\(\int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx\) [55]

Optimal result

Integrand size = 17, antiderivative size = 123 \[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{a^{3/2} d}-\frac {5 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a+i a \text {csch}(c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}} \] Output:

2*arctanh(a^(1/2)*coth(d*x+c)/(a+I*a*csch(d*x+c))^(1/2))/a^(3/2)/d-5/4*arc 
tanh(1/2*a^(1/2)*coth(d*x+c)*2^(1/2)/(a+I*a*csch(d*x+c))^(1/2))*2^(1/2)/a^ 
(3/2)/d-1/2*coth(d*x+c)/d/(a+I*a*csch(d*x+c))^(3/2)
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(327\) vs. \(2(123)=246\).

Time = 3.32 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.66 \[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\frac {\left (-2 \sqrt {a}-8 \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {a}}\right ) \sqrt {i a (i+\text {csch}(c+d x))}+5 \sqrt {2} \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {2} \sqrt {a}}\right ) \sqrt {i a (i+\text {csch}(c+d x))}+i \text {csch}(c+d x) \left (2 \sqrt {a}-8 \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {a}}\right ) \sqrt {i a (i+\text {csch}(c+d x))}+5 \sqrt {2} \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {2} \sqrt {a}}\right ) \sqrt {i a (i+\text {csch}(c+d x))}\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{4 a^{3/2} d (i+\text {csch}(c+d x)) \sqrt {a+i a \text {csch}(c+d x)} \left (\cosh \left (\frac {1}{2} (c+d x)\right )-i \sinh \left (\frac {1}{2} (c+d x)\right )\right )} \] Input:

Integrate[(a + I*a*Csch[c + d*x])^(-3/2),x]
 

Output:

((-2*Sqrt[a] - 8*ArcTan[Sqrt[I*a*(I + Csch[c + d*x])]/Sqrt[a]]*Sqrt[I*a*(I 
 + Csch[c + d*x])] + 5*Sqrt[2]*ArcTan[Sqrt[I*a*(I + Csch[c + d*x])]/(Sqrt[ 
2]*Sqrt[a])]*Sqrt[I*a*(I + Csch[c + d*x])] + I*Csch[c + d*x]*(2*Sqrt[a] - 
8*ArcTan[Sqrt[I*a*(I + Csch[c + d*x])]/Sqrt[a]]*Sqrt[I*a*(I + Csch[c + d*x 
])] + 5*Sqrt[2]*ArcTan[Sqrt[I*a*(I + Csch[c + d*x])]/(Sqrt[2]*Sqrt[a])]*Sq 
rt[I*a*(I + Csch[c + d*x])]))*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2]))/( 
4*a^(3/2)*d*(I + Csch[c + d*x])*Sqrt[a + I*a*Csch[c + d*x]]*(Cosh[(c + d*x 
)/2] - I*Sinh[(c + d*x)/2]))
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {3042, 4264, 27, 3042, 4408, 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.

\(\displaystyle \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a-a \csc (i c+i d x))^{3/2}}dx\)

\(\Big \downarrow \) 4264

\(\displaystyle -\frac {\int -\frac {4 a-i a \text {csch}(c+d x)}{2 \sqrt {i \text {csch}(c+d x) a+a}}dx}{2 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {4 a-i a \text {csch}(c+d x)}{\sqrt {i \text {csch}(c+d x) a+a}}dx}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\csc (i c+i d x) a+4 a}{\sqrt {a-a \csc (i c+i d x)}}dx}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\)

\(\Big \downarrow \) 4408

\(\displaystyle \frac {5 a \int -\frac {i \text {csch}(c+d x)}{\sqrt {i \text {csch}(c+d x) a+a}}dx+4 \int \sqrt {i \text {csch}(c+d x) a+a}dx}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {4 \int \sqrt {i \text {csch}(c+d x) a+a}dx-5 i a \int \frac {\text {csch}(c+d x)}{\sqrt {i \text {csch}(c+d x) a+a}}dx}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {4 \int \sqrt {a-a \csc (i c+i d x)}dx-5 i a \int \frac {i \csc (i c+i d x)}{\sqrt {a-a \csc (i c+i d x)}}dx}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {5 a \int \frac {\csc (i c+i d x)}{\sqrt {a-a \csc (i c+i d x)}}dx+4 \int \sqrt {a-a \csc (i c+i d x)}dx}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\)

\(\Big \downarrow \) 4261

\(\displaystyle \frac {5 a \int \frac {\csc (i c+i d x)}{\sqrt {a-a \csc (i c+i d x)}}dx-\frac {8 i a \int \frac {1}{a-\frac {a^2 \coth ^2(c+d x)}{i \text {csch}(c+d x) a+a}}d\frac {i a \coth (c+d x)}{\sqrt {i \text {csch}(c+d x) a+a}}}{d}}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {5 a \int \frac {\csc (i c+i d x)}{\sqrt {a-a \csc (i c+i d x)}}dx+\frac {8 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\)

\(\Big \downarrow \) 4282

\(\displaystyle \frac {\frac {10 i a \int \frac {1}{2 a-\frac {a^2 \coth ^2(c+d x)}{i \text {csch}(c+d x) a+a}}d\frac {i a \coth (c+d x)}{\sqrt {i \text {csch}(c+d x) a+a}}}{d}+\frac {8 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {8 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}-\frac {5 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\)

Input:

Int[(a + I*a*Csch[c + d*x])^(-3/2),x]
 

Output:

((8*Sqrt[a]*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a + I*a*Csch[c + d*x]]])/ 
d - (5*Sqrt[2]*Sqrt[a]*ArcTanh[(Sqrt[a]*Coth[c + d*x])/(Sqrt[2]*Sqrt[a + I 
*a*Csch[c + d*x]])])/d)/(4*a^2) - Coth[c + d*x]/(2*d*(a + I*a*Csch[c + d*x 
])^(3/2))
 

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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 4264

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

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]
 

rule 4408

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ 
.) + (a_)], x_Symbol] :> Simp[c/a   Int[Sqrt[a + b*Csc[e + f*x]], x], x] - 
Simp[(b*c - a*d)/a   Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; F 
reeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
 

Maple [F]

\[\int \frac {1}{\left (a +i a \,\operatorname {csch}\left (d x +c \right )\right )^{\frac {3}{2}}}d x\]

Input:

int(1/(a+I*a*csch(d*x+c))^(3/2),x)
 

Output:

int(1/(a+I*a*csch(d*x+c))^(3/2),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. 873 vs. \(2 (96) = 192\).

