

Sympy [F]

\[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int \frac {1}{\sqrt {a + b \operatorname {sech}{\left (c + d x \right )}}}\, dx \] Input:

integrate(1/(a+b*sech(d*x+c))**(1/2),x)
 

Output:

Integral(1/sqrt(a + b*sech(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \] Input:

integrate(1/(a+b*sech(d*x+c))^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/sqrt(b*sech(d*x + c) + a), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \] Input:

integrate(1/(a+b*sech(d*x+c))^(1/2),x, algorithm="giac")
 

Output:

integrate(1/sqrt(b*sech(d*x + c) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int \frac {1}{\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}} \,d x \] Input:

int(1/(a + b/cosh(c + d*x))^(1/2),x)
 

Output:

int(1/(a + b/cosh(c + d*x))^(1/2), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int \frac {\sqrt {\mathrm {sech}\left (d x +c \right ) b +a}}{\mathrm {sech}\left (d x +c \right ) b +a}d x \] Input:

int(1/(a+b*sech(d*x+c))^(1/2),x)
 

Output:

int(sqrt(sech(c + d*x)*b + a)/(sech(c + d*x)*b + a),x)