Integrand size = 18, antiderivative size = 83 \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx=\cosh \left (\frac {c}{2}\right ) \sqrt {a+a \cosh (c+d x)} \text {Chi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right )+\sqrt {a+a \cosh (c+d x)} \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right ) \] Output:
cosh(1/2*c)*(a+a*cosh(d*x+c))^(1/2)*Chi(1/2*d*x)*sech(1/2*d*x+1/2*c)+(a+a* cosh(d*x+c))^(1/2)*sech(1/2*d*x+1/2*c)*sinh(1/2*c)*Shi(1/2*d*x)
Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx=\sqrt {a (1+\cosh (c+d x))} \text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (\cosh \left (\frac {c}{2}\right ) \text {Chi}\left (\frac {d x}{2}\right )+\sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right )\right ) \] Input:
Integrate[Sqrt[a + a*Cosh[c + d*x]]/x,x]
Output:
Sqrt[a*(1 + Cosh[c + d*x])]*Sech[(c + d*x)/2]*(Cosh[c/2]*CoshIntegral[(d*x )/2] + Sinh[c/2]*SinhIntegral[(d*x)/2])
Time = 0.49 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.69, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3800, 3042, 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.
\(\displaystyle \int \frac {\sqrt {a \cosh (c+d x)+a}}{x} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {a+a \sin \left (i c+i d x+\frac {\pi }{2}\right )}}{x}dx\) |
\(\Big \downarrow \) 3800 |
\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \int \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{x}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \int \frac {\sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{2}\right )}{x}dx\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\cosh \left (\frac {c}{2}\right ) \int \frac {\cosh \left (\frac {d x}{2}\right )}{x}dx-i \sinh \left (\frac {c}{2}\right ) \int \frac {i \sinh \left (\frac {d x}{2}\right )}{x}dx\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\sinh \left (\frac {c}{2}\right ) \int \frac {\sinh \left (\frac {d x}{2}\right )}{x}dx+\cosh \left (\frac {c}{2}\right ) \int \frac {\cosh \left (\frac {d x}{2}\right )}{x}dx\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\sinh \left (\frac {c}{2}\right ) \int -\frac {i \sin \left (\frac {i d x}{2}\right )}{x}dx+\cosh \left (\frac {c}{2}\right ) \int \frac {\sin \left (\frac {i d x}{2}+\frac {\pi }{2}\right )}{x}dx\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\cosh \left (\frac {c}{2}\right ) \int \frac {\sin \left (\frac {i d x}{2}+\frac {\pi }{2}\right )}{x}dx-i \sinh \left (\frac {c}{2}\right ) \int \frac {\sin \left (\frac {i d x}{2}\right )}{x}dx\right )\) |
\(\Big \downarrow \) 3779 |
\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right )+\cosh \left (\frac {c}{2}\right ) \int \frac {\sin \left (\frac {i d x}{2}+\frac {\pi }{2}\right )}{x}dx\right )\) |
\(\Big \downarrow \) 3782 |
\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\cosh \left (\frac {c}{2}\right ) \text {Chi}\left (\frac {d x}{2}\right )+\sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right )\right )\) |
Input:
Int[Sqrt[a + a*Cosh[c + d*x]]/x,x]
Output:
Sqrt[a + a*Cosh[c + d*x]]*Sech[c/2 + (d*x)/2]*(Cosh[c/2]*CoshIntegral[(d*x )/2] + Sinh[c/2]*SinhIntegral[(d*x)/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[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])
\[\int \frac {\sqrt {a +a \cosh \left (d x +c \right )}}{x}d x\]
Input:
int((a+a*cosh(d*x+c))^(1/2)/x,x)
Output:
int((a+a*cosh(d*x+c))^(1/2)/x,x)
Exception generated. \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*cosh(d*x+c))^(1/2)/x,x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx=\int \frac {\sqrt {a \left (\cosh {\left (c + d x \right )} + 1\right )}}{x}\, dx \] Input:
integrate((a+a*cosh(d*x+c))**(1/2)/x,x)
Output:
Integral(sqrt(a*(cosh(c + d*x) + 1))/x, x)
\[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx=\int { \frac {\sqrt {a \cosh \left (d x + c\right ) + a}}{x} \,d x } \] Input:
integrate((a+a*cosh(d*x+c))^(1/2)/x,x, algorithm="maxima")
Output:
integrate(sqrt(a*cosh(d*x + c) + a)/x, x)
Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx=\frac {1}{2} \, \sqrt {2} {\left (\sqrt {a} {\rm Ei}\left (\frac {1}{2} \, d x\right ) e^{\left (\frac {1}{2} \, c\right )} + \sqrt {a} {\rm Ei}\left (-\frac {1}{2} \, d x\right ) e^{\left (-\frac {1}{2} \, c\right )}\right )} \] Input:
integrate((a+a*cosh(d*x+c))^(1/2)/x,x, algorithm="giac")
Output:
1/2*sqrt(2)*(sqrt(a)*Ei(1/2*d*x)*e^(1/2*c) + sqrt(a)*Ei(-1/2*d*x)*e^(-1/2* c))
Timed out. \[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx=\int \frac {\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}}{x} \,d x \] Input:
int((a + a*cosh(c + d*x))^(1/2)/x,x)
Output:
int((a + a*cosh(c + d*x))^(1/2)/x, x)
\[ \int \frac {\sqrt {a+a \cosh (c+d x)}}{x} \, dx=\sqrt {a}\, \left (\int \frac {\sqrt {\cosh \left (d x +c \right )+1}}{x}d x \right ) \] Input:
int((a+a*cosh(d*x+c))^(1/2)/x,x)
Output:
sqrt(a)*int(sqrt(cosh(c + d*x) + 1)/x,x)