Time = 0.13 (sec) , antiderivative size = 873, normalized size of antiderivative = 7.10 \[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+I*a*csch(d*x+c))^(3/2),x, algorithm="fricas")
 

Output:

-1/4*(5*sqrt(1/2)*(a^2*d*e^(2*d*x + 2*c) + 2*I*a^2*d*e^(d*x + c) - a^2*d)* 
sqrt(1/(a^3*d^2))*log(2*(2*sqrt(1/2)*(a^2*d*e^(2*d*x + 2*c) - a^2*d)*sqrt( 
a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a^3*d^2)) + a*e^(d*x + c) - I*a)*e^(-d*x 
- c)) - 5*sqrt(1/2)*(a^2*d*e^(2*d*x + 2*c) + 2*I*a^2*d*e^(d*x + c) - a^2*d 
)*sqrt(1/(a^3*d^2))*log(-2*(2*sqrt(1/2)*(a^2*d*e^(2*d*x + 2*c) - a^2*d)*sq 
rt(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a^3*d^2)) - a*e^(d*x + c) + I*a)*e^(-d 
*x - c)) - 2*(a^2*d*e^(2*d*x + 2*c) + 2*I*a^2*d*e^(d*x + c) - a^2*d)*sqrt( 
1/(a^3*d^2))*log(2*((a*d*e^(2*d*x + 2*c) - a*d)*sqrt(a/(e^(2*d*x + 2*c) - 
1))*sqrt(1/(a^3*d^2)) + e^(d*x + c) + I)*e^(-d*x - c)/(a*d)) + 2*(a^2*d*e^ 
(2*d*x + 2*c) + 2*I*a^2*d*e^(d*x + c) - a^2*d)*sqrt(1/(a^3*d^2))*log(-2*(( 
a*d*e^(2*d*x + 2*c) - a*d)*sqrt(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a^3*d^2)) 
 - e^(d*x + c) - I)*e^(-d*x - c)/(a*d)) - 2*(a^2*d*e^(2*d*x + 2*c) + 2*I*a 
^2*d*e^(d*x + c) - a^2*d)*sqrt(1/(a^3*d^2))*log(2*((a^2*d*e^(2*d*x + 2*c) 
- I*a^2*d*e^(d*x + c) - 2*a^2*d)*sqrt(1/(a^3*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)/(a*d)) + 2*(a^2*d*e^(2*d*x + 2*c) + 2*I*a^2*d*e^(d*x + c) - a 
^2*d)*sqrt(1/(a^3*d^2))*log(-2*((a^2*d*e^(2*d*x + 2*c) - I*a^2*d*e^(d*x + 
c) - 2*a^2*d)*sqrt(1/(a^3*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)/(a*d)) 
 + 2*sqrt(a/(e^(2*d*x + 2*c) - 1))*(e^(3*d*x + 3*c) - I*e^(2*d*x + 2*c)...
 

Sympy [F]

\[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\int \frac {1}{\left (i a \operatorname {csch}{\left (c + d x \right )} + a\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(a+I*a*csch(d*x+c))**(3/2),x)
 

Output:

Integral((I*a*csch(c + d*x) + a)**(-3/2), x)
 

Maxima [F]

\[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (i \, a \operatorname {csch}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(a+I*a*csch(d*x+c))^(3/2),x, algorithm="maxima")
 

Output:

integrate((I*a*csch(d*x + c) + a)^(-3/2), x)
 

Giac [F]

\[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (i \, a \operatorname {csch}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(a+I*a*csch(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

integrate((I*a*csch(d*x + c) + a)^(-3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (a+\frac {a\,1{}\mathrm {i}}{\mathrm {sinh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:

int(1/(a + (a*1i)/sinh(c + d*x))^(3/2),x)
 

Output:

int(1/(a + (a*1i)/sinh(c + d*x))^(3/2), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\mathrm {csch}\left (d x +c \right ) i +1}}{\mathrm {csch}\left (d x +c \right )^{3} i +\mathrm {csch}\left (d x +c \right )^{2}+\mathrm {csch}\left (d x +c \right ) i +1}d x -\left (\int \frac {\sqrt {\mathrm {csch}\left (d x +c \right ) i +1}\, \mathrm {csch}\left (d x +c \right )}{\mathrm {csch}\left (d x +c \right )^{3} i +\mathrm {csch}\left (d x +c \right )^{2}+\mathrm {csch}\left (d x +c \right ) i +1}d x \right ) i \right )}{a^{2}} \] Input:

int(1/(a+I*a*csch(d*x+c))^(3/2),x)
 

Output:

(sqrt(a)*(int(sqrt(csch(c + d*x)*i + 1)/(csch(c + d*x)**3*i + csch(c + d*x 
)**2 + csch(c + d*x)*i + 1),x) - int((sqrt(csch(c + d*x)*i + 1)*csch(c + d 
*x))/(csch(c + d*x)**3*i + csch(c + d*x)**2 + csch(c + d*x)*i + 1),x)*i))/ 
a**2