\(\int \frac {1}{\sqrt {a+a \text {sech}(c+d x)}} \, dx\) [81]

Optimal result

Integrand size = 14, antiderivative size = 85 \[ \int \frac {1}{\sqrt {a+a \text {sech}(c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {2} \sqrt {a+a \text {sech}(c+d x)}}\right )}{\sqrt {a} d} \] Output:

2*arctanh(a^(1/2)*tanh(d*x+c)/(a+a*sech(d*x+c))^(1/2))/a^(1/2)/d-2^(1/2)*a 
rctanh(1/2*a^(1/2)*tanh(d*x+c)*2^(1/2)/(a+a*sech(d*x+c))^(1/2))/a^(1/2)/d
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.39 \[ \int \frac {1}{\sqrt {a+a \text {sech}(c+d x)}} \, dx=\frac {\left (1+e^{c+d x}\right ) \left (\sqrt {2} \text {arcsinh}\left (e^{c+d x}\right )-2 \text {arctanh}\left (\frac {-1+e^{c+d x}}{\sqrt {2} \sqrt {1+e^{2 (c+d x)}}}\right )-\sqrt {2} \text {arctanh}\left (\sqrt {1+e^{2 (c+d x)}}\right )\right )}{\sqrt {2} d \sqrt {1+e^{2 (c+d x)}} \sqrt {a (1+\text {sech}(c+d x))}} \] Input:

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

Output:

((1 + E^(c + d*x))*(Sqrt[2]*ArcSinh[E^(c + d*x)] - 2*ArcTanh[(-1 + E^(c + 
d*x))/(Sqrt[2]*Sqrt[1 + E^(2*(c + d*x))])] - Sqrt[2]*ArcTanh[Sqrt[1 + E^(2 
*(c + d*x))]]))/(Sqrt[2]*d*Sqrt[1 + E^(2*(c + d*x))]*Sqrt[a*(1 + Sech[c + 
d*x])])
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4263, 3042, 4261, 216, 4282, 216}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4263

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4261

\(\displaystyle \frac {2 i \int \frac {1}{a-\frac {a^2 \tanh ^2(c+d x)}{\text {sech}(c+d x) a+a}}d\left (-\frac {i a \tanh (c+d x)}{\sqrt {\text {sech}(c+d x) a+a}}\right )}{d}-\int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}}dx\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {2 \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{\sqrt {a} d}-\int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}}dx\)

\(\Big \downarrow \) 4282

\(\displaystyle \frac {2 \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{\sqrt {a} d}-\frac {2 i \int \frac {1}{2 a-\frac {a^2 \tanh ^2(c+d x)}{\text {sech}(c+d x) a+a}}d\left (-\frac {i a \tanh (c+d x)}{\sqrt {\text {sech}(c+d x) a+a}}\right )}{d}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {2 \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {2} \sqrt {a \text {sech}(c+d x)+a}}\right )}{\sqrt {a} d}\)

Input:

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

Output:

(2*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + a*Sech[c + d*x]]])/(Sqrt[a]*d) 
 - (Sqrt[2]*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sech[c + d 
*x]])])/(Sqrt[a]*d)
 

Defintions of rubi rules used

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 3042

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

rule 4261

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

rule 4263

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

rule 4282

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S 
ymbol] :> Simp[-2/f   Subst[Int[1/(2*a + x^2), x], x, b*(Cot[e + f*x]/Sqrt[ 
a + b*Csc[e + f*x]])], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 868 vs. \(2 (70) = 140\).

Time = 0.11 (sec) , antiderivative size = 868, normalized size of antiderivative = 10.21 \[ \int \frac {1}{\sqrt {a+a \text {sech}(c+d x)}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(sqrt(2)*sqrt(a)*log(-(3*cosh(d*x + c)^2 + 2*(3*cosh(d*x + c) - 1)*sin 
h(d*x + c) + 3*sinh(d*x + c)^2 - 2*sqrt(2)*(cosh(d*x + c)^3 + (3*cosh(d*x 
+ c) - 1)*sinh(d*x + c)^2 + sinh(d*x + c)^3 - cosh(d*x + c)^2 + (3*cosh(d* 
x + c)^2 - 2*cosh(d*x + c) + 1)*sinh(d*x + c) + cosh(d*x + c) - 1)*sqrt(a/ 
(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1))/s 
qrt(a) - 2*cosh(d*x + c) + 3)/(cosh(d*x + c)^2 + 2*(cosh(d*x + c) + 1)*sin 
h(d*x + c) + sinh(d*x + c)^2 + 2*cosh(d*x + c) + 1)) + sqrt(a)*log(-(a*cos 
h(d*x + c)^4 + a*sinh(d*x + c)^4 - 3*a*cosh(d*x + c)^3 + (4*a*cosh(d*x + c 
) - 3*a)*sinh(d*x + c)^3 + 5*a*cosh(d*x + c)^2 + (6*a*cosh(d*x + c)^2 - 9* 
a*cosh(d*x + c) + 5*a)*sinh(d*x + c)^2 + (cosh(d*x + c)^5 + (5*cosh(d*x + 
c) - 3)*sinh(d*x + c)^4 + sinh(d*x + c)^5 - 3*cosh(d*x + c)^4 + (10*cosh(d 
*x + c)^2 - 12*cosh(d*x + c) + 5)*sinh(d*x + c)^3 + 5*cosh(d*x + c)^3 + (1 
0*cosh(d*x + c)^3 - 18*cosh(d*x + c)^2 + 15*cosh(d*x + c) - 7)*sinh(d*x + 
c)^2 - 7*cosh(d*x + c)^2 + (5*cosh(d*x + c)^4 - 12*cosh(d*x + c)^3 + 15*co 
sh(d*x + c)^2 - 14*cosh(d*x + c) + 4)*sinh(d*x + c) + 4*cosh(d*x + c) - 4) 
*sqrt(a)*sqrt(a/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d* 
x + c)^2 + 1)) - 4*a*cosh(d*x + c) + (4*a*cosh(d*x + c)^3 - 9*a*cosh(d*x + 
 c)^2 + 10*a*cosh(d*x + c) - 4*a)*sinh(d*x + c) + 4*a)/(cosh(d*x + c)^3 + 
3*cosh(d*x + c)^2*sinh(d*x + c) + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d 
*x + c)^3)) + sqrt(a)*log((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + (cos...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {a+a \text {sech}(c+d x)}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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