\(\int (a+i a \text {csch}(c+d x))^{5/2} \, dx\) [51]

Optimal result

Integrand size = 17, antiderivative size = 107 \[ \int (a+i a \text {csch}(c+d x))^{5/2} \, dx=\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}+\frac {14 a^3 \coth (c+d x)}{3 d \sqrt {a+i a \text {csch}(c+d x)}}+\frac {2 a^2 \coth (c+d x) \sqrt {a+i a \text {csch}(c+d x)}}{3 d} \] Output:

2*a^(5/2)*arctanh(a^(1/2)*coth(d*x+c)/(a+I*a*csch(d*x+c))^(1/2))/d+14/3*a^ 
3*coth(d*x+c)/d/(a+I*a*csch(d*x+c))^(1/2)+2/3*a^2*coth(d*x+c)*(a+I*a*csch( 
d*x+c))^(1/2)/d
 

Mathematica [A] (warning: unable to verify)

Time = 1.36 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.27 \[ \int (a+i a \text {csch}(c+d x))^{5/2} \, dx=\frac {2 a^2 \sqrt {a+i a \text {csch}(c+d x)} \left (-7 i+\coth (c+d x)+\frac {3 (-1)^{3/4} \text {arctanh}\left ((-1)^{3/4} \sqrt {i+\text {csch}(c+d x)}\right ) \coth (c+d x)}{(-i+\text {csch}(c+d x)) \sqrt {i+\text {csch}(c+d x)}}+\frac {14 \sinh \left (\frac {1}{2} (c+d x)\right )}{\cosh \left (\frac {1}{2} (c+d x)\right )-i \sinh \left (\frac {1}{2} (c+d x)\right )}\right )}{3 d} \] Input:

Integrate[(a + I*a*Csch[c + d*x])^(5/2),x]
 

Output:

(2*a^2*Sqrt[a + I*a*Csch[c + d*x]]*(-7*I + Coth[c + d*x] + (3*(-1)^(3/4)*A 
rcTanh[(-1)^(3/4)*Sqrt[I + Csch[c + d*x]]]*Coth[c + d*x])/((-I + Csch[c + 
d*x])*Sqrt[I + Csch[c + d*x]]) + (14*Sinh[(c + d*x)/2])/(Cosh[(c + d*x)/2] 
 - I*Sinh[(c + d*x)/2])))/(3*d)
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {3042, 4262, 27, 3042, 4403, 26, 3042, 26, 4261, 216, 4279}

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.

\(\displaystyle \int (a+i a \text {csch}(c+d x))^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a-a \csc (i c+i d x))^{5/2}dx\)

\(\Big \downarrow \) 4262

\(\displaystyle \frac {2}{3} a \int \frac {1}{2} \sqrt {i \text {csch}(c+d x) a+a} (7 i \text {csch}(c+d x) a+3 a)dx+\frac {2 a^2 \coth (c+d x) \sqrt {a+i a \text {csch}(c+d x)}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} a \int \sqrt {i \text {csch}(c+d x) a+a} (7 i \text {csch}(c+d x) a+3 a)dx+\frac {2 a^2 \coth (c+d x) \sqrt {a+i a \text {csch}(c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} a \int (3 a-7 a \csc (i c+i d x)) \sqrt {a-a \csc (i c+i d x)}dx+\frac {2 a^2 \coth (c+d x) \sqrt {a+i a \text {csch}(c+d x)}}{3 d}\)

\(\Big \downarrow \) 4403

\(\displaystyle \frac {1}{3} a \left (3 a \int \sqrt {i \text {csch}(c+d x) a+a}dx-7 a \int -i \text {csch}(c+d x) \sqrt {i \text {csch}(c+d x) a+a}dx\right )+\frac {2 a^2 \coth (c+d x) \sqrt {a+i a \text {csch}(c+d x)}}{3 d}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{3} a \left (3 a \int \sqrt {i \text {csch}(c+d x) a+a}dx+7 i a \int \text {csch}(c+d x) \sqrt {i \text {csch}(c+d x) a+a}dx\right )+\frac {2 a^2 \coth (c+d x) \sqrt {a+i a \text {csch}(c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} a \left (3 a \int \sqrt {a-a \csc (i c+i d x)}dx+7 i a \int i \csc (i c+i d x) \sqrt {a-a \csc (i c+i d x)}dx\right )+\frac {2 a^2 \coth (c+d x) \sqrt {a+i a \text {csch}(c+d x)}}{3 d}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{3} a \left (3 a \int \sqrt {a-a \csc (i c+i d x)}dx-7 a \int \csc (i c+i d x) \sqrt {a-a \csc (i c+i d x)}dx\right )+\frac {2 a^2 \coth (c+d x) \sqrt {a+i a \text {csch}(c+d x)}}{3 d}\)

\(\Big \downarrow \) 4261

\(\displaystyle \frac {1}{3} a \left (-\frac {6 i a^2 \int \frac {1}{a-\frac {a^2 \coth ^2(c+d x)}{i \text {csch}(c+d x) a+a}}d\frac {i a \coth (c+d x)}{\sqrt {i \text {csch}(c+d x) a+a}}}{d}-7 a \int \csc (i c+i d x) \sqrt {a-a \csc (i c+i d x)}dx\right )+\frac {2 a^2 \coth (c+d x) \sqrt {a+i a \text {csch}(c+d x)}}{3 d}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{3} a \left (\frac {6 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}-7 a \int \csc (i c+i d x) \sqrt {a-a \csc (i c+i d x)}dx\right )+\frac {2 a^2 \coth (c+d x) \sqrt {a+i a \text {csch}(c+d x)}}{3 d}\)

