\(\int \frac {(a+i a \sinh (e+f x))^{3/2}}{x} \, dx\) [128]

Optimal result

Integrand size = 21, antiderivative size = 261 \[ \int \frac {(a+i a \sinh (e+f x))^{3/2}}{x} \, dx=\frac {3}{2} i a \text {Chi}\left (\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {1}{4} (2 e-i \pi )\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {1}{2} i a \text {Chi}\left (\frac {3 f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {1}{4} (6 e+i \pi )\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {3}{2} i a \cosh \left (\frac {1}{4} (2 e-i \pi )\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)} \text {Shi}\left (\frac {f x}{2}\right )+\frac {1}{2} i a \cosh \left (\frac {1}{4} (6 e+i \pi )\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)} \text {Shi}\left (\frac {3 f x}{2}\right ) \] Output:

3/2*a*Chi(1/2*f*x)*sech(1/2*e+1/4*I*Pi+1/2*f*x)*cosh(1/2*e+1/4*I*Pi)*(a+I* 
a*sinh(f*x+e))^(1/2)+1/2*I*a*Chi(3/2*f*x)*sech(1/2*e+1/4*I*Pi+1/2*f*x)*sin 
h(3/2*e+1/4*I*Pi)*(a+I*a*sinh(f*x+e))^(1/2)+3/2*a*sinh(1/2*e+1/4*I*Pi)*sec 
h(1/2*e+1/4*I*Pi+1/2*f*x)*(a+I*a*sinh(f*x+e))^(1/2)*Shi(1/2*f*x)+1/2*I*a*c 
osh(3/2*e+1/4*I*Pi)*sech(1/2*e+1/4*I*Pi+1/2*f*x)*(a+I*a*sinh(f*x+e))^(1/2) 
*Shi(3/2*f*x)
 

Mathematica [A] (verified)

Time = 1.61 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.56 \[ \int \frac {(a+i a \sinh (e+f x))^{3/2}}{x} \, dx=\frac {a \sqrt {a+i a \sinh (e+f x)} \left (3 \text {Chi}\left (\frac {f x}{2}\right ) \left (\cosh \left (\frac {e}{2}\right )+i \sinh \left (\frac {e}{2}\right )\right )-\text {Chi}\left (\frac {3 f x}{2}\right ) \left (\cosh \left (\frac {3 e}{2}\right )-i \sinh \left (\frac {3 e}{2}\right )\right )+\left (i \cosh \left (\frac {e}{2}\right )+\sinh \left (\frac {e}{2}\right )\right ) \left (3 \text {Shi}\left (\frac {f x}{2}\right )+(1+2 i \sinh (e)) \text {Shi}\left (\frac {3 f x}{2}\right )\right )\right )}{2 \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )} \] Input:

Integrate[(a + I*a*Sinh[e + f*x])^(3/2)/x,x]
 

Output:

(a*Sqrt[a + I*a*Sinh[e + f*x]]*(3*CoshIntegral[(f*x)/2]*(Cosh[e/2] + I*Sin 
h[e/2]) - CoshIntegral[(3*f*x)/2]*(Cosh[(3*e)/2] - I*Sinh[(3*e)/2]) + (I*C 
osh[e/2] + Sinh[e/2])*(3*SinhIntegral[(f*x)/2] + (1 + (2*I)*Sinh[e])*SinhI 
ntegral[(3*f*x)/2])))/(2*(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2]))
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.57, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3800, 3042, 3793, 2009}

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.

\(\displaystyle \int \frac {(a+i a \sinh (e+f x))^{3/2}}{x} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+a \sin (i e+i f x))^{3/2}}{x}dx\)

\(\Big \downarrow \) 3800

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \int \frac {\cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{x}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \int \frac {\sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3}{x}dx\)

