\(\int \frac {\sqrt {a+i a \sinh (e+f x)}}{x^2} \, dx\) [123]

Optimal result

Integrand size = 21, antiderivative size = 149 \[ \int \frac {\sqrt {a+i a \sinh (e+f x)}}{x^2} \, dx=-\frac {\sqrt {a+i a \sinh (e+f x)}}{x}+\frac {1}{2} f \text {Chi}\left (\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {1}{4} (2 e+i \pi )\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {1}{2} f \cosh \left (\frac {1}{4} (2 e+i \pi )\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)} \text {Shi}\left (\frac {f x}{2}\right ) \] Output:

-(a+I*a*sinh(f*x+e))^(1/2)/x+1/2*f*Chi(1/2*f*x)*sech(1/2*e+1/4*I*Pi+1/2*f* 
x)*sinh(1/2*e+1/4*I*Pi)*(a+I*a*sinh(f*x+e))^(1/2)+1/2*f*cosh(1/2*e+1/4*I*P 
i)*sech(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)
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {a+i a \sinh (e+f x)}}{x^2} \, dx=\frac {\sqrt {a+i a \sinh (e+f x)} \left (f x \text {Chi}\left (\frac {f x}{2}\right ) \left (i \cosh \left (\frac {e}{2}\right )+\sinh \left (\frac {e}{2}\right )\right )-2 \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )+f x \left (\cosh \left (\frac {e}{2}\right )+i \sinh \left (\frac {e}{2}\right )\right ) \text {Shi}\left (\frac {f x}{2}\right )\right )}{2 x \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )} \] Input:

Integrate[Sqrt[a + I*a*Sinh[e + f*x]]/x^2,x]
 

Output:

(Sqrt[a + I*a*Sinh[e + f*x]]*(f*x*CoshIntegral[(f*x)/2]*(I*Cosh[e/2] + Sin 
h[e/2]) - 2*(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2]) + f*x*(Cosh[e/2] + I 
*Sinh[e/2])*SinhIntegral[(f*x)/2]))/(2*x*(Cosh[(e + f*x)/2] + I*Sinh[(e + 
f*x)/2]))
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.82, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3042, 3800, 3042, 3778, 26, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3800

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

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

Input:

Int[Sqrt[a + I*a*Sinh[e + f*x]]/x^2,x]
 

Output:

Sech[e/2 + (I/4)*Pi + (f*x)/2]*Sqrt[a + I*a*Sinh[e + f*x]]*(-(Cosh[e/2 + ( 
I/4)*Pi + (f*x)/2]/x) - (I/2)*f*(I*CoshIntegral[(f*x)/2]*Sinh[(2*e + I*Pi) 
/4] + I*Cosh[(2*e + I*Pi)/4]*SinhIntegral[(f*x)/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 3042

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

rule 3778

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c 
 + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1))   Int[( 
c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 
1]
 

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz 
}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate((a+I*a*sinh(f*x+e))^(1/2)/x^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((a+I*a*sinh(f*x+e))**(1/2)/x**2,x)
 

Output:

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

Maxima [F]

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

integrate((a+I*a*sinh(f*x+e))^(1/2)/x^2,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((a+I*a*sinh(f*x+e))^(1/2)/x^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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