\(\Big \downarrow \) 4279

\(\displaystyle \frac {2 a^2 \coth (c+d x) \sqrt {a+i a \text {csch}(c+d x)}}{3 d}+\frac {1}{3} a \left (\frac {6 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}+\frac {14 a^2 \coth (c+d x)}{d \sqrt {a+i a \text {csch}(c+d x)}}\right )\)

Input:

Int[(a + I*a*Csch[c + d*x])^(5/2),x]
 

Output:

(2*a^2*Coth[c + d*x]*Sqrt[a + I*a*Csch[c + d*x]])/(3*d) + (a*((6*a^(3/2)*A 
rcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a + I*a*Csch[c + d*x]]])/d + (14*a^2*C 
oth[c + d*x])/(d*Sqrt[a + I*a*Csch[c + d*x]])))/3
 

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 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]
 

rule 4279

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

rule 4403

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

Maple [F]

\[\int \left (a +i a \,\operatorname {csch}\left (d x +c \right )\right )^{\frac {5}{2}}d x\]

Input:

int((a+I*a*csch(d*x+c))^(5/2),x)
 

Output:

int((a+I*a*csch(d*x+c))^(5/2),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. 566 vs. \(2 (87) = 174\).

Time = 0.10 (sec) , antiderivative size = 566, normalized size of antiderivative = 5.29 \[ \int (a+i a \text {csch}(c+d x))^{5/2} \, dx =\text {Too large to display} \] Input:

integrate((a+I*a*csch(d*x+c))^(5/2),x, algorithm="fricas")
 

Output:

1/6*(3*sqrt(a^5/d^2)*(d*e^(2*d*x + 2*c) - d)*log(2*(a^3*e^(d*x + c) + I*a^ 
3 + sqrt(a^5/d^2)*(d*e^(2*d*x + 2*c) - d)*sqrt(a/(e^(2*d*x + 2*c) - 1)))*e 
^(-d*x - c)/d) - 3*sqrt(a^5/d^2)*(d*e^(2*d*x + 2*c) - d)*log(2*(a^3*e^(d*x 
 + c) + I*a^3 - sqrt(a^5/d^2)*(d*e^(2*d*x + 2*c) - d)*sqrt(a/(e^(2*d*x + 2 
*c) - 1)))*e^(-d*x - c)/d) + 3*sqrt(a^5/d^2)*(d*e^(2*d*x + 2*c) - d)*log(2 
*(sqrt(a^5/d^2)*(a*d*e^(2*d*x + 2*c) - I*a*d*e^(d*x + c) - 2*a*d) + (a^3*e 
^(3*d*x + 3*c) - 2*I*a^3*e^(2*d*x + 2*c) - a^3*e^(d*x + c) + 2*I*a^3)*sqrt 
(a/(e^(2*d*x + 2*c) - 1)))*e^(-2*d*x - 2*c)/d) - 3*sqrt(a^5/d^2)*(d*e^(2*d 
*x + 2*c) - d)*log(-2*(sqrt(a^5/d^2)*(a*d*e^(2*d*x + 2*c) - I*a*d*e^(d*x + 
 c) - 2*a*d) - (a^3*e^(3*d*x + 3*c) - 2*I*a^3*e^(2*d*x + 2*c) - a^3*e^(d*x 
 + c) + 2*I*a^3)*sqrt(a/(e^(2*d*x + 2*c) - 1)))*e^(-2*d*x - 2*c)/d) + 8*(4 
*a^2*e^(3*d*x + 3*c) - 3*I*a^2*e^(2*d*x + 2*c) - 3*a^2*e^(d*x + c) + 4*I*a 
^2)*sqrt(a/(e^(2*d*x + 2*c) - 1)))/(d*e^(2*d*x + 2*c) - d)
 

Sympy [F]

\[ \int (a+i a \text {csch}(c+d x))^{5/2} \, dx=\int \left (i a \operatorname {csch}{\left (c + d x \right )} + a\right )^{\frac {5}{2}}\, dx \] Input:

integrate((a+I*a*csch(d*x+c))**(5/2),x)
 

Output:

Integral((I*a*csch(c + d*x) + a)**(5/2), x)
 

Maxima [F]

\[ \int (a+i a \text {csch}(c+d x))^{5/2} \, dx=\int { {\left (i \, a \operatorname {csch}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((a+I*a*csch(d*x+c))^(5/2),x, algorithm="maxima")
 

Output:

integrate((I*a*csch(d*x + c) + a)^(5/2), x)
 

Giac [F]

\[ \int (a+i a \text {csch}(c+d x))^{5/2} \, dx=\int { {\left (i \, a \operatorname {csch}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((a+I*a*csch(d*x+c))^(5/2),x, algorithm="giac")
 

Output:

integrate((I*a*csch(d*x + c) + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int (a+i a \text {csch}(c+d x))^{5/2} \, dx=\int {\left (a+\frac {a\,1{}\mathrm {i}}{\mathrm {sinh}\left (c+d\,x\right )}\right )}^{5/2} \,d x \] Input:

int((a + (a*1i)/sinh(c + d*x))^(5/2),x)
 

Output:

int((a + (a*1i)/sinh(c + d*x))^(5/2), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int (a+i a \text {csch}(c+d x))^{5/2} \, dx=\sqrt {a}\, a^{2} \left (\int \sqrt {\mathrm {csch}\left (d x +c \right ) i +1}d x +2 \left (\int \sqrt {\mathrm {csch}\left (d x +c \right ) i +1}\, \mathrm {csch}\left (d x +c \right )d x \right ) i -\left (\int \sqrt {\mathrm {csch}\left (d x +c \right ) i +1}\, \mathrm {csch}\left (d x +c \right )^{2}d x \right )\right ) \] Input:

int((a+I*a*csch(d*x+c))^(5/2),x)
 

Output:

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