\(\Big \downarrow \) 3793

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \int \left (\frac {3 i \sinh \left (\frac {1}{4} (2 e-i \pi )+\frac {f x}{2}\right )}{4 x}+\frac {i \sinh \left (\frac {1}{4} (6 e+i \pi )+\frac {3 f x}{2}\right )}{4 x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {3}{4} i \sinh \left (\frac {1}{4} (2 e-i \pi )\right ) \text {Chi}\left (\frac {f x}{2}\right )+\frac {1}{4} i \sinh \left (\frac {1}{4} (6 e+i \pi )\right ) \text {Chi}\left (\frac {3 f x}{2}\right )+\frac {3}{4} i \cosh \left (\frac {1}{4} (2 e-i \pi )\right ) \text {Shi}\left (\frac {f x}{2}\right )+\frac {1}{4} i \cosh \left (\frac {1}{4} (6 e+i \pi )\right ) \text {Shi}\left (\frac {3 f x}{2}\right )\right )\)

Input:

Int[(a + I*a*Sinh[e + f*x])^(3/2)/x,x]
 

Output:

2*a*Sech[e/2 + (I/4)*Pi + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]]*(((3*I)/4)* 
CoshIntegral[(f*x)/2]*Sinh[(2*e - I*Pi)/4] + (I/4)*CoshIntegral[(3*f*x)/2] 
*Sinh[(6*e + I*Pi)/4] + ((3*I)/4)*Cosh[(2*e - I*Pi)/4]*SinhIntegral[(f*x)/ 
2] + (I/4)*Cosh[(6*e + I*Pi)/4]*SinhIntegral[(3*f*x)/2])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Simp[(2*a)^IntPart[n]*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e 
/2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n]))   Int[(c + d*x)^m*Sin[e/2 + a 
*(Pi/(4*b)) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && 
EqQ[a^2 - b^2, 0] && IntegerQ[n + 1/2] && (GtQ[n, 0] || IGtQ[m, 0])
 

Maple [F]

\[\int \frac {\left (a +i a \sinh \left (f x +e \right )\right )^{\frac {3}{2}}}{x}d x\]

Input:

int((a+I*a*sinh(f*x+e))^(3/2)/x,x)
 

Output:

int((a+I*a*sinh(f*x+e))^(3/2)/x,x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {(a+i a \sinh (e+f x))^{3/2}}{x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+I*a*sinh(f*x+e))^(3/2)/x,x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F]

\[ \int \frac {(a+i a \sinh (e+f x))^{3/2}}{x} \, dx=\int \frac {\left (i a \left (\sinh {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}}}{x}\, dx \] Input:

integrate((a+I*a*sinh(f*x+e))**(3/2)/x,x)
 

Output:

Integral((I*a*(sinh(e + f*x) - I))**(3/2)/x, x)
 

Maxima [F]

\[ \int \frac {(a+i a \sinh (e+f x))^{3/2}}{x} \, dx=\int { \frac {{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{x} \,d x } \] Input:

integrate((a+I*a*sinh(f*x+e))^(3/2)/x,x, algorithm="maxima")
 

Output:

integrate((I*a*sinh(f*x + e) + a)^(3/2)/x, x)
 

Giac [F]

\[ \int \frac {(a+i a \sinh (e+f x))^{3/2}}{x} \, dx=\int { \frac {{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{x} \,d x } \] Input:

integrate((a+I*a*sinh(f*x+e))^(3/2)/x,x, algorithm="giac")
 

Output:

integrate((I*a*sinh(f*x + e) + a)^(3/2)/x, x)
 

Mupad [F(-1)]

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

int((a + a*sinh(e + f*x)*1i)^(3/2)/x,x)
 

Output:

int((a + a*sinh(e + f*x)*1i)^(3/2)/x, x)
 

Reduce [F]

\[ \int \frac {(a+i a \sinh (e+f x))^{3/2}}{x} \, dx=\sqrt {a}\, a \left (\int \frac {\sqrt {\sinh \left (f x +e \right ) i +1}}{x}d x +\left (\int \frac {\sqrt {\sinh \left (f x +e \right ) i +1}\, \sinh \left (f x +e \right )}{x}d x \right ) i \right ) \] Input:

int((a+I*a*sinh(f*x+e))^(3/2)/x,x)
 

Output:

sqrt(a)*a*(int(sqrt(sinh(e + f*x)*i + 1)/x,x) + int((sqrt(sinh(e + f*x)*i 
+ 1)*sinh(e + f*x))/x,x)*i)