Integrand size = 14, antiderivative size = 42 \[ \int \frac {\sqrt {a+a \cosh (x)}}{x^2} \, dx=-\frac {\sqrt {a+a \cosh (x)}}{x}+\frac {1}{2} \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right ) \text {Shi}\left (\frac {x}{2}\right ) \] Output:
-(a+a*cosh(x))^(1/2)/x+1/2*(a+a*cosh(x))^(1/2)*sech(1/2*x)*Shi(1/2*x)
Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {a+a \cosh (x)}}{x^2} \, dx=\frac {\sqrt {a (1+\cosh (x))} \left (-2+x \text {sech}\left (\frac {x}{2}\right ) \text {Shi}\left (\frac {x}{2}\right )\right )}{2 x} \] Input:
Integrate[Sqrt[a + a*Cosh[x]]/x^2,x]
Output:
(Sqrt[a*(1 + Cosh[x])]*(-2 + x*Sech[x/2]*SinhIntegral[x/2]))/(2*x)
Time = 0.40 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3800, 3042, 3778, 26, 3042, 26, 3779}
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.
\(\displaystyle \int \frac {\sqrt {a \cosh (x)+a}}{x^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {a+a \sin \left (\frac {\pi }{2}+i x\right )}}{x^2}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \int \frac {\cosh \left (\frac {x}{2}\right )}{x^2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \int \frac {\sin \left (\frac {i x}{2}+\frac {\pi }{2}\right )}{x^2}dx\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (-\frac {\cosh \left (\frac {x}{2}\right )}{x}+\frac {1}{2} i \int -\frac {i \sinh \left (\frac {x}{2}\right )}{x}dx\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {1}{2} \int \frac {\sinh \left (\frac {x}{2}\right )}{x}dx-\frac {\cosh \left (\frac {x}{2}\right )}{x}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (-\frac {\cosh \left (\frac {x}{2}\right )}{x}+\frac {1}{2} \int -\frac {i \sin \left (\frac {i x}{2}\right )}{x}dx\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (-\frac {\cosh \left (\frac {x}{2}\right )}{x}-\frac {1}{2} i \int \frac {\sin \left (\frac {i x}{2}\right )}{x}dx\right )\) |
\(\Big \downarrow \) 3779 |
\(\displaystyle \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a} \left (\frac {\text {Shi}\left (\frac {x}{2}\right )}{2}-\frac {\cosh \left (\frac {x}{2}\right )}{x}\right )\) |
Input:
Int[Sqrt[a + a*Cosh[x]]/x^2,x]
Output:
Sqrt[a + a*Cosh[x]]*Sech[x/2]*(-(Cosh[x/2]/x) + SinhIntegral[x/2]/2)
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[((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])
\[\int \frac {\sqrt {a +\cosh \left (x \right ) a}}{x^{2}}d x\]
Input:
int((a+cosh(x)*a)^(1/2)/x^2,x)
Output:
int((a+cosh(x)*a)^(1/2)/x^2,x)
Exception generated. \[ \int \frac {\sqrt {a+a \cosh (x)}}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*cosh(x))^(1/2)/x^2,x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int \frac {\sqrt {a+a \cosh (x)}}{x^2} \, dx=\int \frac {\sqrt {a \left (\cosh {\left (x \right )} + 1\right )}}{x^{2}}\, dx \] Input:
integrate((a+a*cosh(x))**(1/2)/x**2,x)
Output:
Integral(sqrt(a*(cosh(x) + 1))/x**2, x)
\[ \int \frac {\sqrt {a+a \cosh (x)}}{x^2} \, dx=\int { \frac {\sqrt {a \cosh \left (x\right ) + a}}{x^{2}} \,d x } \] Input:
integrate((a+a*cosh(x))^(1/2)/x^2,x, algorithm="maxima")
Output:
integrate(sqrt(a*cosh(x) + a)/x^2, x)
\[ \int \frac {\sqrt {a+a \cosh (x)}}{x^2} \, dx=\int { \frac {\sqrt {a \cosh \left (x\right ) + a}}{x^{2}} \,d x } \] Input:
integrate((a+a*cosh(x))^(1/2)/x^2,x, algorithm="giac")
Output:
integrate(sqrt(a*cosh(x) + a)/x^2, x)
Timed out. \[ \int \frac {\sqrt {a+a \cosh (x)}}{x^2} \, dx=\int \frac {\sqrt {a+a\,\mathrm {cosh}\left (x\right )}}{x^2} \,d x \] Input:
int((a + a*cosh(x))^(1/2)/x^2,x)
Output:
int((a + a*cosh(x))^(1/2)/x^2, x)
\[ \int \frac {\sqrt {a+a \cosh (x)}}{x^2} \, dx=\frac {\sqrt {a}\, \left (-2 \sqrt {\cosh \left (x \right )+1}+\left (\int \frac {\sqrt {\cosh \left (x \right )+1}\, \sinh \left (x \right )}{\cosh \left (x \right ) x +x}d x \right ) x \right )}{2 x} \] Input:
int((a+a*cosh(x))^(1/2)/x^2,x)
Output:
(sqrt(a)*( - 2*sqrt(cosh(x) + 1) + int((sqrt(cosh(x) + 1)*sinh(x))/(cosh(x )*x + x),x)*x))/(2*x)