\(\int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx\) [54]

Optimal result

Integrand size = 17, antiderivative size = 91 \[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d} \] Output:

2*arctanh(a^(1/2)*coth(d*x+c)/(a+I*a*csch(d*x+c))^(1/2))/a^(1/2)/d-2^(1/2) 
*arctanh(1/2*a^(1/2)*coth(d*x+c)*2^(1/2)/(a+I*a*csch(d*x+c))^(1/2))/a^(1/2 
)/d
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 1.69 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=\frac {\sqrt {a} \left (2 \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {a}}\right )-\sqrt {2} \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {2} \sqrt {a}}\right )\right ) \coth (c+d x)}{d \sqrt {i a (i+\text {csch}(c+d x))} \sqrt {a+i a \text {csch}(c+d x)}} \] Input:

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

Output:

(Sqrt[a]*(2*ArcTan[Sqrt[I*a*(I + Csch[c + d*x])]/Sqrt[a]] - Sqrt[2]*ArcTan 
[Sqrt[I*a*(I + Csch[c + d*x])]/(Sqrt[2]*Sqrt[a])])*Coth[c + d*x])/(d*Sqrt[ 
I*a*(I + Csch[c + d*x])]*Sqrt[a + I*a*Csch[c + d*x]])
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {3042, 4263, 26, 3042, 26, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sqrt {a-a \csc (i c+i d x)}}dx\)

\(\Big \downarrow \) 4263

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4261

\(\displaystyle \int \frac {\csc (i c+i d x)}{\sqrt {a-a \csc (i c+i d x)}}dx-\frac {2 i \int \frac {1}{a-\frac {a^2 \coth ^2(c+d x)}{i \text {csch}(c+d x) a+a}}d\frac {i a \coth (c+d x)}{\sqrt {i \text {csch}(c+d x) a+a}}}{d}\)

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 4282

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

\(\Big \downarrow \) 216

\(\displaystyle \frac {2 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}\)

Input:

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

Output:

(2*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a + I*a*Csch[c + d*x]]])/(Sqrt[a]* 
d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Coth[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Csch[c 
 + d*x]])])/(Sqrt[a]*d)
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

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]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 551 vs. \(2 (72) = 144\).

Time = 0.13 (sec) , antiderivative size = 551, normalized size of antiderivative = 6.05 \[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*sqrt(2)*sqrt(1/(a*d^2))*log(2*(sqrt(2)*(a*d*e^(2*d*x + 2*c) - a*d)*sq 
rt(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a*d^2)) + a*e^(d*x + c) - I*a)*e^(-d*x 
 - c)) + 1/2*sqrt(2)*sqrt(1/(a*d^2))*log(-2*(sqrt(2)*(a*d*e^(2*d*x + 2*c) 
- a*d)*sqrt(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a*d^2)) - a*e^(d*x + c) + I*a 
)*e^(-d*x - c)) + 1/2*sqrt(1/(a*d^2))*log(2*((d*e^(2*d*x + 2*c) - d)*sqrt( 
a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a*d^2)) + e^(d*x + c) + I)*e^(-d*x - c)/d 
) - 1/2*sqrt(1/(a*d^2))*log(-2*((d*e^(2*d*x + 2*c) - d)*sqrt(a/(e^(2*d*x + 
 2*c) - 1))*sqrt(1/(a*d^2)) - e^(d*x + c) - I)*e^(-d*x - c)/d) + 1/2*sqrt( 
1/(a*d^2))*log(2*((a*d*e^(2*d*x + 2*c) - I*a*d*e^(d*x + c) - 2*a*d)*sqrt(1 
/(a*d^2)) + sqrt(a/(e^(2*d*x + 2*c) - 1))*(e^(3*d*x + 3*c) - 2*I*e^(2*d*x 
+ 2*c) - e^(d*x + c) + 2*I))*e^(-2*d*x - 2*c)/d) - 1/2*sqrt(1/(a*d^2))*log 
(-2*((a*d*e^(2*d*x + 2*c) - I*a*d*e^(d*x + c) - 2*a*d)*sqrt(1/(a*d^2)) - s 
qrt(a/(e^(2*d*x + 2*c) - 1))*(e^(3*d*x + 3*c) - 2*I*e^(2*d*x + 2*c) - e^(d 
*x + c) + 2*I))*e^(-2*d*x - 2*c)/d)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\mathrm {csch}\left (d x +c \right ) i +1}}{\mathrm {csch}\left (d x +c \right )^{2}+1}d x -\left (\int \frac {\sqrt {\mathrm {csch}\left (d x +c \right ) i +1}\, \mathrm {csch}\left (d x +c \right )}{\mathrm {csch}\left (d x +c \right )^{2}+1}d x \right ) i \right )}{a} \] Input:

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

Output:

(sqrt(a)*(int(sqrt(csch(c + d*x)*i + 1)/(csch(c + d*x)**2 + 1),x) - int((s 
qrt(csch(c + d*x)*i + 1)*csch(c + d*x))/(csch(c + d*x)**2 + 1),x)*i))/a