Integrand size = 21, antiderivative size = 149 \[ \int \frac {\sqrt {a+i a \sinh (e+f x)}}{x^2} \, dx=-\frac {\sqrt {a+i a \sinh (e+f x)}}{x}+\frac {1}{2} f \text {Chi}\left (\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {1}{4} (2 e+i \pi )\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {1}{2} f \cosh \left (\frac {1}{4} (2 e+i \pi )\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)} \text {Shi}\left (\frac {f x}{2}\right ) \]
-(a+I*a*sinh(f*x+e))^(1/2)/x+1/2*f*cosh(1/2*e+1/4*I*Pi)*sech(1/2*e+1/4*I*P i+1/2*f*x)*Shi(1/2*f*x)*(a+I*a*sinh(f*x+e))^(1/2)+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)
Time = 0.48 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {a+i a \sinh (e+f x)}}{x^2} \, dx=\frac {\sqrt {a+i a \sinh (e+f x)} \left (f x \text {Chi}\left (\frac {f x}{2}\right ) \left (i \cosh \left (\frac {e}{2}\right )+\sinh \left (\frac {e}{2}\right )\right )-2 \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )+f x \left (\cosh \left (\frac {e}{2}\right )+i \sinh \left (\frac {e}{2}\right )\right ) \text {Shi}\left (\frac {f x}{2}\right )\right )}{2 x \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )} \]
(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]))
Time = 0.61 (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 |
\(\displaystyle \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (-\frac {1}{2} i f \left (i \sinh \left (\frac {1}{4} (2 e+i \pi )\right ) \text {Chi}\left (\frac {f x}{2}\right )+i \cosh \left (\frac {1}{4} (2 e+i \pi )\right ) \text {Shi}\left (\frac {f x}{2}\right )\right )-\frac {\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{x}\right )\) |
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]))
3.2.23.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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]
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]
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]
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]
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])
\[\int \frac {\sqrt {a +i a \sinh \left (f x +e \right )}}{x^{2}}d x\]
Exception generated. \[ \int \frac {\sqrt {a+i a \sinh (e+f x)}}{x^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int \frac {\sqrt {a+i a \sinh (e+f x)}}{x^2} \, dx=\int \frac {\sqrt {i a \left (\sinh {\left (e + f x \right )} - i\right )}}{x^{2}}\, dx \]
\[ \int \frac {\sqrt {a+i a \sinh (e+f x)}}{x^2} \, dx=\int { \frac {\sqrt {i \, a \sinh \left (f x + e\right ) + a}}{x^{2}} \,d x } \]
\[ \int \frac {\sqrt {a+i a \sinh (e+f x)}}{x^2} \, dx=\int { \frac {\sqrt {i \, a \sinh \left (f x + e\right ) + a}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+i a \sinh (e+f x)}}{x^2} \, dx=\int \frac {\sqrt {a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}}}{x^2} \,d x \]