\(\int (a+i a \text {csch}(c+d x))^{3/2} \, dx\) [52]

Optimal result

Integrand size = 17, antiderivative size = 72 \[ \int (a+i a \text {csch}(c+d x))^{3/2} \, dx=\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}+\frac {2 a^2 \coth (c+d x)}{d \sqrt {a+i a \text {csch}(c+d x)}} \] Output:

2*a^(3/2)*arctanh(a^(1/2)*coth(d*x+c)/(a+I*a*csch(d*x+c))^(1/2))/d+2*a^2*c 
oth(d*x+c)/d/(a+I*a*csch(d*x+c))^(1/2)
 

Mathematica [A] (verified)

Time = 1.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.39 \[ \int (a+i a \text {csch}(c+d x))^{3/2} \, dx=-\frac {2 i a \coth (c+d x) \sqrt {a+i a \text {csch}(c+d x)} \left (-\sqrt [4]{-1} \text {arctanh}\left ((-1)^{3/4} \sqrt {i+\text {csch}(c+d x)}\right )+\sqrt {i+\text {csch}(c+d x)}\right )}{d (-i+\text {csch}(c+d x)) \sqrt {i+\text {csch}(c+d x)}} \] Input:

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

Output:

((-2*I)*a*Coth[c + d*x]*Sqrt[a + I*a*Csch[c + d*x]]*(-((-1)^(1/4)*ArcTanh[ 
(-1)^(3/4)*Sqrt[I + Csch[c + d*x]]]) + Sqrt[I + Csch[c + d*x]]))/(d*(-I + 
Csch[c + d*x])*Sqrt[I + Csch[c + d*x]])
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3042, 4262, 27, 3042, 4261, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4262

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4261

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

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

Defintions of rubi rules used

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 4262

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-b^2)*C 
ot[c + d*x]*((a + b*Csc[c + d*x])^(n - 2)/(d*(n - 1))), x] + Simp[a/(n - 1) 
   Int[(a + b*Csc[c + d*x])^(n - 2)*(a*(n - 1) + b*(3*n - 4)*Csc[c + d*x]), 
 x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 1] && Inte 
gerQ[2*n]
 

Maple [F]

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

Input:

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

Output:

int((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. 461 vs. \(2 (60) = 120\).

Time = 0.10 (sec) , antiderivative size = 461, normalized size of antiderivative = 6.40 \[ \int (a+i a \text {csch}(c+d x))^{3/2} \, dx=\frac {\sqrt {\frac {a^{3}}{d^{2}}} d \log \left (\frac {2 \, {\left (a^{2} e^{\left (d x + c\right )} + i \, a^{2} + {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}\right )} e^{\left (-d x - c\right )}}{d}\right ) - \sqrt {\frac {a^{3}}{d^{2}}} d \log \left (\frac {2 \, {\left (a^{2} e^{\left (d x + c\right )} + i \, a^{2} - {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}\right )} e^{\left (-d x - c\right )}}{d}\right ) + \sqrt {\frac {a^{3}}{d^{2}}} d \log \left (\frac {2 \, {\left ({\left (a d e^{\left (2 \, d x + 2 \, c\right )} - i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {a^{3}}{d^{2}}} + {\left (a^{2} e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - a^{2} e^{\left (d x + c\right )} + 2 i \, a^{2}\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) - \sqrt {\frac {a^{3}}{d^{2}}} d \log \left (-\frac {2 \, {\left ({\left (a d e^{\left (2 \, d x + 2 \, c\right )} - i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {a^{3}}{d^{2}}} - {\left (a^{2} e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - a^{2} e^{\left (d x + c\right )} + 2 i \, a^{2}\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) + 4 \, {\left (a e^{\left (d x + c\right )} - i \, a\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}}{2 \, d} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((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]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

sqrt(a)*a*(int(sqrt(csch(c + d*x)*i + 1),x) + int(sqrt(csch(c + d*x)*i + 1 
)*csch(c + d*x),x)*i)